Department of Precision and Microsystems Engineering Maximization of the geometric non-linearities of a thin-walled structure in resonance N.K. Teunisse Report no Coaches Professor Specialisation Type of report Date : : : : : : EM 2015.034 prof.dr.ir. W. Lacarbonara, dr.ir. P. Tiso and dr.ir. J.F.L Goosen prof.dr.ir. F. van Keulen Engineering Mechanics Master Thesis 30 November 2015 Precision and Microsystems Engineering Maximization of the geometric non-linearities of a thin-walled structure in resonance by N.K. Teunisse in partial fulfillment of the requirements for the degree of Master of Science in Mechanical Engineering at the Delft University of Technology, to be defended publicly on 30 November 2015 Student number: Supervisor: Supervisor: Supervisor: 1522086 dr.ir. J.F.L. Goosen, prof.dr.ir. W. Lacarbonara, dr.ir. P. Tiso, Delft University of Technology Sapienza - Università di Roma Eidgenössische Technische Hochschule Zürich An electronic version of this thesis is available at http://repository.tudelft.nl/. c Precision and Microsystems Engineering (PME) Copyright All rights reserved. Abstract Micro air vehicles (MAVs) are small aircraft that are promising for search and rescue missions or remote observations of hazardous or inaccessible areas. Currently one of the problems MAVs face is the limited action radius. To increase the action radius, a more energy efficient MAV is needed. This can be achieved by exploiting the resonance of the structure that couples the actuator and the wings. A MAV that brought this concept to practice was designed and manufactured by C.T. Bolsman [1]. The developed prototype uses a flapping wing actuation mechanism producing enough lift to overcome the mass of the resonant structure, but not enough to also lift the actuator. A way to increase lift is to create a more effective flapping wing motion. Investigations carried out within the Department of Precision and Microsystems Engineering (PME) at Delft University of Technology show that a constant velocity of the sweeping (flutter) motion between stroke reversals increases the effectiveness. The function that ideally matches this velocity profile is a triangular-shaped displacement function in time. This work focuses on obtaining a triangular-shaped function for the angular displacement at the end of a resonating beam. This is not a priori granted, because the beam is subject to a harmonic forcing in order to be excited and kept in resonance. Provided that the system operates in its linear regime, the response of the beam, over time, will be a harmonic motion. Yet, by exploiting the geometric non-linearities, it is possible to retrieve a triangular motion as displacement profile, while the input remains a harmonic torque. In this work, the answers are given to the following questions: What causes the geometric non-linearities? Which beam cross-section maximizes the geometric non-linearities? Past work showed that geometric non-linearities arise from the longitudinal stresses induced by the axial non-uniform shortening of the structure over the cross-section. In this work the formulation of the non-linear stiffness is provided, together with the formulation of the uniform torsion stiffness and the stiffness caused by warping. Collectively, they yield the total torsional stiffness of the beam. With this knowledge, the author proposes an analytical model of a beam under torsion and an asymptotic approximation to its solution with the help of a perturbation method, the so-called method of multiple scales. This analytical model is derived for optimization purposes due to the low computational costs associated with the delivered dynamic responses. However, the derived solution of the analytical model is an approximation, so the results are verified with a more reliable, but computational expensive method: the finite element method. Consequently, a finite element code is developed and a beam element that can handle warping and geometric non-linearities is implemented. Next, both models are verified with the finite element program ANSYS. Master of Science Thesis N.K. Teunisse ii In the last part of the work, the analytical model is used as a basis for the optimization. Some approximations are made to enable the computation of a fine grid of the various crosssectional features and their corresponding degree of triangularity of the response over time. Finally, the optimal cross-sectional features are found. The corresponding dynamic response is, as expected, a triangular-shaped function. Next, the approximations are verified and some recommendations are provided regarding an already manufactured MAV based on torsion. Lastly, recommendations for future investigations, although outside the scope of this project are provided and a point of attention is noted regarding buckling of thin-walled structures. N.K. Teunisse Master of Science Thesis Table of Contents Preface and acknowledgments xi 1 Introduction 1-1 Research context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-1-1 Introduction to non-linearities . . . . . . . . . . . . . . . . . . . . . . . . 1-2 Research goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 2 3 1-3 Project outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2 Torsion of thin-walled structures 2-1 Introduction to thin-walled structures . . . . . . . . . . . . . . . . . . . . . . . . 2-2 Definition of warping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 5 6 2-3 2-4 2-5 2-6 Definition of uniform torsion . . . . . . . Definition of non-uniform torsion . . . . . Definition of non-linear torsion . . . . . . Introduction to flexural-torsional coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 8 9 10 2-7 Conclusions and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3 Analytical model 11 3-1 Derivation of the initial-boundary value problem . . . . . . . . . . . . . . . . . . 11 3-1-1 Non-reduced governing equations . . . . . . . . . . . . . . . . . . . . . . 11 3-1-2 Reduced governing equations . . . . . . . . . . . . . . . . . . . . . . . . 13 3-1-3 Non-dimensionalization of the reduced governing equations . . . . . . . . 14 3-1-4 Final initial-boundary value problem . . . . . . . . . . . . . . . . . . . . 15 3-2 Perturbation analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3-2-1 Choice of the perturbation method . . . . . . . . . . . . . . . . . . . . . 16 3-2-2 First order approximation of the analytical solution . . . . . . . . . . . . 19 3-2-3 Higher order approximations of the analytical solution . . . . . . . . . . . 26 3-3 Conclusions and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Master of Science Thesis N.K. Teunisse iv 4 Finite element method 4-1 Introduction to the finite element method . . . . . . . . . 4-1-1 The method . . . . . . . . . . . . . . . . . . . . . 4-1-2 The stiffness, damping and mass matrix . . . . . . 4-2 Choice of the finite element software . . . . . . . . . . . . 4-2-1 Choice of developing custom finite element software 4-2-2 Erroneous assumption . . . . . . . . . . . . . . . . 4-3 Implementation of the custom finite element software . . . 4-3-1 Choice and definition of Rayleigh damping . . . . . 4-4 Conclusions and outlook . . . . . . . . . . . . . . . . . . . Table of Contents . . . . . . . . . . . . . . . . . . 5 Model verification 5-1 Description of the evaluated models and elements . . . . . . . 5-2 Static evaluation of the developed finite element model . . . . 5-2-1 Evaluated geometries . . . . . . . . . . . . . . . . . . 5-2-2 Limitations of the developed finite element model . . . 5-3 Comparison of the natural frequencies and modes . . . . . . . 5-4 Dynamic evaluation of the developed finite element model . . 5-4-1 Computational procedures regarding dynamic analysis . 5-4-2 Evaluated geometries . . . . . . . . . . . . . . . . . . 5-4-3 Summary of the evaluation of the finite element model 5-5 Dynamic evaluation of the derived analytical model . . . . . . 5-5-1 Implementation of the analytical model in MATLAB . 5-5-2 Obtaining the correct damping coefficient . . . . . . . 5-5-3 Evaluated geometries . . . . . . . . . . . . . . . . . . 5-5-4 Summary of the evaluation of the analytical model . . 5-6 Reduction of the transient using the analytical model . . . . . 5-7 Conclusions and outlook . . . . . . . . . . . . . . . . . . . . . 6 Optimization 6-1 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . 6-1-1 Objective function . . . . . . . . . . . . . . . . . . . . 6-1-2 Constraints and bounds . . . . . . . . . . . . . . . . . 6-1-3 Negative null form . . . . . . . . . . . . . . . . . . . . 6-2 Implementation of the optimization objective and constraints . 6-2-1 Derivation of the approximate model . . . . . . . . . . 6-2-2 Choice of the optimization algorithm . . . . . . . . . . 6-3 The design space . . . . . . . . . . . . . . . . . . . . . . . . 6-4 Optimal cross-section and corresponding dynamic response . . 6-5 Conclusions regarding rectangular and X-shaped cross-section . 6-6 Validation of the approximate model . . . . . . . . . . . . . . 6-7 Improving the manufactured designs . . . . . . . . . . . . . . 6-8 Optimized beam in ANSYS . . . . . . . . . . . . . . . . . . . 6-9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . N.K. Teunisse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 27 27 28 28 28 28 28 29 29 . . . . . . . . . . . . . . . . 31 31 32 32 37 37 38 38 39 40 44 44 45 46 50 50 52 . . . . . . . . . . . . . . 53 53 54 55 55 56 56 56 57 60 60 62 63 63 64 Master of Science Thesis Table of Contents v 7 Conclusions and recommendations 7-1 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 66 A Description of the non-linear elastic non-uniform torsional beam element 67 A-1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-2 Equilibrium equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 69 A-3 Finite element discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-3-1 Solution strategy in nonlinear context . . . . . . . . . . . . . . . . . . . 73 75 A-3-2 Mass matrix including rotary inertia . . . . . . . . . . . . . . . . . . . . 77 B Approximation of a triangular function 79 C Higher-order approximations of the analytical solution 81 C-1 Third order solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-2 The higher-order problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 84 C-3 Finding slow time scale coefficients for the third order solution . . . . . . . . . . 85 D Analytical solutions of the cross-sectional properties 87 E Optimization algorithms 89 E-1 Different optimization solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 E-1-1 Trust region reflective algorithm . . . . . . . . . . . . . . . . . . . . . . 90 E-1-2 Interior point algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 E-1-3 Active Set algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 E-1-4 SQP algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 Master of Science Thesis N.K. Teunisse vi N.K. Teunisse Table of Contents Master of Science Thesis List of Figures 1-1 Example of a deflection curve where the non-linear system exhibits a hardening effect. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-2 A manufactured design based on torsion within the Atalanta project. . . . . . . . 3 3 2-1 The different types of thin-walled cross-sections [2]. . . . . . . . . . . . . . . . . 6 2-2 Concept of sectorial coordinates based on image of Shama [3]. . . . . . . . . . . 7 2-3 Uniform torsion of I-beam [2]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2-4 St. Venant stresses within various open cross-sections [2]. . . . . . . . . . . . . . 8 2-5 Normal and shear stresses due to warping [4]. . . . . . . . . . . . . . . . . . . . 9 2-6 Non-uniform torsion of I-beam [2]. . . . . . . . . . . . . . . . . . . . . . . . . . 9 2-7 Explanation of non-linear torsion. . . . . . . . . . . . . . . . . . . . . . . . . . . 10 5-1 Comparison of the static deformation of a free-clamped plate in both MATLAB and ANSYS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-2 Comparison of the static deformation of a linear and non-linear free-clamped Ibeam in both MATLAB and ANSYS. . . . . . . . . . . . . . . . . . . . . . . . . 5-3 The deformed cross-section at the end of a clamped-free I-beam subject to torque. 34 36 37 5-4 Comparison of the fundamental torsional mode using three different models: MATLAB FEM, analytical model, ANSYS beam. . . . . . . . . . . . . . . . . . . . . 38 5-5 Comparison of the dynamic response of profile Id 1 subject to 5000Nm driven at the first torsional natural frequency. . . . . . . . . . . . . . . . . . . . . . . . . 42 5-6 Comparison of the dynamic response of profile Id 1 subject to 15000Nm driven at the first torsional natural frequency. . . . . . . . . . . . . . . . . . . . . . . . . 42 5-7 Comparison of the dynamic response of profile Id 2 subject to 2000Nm driven at the first torsional natural frequency. . . . . . . . . . . . . . . . . . . . . . . . . 42 5-8 Comparison of the dynamic response of profile Id 2 subject to 10000Nm driven at the first torsional natural frequency. . . . . . . . . . . . . . . . . . . . . . . . . 42 5-9 Comparison of the dynamic response of profile Id 3 subject to 1000Nm driven at the first torsional natural frequency. . . . . . . . . . . . . . . . . . . . . . . . . 43 5-10 Comparison of the dynamic response of profile Id 3 subject to 3000Nm driven at the first torsional natural frequency. . . . . . . . . . . . . . . . . . . . . . . . . 43 Master of Science Thesis N.K. Teunisse viii List of Figures 5-11 Comparison of the dynamic response of profile Id 4 subject to 500Nm driven at the first torsional natural frequency. . . . . . . . . . . . . . . . . . . . . . . . . 5-12 The slightly more triangular steady-state response of profile Id 3 subject to 5000Nm compared with a sine. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-13 Comparison of the steady-state response of profile Id 1 subject to 5000Nm driven at the first torsional natural frequency. . . . . . . . . . . . . . . . . . . . . . . . 5-14 Comparison of the steady-state response of profile Id 1 subject to 70000Nm driven at the first torsional natural frequency. . . . . . . . . . . . . . . . . . . . . . . . 5-15 Comparison of the steady-state response of profile Id 2 subject to 1000Nm driven at the first torsional natural frequency. . . . . . . . . . . . . . . . . . . . . . . . 5-16 Comparison of the steady-state response of profile Id 2 subject to 20000Nm driven at the first torsional natural frequency. . . . . . . . . . . . . . . . . . . . . . . . 5-17 Comparison of the steady-state response of profile Id 3 subject to 1000Nm driven at the first torsional natural frequency. . . . . . . . . . . . . . . . . . . . . . . . 5-18 Comparison of the steady-state response of profile Id 3 subject to 5000Nm driven at the first torsional natural frequency. . . . . . . . . . . . . . . . . . . . . . . . 5-19 Comparison of the steady-state response of profile Id 4 subject to 2000Nm driven at the first torsional natural frequency. . . . . . . . . . . . . . . . . . . . . . . . 5-20 Comparison of the steady-state response of profile Id 4 subject to 10000Nm driven at the first torsional natural frequency. . . . . . . . . . . . . . . . . . . . . . . . 5-21 The dynamic response of profile Id 1 with initial conditions equal to the analytical model at t = 0. The beam is subject to 70000Nm and driven at its first torsional natural frequency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-1 The values of ω̄1 for varying ηw computed by a root-finding algorithm compared with the fitted function, described in eq.6-12. . . . . . . . . . . . . . . . . . . . 6-2 The objective function plotted against the width (b) and height (h). The black asterisks represent the points where the first torsional eigenfrequency equals 60Hz, the asterisk with the red circle is the optimum. The flange thickness (tf ) is 0.0001m, the wall thickness (tw ) is 0.0001m and the length is 0.1m. . . . . . . . 6-3 The objective function plotted against the flange thickness (tf ) and wall thickness (tw ). The black asterisks represent the points where the first torsional eigenfrequency equals 60Hz, the asterisk with the red circle is the optimum. The height (h) is 0.01m, the width (b) is 0.0028m and the length is 0.1m. . . . . . . . . . . 6-4 The profile that results in the most non-linear dynamic response subject to the upper and lower bounds, a length of 0.1m and a first torsional linear eigenfrequency of 60Hz. Its cross-sectional features are b=0.0028m, h=0.01m, tf =0.0001m and tw =0.0001m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-5 Angular rotation of the optimal cross-section that is subject to 0.0005Nm driven at the first torsional natural frequency. The optimized response is computed by the MATLAB FEM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-6 Angular velocity of the optimal cross-section that is subject to 0.0005Nm driven at the first torsional natural frequency. The optimized response is computed by the MATLAB FEM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-7 The error of the objective function by the approximate model compared with ANSYS. 6-8 Schematic view of the base plate of the manufactured design seen in fig.1-2. The bright red squares are the slats seen from top view. The dark red squares are added to maximize the non-linear behavior of the structure. . . . . . . . . . . . . . . . B-1 Approximation of a triangular function in time using one, two and three cosines. N.K. Teunisse 43 47 48 48 48 48 49 49 49 49 52 57 59 59 60 61 61 63 64 80 Master of Science Thesis List of Tables 3-1 Symbol overview with units and description. . . . . . . . . . . . . . . . . . . . . 13 5-1 Rotational natural frequencies for different models of an free-clamped I-beam. . . 38 5-2 The different geometries used to compare the dynamic results of ANSYS with the implemented MATLAB FEM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 6-1 Comparison of the linear and non-linear deflections of a flat plate and a beam with the optimal cross-section. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 E-1 Solvers available in MATLAB. 91 Master of Science Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . N.K. Teunisse x N.K. Teunisse List of Tables Master of Science Thesis Preface and acknowledgments This work is my Master of Science graduation thesis and represents the most challenging and rewarding academic work I have delivered so far. The topic came rather unexpected as I originally planned to pursue a topic in the field of Model Order Reduction techniques. However after doing half a year of research in this field at San Diego State University, I decided to make a fresh start and searched for another challenge. Paolo Tiso helped me with this by coming up with this project, which was exactly what I was looking for. It combines dynamics, non-linearities, finite element methods and a touch of optimization, all the fields that caught my attention during my Master program. I would like to thank my supervisors Walter Lacarbonara, Paolo Tiso and Hans Goosen for supervising and supporting me in this project. Furthermore, I would like to thank my friends for their support and the unforgeable moments throughout my graduation year. Above all, I would like to thank my parents and sister for their continuous support and faith in me. Delft, University of Technology 30 November 2015 Master of Science Thesis N.K. Teunisse N.K. Teunisse xii N.K. Teunisse Preface and acknowledgments Master of Science Thesis 1 Introduction For more than a decade there has been an increasing interest in drones, but only recently did it also become popular outside the scientific community and military sector. Due to the amount of research spent on drones they are now affordable for the consumers. Robotic insects, on the other hand, are still far from being ready for the consumer market. Until recently they only existed in movies, but in 2013 engineers from Harvard [5] managed to make a robotic fly, being the world’s smallest man-made aircraft. These small aircraft are all grouped under the name micro air vehicles (MAVs). 1-1 Research context MAVs belong to the category of unmanned aerial vehicles (UAV), which encompasses all aircraft without a human pilot aboard. For an aircraft to be identified as MAV the full span of the flying vehicle must be smaller or equal to 150mm. MAVs have gained interest for search and rescue missions or remote observations of hazardous or inaccessible areas. Therefore, they are gaining a lot of interest from the military sector, but also from organizations specialized in managing disasters. Most MAVs suffer from a low flight radius due to their limited energy storage. After all, there is only a small amount of weight that can be lifted by the aircraft and batteries or filled fuel tanks are heavy. So one can develop a MAV that produces more lift allowing it to carry more load. Yet, producing more lift might not be the best strategy since a MAV that generates more lift is most likely going to use more energy too. Consequently, the best strategy is to make the MAV more energy efficient. This work expands some directions tackled by the dissertation of C.T. Bolsman [1] who designed and manufactured an insect-inspired flapping wing actuation mechanism. The project is named the Atalanta project. The MAV developed by Bolsman is unique in the sense that it is very energy efficient. Its design is built around the concept of resonance. When the structure resonates in its so-called eigenfrequency the elastic and inertia forces balance each other, so the energy put into the system solely has to overcome the damping (assuming a linear system). Bolsman created an elastic compliant structure that is excited into resonance by a linear actuator. The structure is designed in such a way that the wings attached to it make a flapping wing motion, resulting in a MAV with a minimized energy consumption Master of Science Thesis N.K. Teunisse 2 Introduction during flight. The developed prototype produced enough lift to overcome the mass of the resonant structure, but unfortunately not enough to also lift the actuator. There are several ways to solve this problem. Some, but definitely not all of them, are listed below 1. Reduction of the mass of the MAV (a) Reduce the mass of the actuator (b) Reduce the mass of the structure 2. Create larger aerodynamic forces (a) Resize the wings (b) Create a more effective flapping wing motion (c) Change the shape of the wings (d) Change number of wings Within the Department Precision and Microsystems Engineering (PME) at Delft University of Technology, research has been carried out to investigate a more effective flapping wing motion, especially by Qi Wang. Simulations showed that a constant velocity of the sweeping (flutter) motion between its stroke reversals increases the effectiveness. Furthermore, Bos et al. [6] showed that the effect of a non-harmonic actuation of the wings strongly influences the lift generated and the efficiency of the flapping wing motion. A function that ideally describes the desired velocity profile is a triangular motion, because the velocity between the minima and maxima is constant and the time for the reversal is infinitely small. An ideal triangular motion is not realistically achievable, but it can be well approximated. Driving the sweeping motion of the wings with a nearly triangular signal is not hard for an actuator. Yet, it is not the actuator driving the wing motion, but the resonating body of the MAV. To excite and keep a structure in resonance, it must be subject to a harmonic forcing. Note that the structure needs to be in a state of resonance, because this minimizes the energy consumption. Since the actuator provides a harmonic load, the dynamic response of the structure will be a harmonic motion provided that the system operates in its linear regime. To transform this harmonic motion to a nearly triangular motion, non-linearities can be advantageously exploited. 1-1-1 Introduction to non-linearities Non-linearities appear in virtually any field. Generally speaking, we say that we are in the presence of non-linearities when the effects are not proportional to the causes. Applying this concept to statics, an example of a system with non-linearities is a system in which doubling the applied force does not cause the displacement to become twice as large as before. There are many sources of non-linearities, which can be divided into 3 categories within the field of mechanics [7]. • Geometric non-linearity: The change in geometry as the structure deforms is taken into account in the strain-displacement and equilibrium equations. In general, it requires large deformation to become noticeable. N.K. Teunisse Master of Science Thesis 1-2 Research goals 3 Load Linear Non-linear Deflection Figure 1-1: Example of a deflection curve where the non-linear system exhibits a hardening effect. Figure 1-2: A manufactured design based on torsion within the Atalanta project. • Material non-linearity: Material behavior depends on the current deformation state and possibly past history of the deformation. Other constitutive aspects (prestress, temperature, time, moisture, electromagnetic fields, etc.) may be involved. • Non-linear boundary conditions: The applied forces and geometric boundary conditions depend on the deformation. In this work only geometric non-linearities are considered. An important effect that is caused by geometric non-linearities is hardening. This implies that the response of a structure shows an increasing stiffness when the deformations grow (see fig.1-1 for an example). The opposite can also take place and is defined as softening. The hardening effect can be used to transform a linear sinusoidal response to a non-linear triangular response. This can be explained with the help of an undamped mass-spring system. For a linear spring the conversion of kinetic energy to potential energy happens gradually over the entire length of the spring. However, for a spring that gets much stiffer for larger deflections, the conversion of energy happens in a significant shorter range. This causes a more abrupt change in velocity and a more spiky displacement profile in time around the neighborhood of the displacements peaks. Moreover, the hardening effect means a decrease in maximum amplitude of the displacement. The velocity for small spring deflections, the region between the successive peaks, remains virtually constant, because the converted amount of kinetic energy is still so low that the velocity is hardly altered. 1-2 Research goals As mentioned earlier, this work aims to have a structure in resonance that provides a triangular motion over time subject to a harmonic input. Such a displacement profile should be suitable for actuating the wings. Furthermore we know that we can make use of the hardening effect, caused by geometric non-linearities, to transform a harmonic motion to a triangular motion. We aim to identify a thin-walled structure since lightweight elements are required. Moreover, the structure is subject to twisting and not to bending, because this makes lighter Master of Science Thesis N.K. Teunisse 4 Introduction designs possible. Several designs were already produced and one of them is shown in fig.1-2. Moreover, to generate a flapping wing motion the structure needs to handle 90 degrees angles of twist resulting in a more slender beam-like structure. In summary, we aim to find the optimal thin-walled beam that maximizes its non-linearities such that the twist angle at end of the beam describes a triangular motion while the beam is excited in its resonance frequency by a harmonically varying torque. The restrictions are that the structure should be smaller than 15cm and must be actuated around 60Hz. 1-3 Project outline This thesis starts with a description of the different types of torsion with a focus on thin-walled structures in chapter 2. Next, the analytical model is derived in chapter 3. Subsequently, the choice of implemented finite element method is explained in chapter 4. The analytical as well as the numerical model are verified in chapter 5 with a commercial finite element package, ANSYS. Hereafter, an optimization is performed to find the optimal cross-section. Both the optimum and the effect of the different parameters on the non-linearities are discussed in chapter 6. Finally the conclusions are drawn in chapter 7. N.K. Teunisse Master of Science Thesis 2 Torsion of thin-walled structures This work focuses on a thin-walled structure that is twisted by an actuator. A good understanding of torsion is required to predict the degree of triangularity of the structure’s dynamic response. Therefore, the different types of torsion are discussed in this chapter and are applied to thin-walled structures. A brief introduction to thin-walled structures and reduction of the research scope are provided in section 2-1. Next, the warping phenomenon is discussed in section 2-2. Uniform torsion, non-uniform torsion and non-linear torsion are explained in respectively section 2-3, 2-4 and 2-5. Lastly, the coupling between torsion and flexure together with its effect on the thesis’ scope are briefly discussed in section 2-6. 2-1 Introduction to thin-walled structures The basic idea of being thin-walled is that the thickness of a component section should be small relative to the other cross-sectional dimensions. There is, however, no clearly defined distinction between sections that should be treated as thin and thick [8]. Thin-walled structures are often used as beams and columns in engineering applications, ranging from buildings to aerospace, when weight or cost savings are of main importance [9]. In this work, the required lightweight elements lead to the use of a thin-walled structure. The cross-section of a thin-walled structure is either an open cross-section or a closed cross-section. It is characterized as a closed cross-section when it is impossible to find an end point while looping through the cross-section. In all the other cases, the cross-section is defined as open, see fig.2-1. Regarding the research scope This work considers solely open cross-sections since open cross-sections are more sensitive to warping [10], which increases the linear stiffness. This was considered beneficial at the start of this work. Master of Science Thesis N.K. Teunisse 6 Torsion of thin-walled structures (a) Open cross-section. (b) Closed cross-section. Figure 2-1: The different types of thin-walled cross-sections [2]. 2-2 Definition of warping Warping is a phenomenon that only appears in torsional problems. It is defined as the outof-plane displacement due to twist and is caused by shear stresses. The most known and implemented description of warping for thin-walled structures originates from Vlasov [11]. Vlasov formulated warping as uw = φps (y, z) dθ (x) dx (2-1) with uw as the out-of-plane displacement due to twist (warping), φps (y, z) as sectorial coordθ dinate or warping function and dx (x) as rate of twist or torsional curvature. The sectorial coordinate completely depends on the geometry, and is not uniform over the cross-section. The computation of the sectorial coordinate for an arbitrary point A holds [10] φps (y, z) = Z QA O r dQ (2-2) where QA is the position of A in the curvilinear coordinate system Q, O represents the sectorial origin, r is the perpendicular distance from the shear center S to the tangent line of a point on Q. The problem is illustrated in fig.2-2. Note that the area swept by the radius vector in fig.2-2 equals half the value of φps . Moreover, the sign convention for the warping function defines an increase of the sectorial coordinate when the radius vector rotates clockwise and a decrease when it rotates counterclockwise. Next, the position of the shear center is determined using the following conditions [10] S(φps )y = S(φps )z = Z ZΩ Ω φps (y, z)z dΩ = 0 (2-3) φps (y, z)y dΩ = 0 (2-4) Finally, the position of the sectorial origin O(φps ) is obtained with the following condition [10] O(φps ) = N.K. Teunisse Z Ω φps (y, z)dΩ = 0 (2-5) Master of Science Thesis 2-3 Definition of uniform torsion 7 S r 1 p 2dφs dQ A Q O Figure 2-2: Concept of sectorial coordinates based on image of Shama [3]. 2-3 Definition of uniform torsion Uniform torsion, also known as St. Venant torsion or primary torsion, occurs when a structure is subject to uniform torque, the warping of the structure is not constrained, and the undeformed cross-section is uniform over the structure’s span. Uniform torsion yields uniform dθ warping, which results in a constant torsional curvature ( dx ). The torsional curvature, or the rate of twist, is defined as the change in the angle of the cross-section around the axis of twist per unit length. Uniform torsion for an I-beam is shown in fig.2-3. Note that the top flange is still straight due to the constant rate of twist and the warping is uniform along the axial direction. Structures experiencing uniform torsion deal only with St. Venant stresses. Moreover, no warping stresses are present due to ability of the structures to warp freely. Warping originates from the distribution of the St. Venant stresses and is amplified by the low torsional rigidity of the individual plates of thin-walled structure. Several examples of the St. Venant stress distributions in thin-walled structures are shown in fig.2-4. The relationship between the applied moment and the torsional curvature for uniform warping and small twist angles, is written as Mx = GIt dθ dx (2-6) where Mx represents the applied moment around the axial axis, G the shear modulus and It the St. Venant torsion constant [4]. Master of Science Thesis N.K. Teunisse 8 Torsion of thin-walled structures Figure 2-3: Uniform torsion of I-beam [2]. 2-4 Figure 2-4: St. Venant stresses within various open cross-sections [2]. Definition of non-uniform torsion Non-uniform torsion, also known as warping torsion or secondary torsion, occurs when the warping of the cross-section is constrained. This is caused by geometric boundary conditions, non-uniform cross-section over the cantilever span, or non-uniform applied torques. Constrained warping generates axial stresses, which are shown in fig.2-5(a). Equilibrium dictates that the axial stresses cause shear stresses as shown in fig.2-5(b). Collectively, these stresses yield the warping stress. Fig.2-5(c) shows that the additional shear stresses generate a moment that partly resists the torsion, resulting in a higher torsional stiffness. This added moment is called the warping torque. The warping torque is not constant over the beam span. Consequently, the torsional curvature is also not uniform over the beam span. This is shown in fig.2-6. Warping varies in both magnitude and direction over a cross-section. Therefore, the reaction forces associated with the warping stresses also vary in magnitude and direction over the cross-section. Moreover, these reaction forces can act like a torque within an individual segment of the thin-walled structure. It is possible to have opposite torques in different segments. For example, a doubly-symmetric I-beam has an opposite, but equal, torque in the flanges due to the constrained warping. This opposite, but equal, torque in different segments of a thin-walled structure is called the bimoment. The value of a bimoment is defined as the magnitude of the torques multiplied with the distance between them. The relationship between the applied moment and the torsional curvature for non-uniform warping and small twist angles, holds Mx = GIt dθ d3 θ − ECs 3 dx dx (2-7) where Mx represents the applied moment around the axial axis, E the Young’s modulus and Cs the warping constant [4]. The latter is calculated as Cs = Z Ω (φps )2 dΩ Note that the warping term disappears in equation 2-7 when N.K. Teunisse (2-8) dθ dx is constant. Master of Science Thesis 2-5 Definition of non-linear torsion 9 Figure 2-5: Normal and shear stresses due to warping [4]. 2-5 Figure 2-6: Non-uniform torsion of I-beam [2]. Definition of non-linear torsion Beside uniform and non-uniform torsion, there is also non-linear torsion. Non-linear torsion becomes significant when the twist angle is not small. To understand this, one has to see the material as a collection of axially directed fibers [12]. When the structure is twisted, the fibers farthest away from the twisting axis will shorten the most. The shortening happens due to the longer helical path the fibers have to describe after twist compared with the straight path before twist. The fibers close to the axis of twist will shorten significantly less. This is shown for a 180 degrees twisted thin plate in fig.2-7a. The blue line represents the contour of the plate when it is represented by a collection of fibers. Yet, the structure is not a collection of loose fibers, so shear stresses between the fibers are present. Therefore, the cross-section tends to remain plane and the fibers are elongated such that the total length of the beam remains the same after twist. Subsequently, this causes a longitudinal axial force resultant at the beam end and equilibrium dictates that this is not possible without an externally applied axial force. Consequently, the beam shortens to lower the stress in the longitudinal direction until the beam is in equilibrium again. Finally, this results in the orange shape provided by fig.2-7a. Since the fibers are helical, the stresses inside these fibers generate both an axial as a tangent force resultant (Ft ). This is shown in fig.2-7b. The tangent component of all fibers leads to an additional torque, referred to as the non-linear Wagner torque [12]. The equation describing both the linear and the non-linear torsion holds Mx = GIt dθ d3 θ 1 dθ − ECs 3 + EIn dx dx 2 dx 3 (2-9) where In represents the non-linear Wagner section constant In = Ipp − Ip2 A (2-10) with Ipp = Ω (x2 + y 2 )2 dΩ as fourth moment of area and Ip = Ω (x2 + y 2 )dΩ as polar moment of area [12]. The first term of In represents the elongation of the fibers to their original longitudinal position while being twisted. The second term represents the contraction of the beam end on its whole. The non-linear term was not taken into account by Vlasov [11], however its importance is shown by Ghobarah [13] and many other works when the angle of twist is not small anymore. R Master of Science Thesis R N.K. Teunisse 10 Torsion of thin-walled structures Before twist After twist with no shear stress After twist with shear stress a0 Axis of twist a0 δz After twisting Ft Axis of twist Longitudinal fibre before twisting (a) Shortening effect of 180 degrees twisted plate based on figure by Trahair [12]. The lines represent the contour of the plate seen from a top view. σn δA (b) Rotation during twist and the corresponding tangent force component. Based on figure by Trahair [12]. Figure 2-7: Explanation of non-linear torsion. 2-6 Introduction to flexural-torsional coupling A structure that is solely subject to a torque, but both twists and bends, shows flexuraltorsional coupling. The flexural-torsional coupling is induced by the non-uniformity of warping as shown by Kim and Kim [14]. Regarding the research scope In this work, flexural-torsional coupling must be prevented since the flexural displacements do not contribute in driving the wings, thereby reducing the energy efficiency of the MAV. In general, flexure and torsion of bars are coupled, unless the centroid and shear center correspond with each other. The latter only holds for doubly symmetric cross-sections. Therefore, only doubly-symmetric cross-sections are studied in this work. 2-7 Conclusions and outlook Thin-walled structures are subject to warping, uniform torsion, non-uniform torsion and nonlinear torsion. This work considers solely thin-walled beams with an open doubly symmetric cross-section, because flexural-torsional coupling must be prevented. In the next chapter an analytical model is derived based on the definitions and scope discussed in this chapter. N.K. Teunisse Master of Science Thesis 3 Analytical model An analytical model is derived in this chapter to gain more insight in the behavior of beams under torsional loads and to obtain the dynamic responses with low computational costs. Low computational costs are important since the analytical model is derived for optimization purposes. The analytical model constructs the dynamic responses with low computation effort, because solely simple calculus is used to compute the approximated solutions. The set of equations that describes a beam subject to torsion and its corresponding initial and boundary conditions, is provided in section 3-1. Next, the choice of the perturbation method to approximate the solution of the governing equations is explained in section 3-2-1. Subsequently, a first order approximation of the analytical solution is derived in section 3-22. Lastly, remarks regarding the higher order approximations of the analytical solution are provided in section 3-2-3. 3-1 3-1-1 Derivation of the initial-boundary value problem Non-reduced governing equations The governing equations of a beam subject to non-uniform non-linear torsion are provided by Sapountzakis [15] and with a totally different approach confirmed by Sina [16]. The equations hold for arbitrary doubly-symmetric cross-sections, which is suited to this work since no flexural-torsional coupling is desired. In case the governing equations are needed for an arbitrary cross-section that is not necessarily double symmetric, one can find the equations in the work of Sapountzakis [17]. The partial differential equations of the initial boundary value problem of the beam used in this work, are formulated as ρAüm − EAu00m − EIp θ0 θ00 = n(x, t) (3-1) 3 ρIp θ̈ − ρCs θ̈00 − GIt θ00 + ECs θ0000 − EIpp (θ0 )2 θ00 − EIp u0m θ00 2 − EIp u00m θ0 = mt (x, t) + Master of Science Thesis ∂ [mw (x, t)] (3-2) ∂x N.K. Teunisse 12 Analytical model subject to the initial conditions (x ∈ (0, L)) um (x, 0) = ūm0 (x) u̇m (x, 0) = ū˙ m0 (x) (3-3) θ(x, 0) = θ̄0 (x) θ̇(x, 0) = θ̄˙ (x) (3-5) (3-4) (3-6) 0 together with the boundary conditions at the beam ends x = 0, L α1 (t)N + α2 (t)um = α3 (t) (3-7) β1 (t)Mt + β2 (t)θ = β3 (t) (3-8) β̄1 (t)Mw + β̄2 (t)θ 0 = β̄3 (t) (3-9) with αi , βi , β̄i (i = 1, 2, 3) as time dependent functions specified at the boundary of the bar. Furthermore N, Mt , Mw are the axial force, twisting and warping moments at the beam ends, respectively, and their lower case symbols in the governing equation are their distributed counterparts. A description of the remaining symbols is provided in table 3-1. The force and moments are described by N 1 = EAu0m + EIp (θ0 )2 2 (3-10) 1 Mt = GIt θ0 − ECs θ000 + EIp u0m θ0 + EIpp (θ0 )3 2 Mw = −ECs θ00 (3-11) (3-12) Note that um , the "average" axial displacement, is used instead of u, the axial displacement. This is done to decouple the axial displacement and the twist angle to its greatest extend. The definition of the "average" axial displacement is um = R Ω udΩ (3-13) A where Ω is the cross-section of the bar. In addition, the cross-section properties are calculated as follows It = Cs = Iy = Iz = Ip = Ipp = ∂φp ∂φp y + z + y s − z s dΩ ∂z ∂y Z ZΩ ZΩ ZΩ Ω 2 (3-14) (φps )2 dΩ (3-15) z 2 dΩ (3-16) y 2 dΩ (3-17) Z ZΩ Ω 2 y 2 + z 2 dΩ = Iy + Iz y2 + z2 2 dΩ (3-18) (3-19) where φps represents the primary warping function, also known as the sectorial coordinate. More information on this can be found in section 2-2. N.K. Teunisse Master of Science Thesis 3-1 Derivation of the initial-boundary value problem Symbol Unit um m G kg · s−2 · m−1 ρ kg · m−3 It Cs Description 13 Symbol Unit Description "Average" axial displacement θ rad Angle of twist Shear modulus E kg · s−2 · m−1 Young’s modulus Density Ip m4 Polar moment of area m4 St. Venant torsion constant Ipp m6 Fourth moment of area about the shear center m6 Warping stant A m2 Area con- Table 3-1: Symbol overview with units and description. 3-1-2 Reduced governing equations A common assumption among dynamic beam formulations is the neglect of the axial inertia term to reduce the set of governing equations. This assumption is proven justified by Sapountzakis [15], who compared the results with and without this assumption. Furthermore, the distributed axial load can be ignored since this is not applicable to this work. These assumptions turn equation 3-1 into Ip 0 00 θθ ∀x ∈ [0, L] A Furthermore, due to the absence of an applied axial load eq.3-10 can be written as u00m = − 1 Ip 0 2 (θ ) ∀x ∈ [0, L] 2A Consequently, the reduced initial boundary value problem, eq.3-2, is written as u0m = − (3-21) 3 ∂ ρIp θ̈ − ρCs θ̈00 − GIt θ00 + ECs θ0000 − EIn (θ0 )2 θ00 = mt (x, t) + [mw (x, t)] 2 ∂x subject to the initial conditions (x ∈ (0, L)) θ(x, 0) = θ̄0 (x) θ̇(x, 0) = θ̄˙ (x) 0 (3-20) (3-22) (3-23) (3-24) together with the boundary conditions at the beam ends x = 0, L β1 Mt + β2 θ = β3 β̄1 Mw + β̄2 θ where Master of Science Thesis 0 = β̄3 1 Mt = GIt θ0 − ECs θ000 + EIn (θ0 )3 2 Mw = −ECs θ00 Ip2 In = Ipp − A (3-25) (3-26) (3-27) (3-28) (3-29) N.K. Teunisse 14 Analytical model More information about In is provided in section 2-5. Moreover, the governing equation (eq.3-22), is compared with the torque-twist relationship in Trahair’s work [12] and they are equal to each other when considering a static case and no applied distributed bi-moment. 3-1-3 Non-dimensionalization of the reduced governing equations The reduced governing equation, eq.3-22, can be written as ∂2θ ∂4θ ∂4θ ∂2θ 3 ∂θ ρIp 2 − ρCs 2 2 − GIt 2 + ECs 4 − EIn ∂ x́ ∂ x́ 2 ∂ x́ ∂ t́ ∂ x́ ∂ t́ 2 ∂2θ ∂mw = mt + 2 ∂ x́ ∂ x́ (3-30) with x́ and t́ respectively as axial position and time. Making these parameters dimensionless yields t = ω t́ x́ x = L (3-31) (3-32) where ω is a characteristic frequency to be defined later and L is the length of the bar. This leads to the following conclusions regarding derivatives ∂n ∂ t́n ∂n ∂ x́n = = ∂n n ω ∂tn ∂n 1 ∂xn Ln (3-33) (3-34) Substituting eq. 3-33 and eq. 3-34 in eq. 3-30 yields ρIp ω 2 θ̈ − 1 ρCs ω 2 00 GIt 00 ECs 0000 3EIn 0 2 00 θ̈ − 2 θ + 4 θ − (θ ) θ = mt + m0w L2 L L 2L4 L (3-35) The prime denotes the derivative with respect to non-dimensional coordinate x and the dot the derivative with respect to the non-dimensional coordinate t. The linear non-dimensional free vibration problem for uniform torsion yields 2 ρIp ω θ̈ − GIt 00 θ =0 L2 (3-36) which results in θ̈ − 1 GIt 00 θ =0 ω 2 ρIp L2 (3-37) To keep the linear natural frequency for uniform torsion unity, ω is defined as the linear natural frequency for uniform torsion ω= s GIt ρIp L2 (3-38) Substitution of eq.3-38 in eq.3-35 and division by (GIt )/L2 yields θ̈ − N.K. Teunisse Cs 00 ECs 0000 3 EIn 0 2 00 L2 L 0 00 θ̈ − θ + θ − (θ ) θ = mt + m 2 2 2 Ip L GIt L 2 GIt L GIt GIt w (3-39) Master of Science Thesis 3-1 Derivation of the initial-boundary value problem 15 The first term represents the rotary inertia. The coefficients of the second term defines the magnitude of the warping inertia with respect to the rotary inertia. These coefficients will be small since the inertia generated by rotation is significant larger than the axial inertia caused by warping. The third term represents the linear torsional stiffness. The coefficients of the fourth term holds the ratio of the warping rigidity (ECs ) and the torsional rigidity (GIt ) of the cross-section. Therefore, it shows the importance of non-uniform torsion in comparison with uniform torsion. The coefficient of the last term at the left hand side expresses the magnitude of the non-linear stiffness with respect to the linear uniform torsional stiffness. The non-linear and warping terms scale with the length. Thus, an increase in the length of the beam yields a decrease in non-linear and warping terms. To make eq.3-39 more compact it is written as with θ̈ − µθ̈00 + ηw θ0000 − θ00 − ηn (θ0 )2 θ00 = κ1 mt + κ2 · m0w µ= Cs , Ip L2 ηw = ECs , GIt L2 κ1 = 3 EIn 2 GIt L2 L κ2 = GIt ηn = L2 , GIt (3-40) (3-41) (3-42) The boundary conditions at the beam ends x = 0, 1 are rewritten as β1 Mt + κ2 β2 θ = κ2 β3 (3-43) β̄1 Mw + κ3 β̄2 θ0 = κ3 β̄3 (3-44) with 1 Mt = θ0 − ηw θ000 + ηn (θ0 )3 3 Mw = −ηw θ00 1 κ3 = GIt (3-45) (3-46) (3-47) Note that the degree of non-linearities is only determined by the term ηn . This fact will be used during the optimization, see chapter 6, to make the response as non-linear as possible. Lastly, it is not expected that the axial inertia due to twist µ will be significant since the axial inertia was already be ignored. Therefore, µ is ignored in the rest of this work. 3-1-4 Final initial-boundary value problem This work considers a clamped-free beam with an applied torque at the free end. The final initial boundary value problem of the beam is formulated as θ̈ + 2ζ θ̇ + ηw θ0000 − θ00 − ηn (θ0 )2 θ00 = 0 subject to the initial conditions (x ∈ (0, 1)) Master of Science Thesis (3-48) θ(x, 0) = 0 (3-49) θ̇(x, 0) = 0 (3-50) N.K. Teunisse 16 Analytical model together with the boundary conditions at the beam ends x=0 x=1 → → θ = 0, θ0 = 0 1 θ0 − ηw θ000 + ηn (θ0 )3 = κ2 M̄ cos(Ωt), 3 (3-51) θ00 = 0 (3-52) where M̄ is the magnitude of the maximum applied torque and Ω is the frequency of the applied torque divided by ω to make it dimensionless. Note that in eq. 3-48 damping is added with ζ as damping ratio. 3-2 Perturbation analysis In section 3-1 the non-linear partial differential equation with the corresponding initial was found and boundary conditions that describes this problem. There are various ways to solve the problem. A first option is to numerically calculate the solution. However, the solution needs to be obtained with low computational costs due to the optimization that is performed later. A second option is to find the closed form solution of the problem, but this was found to be hard, if not impossible. Another option is to perform a perturbation analysis. This analysis is applicable if the problem cannot be solved exactly, but can be formulated by adding a "small" term to the mathematical description of an exactly solvable problem. This holds for the problem in this work since the exact solution of a related problem, the linear problem, is known. The non-linear term will be the added term. The perturbation method is chosen, because it is applicable to the problem in this work and the method is well described in literature. 3-2-1 Choice of the perturbation method An overview of the most commonly used perturbation methods is provided in this section. The discussed methods are the straight forward expansion, the Lindstedt-Poincaré technique, the method of renormalization, the method of multiple scales, variation of parameters and the method of averaging. All methods start with the unperturbed problem, meaning that the perturbation parameter equals zero and the governing equation represents the linear problem. Straight forward expansion The straight forward expansion is exactly what the name implies, namely an expansion of the solution in which is subsequently substituted in the differential equation. The straight forward expansion does not work for non-linear oscillators, because the correction terms grow in time. Resulting that the small correction term becomes the order of the main term or even larger, which does not agree with the conditions of the expansion. Such a straightforward expansion is valid only for times such that t < O(1) or t < O(− 1). The problem is that the expansion does not account for the dependency of the eigenfrequency on the non-linearities. N.K. Teunisse Master of Science Thesis 3-2 Perturbation analysis 17 Lindstedt-Poincaré The issue with the straight forward expansion is solved in the Lindstedt-Poincaré technique since the angular frequency in this method is a function of the non-linearity. Both the displacement and the angular frequency are expanded in which makes the angular frequency appear explicitly in the differential equation. After collecting the equations for each order of and solving them subsequently, the angular frequency is chosen such that the secular terms are annihilated. Method of renormalization This method is similar to the Lindstedt-Poincaré technique, however it introduces the nonlinear dependence of the frequency in the non-uniform expansion of the general solution obtained with the previously mentioned straight forward expansion technique. Variation of parameters This technique rewrites the equation, but does not make assumptions or expansions. It makes certain parameters, in the unperturbed equation, a function of time. This transforms the original higher-order equation to a set of first-order equations for the chosen parameters. The advantage of this transformation depends on the value of . If is small, the major parts of the chosen parameters vary more slowly than general solution. This makes it numerically advantageous to solve the transformed equations instead of the original equations, because a large step size can be used in the integration. The analytical advantage is utilized in the method of averaging. Method of averaging The method of averaging continues with the transformed equations obtained using the variation of parameters method. Because the chosen parameters vary slowly in time, they hardly change during the period of their circular functions. So they can be averaged over the period of the circular function making them constant. The averaging technique is usually referred to as the Van der Pol method or the Krylov-Bogoliubov method [18]. Method of multiple scales This method is used to obtain the analytical approximation of the solution of the non-linear system. This method is chosen, because it is significantly less limited than the LindstedtPoincaré method [19] and the method of renormalization. Moreover, not only does this method provide a uniform expansion, it also provides all the various nonlinear resonance phenomena [18]. Withal, it provides a good physical insight, contrary to the method of averaging which remains a more mathematical technique. Lastly the variation of parameters is a numerical technique making it unsuitable to obtain the approximated solution analytically. The method of multiple scales (MOMS) assumes that variations in the solution happen at different time scales. These different time scales are characterized by the symbols T0 , T1 , T2 , ... Master of Science Thesis N.K. Teunisse 18 Analytical model and are defined by the formula Tn = n t where t represents the time and is a small parameter. Applying this principle to the problem regarding torsion means that the function θ is not a function of x and t anymore, but of x, T0 , T1 , T2 , .... Using the chain rule, yields d dt d2 dt2 = = ∂ ∂ ∂ + + 2 + ... ∂T0 ∂T1 ∂T2 ! ∂2 ∂2 ∂2 ∂2 2 + 2 + ... + 2 + ∂T0 ∂T1 ∂T0 ∂T2 ∂T12 ∂T02 (3-53) (3-54) Next, an uniform approximate solution is sought in the form θ = θ0 (T0 T1 , T2 , ...) + θ1 (T0 , T1 , T2 , ...) + ... (3-55) After which the expanded derivatives are substituted in the original problem and the expansion of θ is substituted in the result. Subsequently, the coefficients of like powers of are collected and equated to zero. The equations are solved from θ0 to θi , where i is the desired order. Important is that the higher-order equations are solved by diminishing secular terms in the solution. Secular terms are terms that go to infinite when t → ∞. Secular terms will exist when the right-hand side contains terms that are equal to the solutions of the homogeneous problem. Secular terms can be eliminated in two ways: 1. Set the coefficient of each of the secular terms equal to zero. 2. Seek a particular solution with undetermined coefficients containing the secular functions. After substitution let the undetermined coefficients balance out the coefficients of the secular terms on the RHS. The process of eliminating secular terms is, essentially, ensuring that the right-hand side of the equation for θ0 is orthogonal to the periodic solutions (’eigensolutions’) of the differential equation on the left hand side. Next to secular terms there are also small divisor terms. These terms tend to infinite when the system is excited at the sub and super harmonic frequencies and the natural frequencies. One can deal with these frequencies by introducing the detuning parameter σ. This parameter is used as follows: ω = ωr + σ with ω as excitation frequency and ωr as the resonance frequency that causes small divisor terms. The small parameter ensures that ω stays close to ωr . The addition of the detuning parameter converts the small divisor terms to secular terms. When the system is excited in its primary resonance, θ and θ0 become large. The latter leads to substantial non-linear terms which are of the same or higher order as the linear terms. This causes problems with the order of the expansion and there are two ways to continue: 1. Make the non-linearities of O(0 ) → 2. Make the excitation force appear at O() Strong non-linear system → Weak non-linear system This same holds for the damping term. N.K. Teunisse Master of Science Thesis 3-2 Perturbation analysis 3-2-2 19 First order approximation of the analytical solution The goal of the perturbation method is to find a first order approximation of the analytical solution. The first order approximation yields a solution close to the linear solution. The solution differs in phase and amplitude depending on the magnitude of the applied torque due to the non-linearities. The desired triangular response over time emerges from the harmonic response thanks to the increased non-linearities at large twist angles. The system is a weakly non-linear system since the non-linear system heavily relies on the (unperturbed) linear system. Therefore, the twist angle is redefined as θ(x, t) = θ̂(x, t) (3-56) The governing equations (eq.3-48) becomes ¨ ˙ θ̂ + 2ζ θ̂ + ηw θ̂0000 − θ̂00 − 2 ηn (θ̂0 )2 θ̂00 = 0 (3-57) A quadratic expansion for θ̂ is used θ̂(x, t) = θ̂1 (x, t0 , t2 ) + 2 θ̂3 (x, t0 , t2 ) + . . . (3-58) Resulting in the expansion of θ θ(x, t) = θ1 (x, t0 , t2 ) + 3 θ3 (x, t0 , t2 ) + . . . (3-59) where t0 = t is the fast scale and t2 = 2 t is the slow time scale due to the quadratic expansion. The time scales lead to the expansions of the time derivatives d/dt = D0 + 2 D2 + . . . 2 2 d /dt = D02 2 + 2 D0 D2 + . . . (3-60) (3-61) with D0 = ∂/∂t0 and D2 = ∂/∂t2 . The damping factor ζ is rescaled as 2 ζ and the moment amplitude M̄ as 3 M̄ . Note that rescaling the moment amplitude and damping is only valid for weakly non-linear systems [18]. Thus, substituting the assumed expansion for θ, the rescaled damping, the rescaled forcing and the time derivatives into the governing equation, yields: Order D02 θ̂1 + ηw θ̂10000 − θ̂100 = 0 (3-62) subject to the boundary conditions θ̂1 = θ̂10 = 0 at x = 0 θ̂10 Order 3 − ηw θ̂1000 = 0 and θ̂100 = 0 at x = 1 D02 θ̂3 + ηw θ̂30000 − θ̂300 = −2D0 D2 θ̂1 − 2ζD0 θ̂1 + ηn (θ̂10 )2 θ̂100 (3-63) (3-64) (3-65) subject to the boundary conditions 1 (θ̂30 − ηw θ̂3000 ) = κ2 M̄ cos Ωt0 − ηn (θ̂10 )3 3 Master of Science Thesis θ̂3 = θ̂30 = 0 at x = 0 and θ̂300 = 0 at x = 1 (3-66) (3-67) N.K. Teunisse 20 Analytical model Solution of order - Free vibration problem The next step is to find the solution of the order which equals the free vibration problem D02 θ̂1 + ηw θ̂10000 − θ̂100 = 0 (3-68) subject to the boundary conditions θ̂1 = θ̂10 = 0 at x = 0 θ̂10 − ηw θ̂1000 = 0 and θ̂100 (3-69) = 0 at x = 1 (3-70) Let θ̂1 (x, t0 ) = eiωn t Θn (x) with ωn as the natural frequency of the nth mode. Consequently, eq.3-68 becomes 00 − ωn2 Θn + ηw Θ0000 (3-71) n − Θn = 0 Let Θn (x) = Aeλn x to obtain ηw λ4n − λ2n − ωn2 = 0 (3-72) The roots of eq.3-72 are λn = (±iλn1 , ±λn2 ) where s λn1 = −1 + 1 + 4ηw ωn2 , 2ηw p s λn2 = 1+ p 1 + 4ηw ωn2 2ηw (3-73) Thus the eigenfunctions are Θn (x) = An1 sin(λn1 x) + An2 cos(λn1 x) + An3 sinh(λn2 x) + An4 cosh(λn2 x) (3-74) They are subject to the boundary conditions in eq.3-69 and eq.3-70. This results in the following system of equations that needs te be solved [C]{a} = {0} (3-75) with [C] = [[C1 ] [C2 ]] (3-76) 0 1 λn1 0 [C1 ] = 2 2 −λn1 sin(λn1 ) −λn1 cos(λn1 ) ηw λ3n1 cos(λn1 ) + cos(λn1 )λn1 −ηw λ3n1 sin(λn1 ) − sin(λn1 )λn1 0 1 λn2 0 [C2 ] = 2 sinh(λ ) 2 cosh(λ ) λ λ n2 n2 n2 n2 3 3 −ηw λn2 cosh(λn2 ) + cosh(λn2 )λn2 −λn2 sinh(λn2 )ηw + sinh(λn2 )λn2 {a}T = An1 An2 An3 An4 n N.K. Teunisse o (3-77) Master of Science Thesis 3-2 Perturbation analysis 21 To obtain a non-trivial solution the matrix needs to have a determinant equal to zero det(C) = 0 (3-78) This results in a characteristic equation dependent of ωn and ηw which cannot be isolated for ωn . To determine ωn , the roots are sought for a specific ηw . Next, the eigenfrequency (ωn ) and the warping ratio (ηw ) are substituted in the coefficient matrix (eq.3-76). Since the determinant of the matrix is zero, the matrix is singular. To find a solution, An4 is set to unity. This is justified since the modes are rescaled later. Subsequently, the modes are determined and normalized. Therefore, they satisfy the condition ΘTn (x)Θn (x) = 1 (3-79) Finally, the solution of the homogeneous problem is θ̂1 = ∞ X eiωn t0 Θn (x) + n=1 ∞ X e−iωn t0 Θn (x) (3-80) n=1 Yet, the system is only excited in the nth torsional eigenfrequency which results in θ̂1 = eiωn t0 Θn (x) + e−iωn t0 Θn (x) (3-81) Rewrite order 3 Before the solution of the first order (eq.3-81) is substituted in the third order, it needs to be scaled by an unknown complex function to make the phase and amplitude dependent of the slower time scale t2 θ̂1 = An (t2 )eiωn t0 Θn (x) + A¯n (t2 )e−iωn t0 Θn (x) (3-82) Substituting the generated solution into the third order problem and rewriting the forcing frequency as Ω = ωn + 2 σ (fundamental resonance of the nth torsional mode), yields D02 θ̂3 + ηw θ̂30000 − θ̂300 = [−2iωn ζAn Θn − 2iωn D2 An Θn + 3ηn A2n Ān (Θ0n )2 Θ00n ]eiωn t0 + [ηn A3n (Θ0 )2 Θ00 ]e3iωn t0 + cc (3-83) subject to the boundary conditions θ̂3 = θ̂30 = 0 at x = 0 (3-84) 1 1 (θ̂30 − ηw θ̂3000 ) = [ κ2 M̄ eiσt2 − ηn A2n Ān (Θ0n )3 ]eiωn t0 − [ ηn A3n (Θ0n )3 ]e3iωn t0 + cc at x = 1 2 3 (3-85) θ̂300 = 0 at x = 1 with σ as the detuning parameter. To find an expression for An , the secular terms in eq.3-83 need to vanish. Moreover, the solution needs to satisfy the in-homogeneous boundary condition. Therefore, the inhomogeneous boundary conditions are transferred to the governing equation. Master of Science Thesis N.K. Teunisse 22 Analytical model Creating a homogeneous problem The easiest way to transfer the inhomogeneous boundary condition to the differential equation is to assume θ̂3 (x, t0 , t2 ) as θ̂3 (x, t0 , t2 ) = a0 (t0 , t2 ) + a1 (t0 , t2 )x + a2 (t0 , t2 )x2 + a3 (t0 , t2 )x3 + θ̃ˆ3 (x, t0 , t2 ) (3-86) where θ̃ˆ3 (x, t0 , t2 ) satisfies the homogeneous boundary conditions. Substituting eq.3-86 in the homogeneous boundary conditions of eq.3-84 and eq.3-85 yields θ̂3 = −3a3 x2 + a3 x3 + θ̃ˆ3 (3-87) Using this new θ̂3 in the non-homogeneous boundary equation (eq.3-85), results in a3 = [ 12 κ2 M̄ eiσt2 − ηn A2n Ān (Θ0n (1))3 ]eiωn t0 − [ 13 ηn A3n (Θ0n (1))3 ]e3iωn t0 + cc −3 − 6ηw (3-88) Note that a3 is not depending on spatial parameters since the boundary condition is evaluated at x = 1. So θ̂3 in eq.3-87 consists of θ̃ˆ3 that satisfies the homogeneous boundary conditions and the other terms that transforms the non-homogeneous boundary conditions to homogeneous boundary conditions. The governing equation becomes D02 θ̃ˆ3 + ηw θ̃ˆ30000 − θ̃ˆ300 = β1 (x, t2 )eiωn t0 + β2 (x, t2 )e3iωn t0 + with β̄1 (x, t2 )e−iωn t0 + β̄2 (x, t2 )e−3iωn t0 − D02 a3 x3 + 3D02 a3 x2 + 6a3 x − 6a3 (3-89) β1 (x, t2 ) = −2iωn ζAn Θn − 2iωn D2 An Θn + 3ηn A2n Ān (Θ0n )2 Θ00n β2 (x, t2 ) = ηn A3n (Θ0n )2 Θ00n (3-90) (3-91) To make it more clear, a3 is written as with a3 = γ1 (t2 )eiωn t0 − γ2 (t2 )e3iωn t0 + cc γ1 (t2 ) = γ2 (t2 ) = (3-92) 1 iσt2 2 κ2 M̄ e − ηn A2n Ān (Θ0n (1))3 −3 − 6ηw 1 3 0 3 3 ηn An (Θn (1)) −3 − 6ηw (3-93) (3-94) After rewriting eq.3-89 and some simple calculations, the governing equation yields D02 θ̃ˆ3 + ηw θ̃ˆ30000 − θ̃ˆ300 = [γ1 (ωn2 x3 − 3ωn2 x2 + 6x − 6) + β1 ]eiωn t0 subject to + [γ2 (−9ωn2 x3 + 27ωn2 x2 − 6x + 6) + β2 ]e3iωn t0 + cc (3-95) θ̃ˆ3 = θ̃ˆ30 = 0 at x = 0 θ̃ˆ30 − ηw θ̃ˆ3000 = 0 and θ̃ˆ300 = 0 at x = 1 N.K. Teunisse (3-96) (3-97) Master of Science Thesis 3-2 Perturbation analysis 23 Using mode orthogonality Since θ̃ˆ3 is subject to homogeneous boundary conditions, it can be expressed as a summation of all the modes ∞ X θ̃ˆ3 (x, t0 , t2 ) = gm (t0 , t2 )Θm (x) (3-98) m=1 with gm (t0 , t2 ) as the time dependent function that scales every mode accordingly. This, combined with eq.3-71, turns eq.3-95 into ∞ X 2 [D02 gm Θm + ωm gm Θm ] = F (x, t0 , t2 ) (3-99) m=1 where F (x, t0 , t2 ) represents the right hand side of eq.3-95. Next one can use the mode orthogonality to get rid of the infinite series D02 gn + ωn2 gn = Z 1 0 F · Θn dx (3-100) Consequently, the differential equation yields D02 gn + ωn2 gn = Z 1 0 γ1 (ωn2 x3 Z 1 0 − 3ωn2 x2 γ2 (−9ωn2 x3 + + 6x − 6)Θn + β1 Θn dx eiωn t0 + 27ωn2 x2 − 6x + 6)Θn + β2 Θn dx e3iωn t0 + cc (3-101) Eq.3-101 shows that eiωn t0 is the homogeneous solution of the left-hand side of eq.3-101. Removal of secular terms The secular terms of θ̃ˆ3 are eliminated to find An (t2 ). Secular terms appear when the homogeneous solution exists in the forcing term. To make the secular term vanish, the coefficient of eiωn t0 in eq.3-101 is set to zero Z 1 0 γ1 (ωn2 x3 − 3ωn2 x2 + 6x − 6)Θn dx + Z 1 0 β1 Θn dx = 0 (3-102) Filling in γ1 and β1 and using the orthogonality property of normalized modes, provide the following 1 iσt2 2 κ2 M̄ e − ηn A2n Ān (Θ0n (1))3 −3 − 6ηw Z 1 0 (ωn2 x3 − 3ωn2 x2 + 6x − 6)Θn dx − 2iωn ζAn − 2iωn D2 An + 3ηn A2n Ān Z 1 0 (Θ0n )2 Θ00n Θn dx = 0 (3-103) This equation contains only one unknown, namely the complex function An (t2 ). Therefore, An (t2 ) can be determined. Master of Science Thesis N.K. Teunisse 24 Analytical model Finding An To find An (t2 ), eq.3-103 is written as with κ2 cx1 M̄ iσt2 ηn cx1 (Θ0n (1))3 2 e − An Ān − 2iωn (ζAn + D2 An ) + 3ηn cx2 A2n Ān = 0 (3-104) −6 − 12ηw −3 − 6ηw cx1 = cx2 = Z 1 0 Z 1 0 (ωn2 x3 − 3ωn2 x2 + 6x − 6)Θn dx (3-105) (Θ0n )2 Θ00n Θn dx (3-106) To find a solution, one can write An in polar coordinates An (t2 ) = a(t2 )eib(t2 ) (3-107) Substitute this equation into eq.3-104 and multiply it by e−ib yields κ2 cx1 M̄ i(σt2 −b) ηn cx1 (Θ0n (1))3 e + 3ηn cx2 − −6 − 12ηw −3 − 6ηw ! a3 − 2iωn (ζa + ȧ) + 2ωn aḃ = 0 (3-108) with (˙) indicating the derivative with respect to t2 . Subsequently, the exponential term is converted to a trigonometric term κ2 cx1 M̄ (cos(σt2 − b) + i sin(σt2 − b)) −6 − 12ηw ηn cx1 (Θ0n (1))3 + 3ηn cx2 − −3 − 6ηw ! a3 − 2iωn (ζa + ȧ) + 2ωn aḃ = 0 (3-109) Separating the real and imaginary parts and rewriting the formula yields 3ηn cx2 ηn cx1 (Θ0n (1))3 + 2ωn 6ωn + 12ηw ωn κ2 cx1 M̄ cos(σt2 − b) − 12ωn + 24ηw ωn ! a3 = aḃ κ2 cx1 M̄ sin(σt2 − b) − ζa = ȧ −12ωn − 24ηw ωn (3-110) Next, this system is transformed into an autonomous system by introducing the transformation Consequently, ν(t2 ) = σt2 − b (3-111) ḃ = σ − ν̇ (3-112) ȧ and ν̇ are set to zero since this work solely considers steady state. After the substitution of b and the solely interest in the steady state situation, the system of equations in eq.3-110 holds 3ηn cx2 ηn cx1 (Θ0n (1))3 + 2ωn 6ωn + 12ηw ωn N.K. Teunisse ! a3 + σa = κ2 cx1 M̄ cos(ν) 12ωn + 24ηw ωn (3-113) −ζa = κ2 cx1 M̄ sin(ν) 12ωn + 24ηw ωn (3-114) Master of Science Thesis 3-2 Perturbation analysis 25 Squaring eq.3-113 and eq.3-114, and adding the results yields c2a1 a6 + 2ca1 σa4 + ζ 2 a2 + σ 2 a2 − c2a2 = 0 (3-115) with ca1 = ca2 = 3ηn cx2 ηn cx1 (Θ0n (1))3 + 2ωn 6ωn + 12ηw ωn κ2 cx1 M̄ 12ωn + 24ηw ωn (3-116) (3-117) The problem, eq.3-115, can be further reduced to a cubic polynomial problem by letting α = a2 . This results in c2a1 α3 + 2ca1 σα2 + (σ 2 + ζ 2 )α − c2a2 = 0 (3-118) This third order polynomial can be easily solved and the real positive values for a are applicable to the problem in this work. Once a is known, one can compute ν based on eq.3-114, namely ν = arcsin − aζ ca2 (3-119) Hereafter b is defined as b = σt2 − ν (3-120) Using eq.3-107, one can finally define An An (t2 ) = aei(σt2 −ν) (3-121) This was the missing piece to obtain the first order approximation, which can now be constructed. The solution Finally θ̂1 can be formulated and it holds 2 +ω )t −ν) n 0 θ̂1 = aei((σ 2 +ω )t −ν) n 0 Θn (x) + ae−i((σ Θn (x) (3-122) Which can be rewritten as θ̂1 = 2a cos((σ2 + ωn )t0 − ν)Θn (x) = 2a cos(Ωt0 − ν)Θn (x) (3-123) Resulting in the first order approximation of θ(x, t) θ(x, t) ≈ 2a cos(Ωt0 − ν)Θn (x) (3-124) Above derivation is repeated using trigonometric functions instead of exponential functions to minimize the chance of errors in the math and verifying the derived formulas for the first order approximation. The newly obtained approximation gave exactly the same results, so the likelihood of errors in the calculation is small. Master of Science Thesis N.K. Teunisse 26 3-2-3 Analytical model Higher order approximations of the analytical solution So far, the first order approximation of the analytical model is found. It consists of a cosine with a phase and amplitude which are scaled by the damping, the detuning parameter and the magnitude of the applied torque. In addition, at least two higher-order approximations are required to describe an approximated triangular function before it is suited to the optimization. More information on the approximation of a triangular function is provided in appendix B. The solution of the third order problem (eq.3-95), excluding the dependency of the slower time scales, is derived in appendix C. 3-3 Conclusions and outlook The first order approximation of the analytic solution of a beam subject to non-uniform nonlinear torsion is successfully derived. However, the higher-order approximations are necessary to optimize the response over time to a triangular function. In the subsequent chapter, a numerical method is developed that can describe a triangular dynamic response. N.K. Teunisse Master of Science Thesis 4 Finite element method In the previous chapter, an approximation of the analytical solution of a beam subjected to torsion was derived. Since it is an approximation, the results are verified with a more reliable, but computationally expensive, method: the finite element method. A brief introduction to the finite element method is provided in section 4-1. After this, the choice of the finite element package is explained in section 4-2. Lastly, the implementation of the finite element method is discussed in section 4-3. 4-1 Introduction to the finite element method The distribution of temperature in the piston of an engine, the distribution of stresses in a paving slab or the distribution of displacements of a thin-walled beam, are examples of field problems encountered in the field of engineering. The finite element method is used to find a numerical solution for field problems. 4-1-1 The method The finite element method solves the field problem by dividing the structure into small pieces, the so-called finite elements. As we transform the continuous system into a discrete system, we call this process ’discretization’. In each finite element a so-called field quantity, for example displacements, is allowed to have a simple spatial variation. The actual variation in the region spanned by an element is almost certainly more complicated. Yet, by decreasing the size of the elements, the error becomes negligible. This is under the condition that the element holds a correct description of the approximated field quantity. One can imagine that an incorrect relation between loads and displacements results in an overall erroneous solution. Every element contains a finite number of spatial points (nodes) that are all related to each other. Next, the field quantity is approximated for every node. The values between the nodes are interpolated. Collectively, the nodes represent the field quantity in space. Summarized, the essence of the finite element method is an approximation by piecewise interpolation of a field quantity [20]. Master of Science Thesis N.K. Teunisse 28 4-1-2 Finite element method The stiffness, damping and mass matrix By dividing the system into finite elements, the associated properties are also discretized. In this thesis, the discretized properties are the stiffness, inertia and damping of an element. The stiffness of an element relates the loads on the nodes to the displacements of the nodes and vice versa. The damping relates the loads on the nodes to the velocity of the nodes and vice versa. Lastly, the inertia relates the loads on the nodes to the acceleration of the node and vice versa. The stiffness of an element is represented by the element stiffness matrix. The sum of all elements stiffness matrices yields the global stiffness matrix. The same holds for the inertia and damping of the system, leading to the mass and damping matrix, respectively. 4-2 Choice of the finite element software The choice of the finite element software depends on their elements. First the beam elements of commercial finite element packages such as COMSOL, ANSYS and Abaqus were evaluated. The beam elements of COMSOL do not support constrained warping. The beam elements of ANSYS and Abaqus do not support non-linear warping. Therefore, the beam elements of the available commercial finite element software cannot be used for the problem in this work. Next, shell elements were considered to be inaccurate, because the warping of thin-walled structures is not taken into account correctly. This was stated in the work of Sapountzakis and Mokos [21] and mentioned again in the review paper by Sapountzakis [4]. Lastly, solid elements were evaluated. It was assumed that the thin-walled structure needs at least three solid elements in the thickness to calculate the shear deformation accurately and a large amount of elements in the length due to its slenderness. Therefore, solid elements were not used, because it was expected that this would increase the computational expenses too much. 4-2-1 Choice of developing custom finite element software Since the beam, shell and solid elements of COMSOL, ANSYS and Abaqus are not suited for the torsional problem in this work, a custom finite element package was developed using MATLAB. This enables the use of non-linear beam elements to minimize the number of required elements and computational expenses. 4-2-2 Erroneous assumption After developing the finite element method in MATLAB, it was found that the tapered structure used in the original article [21] could not be considered thin-walled at its small end. Modeling the tapered structure with shell elements led to the erroneous conclusion that shell elements cannot capture warping correctly. 4-3 Implementation of the custom finite element software The finite element method (FEM) package in MATLAB is based on a FEM code written by Paolo Tiso [22]. The implemented beam element is derived by Mohri [9] and supports warping, N.K. Teunisse Master of Science Thesis 4-4 Conclusions and outlook 29 large rotations (non-linear torsion) and flexural-torsional coupling. To capture the warping of the beam element, an additional (7th) degree of freedom (DOF) is added to the standard three translations and three rotations of each node. The additional DOF represents the torsional curvature ( dφ dx ). This leads to a 14x14 non-linear element stiffness matrix. The derivation of this matrix is completely redone to verify its correctness and provided by appendix A. Furthermore, a mass matrix is constructed to obtain the dynamic behavior of the system. The description of the mass matrix, including rotary inertia, is provided by appendix A-3-2. Finally, a damping matrix is required to include damping in the system. 4-3-1 Choice and definition of Rayleigh damping Rayleigh damping is chosen because of two reasons. First, Rayleigh damping is related to the damping ratio used in the analytical model. Therefore, the two models can be compared. Second, ANSYS uses Rayleigh damping, which makes it possible to compare ANSYS with the FEM code written in MATLAB. The Rayleigh damping matrix [C] is constructed by multiplying the two Rayleigh damping constants α and β with the mass and stiffness matrix, respectively. Consequently, the Rayleigh damping matrix [C] is written as [C] = α[M ] + β[K] (4-1) with [M ] as mass matrix and [K] as stiffness matrix. The first term on the right hand side of eq.4-1 is called the mass proportional damping and the second term is called the stiffness proportional damping. The values of α and β are related to the modal damping ratio ξi . ξi represents the ratio of the actual damping to the critical damping for a particular vibration mode i. If ωi is the natural circular frequency of mode i, α and β satisfy the relation [23] ξi = α ωi +β 2ωi 2 (4-2) Solely use of mass proportional damping throughout this work The stiffness matrix [K] changes during a non-linear analysis if geometric non-linearities are the only source of non-linearity. Therefore, the damping matrix [C] also varies during a nonlinear analysis if β 6= 0. On the contrary, the damping ratio of the analytical model remains constant. To make valid comparison between the analytical and finite element model, β must equal zero. In addition, the mass matrix [M ] remains constant during a non-linear analysis. Consequently, solely mass proportional damping is considered in this work. 4-4 Conclusions and outlook A finite element code is developed in MATLAB. In addition, it is concluded that solely mass proportional damping will be used as damping for the finite element method. In the next chapter, both the approximated solution of the analytical model as the results from the finite element model are compared with the responses obtained from ANSYS. Master of Science Thesis N.K. Teunisse 30 N.K. Teunisse Finite element method Master of Science Thesis 5 Model verification In the previous chapters, a finite element method (FEM) was implemented in MATLAB and a first order approximation of an analytical solution was derived. Both models aim to predict the dynamic response of a beam under torsion, each with their own advantages and disadvantages. To validate the correctness of the FEM in MATLAB, it is evaluated with the reliable finite element software ANSYS. The evaluation is performed for both static and dynamic cases. Although this work focuses on dynamics, static cases are also included to solely evaluate the stiffness of the implemented beam element. Next, the analytical model is evaluated with the validated MATLAB FEM. The evaluation is merely performed for dynamic cases since the analytical model is derived for dynamic problems. A brief description of the analytical model, the finite element method in MATLAB and the used ANSYS elements is provided in section 5-1. Subsequently, the FEM in MATLAB is compared with ANSYS for various static problems in section 5-2. The comparison identifies the limitations of the MATLAB FEM. Moreover, the limitations are explained in section 52-2. Next, the first two natural frequencies and corresponding modes of a thin-walled beam are determined by the FEM in MATLAB and the various elements in ANSYS. The obtained results are compared in section 5-3. The dynamic results of both finite element models are compared in section 5-4 and the conclusions are drawn in section 5-4-3. Section 5-5 deals with the analytical model. The implementation of the analytical model is explained in section 5-5-1 and obtaining the correct damping coefficient is discussed in section 5-5-2. The verification of the analytical model is discussed in section 5-5-3. It is investigated in section 5-6, whether the analytical model can provide the initial conditions for the time integration in MATLAB such that the response of the structure is (almost) immediately situated in steady state. Finally, the conclusion regarding the analytical model are provided in section 5-5-4. 5-1 Description of the evaluated models and elements The developed finite element code in MATLAB is evaluated in this chapter using ANSYS. Subsequently, the derived analytical model is evaluated in this chapter using the verified finite element code in MATLAB. A brief summary of the compared models and the various elements from ANSYS is provided below Master of Science Thesis N.K. Teunisse 32 Model verification • MATLAB non-linear beam element (MATLAB FEM): A beam element derived by Mohri [9] that supports non-linearities is implemented in a FEM package written in MATLAB. The element holds two nodes with each 7 degrees of freedom (DOF). More information is provided in chapter 4. • ANSYS BEAM188 element (ANSYS Beam): A beam element available in the commercial FEM package ANSYS, defined as BEAM188. The options of the element are set such that warping is supported, so it contains 7 DOF per node and 2 nodes in total. This element does not support geometric non-linearities. More information can be found in the element reference of ANSYS [24] • ANSYS SHELL181 element (ANSYS Shell): A shell element available in the commercial FEM package ANSYS, defined as SHELL181. It is used in its default configuration and it has 4 nodes with 6DOF per node. It supports geometric non-linearities and more information can be found in the element reference of ANSYS [24]. • ANSYS SOLID186 element (ANSYS Solid): A solid element available in the commercial FEM package ANSYS, defined as SOLID186. It is used in its default configuration and it has 20 nodes with 3DOF per node. It supports geometric non-linearities and more information can be found in the element reference of ANSYS [24]. • Analytical model: A first order approximation of the analytical solution of a beam subject to torsion. This approximation is derived with the help of a perturbation method. The model supports non-linearities and warping. More information can be found in chapter 3. 5-2 Static evaluation of the developed finite element model Static cases are studied in this section to solely evaluate the stiffness of the MATLAB nonlinear beam element with the elements in ANSYS is by focusing on static cases. Since damping and inertia are not present in statics, the deflections solely dependent on the stiffness. An advantage of considering static cases is that errors are identified with less effort. This is driven by less variables that are dealt with, thanks to the absence of damping and inertia. Furthermore, lesser code is required for the static calculations than for the dynamic calculations. 5-2-1 Evaluated geometries A rectangular and an I-shaped cross-section are considered for evaluation. First a free-clamped plate used by Trahair [12] is highlighted to illustrate the problem. Next, several plates with different geometric features are studied by varying the thickness, width and length. A focus was set on varying the length/width ratio and the width/thickness ratio. The conclusions regarding the validity of the MATLAB FEM for the various geometries are presented. Subsequently, an I-shaped cross-section is studied. First, an I-beam is studied that is also used in the dynamic comparison. Next, the flange thickness, the wall thickness and the length of the beam are varied. Again, the conclusions regarding the validity of the MATLAB FEM for the various geometries are presented. In addition, the evaluated beams are made of structural N.K. Teunisse Master of Science Thesis 5-2 Static evaluation of the developed finite element model 33 steel. Therefore, the beam has a Young’s modulus of 200GPa, a shear modulus of 77GPa and a density of 7800kg/m3 . Geometry: Plate A rectangular plate with a width of 0.2m, a thickness of 0.01m and a length of 1m is modeled with MATLAB and ANSYS for both the linear and non-linear case. The plate is clamped at one side and free at the other. A torque is applied to the free end. The clamp is modeled such that the warping can be constrained or unconstrained, so both cases can be tested. The results are shown in fig.5-1a and fig.5-1b. The linear beam with unconstrained warping provides an excellent match between the ANSYS beam, ANSYS shell and MATLAB model. The difference increases when non-linearities are considered. The angle of twist of the MATLAB beam element differs 3% with the shell elements in ANSYS for an applied moment of 9000Nm. Next, the shell and solid elements match really well for both the linear and non-linear case when the warping is constrained. Moreover, the ANSYS beam element and the MATLAB model match well for the linear case. However, there is a larger difference between the beam elements and the solid and shell elements, namely around 3% for the linear case and 6% for the non-linear case. In addition, the deflection curve in fig.5-1b is compared with the static response of Trahair’s work [12], which uses the same geometry. Both deflection curves match greatly. Different geometries are evaluated by varying the length/width and width/thickness ratio. The following conclusions can be drawn regarding the tip deflection when a beam represented by MATLAB elements is compared with a beam that consists of ANSYS shell elements: 1. A more slender beam provides a better match. 2. All linear results between the ANSYS and MATLAB elements show a good match (≤3% difference). 3. The non-linear results for an plate with unconstrained warping show a good match (≤4% difference) 4. For constrained warping and a plate with that is almost not slender anymore ( length width = 10), the difference varies from acceptable (6%) till great (1%). 5. If the length/width ratio is 5 and the warping is constrained, differences between 5% and 8% are found. This is in agreement with the assumption that the non-linear beam model must be slender. Furthermore, the (linear) ANSYS beam and the linear MATLAB beam gave a perfect match (<1% difference) for every tried configuration. Geometry: I-beam A free-clamped I-beam with a length of 6.1m, width of 0.43m, height of 0.549m, flange thickness of 0.02m and a wall thickness of 0.03m has a moment applied at its free end. The twist angle is measured at the free end and shown in fig.5-2 for the different models. The Master of Science Thesis N.K. Teunisse 34 M [Nm] - Moment applied at free end 1 ·104 0.8 0.6 0.4 MATLAB Beam elements - Linear MATLAB Beam elements - Nonlinear ANSYS Beam element - Linear ANSYS Shell element - Linear ANSYS Shell element - Nonlinear 0.2 0 1 M [Nm] - Moment applied at free end Model verification 0 0.2 0.4 0.6 0.8 1 1.2 θ [rad] - Angle of twist at free end 1.4 1.6 1.8 (a) No constrained warping. ·104 0.8 0.6 MATLAB Beam elements - Linear MATLAB Beam elements - Nonlinear ANSYS Beam element - Linear ANSYS Shell element - Linear ANSYS Shell element - Nonlinear ANSYS Solid element - Linear ANSYS Solid element - Nonlinear 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 1.2 θ [rad] - Angle of twist at free end 1.4 1.6 (b) Constrained warping. Figure 5-1: Comparison of the static deformation of a free-clamped plate in both MATLAB and ANSYS. N.K. Teunisse Master of Science Thesis 1.8 5-2 Static evaluation of the developed finite element model 35 linear results show that the MATLAB beam element and the ANSYS beam element match excellently (<1%). The same holds for a comparison of the solid elements (<1%) and shell elements (<2%) with the beam elements. The twist angle at the beam’s end differs 1% for a load of 3 · 105 Nm between the solid elements of ANSYS and the beam elements in MATLAB when non-linearities are considered. The use of MATLAB beam elements, including nonlinearities, yields an error of 4% at a load of 3 · 105 Nm compared to the shell elements in ANSYS. In addition, a comparison is made with the work of Mohri [9] to verify the model. The deflection curve of both works are nearly equal for a free-clamped I-beam with a length of 6.1m, width of 0.21387m, height of 0.549m, flange thickness of 0.0236m and a wall thickness of 0.01473m. To investigate the limits of the MATLAB model, the wall and flange thickness are varied. The obtained conclusions of the variation in thicknesses are as follows 1. As long as the wall is thicker than the flanges and warping is unconstrained, the linear and non-linear results are accurate (≤3%). 2. When the wall is thicker than the flanges and warping is constrained, the linear results (≤5%) and non-linear results (≤5%) are good. 3. When the wall and flanges have equal thickness and warping is unconstrained, the linear and non-linear results vary, but are still good (≤5%). 4. When the wall and flanges have equal thickness and warping is constrained, the linear results might become inaccurate (≤7%), but the non-linear results are still good (≤5%). 5. When the flanges are thicker than the wall, the linear and non-linear results become inaccurate (>8%) for both constrained and unconstrained warping. The percentages in the above results represent the difference in twist angle at the end of the beam between the different models. Furthermore, the results are obtained for a length/width ratio of 11 since this can be expected in the final design of the studied resonant body. In addition, a length/width ratio of 18 is tested and yields similar conclusions as for the length/width ratio of 11. Yet, the difference are significantly lower in comparison with the beam having a length/width ratio of 11. Besides, the models in ANSYS and MATLAB were all checked for mesh convergence. Master of Science Thesis N.K. Teunisse M[Nm] - Moment applied at free end 36 3 Model verification ·105 2.5 2 MATLAB Beam element - Linear MATLAB Beam element - Nonlinear ANSYS Shell elements - Linear ANSYS Shell elements - Nonlinear ANSYS Solid elements - Linear ANSYS Solid elements - Nonlinear ANSYS Beam elements - Linear 1.5 1 0.5 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 θ[rad] - Angle of twist at free end 1.6 1.8 Figure 5-2: Comparison of the static deformation of a linear and non-linear free-clamped I-beam in both MATLAB and ANSYS. N.K. Teunisse Master of Science Thesis 2 5-3 Comparison of the natural frequencies and modes (a) Wall thicker than flanges. Wall is straight. 37 (b) Flanges thicker than wall. The wall has slight S shape and flanges are not orthogonal to the mid-line of the wall. Figure 5-3: The deformed cross-section at the end of a clamped-free I-beam subject to torque. 5-2-2 Limitations of the developed finite element model The experiments showed that the wall of an I-beam should be thicker than the flanges to achieve a good match between the ANSYS shell elements and the MATLAB non-linear beam elements. This is explained by comparing the two opposite scenario’s, namely the first scenario where the beam has a thick wall with thin flanges and a second scenario with thick flanges and a thin wall. When a torque is applied to the I-beam, the wall will rotate in the first scenario and the flanges will have to follow. Yet, in the second scenario, the flanges are stiffer than the wall. Therefore, the flanges determine the motion and the wall has to follow. This will change the cross-sectional shape of the wall from a straight line, shown in fig.5-3a, to a S-shape, as can be slightly seen in fig.5-3b. More obvious is the twist of the flanges with respect to the mid-line of the wall center. Consequently, the cross-section deforms when angles of twist are large and the flanges are thicker than the wall. This is in contrast with the assumptions of the non-linear beam element in MATLAB, provided in appendix A, which state that the contour of the cross-section should be rigid in its own plane. Since this assumption does not hold anymore, it will make the beam model invalid and generate erroneous results. 5-3 Comparison of the natural frequencies and modes In this section the eigenfrequencies and eigenmodes are compared since this represents the ratio of the stiffness and the inertia of each models. The same I-beam as in previous section is used, so a length of 6.1m, width of 0.43m, height of 0.549, flange thickness of 0.02m and a wall thickness of 0.03m. This comparison extracts the eigenvalue information from an undeformed system. Therefore, only linear eigenmodes and eigenfrequencies are considered. The eigenfrequencies are provided in table 5-1 for both the analytical, MATLAB and ANSYS model. The results between the MATLAB and analytical model match perfectly, which is not surprising as the two models are based on the same governing equation. Furthermore, the ANSYS beam element shows an excellent match (<1% difference) with the other two beam models. The match between the beam models continues for the virtually identical mass normalized eigenmodes, which are shown in fig.5-4. On the contrary, the shell elements in ANSYS show an error of 4-5% in the eigenfrequencies and the solid elements lead to errors of 1% and 4%. Master of Science Thesis N.K. Teunisse 38 Model verification Rotational natural frequencies for free-clamped I-beam[Hz] MATLAB Analytic ANSYS Beam ANSYS Shell ANSYS Solid 12.995 56.357 12.995 56.357 13.024 55.870 12.721 54.479 12.899 55.053 Table 5-1: Rotational natural frequencies for different models of an free-clamped I-beam. θ [rad] at free end 0.3 0.2 0.1 0 MATLAB FEM Analytical model ANSYS Beam 0 0.5 1 1.5 2 2.5 3 3.5 Axial position[m] 4 4.5 5 5.5 Figure 5-4: Comparison of the fundamental torsional mode using three different models: MATLAB FEM, analytical model, ANSYS beam. 5-4 Dynamic evaluation of the developed finite element model Previous sections compared the stiffness and the torsional eigenfrequencies of the MATLAB FEM and the different elements in ANSYS. This gave satisfying results, so the dynamic responses of the models are compared in this section. First is described which methods were contemplated to obtain the dynamic response of a beam in the MATLAB FEM while keeping the optimization in mind. Next, a comparison is made between the results of ANSYS and MATLAB FEM for different cross-sections. Finally, a conclusion is drawn about the validity of the MATLAB FEM in the field of dynamics. 5-4-1 Computational procedures regarding dynamic analysis Originally the dynamic response of the FEM would be used in the optimization, however at the end another method was chosen. Yet, the choice of the method that generates the dynamic response is closely connected to the optimization. There are several methods that are used to obtain the dynamic behavior of a beam subject to a harmonic torsional load. The most beneficial method is using a response spectrum analysis, because this method directly obtains the amplitudes and frequencies of the different sines that shape the steadystate response. In the optimization the amplitudes and frequencies can be monitored, so the steady-state response can efficiently be optimized towards a triangular dynamic response. A less efficient way is to perform a harmonic analysis to obtain the steady-state response, thereby N.K. Teunisse Master of Science Thesis 6 5-4 Dynamic evaluation of the developed finite element model 39 skipping the transient state. Subsequently, a fast Fourier transformation can be applied to the dynamic response. This yields, again, the amplitudes and frequencies of the steady-state response. Yet, both methods are not applicable to the problem in this work since geometric non-linearities are included. The stiffness of the studied structure in this work is dependent on the displacements, thus time history is needed. Therefore, the time-integration is used. The time-integration is performed with help of the Newmark scheme employing the average constant acceleration algorithm without damping (γ = 12 , β = 14 ). 5-4-2 Evaluated geometries Several geometries are used for the comparison between MATLAB and ANSYS when considering dynamics. The different geometries are provided in table 5-2. All simulations used for the comparison make use of a Rayleigh damping of α = 2 and β = 0, more information regarding Rayleigh damping is provided chapter 4. The geometries in ANSYS are meshed with shell elements. All beams are made of steel, thus a density of 7800kg/m3 and a Young’s modulus of 200GPa. To enable a good comparison between the results obtained with ANSYS and MATLAB the eigenfrequency are equaled. Therefore, the shear modulus is tuned in the MATLAB model which is originally set at 77GPa. The effect on the simulations is small since the maximum error in the fundamental resonance frequency of the different profiles in both models is 2%. Yet, it is necessary as the simulations can take up to 5s before steady state is reached and an one percent difference for a frequency of 13Hz causes a big mismatch at the end of the simulation. Equalizing the linear eigenfrequencies does not force both models to generate the same results since the non-linearities play an important role in shaping the dynamic results. Consequently, even though both models possess the same fundamental linear torsional resonance frequency, the comparison of the two different models is still a good test to see whether the MATLAB model produces accurate results. Geometry: I-beam The geometric features of the first considered I-beam are provided in table 5-2 and marked as Id 1. The beam is evaluated for the load amplitudes 5000Nm and 15000Nm. The results for ANSYS and MATLAB for a load of 5000Nm are provided by fig.5-5. It shows an excellent match in amplitude and phase. This match continues for a load of 15000Nm, see fig.5-6. In addition, the results for the profile marked with Id 2 also shows an excellent match in both the transient and steady state for the loads 2000Nm and 10000Nm. The results are shown in fig.5-7 and fig.5-8. A load amplitude of 10000Nm and 5000Nm was also tested for profile 1 and profile 2 respectively and both showed an equally good matches. Geometry: X-beam To challenge the FEM in MATLAB, a beam with an X-shaped cross-section is evaluated. This profile was chosen since it is doubly symmetric, but totally different than an I-profile. The geometric features are provided in table 5-2 marked as Id 3. The dynamic response of the twist angle at the free end, while the beam is subject to 1000Nm and 3000Nm driven at the first torsional frequency, is provided by fig.5-9 and fig.5-10. Both show an excellent Master of Science Thesis N.K. Teunisse 40 Model verification match in amplitude and phase during both the transient and steady state. Furthermore, a load amplitude of 2000Nm was tested and provided an equally good match. Geometry: Plate Finally, a plate is examined in both MATLAB and ANSYS. In ANSYS this gave some complications since the stiffness in the radial direction is minimal, especially for slender plates. Furthermore, ANSYS defines the shear center of this doubly symmetric profile not exactly at the centroid, but at 6 · 10−18 m next to the centroid. As explained in section 2-6, cross-sections that have a shear center and centroid that do not coincide, will exhibit flexural-torsional coupling. In general, this numerical mistake does not cause any problem, because the stiffness and damping make sure that this extremely small coupling stays unnoticed. However, in this case we have low damping and low stiffness in the direction of the flexural movement. This creates a positive feedback loop since the flexural displacements strengthens the flexural-torsional coupling. Over time, this loop excites the flexural modes and thereby altering the response. High values of the mass proportional damping (see chapter 4) are necessary to diminish the coupling. Yet, high values of the mass proportional damping also cause a mismatch between ANSYS and MATLAB. It is still possible to compare both models for the transient part until the flexural-torsional coupling becomes significant in. The results of a short simulation of profile Id 4, see table 5-2, is presented in fig.5-11. The results show an moderate match in amplitude and a slight phase shift. For this simulation a damping factor (α) of 2 was used, like in previous simulations, and a moment of 500Nm driven at the first torsional natural frequency was exerted at the free end of the beam. 5-4-3 Summary of the evaluation of the finite element model The MATLAB FEM is an accurate predictor of the dynamic behavior of doubly symmetric beams subject to an external torque driven at the first torsional natural frequency. During this comparison ANSYS is taken as a benchmark and all tests used Rayleigh damping with α=2 and β=0. Various load amplitudes and cross-sections were tested and the I-profiles and X-shaped profile gave an excellent match. Furthermore, a plate was tested too, but showed a lesser match. This is caused by the torsional-flexural coupling in the structure due to a numerical error in ANSYS. N.K. Teunisse Master of Science Thesis 5-4 Dynamic evaluation of the developed finite element model Id Cross-section profile 41 Specifications[m] • Height: 0.549 • Width: 0.43 • Flange thickness: 0.02 1 • Wall thickness: 0.03 • Length: 6.1 • Height: 0.4 • Width: 0.4 • Flange thickness: 0.015 2 • Wall thickness: 0.025 • Length: 6.1 • Flange length: 0.2 • Flange thickness: 0.02 • Total width: 0.42 3 • Total height: 0.42 • Length: 4.5 • Width: 0.3 • Height: 0.03 4 • Length: 3 Table 5-2: The different geometries used to compare the dynamic results of ANSYS with the implemented MATLAB FEM. Master of Science Thesis N.K. Teunisse θ[rad] at free end 42 Model verification ANSYS MATLAB 0.5 0 −0.5 0 0.5 1 1.5 2 Time [s] 2.5 3 3.5 4 θ[rad] at free end Figure 5-5: Comparison of the dynamic response of profile Id 1 subject to 5000Nm driven at the first torsional natural frequency. 1 ANSYS MATLAB 0 −1 0 0.5 1 1.5 2 Time [s] 2.5 3 3.5 4 θ[rad] at free end Figure 5-6: Comparison of the dynamic response of profile Id 1 subject to 15000Nm driven at the first torsional natural frequency. ANSYS MATLAB 0.5 0 −0.5 0 0.5 1 1.5 2 Time [s] 2.5 3 3.5 4 θ[rad] at free end Figure 5-7: Comparison of the dynamic response of profile Id 2 subject to 2000Nm driven at the first torsional natural frequency. ANSYS MATLAB 1 0 −1 0 0.5 1 1.5 2 Time [s] 2.5 3 3.5 Figure 5-8: Comparison of the dynamic response of profile Id 2 subject to 10000Nm driven at the first torsional natural frequency. N.K. Teunisse Master of Science Thesis 4 θ[rad] at free end 5-4 Dynamic evaluation of the developed finite element model 43 ANSYS MATLAB 0.5 0 −0.5 0 0.5 1 1.5 2 Time[s] 2.5 3 3.5 4 θ[rad] at free end Figure 5-9: Comparison of the dynamic response of profile Id 3 subject to 1000Nm driven at the first torsional natural frequency. 1 ANSYS MATLAB 0 −1 0 0.5 1 1.5 2 Time[s] 2.5 3 3.5 4 θ[rad] at free end Figure 5-10: Comparison of the dynamic response of profile Id 3 subject to 3000Nm driven at the first torsional natural frequency. 0.5 ANSYS MATLAB 0 −0.5 0 0.1 0.2 0.3 0.4 0.5 Time [s] 0.6 0.7 0.8 0.9 Figure 5-11: Comparison of the dynamic response of profile Id 4 subject to 500Nm driven at the first torsional natural frequency. Master of Science Thesis N.K. Teunisse 1 44 5-5 Model verification Dynamic evaluation of the derived analytical model It was concluded in previous section that the MATLAB FEM generates accurate dynamic results. In this section, the quality of the analytical model is verified with the help of the MATLAB FEM. Being a first order approximation, the analytical model can only describe a sine or cosine. Therefore, the focus in the verification lies on the correct amplitude and phase shift caused by the non-linearities. This section starts with the implementation of the model in MATLAB. Next, the determination of the correct damping ratio is discussed. This is an issue since the Rayleigh damping does not convert itself to the correct damping ratio. Subsequently, the analytical model is compared with the MATLAB FEM for different geometries. Finally, a conclusion follows whether the analytical model provides a good approximation of the dynamic response. 5-5-1 Implementation of the analytical model in MATLAB The implementation of the analytical model in MATLAB is straight forward as the equations are already derived in chapter 3. The first step in MATLAB is to retrieve the fundamental eigenfrequency and corresponding mode. This is already computed by the MATLAB FEM 2 by solving the eigen problem [K] − ω1 [M ] {Θ1 } = {0} with [K] as stiffness matrix, [M ] as mass matrix, ω as first torsional eigenfrequency and {Θ1 } as vector holding the discretized fundamental torsional mode. The finite element code normalizes the eigenmode {Θ1 } with respect to the mass matrix. The analytical model requires the modes to have a length of unity. q To solve this the modes are multiplied with a factor (ρIp L). Also the natural frequencies are rescaled since the analytical model uses dimensionless frequencies. Therefore, all frequencies q GIt are divided by the linear natural frequency for uniform torsion ωL = ρIp L2 . Note that above steps skip the need to find and discretize the analytical modes. The next step is to obtain the analytical mode shape coefficients to compute the integrals cx1 and cx2 . This requires no root-finding methods since the first torsional frequency ω1 and the warping parameter ηw are known. λ1 and λ2 are written as λ1 = v q u u −1 + 1 + 4η ω 2 w 1 t 2ηw , λ2 = v q u u 1 + 1 + 4η ω 2 w 1 t 2ηw (5-1) Next the mode shape coefficients A1 , A2 , A3 , A4 are collected 2λ21 cos(λ1 )ηw + cosh(α) 4ηw ω12 + 1 + cosh(α) q A1 = A2 = −1 A3 = − A4 = 1 2λ1 ηw (sin(λ1 )λ1 + sinh(λ2 )λ2 ) (5-2) (5-3) 2λ21 cos(λ1 )ηw + cosh(α) 4ηw ω12 + 1 + cosh(α) q 2λ2 ηw (sin(λ1 )λ1 + sinh(λ2 )λ2 ) (5-4) (5-5) v q u u 1 + 4η ω 2 + 1 w 1 1 t α= √ N.K. Teunisse 2 ηw Master of Science Thesis 5-5 Dynamic evaluation of the derived analytical model 45 Above equations are the algebraic solution of eq.3-75 and are rescaled in MATLAB such that the corresponding mode is normalized. Finally, the spatial integrals cx1 (eq.3-105) and cx2 (eq.3-106) are algebraically pre-computed and filled in by MATLAB, because MATLAB is fast in discrete operations, but is slow in integral calculus. Next, the coefficients of the polynomial (ca1 ,ca2 ), provided in eq.3-115, are calculated together with the damping ratio ζ. The value of the detuning parameter σ is equaled to zero, because the beam is excited exactly at its fundamental linear torsional natural frequency. Large deflections may cause the eigenfrequency of the system to slightly shift, which results in a system that is not exactly excited at its new eigenfrequency. Yet, this work considers a first order approximation based on the complexly scaled first linear torsional mode. Therefore, the detuning parameter is zero. Subsequently, the positive real root of the polynomial is determined by MATLAB and the amplitude factor a of the response is computed. Next, the phase difference ν is calculated with the help of eq.3-119. Finally, the approximated solution in MATLAB is constructed as follows {θ} = 2a sin(ω1 ωL t − ν){Θ1 } (5-6) where the vector {θ} holds the displacements of the entire beam. The analytical solution is a sine instead of the cosine provided by chapter 3, because the beam is also excited by a sinusoidal varying load. A cosine excites a lot of modes at t = 0 since the maximum load is applied to an undeformed structure. This increases the transient. Therefore, a sine is chosen as forcing function. The above implementation of the analytical model in MATLAB breaks down when the warping parameter ηw becomes negligible (< 1 · 10−3 ) since this will blow up λ1 and λ2 , see eq.5-1. Very low values of ηw are found for plates and X-shaped cross-section. The solution is to neglect the warping. Consequently ηw diminishes and the free vibration problem (eq.3-68) reduces to D02 θ̂1 − θ̂100 = 0 subject to the boundary conditions (5-7) θ̂1 = θ̂10 = 0 at x = 0 θ̂10 = θ̂100 (5-8) = 0 at x = 1 (5-9) This problem is easy to solve and has the following solution θ̂1 = eiωn t Θn with Θn = A1 sin(ωn x) and 1 3 ωn = 0, π, π, . . . 2 2 (5-10) √ Next, the mode is normalized. This yields A = 2. Hereafter, MATLAB can continue its 1 √ normal routine with the values A1 = 2, A2 = A3 = A4 λ2 = 0 and λ1 = 12 π. 5-5-2 Obtaining the correct damping coefficient In chapter 4 we discussed the relation between the modal damping ratio and the Rayleigh damping parameters ξi = Master of Science Thesis α ωi +β 2ωi 2 (5-11) N.K. Teunisse 46 Model verification Since only the first torsional mode is triggered, it is expected that ξ1 is sufficient as overall damping ratio. Nonetheless, the linear and non-linear simulations do not match between the analytical model and FEM for the computed damping ratio. To find the correct damping ratio, the system is studied in its linear regime. The system behaves linear when the maximum applied torque is drastically lowered or the non-linear parameter In (eq.3-22) is set to zero. This is achieved in the MATLAB FEM by setting I2 Ipp = Ap . The maximum amplitude of this system in steady state solely depends on the torque amplitude and the damping ratio. Next, a linear-damped beam is excited in its first torsional frequency and the damping ratio is tuned such that it matches the solution of the MATLAB FEM. Calculating the dynamic response of the linear-damped beam takes time. To speed this up a reduced order model can be made by projecting the stiffness, mass and damping matrix on the first torsional mode. Finally, the linear dynamic response of a lightly damped beam can be calculated with almost no effort after which the damping ratio ζ can be scaled such that the systems posses the same damping properties. 5-5-3 Evaluated geometries After the implementation of the analytical model implemented in MATLAB, the results between the MATLAB FEM and the analytical model can be compared. The same geometries are evaluated as in the comparison between MATLAB and ANSYS. The different geometries are provided in table 5-2. All simulations, except the ones of the flat plate, make use of Rayleigh damping with α = 2 and β = 0. The dynamic responses of the plate used α = 10 and β = 0 as Rayleigh damping parameters. Although the Rayleigh damping was constant for the other three geometries, the damping ratio ζ differed for the different geometries. However, the damping ratio was determined for each cross-section with the technique described in section 5-5-2. All beams in this comparison are made of steel, resulting in a density of 7800kg/m3 , a shear modulus of 77GPa and a Young’s modulus of 200GPa. Lastly, note that only the steady-state responses are shown since the analytical model does not generate the transient. Therefore, there is no point comparing the transient. Geometry: I-beam The first geometry used for comparison is the I-profile indicated by Id 1 in table 5-2. The geometry is tested for a variety of load amplitudes causing small and big angles of twist. The solutions of two load amplitudes, 5000Nm and 70000Nm, are provided in this work, because the corresponding maximum angles of twist in steady state are 0.6rad and 1.5rad, respectively. The corresponding solutions are shown in figure 5-13 and figure 5-14. Both figures show an excellent match between the analytical approximation and the MATLAB FEM. This means the analytical model performs well for both smaller and bigger angles of twist. We can also conclude that the non-linearities are captured well by the analytical model. A similar great match is found for the I-profile marked as Id 2 in table 5-2. The dynamic response of the analytical model and MATLAB FEM are shown in figure 5-15 and figure 5-16 for a maximum torque of 1000Nm and 20000Nm. Summarized, the analytical model excellently captures both the linear as non-linear dynamic behavior of the tested I-profiles. This proves that the governing equations derived at the beginning of chapter 3 are correct. N.K. Teunisse Master of Science Thesis 5-5 Dynamic evaluation of the derived analytical model 47 θ[rad] at free end 1 0.5 0 −0.5 −1 5.7 MATLAB FEM Sine 5.71 5.72 5.73 5.74 Time[s] 5.75 5.76 Figure 5-12: The slightly more triangular steady-state response of profile Id 3 subject to 5000Nm compared with a sine. Geometry: X-beam The third profile that is examined, is an X-shaped profile. The geometric features are provided in table 5-2 (Id 3). This profile yields a low warping coefficient Cs and therefore also a low warping parameter ηw . To deal with this, the warping is ignored in the analytical model. The dynamic responses of the different models for a torque of 1000Nm and 5000Nm are shown in figure 5-17 and figure 5-18. The dynamic responses show a great match in phase, however the maximum amplitude of the analytical model is about 5% lower than the results of the FEM. Further investigation showed that the response of the MATLAB FEM is not exactly a sine anymore, but a slightly more pointed version of it. To make it more clear one period is taken from the latest response and compared to a sine with an equal amplitude in figure 5-12. The more pointed shape is caused by the non-linearities and represents the start of a more triangular signal. Since the analytical model is just a first-order approximation, it cannot capture these extra effects, which results in the generated error. Geometry: Plate Lastly a plate is investigated. Its geometric features can be found in the last row of table 5-2. The Rayleigh damping parameter α is set to 10 to make sure the plate reaches its steady state in a reasonable amount of time without changing the maximum twist angle in steady state too much. This step was not necessary for the other profiles, because their inertia, and therefore the mass proportional damping, is higher. Next, two load amplitudes are chosen that both result in a significant different steady-state response amplitude. A torque of 2000Nm and 10000Nm are applied to the end of the plate. The corresponding steady-state responses are shown in figure 5-19 and figure 5-20. Similar to the X-beam, the phase matches perfectly, but the amplitude of the analytical model is slightly lower than the MATLAB FEM. The differences are 4% and 6% for a torque of 2000Nm and 10000Nm, respectively. Further analysis showed that the non-linearities cause a more triangular dynamic response. Therefore, the first order approximation is not sufficient, which explains the small error. Master of Science Thesis N.K. Teunisse θ[rad] at free end 48 Model verification 0.5 MATLAB FEM Analytic model 0 −0.5 4 4.1 4.2 4.3 4.4 4.5 Time[s] 4.6 4.7 4.8 4.9 5 θ[rad] at free end Figure 5-13: Comparison of the steady-state response of profile Id 1 subject to 5000Nm driven at the first torsional natural frequency. MATLAB FEM Analytic model 1 0 −1 4 4.1 4.2 4.3 4.4 4.5 Time[s] 4.6 4.7 4.8 4.9 5 θ[rad] at free end Figure 5-14: Comparison of the steady-state response of profile Id 1 subject to 70000Nm driven at the first torsional natural frequency. 0.4 0.2 0 −0.2 −0.4 MATLAB FEM Analytic model 6 6.1 6.2 6.3 6.4 6.5 Time[s] 6.6 6.7 6.8 6.9 7 θ[rad] at free end Figure 5-15: Comparison of the steady-state response of profile Id 2 subject to 1000Nm driven at the first torsional natural frequency. MATLAB FEM Analytic model 1 0 −1 6 6.1 6.2 6.3 6.4 6.5 Time[s] 6.6 6.7 6.8 6.9 Figure 5-16: Comparison of the steady-state response of profile Id 2 subject to 20000Nm driven at the first torsional natural frequency. N.K. Teunisse Master of Science Thesis 7 θ[rad] at free end 5-5 Dynamic evaluation of the derived analytical model 49 0.5 MATLAB FEM Analytic model 0 −0.5 7 7.1 7.2 7.3 7.4 7.5 Time[s] 7.6 7.7 7.8 7.9 8 θ[rad] at free end Figure 5-17: Comparison of the steady-state response of profile Id 3 subject to 1000Nm driven at the first torsional natural frequency. 1 MATLAB FEM Analytic model 0.5 0 −0.5 −1 5 5.1 5.2 5.3 5.4 5.5 Time[s] 5.6 5.7 5.8 5.9 6 θ[rad] at free end Figure 5-18: Comparison of the steady-state response of profile Id 3 subject to 5000Nm driven at the first torsional natural frequency. MATLAB FEM Analytic model 0.5 0 −0.5 1.76 1.78 1.8 1.82 1.84 1.86 1.88 Time[s] 1.9 1.92 1.94 1.96 1.98 2 θ[rad] at free end Figure 5-19: Comparison of the steady-state response of profile Id 4 subject to 2000Nm driven at the first torsional natural frequency. MATLAB FEM Analytic model 1 0 −1 1 1.02 1.04 1.06 1.08 1.1 1.12 1.14 Time[s] 1.16 1.18 1.2 1.22 1.24 Figure 5-20: Comparison of the steady-state response of profile Id 4 subject to 10000Nm driven at the first torsional natural frequency. Master of Science Thesis N.K. Teunisse 50 5-5-4 Model verification Summary of the evaluation of the analytical model The analytical model is successfully implemented in MATLAB. The correct damping coefficient is determined by comparing the linear response of the structure between ANSYS and the MATLAB FEM. Next, the non-linear dynamic response of the analytical model is compared with the finite element model in MATLAB. An excellent match was found for the studied I-profiles, which proves that the governing equations derived at the beginning of chapter 3 are correct. A lesser, but still good, match was found for the X-shaped profile and the plate. The small mismatch in amplitude between the models is caused by the non-linearities that make the dynamic response more triangular. A triangular response cannot be captured by the analytic model since it can only describe a sine or cosine. The derived conclusions hold for both small and big angles of twist. 5-6 Reduction of the transient using the analytical model In this work, a beam is excited at its fundamental torsional resonance frequency by a forcing function, which causes the beam to vibrate. In the beginning, the deformations are small, however the deformations grow over time. Eventually, the deformations will show a periodic pattern if the forcing function remains the same harmonic. When the structure shows a periodic pattern in its deformations it is in steady state. The time period previously to the steady state, is called the transient. The transient is not of interest in this work, because we need a periodic deformation pattern, namely a triangular function with a constant maximum amplitude. Therefore, computing the transient part is a waste of resources. Yet, the stiffness of the structure at a certain time instance is dependent on the deformations of the previous time instance due to the non-linearities. Therefore, the deformations at time instance t cannot be calculated without the availability of the deformations at the previous time instance. If the analytical model is used to calculate the initial conditions for the time integration of the MATLAB FEM, MATLAB gets the required the information of the previous time instance. Consequently, the time integration can start from a point where the system is already in steady state and thus stays in steady state. Important is that the steady state of the analytical model equals the steady state computed by the FEM. Previous section showed that the steady-state responses of the analytical model match the responses of the MATLAB FEM excellently for the I-profiles. So the analytical model is capable of strongly reducing the transient in the dynamic response of the tested I-beams by providing the correct initial conditions to the time integration. When the steady-state response of a profile becomes more triangular, like the plate or the X-profile the match will be less and the dynamic response of the structure will start with a (small) transient again. However, the time to reach its steady state is still significantly less than if the time integration starts with an undeformed structure in rest. This method is only efficient if the analytical model computes the initial conditions with low computational costs, which is the case for the derived analytical model. The degrees of freedom that are important for the torsional problem are the twist angle θ(x, t), the torsional curvature θ0 (x, t) and the "average" axial displacement um (x, t). These degrees of freedom are explained in chapter 3 and are required as initial conditions of the time integration to skip the transient. θ(x, t) is determined by the analytical model and once known, its spatial derivative θ0 (x, t) is easily found. However, the "average" axial displacement um (x, t) is not N.K. Teunisse Master of Science Thesis 5-6 Reduction of the transient using the analytical model 51 immediately available. First the axial velocity, provided by eq.3-21, is integrated um = − 1 Ip 2A Z x 0 (θ0 )2 dx (5-12) θ(x, t) consists of a spatial and temporal part. Therefore, θ(x, t) holds θ(x, t) = θt (t)θs (x) (5-13) Subsequently, eq.5-12 is rewritten um = − 1 Ip (θt )2 2A Z x 0 (θs0 )2 dx (5-14) For the derived first order approximation the axial displacement yields um = − i 1 Ip h 2 2 4a sin (ω1 ωL t − ν) ψ(x) 2A (5-15) with ψ(x) = Z x 0 (θs0 )2 dx = Z x 0 (Θ0 (x))2 dx (5-16) Finally the necessary displacements, velocities and accelerations of the analytical model for the FEM at t = 0 are θ(x, 0) = 2a sin(−ν)Θ(x) (5-17) θ̇(x, 0) = 2aω1 ωL cos(−ν)Θ(x) (5-18) θ̈(x, 0) = −2a(ω1 ωL )2 sin(−ν)Θ(x) (5-19) θ0 (x, 0) = 2a sin(−ν)Θ0 (x) (5-20) θ̇0 (x, 0) = 2aω1 ωL cos(−ν)Θ0 (x) (5-21) θ̈0 (x, 0) = −2a(ω1 ωL )2 sin(−ν)Θ0 (x) i 1 Ip h 2 2 um (x, 0) = − 4a sin (−ν) ψ(x) 2A i 1 Ip h 2 u̇m (x, 0) = − 8a ω1 ωL sin(−ν) cos(−ν) ψ(x) 2A i 1 Ip h 2 üm (x, 0) = − 8a (ω1 ωL )2 cos2 (−ν) − 8a2 (ω1 ωL )2 sin2 (−ν) ψ(x) 2A (5-22) (5-23) (5-24) (5-25) Next, the displacements, velocities and accelerations of the nodes are calculated since the finite element method is a discrete method. The result is shown in fig. 5-21 for the I-profile marked as Id 1 in table 5-2. Fig. 5-21 shows that the system is almost immediately in steady state. A slight variation in maximum amplitude is visible in the first second, but this biggest error is less than 2% of the maximum amplitude in steady state. Similar good results are found for the I-profile marked as Id 2 in table 5-2. In conclusion, the method proves to be very effective in reducing the transient. Master of Science Thesis N.K. Teunisse θ[rad] at free end 52 Model verification 1 0 −1 MATLAB FEM with analytic ICs 0 0.1 0.2 0.3 0.4 0.5 Time[s] 0.6 0.7 0.8 0.9 Figure 5-21: The dynamic response of profile Id 1 with initial conditions equal to the analytical model at t = 0. The beam is subject to 70000Nm and driven at its first torsional natural frequency. 5-7 Conclusions and outlook The finite element method in MATLAB accurately predicts the dynamic behavior of both I-beams and X-beams subject to an external torque driven at the first torsional natural frequency. Next, the analytical model provides a correct dynamic response if the non-linearities are small. An error in the amplitude arises when the non-linearities cause a more triangular dynamic response. Furthermore, the analytical model can be effectively used to reduce the transient of the dynamic response obtained with the finite element method in MATLAB. Above all, it is proven that the governing equations derived at the beginning of chapter 3 are correct. Consequently, the governing equation can be used for the optimization instead of the computationally expensive finite element model in MATLAB. The optimization, including the implementation of the governing equation in the optimization, is discussed in the subsequent chapter. N.K. Teunisse Master of Science Thesis 1 6 Optimization The optimization is the last step in this work and provides the answer to the key question: which cross-sectional features make the response over time as triangular as possible? The answer this, the formulation of the optimization problem is provided in section 6-1. Next, the approximations and the choice of the optimization algorithm are explained in section 6-2. The variation of the objective function in the design space is discussed in section 6-3. Subsequently, the optimal profile and the corresponding dynamic response are obtained in section 6-4. In addition, conclusions are drawn for the flat plate and X-shaped cross-sections based on the optimization results of the I-beam in section 6-5. The used approximations are validated in section 6-6 and improvements are suggested for an existing structure based on the optimization results in section 6-7. Finally, the optimized beam is tested in ANSYS in section 6-8. 6-1 Problem formulation The optimization is carried out for a clamped-free I-beam with a harmonically varying torque applied at the free end. The cross-sectional features are width b, height h, flange thickness tf and wall thickness tw . The governing equation of the torsional problem was found in chapter 3 and is written as θ̈ + 2ζ θ̇ + ηw θ0000 − θ00 − ηn (θ0 )2 θ00 = 0 subject to the boundary conditions at the bar ends x=0 x=1 → → θ = 0, θ0 = 0 1 θ0 − ηw θ000 + ηn (θ0 )3 = κ2 M̄ cos(Ωt), 3 θ00 = 0 with ηw = Master of Science Thesis ECs , GIt L2 ηn = 3 EIn , 2 GIt L2 κ2 = L GIt N.K. Teunisse 54 Optimization The coefficients ηw , ηn and κ2 depend on the length of the beam. To decouple the optimization results from its length, the cross-sectional features are made dimensionless by dividing them by the length. This yields b̃ = b , L h̃ = h , L t̃f = tf , L tw , L t̃w = ηn = 3 E I˜n , 2 GI˜t ηw = E C̃s , GI˜t κ̃2 = 1 ˜ GIt L3 with I˜n = I˜pp − I˜pp = I˜p = I˜y = I˜z = I˜t = C̃s = I˜p2 (6-1) Z Ã 2 ZΩ̃ ZΩ̃ ZΩ̃ Ω̃ ỹ 2 + z̃ 2 (6-2) ỹ 2 + z̃ 2 dΩ̃ = I˜y + I˜z (6-3) z̃ 2 dΩ̃ (6-4) ỹ 2 dΩ̃ Z Ω̃ Z Ω̃ dΩ̃ (6-5) ỹ 2 + z̃ 2 + ỹ φ̃ps 2 ∂ φ̃ps ∂ φ̃p − z̃ s ∂ z̃ ∂ ỹ dΩ̃ ! dΩ̃ (6-6) (6-7) where φ̃ps is the dimensionless warping function and Ω̃ the dimensionless area. During the optimization only the cross-sectional features of the I-profile are varied, so the Young’s modulus E, the shear modulus G and length L are kept constant. 6-1-1 Objective function The goal of the optimization is to obtain a triangular function over time for the twist angle at the free end of the beam in response to a harmonic excitation. Since a triangular function over time has a constant velocity between the successive peaks, it leads to a more energy-efficient flapping wing motion. A triangular response over time is possible thanks to the non-linearities that are inherent in the beam and contribute to a stiffer beam for higher twist angles. This is known as the hardening effect. The non-linear stiffness in the governing equation is represented by ηn and is defined with respect to the linear uniform torsional stiffness. Thus not with respect to the total linear stiffness since the warping stiffness is not included in ηn . To obtain a triangular response over time, the non-linear stiffness compared to the total linear stiffness needs to be high. Therefore, solely maximizing ηn is insufficient to obtain a triangular response over time. Also, maximizing the ratio of ηn and ηw , does not lead to the optimal cross-section, because both ηn and ηw are normalized with respect to the linear uniform torsional stiffness. So the ratio ηn /ηw does not account for the linear uniform torsional stiffness. An objective function f that maximizes ηn with respect to the uniform and non-uniform torsional stiffness, is f= N.K. Teunisse ηn 1 + ηw (6-8) Master of Science Thesis 6-1 Problem formulation 55 The 1 in the denominator of eq.6-8 represents the uniform stiffness. Note that eq.6-8 maximizes ηn , for small ηw , and maximizes the ratio between ηn and ηw , if warping dominates the linear stiffness. 6-1-2 Constraints and bounds The beam needs to satisfy the design constraints. The first constraint is a first torsional natural frequency Ω of 60Hz. A division by the first linear eigenfrequency for uniform torsion expresses Ω as the non-dimensional frequency Ω̄ s Ω̄ = Ω ρIp L2 GIt (6-9) Rewriting Ω̄ in length independent parameters yields s Ω̄ = Ω ρI˜p L2 GI˜t (6-10) Note that Ω̄ is length dependent since the first natural frequency depends on the length of the beam. Although the unconstrained optimum is valid for any beam length, the equality constraint restricts the length of the beam if the values of the cross-sectional features are bounded, as is the case in this work. The maximum of b̃ and h̃ is set to 10% of the beam length to ensure the beam’s slenderness, which is an assumption made in the MATLAB FEM. b̃ and h̃ are bounded below by 0.1% of the beam length. The minimum wall thickness t̃w and flange thickness t̃f is set to 0.01% of the beam length. This bound originates from the design requirement that a beam of 10cm need to have a minimal thickness of 0.1mm due to the feasibility of manufacturing the beam. The maximum thickness is set to 1% of the length. Furthermore, there are the constraints that prevent an impossible geometry, namely b̃ ≥ t̃w and h̃ ≥ 2t̃f . 6-1-3 Negative null form Finally, the optimization problem can be summarized in the negative null form i h ηn (q) minimize − q = b̃ h̃ t̃f t̃w q 1 + ηw (q) subject to g(q) ≤ 0 h(q) = 0 q ∈ χ ⊂ R4 q ≤ q ≤ q̄ ¯ " # t̃w − b̃ g(q) = 2t̃f − h̃ h(q) = ω̄1 − Ω̄ q = 0.001 0.001 0.001 0.001 ¯ h i q̄ = 0.1 0.1 0.01 0.01 h Master of Science Thesis i N.K. Teunisse 56 Optimization with ω̄1 as the non-dimensional first eigenfrequency of the system. 6-2 Implementation of the optimization objective and constraints In section 6-1 the objective, constraints and bounds of the optimization are set. This section discusses the implementation of the optimization objective and constraints. Furthermore, the choice of the optimization algorithm is discussed. 6-2-1 Derivation of the approximate model Until now, the cross-sectional properties were obtained with the help of ANSYS. Solving eq.6-1 until eq.6-7 with ANSYS is slow compared to solving them analytically. Therefore, the analytical solutions of the cross-sectional properties I˜n , I˜pp , I˜p , I˜y , I˜z and C̃s for an Ishaped cross-section are obtained. They are provided in appendix D. The St. Venant torsion constant I˜t does not have a closed form solution. An approximation of the torsion constant for thin-walled I-profiles yields [8] 2b̃t˜f 3 + h̃t˜w 3 I˜t = 3 (6-11) Eq.6-11 holds as long as the flanges are thin compared to their width (b̃/t̃f >10) and the wall is thin compared to its height (h̃/t̃w >10). To obtain the eigenfrequency of the beam without applying numerical root-finding techniques on the characteristic equation (eq.3-78), a fit is made of the curve relating ηw to the first nondimensional eigenfrequency ω̄1 . Both the computed points and the fit are shown in fig.6-1. The following power function is used for the fit ω̄1 = 2.495(ηw )0.6014 + 1.622 h i (6-12) The largest error in the range of interest (1 · 10−5 ≤ ηw ≤ 4) for ω̄1 is 3%. The largest error is less than 1% if 0.01 ≤ ηw ≤ 4. 6-2-2 Choice of the optimization algorithm Extensive research has been carried out to investigate the best algorithm for an optimization with and without the use of ANSYS. The conclusions are provided in appendix E. The investigation is superfluous, because the approximate model makes it feasible to calculate the objective function f for a fine grid of possibilities spanned by the lower and upper bounds of all four optimization parameters in a matter of seconds. The optimum is the lowest value in the objective grid and the optimal cross-sectional features are the grid coordinates. This "brute force" method was chosen, because it is simple to find the global optimum and the required time is low (<10s). Furthermore, it provides a good insight into the problem. The choice of the "brute force" method is under the assumption that the approximate model yields a good approximation. In section 6-6, the approximate model is verified. N.K. Teunisse Master of Science Thesis 6-3 The design space 57 ω̄1 6 4 Computed points Fitted function 2 0 0.5 1 1.5 2 ηw 2.5 3 3.5 4 Figure 6-1: The values of ω̄1 for varying ηw computed by a root-finding algorithm compared with the fitted function, described in eq.6-12. 6-3 The design space A grid is made of the feasible design space by dividing the span between the lower bound and upper bound of each parameter into 60 parts. Since there are 4 parameters (b̃, h̃, t̃f , t̃w ) the complete grid becomes a 4-dimensional tensor of 60x60x60x60. Next, a value of the objective function is calculated for each of these combinations. It is not possible to clearly plot the objective function for all the combinations of the optimization parameters, because one cannot visualize the complete 5D dataset. Therefore, the results are plotted in 3-dimensional figures with two parameters on the x-axis and y-axis and the objective function on the z-axis. The other optimization parameters are kept constant and are mentioned in the caption of the figure. The constant parameters are set to the values of the optimal profile, which is obtained in the next section. In this manner, it is easy to see how the objective function varies around the optimal point. Black asterisks are provided in the plotted meshes and represent the points that satisfy the eigenfrequency constraint. The constrained optimum is defined in every plot as the black asterisk with a red circle around it. The degree of non-linearity depends on the axial non-uniform shortening of the structure over the cross-section due to twist, see section 2-5. The axial shortening of a point on the cross-section increases for a larger distance between the point on the cross-section and the shear center. This is achieved by increasing either b̃ or h̃, or both. Fig. 6-2 shows the effect of b̃ and h̃ on the negative objective function. The following can be concluded • A wide and low I-profile drastically reduces the warping stiffness, because the perpendicular distance between the midline of the flange and the shear center is very low. The non-linearities on the other hand reduce less strongly, because the distance between the furthest points on the cross-section (the flange tips) remain high. This explains the local minimum. • A tall and narrow I-profile has a negligible amount of warping. Also, its uniform torsional stiffness is low due to the small flanges. This results in a low linear torsional stiffness. On the contrary, the non-linearities remain stronger due to the amount of maMaster of Science Thesis N.K. Teunisse 58 Optimization terial far away from the shear center. Consequently, it will dominate the linear torsion leading to a strong decrease of the negative objective function and the global minimum. • A tall and wide I-profile increases the warping stiffness strongly, because the warping function (φps ) depends linearly on the height and the width of the flanges. In addition, the warping constant depends quadratically on the warping function. So a wide and tall profile has a significant linear stiffness due to warping. Therefore, the non-linear to linear stiffness ratio is low and the negative objective function is high. Fig.6-2 shows that the global optimum is found for a narrow I-profile and not for a flat plate (which is the bounded minimum of b). A flat plat has negligible warping stiffness and a low uniform torsional stiffness. Adding flanges increases the warping stiffness, but it remains low for small flanges compared to the linear uniform torsional stiffness, which grows linearly. This makes the denominator of the objective function, eq.6-8, almost solely dependent on the linear uniform torsional stiffness. On the contrary, the non-linearities cannot be ignored for a flat plate and they grow faster than the uniform torsion stiffness. Thus, the non-linearities dominate the regime of narrow flanges. This leads to a rapid decrease of the negative objective function. A further increase in the flange width, results in a significant warping stiffness, thereby dictating the linear stiffness. Consequently, the negative objective function increases again, because the two linear stiffness components grow faster than the non-linear stiffness. Finally, this causes a global minimum at b̃ = 1.442 · 10−2 . The optimal unconstrained profile for a length of 0.1m exhibits the following cross-sectional features: a width of 1.442 · 10−3 m, a height of 0.01m, a flange thickness of 0.0001m and a wall thickness of 0.0001m. Note that the global minimum also holds for I-beams of other lengths if the non-dimensional bounds remain the same and the scaled beam is not subject to any constraints. Fig.6-3 shows the wall thickness and warping thickness against the negative objective function. The negative objective is minimized when the wall and flanges are as thin as possible. This is driven by the strong dependency (cubic) of the St. Venant torsion constant, eq.6-11, on the wall and flange thickness. The warping computed in this work is not affected by the thicknesses, because the warping function is only defined at the midline of the cross-section, thereby making it independent of the thickness. This is allowed because only thin-walled structures are considered in this work. The non-linearities are influenced by the distance between the points on the cross-section and the axis of twist. This distance is not heavily altered by changing the thicknesses. On the other hand, the change in thickness still has a firm effect on the non-linearities, because there is less material causing the hardening effect. Yet, the influence on the linear stiffness is more drastic. This results in an optimum at the lower bounds of the wall and flange thickness. N.K. Teunisse Master of Science Thesis 6-3 The design space 59 -Objective[-] 0 −1 −2 0 0.002 0.004 0.006 h[m] 0.008 0.01 0.01 0.008 0.006 0.004 0.002 0 b[m] Figure 6-2: The objective function plotted against the width (b) and height (h). The black asterisks represent the points where the first torsional eigenfrequency equals 60Hz, the asterisk with the red circle is the optimum. The flange thickness (tf ) is 0.0001m, the wall thickness (tw ) is 0.0001m and the length is 0.1m. -Objective[-] 0 −1 −2 0.001 0.0008 0.0006 tw [m] 0.0004 0.0002 0.0002 0.0004 0.0006 0.0008 0.001 tf [m] Figure 6-3: The objective function plotted against the flange thickness (tf ) and wall thickness (tw ). The black asterisks represent the points where the first torsional eigenfrequency equals 60Hz, the asterisk with the red circle is the optimum. The height (h) is 0.01m, the width (b) is 0.0028m and the length is 0.1m. Master of Science Thesis N.K. Teunisse 60 Optimization Figure 6-4: The profile that results in the most non-linear dynamic response subject to the upper and lower bounds, a length of 0.1m and a first torsional linear eigenfrequency of 60Hz. Its cross-sectional features are b=0.0028m, h=0.01m, tf =0.0001m and tw =0.0001m. 6-4 Optimal cross-section and corresponding dynamic response As described in the previous section, every point on the 4-dimensional grid represents a different cross-section since the grid coordinates are the cross-sectional features of an I-beam. Both the value of the objective function as the eigenfrequency of the cross-section are computed for every point on the grid. Every point that has an eigenfrequency between 58.2Hz and 61.8Hz is considered valid. Next, a minimum is found within this set of valid points. The corresponding grid coordinates represent the optimal cross-sectional features. Fig.6-4 shows the optimal cross-section for a structure that is made of polyethylene (E=2.5GPa, G=0.9GPa, ρ=1600kg/m3 ) and has a length of 0.1m. Polyethylene was chosen to reduce the eigenfrequency. The corresponding dynamic response, computed with the MATLAB FEM, is shown in fig.6-5 and is close to triangular. For ease of comparison, a sine is added to the figure and it is clearly visible that the dynamic response of the optimal cross-section is not sinusoidal. Fig.6-6 shows the velocity over time of the optimized cross-section and reveals that the velocity is more constant between successive peaks than it is for a cosine. The latter is the derivative of the sine in fig.6-5. Fig.6-6 also shows the region in which the amplitude of the angular velocity is within its top 30%. Since the velocity is symmetric in both directions, the top 30% in velocity of the optimized profile is plotted for the positive velocities and the top 30% in velocity of the cosine profile is plotted for the negative velocities. A quantitative analysis shows that the absolute velocity of the optimal profile is within its top 30% during 78% of its period. The absolute value of the cosine is within its top 30% during 48% of its period. Consequently, the time in which the absolute angular velocity remains within its top 30%, increases with 63% for the optimal profile compared to a profile with negligible non-linearities. Note that the optimal profile is tested for twist angles of 90 degrees, which is according to the project requirements. The results in fig.6-5 and fig.6-6 are retrieved for a high Rayleigh damping, α = 100. This converts to a damping ratio of 0.1347 using the theoretical relation described in eq.4-2. Yet, the fitted damping ratio, as described in section 5-5-2, is found to be 0.4358. Smaller damping values were not possible, because they gave non-symmetric or noisy responses. 6-5 Conclusions regarding rectangular and X-shaped cross-section The optimization and fig.6-2 showed that a flat plate is not the optimal solution if one aims to maximize the non-linearities. This is verified by a static comparison. A torque is applied to N.K. Teunisse Master of Science Thesis θ[rad] at free end 6-5 Conclusions regarding rectangular and X-shaped cross-section 61 1.5 0 −1.5 Optimal cross-section Sine 0.45 0.46 0.47 Time[s] 0.48 0.49 0.5 θ̇[rad/s] at free end Figure 6-5: Angular rotation of the optimal cross-section that is subject to 0.0005Nm driven at the first torsional natural frequency. The optimized response is computed by the MATLAB FEM. 500 Optimal cross-section Cosine Top 30%, optimal cross-section Top 30%, cosine 0 −500 0.45 0.46 0.47 Time[s] 0.48 0.49 Figure 6-6: Angular velocity of the optimal cross-section that is subject to 0.0005Nm driven at the first torsional natural frequency. The optimized response is computed by the MATLAB FEM. Master of Science Thesis N.K. Teunisse 0.5 62 Optimization Geometry Linear deflection Non-linear deflection Non-linearity ratio Optimal profile Flat plate 1.895 1.893 0.7027 0.7723 2.70 2.45 Table 6-1: Comparison of the linear and non-linear deflections of a flat plate and a beam with the optimal cross-section. the end of a beam with the optimal cross-section of the unconstrained optimization problem. The torque amplitude is set such that the linear response, including warping, results in a twist angle of approximately 1.9 rad. Next, the deflection of the non-linear response is determined for that same torque. The ratio of those deflections, defined here as the non-linearity ratio, provides the degree of the non-linearity in the response. The experiment is repeated for a flat plate with the same height, thickness and length. A different torque is applied to the flat plate to obtain a linear deflection of approximately 1.9 rad again. The deflections and the ratios are provided in table 6-1. The table clearly shows that the optimal profile behaves more non-linearly than the flat plate. Consequently, a rectangular cross-section is verified to not be the optimal cross-section. Since a flat plate is not the optimal structure to obtain a triangular dynamic response, it can be concluded that a -shaped cross-section is also not the optimal cross-section. After all, a beam with a -shaped cross-section consists of two perpendicular plates attached to each other in the middle. A more optimal beam would, for example, have a cross-section similar to a -shape. 6-6 Validation of the approximate model Thus far, the results have been obtained with the help of a model that approximates the St. Venant torsion constant It and the warping constant Cs . ANSYS is able to calculate these cross-sectional properties more accurately, because it discretizes the cross-section and solves the differential equations with a finite element method. Obtaining the cross-sectional properties from ANSYS takes significantly more time than is required by the approximate model. Consequently, a coarser mesh is used for the computation of the error between the approximate model and ANSYS. The allowed span of each parameter is divided into 10 steps. The relative error of the objective function between the two models is computed and shown in fig.6-7. For a varying width and height, the errors remain below 4%, except when the width (b) and height (h) equals 0.0001m and equals 0.0012m, respectively. For this case, the width is the same as the wall thickness, so the cross-section is rectangular. The height to thickness ratio of this cross-section is 12. This is near the limit of being considered thin-walled (height/thickness=10) and thus results in an error of 5.2%. Furthermore, the flanges are not thin-walled for a 0.0012m wide I-profile, because the width to thickness ratio of the flanges is 5.5. This slightly increases the error as is visible in fig.6-7a. Moreover, in the neighborhood of the global and local minima the error is less than 0.5%. Even for a flat plate that has a height h of 0.01m the approximate model is just 0.5% off. The error in the objective function is also calculated for a varying flange and wall thickness, see fig.6-7b. Again, as long as the N.K. Teunisse Master of Science Thesis 63 0 −5 0.01 0.005 h[m] 0 0.005 0.01 Error in objective[%] Error in objective[%] 6-7 Improving the manufactured designs 0 −20 0 b[m] (a) Varying width (b) and height (h) of an Ibeam. The flange thickness (tf ) is 0.0001m, the wall thickness (tw ) is 0.0001m and the length is 0.1m. 0.0005 tw[m] 0.0005 0.001 0.001 0 tf[m] (b) Varying the flange thickness (tf ) and wall thickness (tw ) of an I-beam. The height (h) is 0.01m, the width (b) is 0.0028m and the length is 0.1m. Figure 6-7: The error of the objective function by the approximate model compared with ANSYS. width thickness ratio of the flanges and wall is higher than 10 the error is low (<5%). For lower ratios the error grows rapidly. 6-7 Improving the manufactured designs In the introduction of this thesis, it is mentioned that some designs of the Atalanta project are manufactured. One of them is shown in the introduction, see fig.1-2. The structure consists of 4 slats attached to a base plate at one side. At the other side, the opposite slats are connected to each other. A schematic view of the base plate is shown in fig.6-8a. The cross-section is similar to an X-shaped cross-section, except for the removed inner part. The inner part of the X-shaped cross-section can be largely neglected, because it contributes significantly less to the torsional stiffness than the outer part of the cross-section. Next, flanges are added to obtain a more triangular dynamic response of the structure. An optimization is performed for a fixed flange and wall thickness of 0.0002m to find the optimal flange width. The upper bound of the height is kept at 10% of the length, to not violate the slenderness assumption, and no restrictions are set to the eigenfrequency. The optimization to maximize the non-linearities yields a height width ratio of 4:1. The optimized slats are shown in fig.6-8b. 6-8 Optimized beam in ANSYS Attempts were performed to obtain the dynamic results from ANSYS using the profile shown in fig.6-4. Yet, ANSYS did not succeed in converging to a solution. The profile was also tested statically, but again ANSYS did not converge to a solution unless the twist angle was small (<5 degrees). It could be that the warping causes the flanges to buckle. Thin-walled structures are namely very susceptible to buckling and the top and bottom flanges are subject to in-plane compressive stresses due to warping. Master of Science Thesis N.K. Teunisse 64 Optimization (a) Original base plate. (b) Optimized design. Figure 6-8: Schematic view of the base plate of the manufactured design seen in fig.1-2. The bright red squares are the slats seen from top view. The dark red squares are added to maximize the non-linear behavior of the structure. 6-9 Conclusions The optimal profile is found that satisfies all the design requirements and, above all, yields a triangular response over time for the rotation of the free end when it is excited in its first torsional natural frequency. In addition, attempts were done to evaluate the optimal profile in ANSYS, but ANSYS did not converge to a solution. N.K. Teunisse Master of Science Thesis 7 Conclusions and recommendations The aim of this thesis was to find the optimal thin-walled beam that maximizes its nonlinearities such that the twist angle at end of the beam describes a triangular motion while the beam is excited in its fundamental resonance frequency by a harmonically varying torque. Before the goal could be pursued, it was necessary to analyze torsion of thin-walled structures. Therefore, both the causes and effects of the linear and non-linear torsion are investigated. In particular we found that the elongation of the "fibers", due to twist, causes the hardening behavior. This effect is strengthened if more "fibers" are located further away from the shear center. Moreover, the cross-section must be doubly symmetric to ensure that the harmonic varying torque does not excite any flexural modes. Next, a thorough investigation into finite element method packages that can deal with warping and non-linear torsion of thin-walled beams was carried out. Due to the incorrect conclusions in the work of Sapountzakis and Mokos [21] regarding the ability of shell elements to capture warping, the author concluded that the commercial FEM packages ANSYS, Abaqus and COMSOL cannot compute reliable results within a reasonable amount of time. Consequently, a FEM program is developed based on a FEM code written by Paolo Tiso [22] and used a beam element derived by Mohri [9] that can handle warping and non-linear torsion. Later, the author found that shell elements can also be employed, so the static and dynamic results of the implemented FEM package were compared with the results generated by ANSYS. An extensive validation followed, using various elements of ANSYS. The results between the two different FEM packages match well for both the static and dynamic benchmarks, provided that for an I-profile the wall is as thick or thicker than the flanges. Although the development of the implemented FEM package can thence be seen as useless, the opposite is true. The most valuable advantage of the developed FEM code is the lower computation time compared with ANSYS. Namely, the developed FEM code uses beam elements instead of shell elements, hereby reducing the degrees of freedom and therewith the computation time drastically. Another advantage that is less measurable is the extra insight gained from developing the code. Next, an analytical model was derived for optimization purposes. The reasoning behind this is the low computation cost compared to the finite element model. This matters because an optimization contains a lot of iterations and the required time is desired to be as low as possible. The governing equation of the torsional problem including warping and non-linearities was obtained from the literature and the boundary conditions stem from the same research context. Due to the nonlinearities, no closed-form solution was found, so a perturbation method namely, the Method Master of Science Thesis N.K. Teunisse 66 Conclusions and recommendations of Multiple Scales was applied to the equations. The first order approximation was successfully derived, however higher-order approximations are necessary to optimize the response over time to a triangular function. Subsequently, the analytical first order approximation was used to reduce the transient of the dynamic simulations by the MATLAB FEM. This was achieved by estimating the steady-state conditions with the help of the analytical model and passing this to the MATLAB FEM as initial conditions for the time integration. This concept worked very well and the transient almost completely disappeared. Yet, more important is the fact that these matching results also proved that the obtained governing equations are correct. The latter is important, because the optimization relies on the non-dimensional governing equation derived from the analytical model. Instead of performing an expensive optimization, several approximations were made and in a "brute force" manner, a fine grid of combinations was calculated. Research was conducted to investigate the best optimization algorithm, but there is no point in using optimizations algorithms when the objective function for the complete design space can be computed in a matter of seconds. This is of course under the conditions that the introduced approximations are valid and a comparison with ANSYS showed that this is indeed the case. The optimal profile found in this work satisfies all the design requirements and, above all, yields a triangular response over time for the rotation of the free end when it is excited in its first torsional natural frequency. Moreover, the time in which the absolute angular velocity remains within its top 30%, increases with 63% for the optimal profile compared to a profile with negligible non-linearities. Finally, a suggestion was made for an already-made structure to increase its geometric non-linearities, thereby achieving a more triangular response over time. 7-1 Recommendations As for every project a scope was set in this work to ensure that solutions are sought into the right direction and the project stays on track. One bound of the scope is that the cross-section is considered uniform over the cantilever span. However it would be interesting to extend the optimization to a beam with a varying cross-section over its length. Some more interesting topics regarding cross-section geometry that were not investigated in this project are listed below • Which optimal profiles and dynamic responses would arise considering other profiles than I-profiles or flat plates. • The effect of removing the inner part of the cross-section to save weight, especially for more complicated structures than flat plates. Moreover, it would be very interesting to see the results of a topology optimization over the entire beam. This will be very hard as both open and closed cross-sections need to be considered together with complexly varying cross-sections over the entire length of the beam. Another recommendation would be to actually manufacture the optimal I-beam and see how close the dynamic response approximates the desired triangular response over time. There is a chance the beam breaks before it reaches the 90 degrees of twist as the developed models does not consider buckling and maximum allowed stresses in the beam. That is definitely a point of attention if this work is to be continued. N.K. Teunisse Master of Science Thesis A Description of the non-linear elastic non-uniform torsional beam element This appendix describes the derivation of a non-linear elastic non-uniform torsional beam element developed by Mohri [9]. This element is used by the finite element method developed in MATLAB to calculate the response of the structure subject to torsional loads. The reason why this appendix is created, and not a simple reference is made to Mohri [9], is because certain steps are explained in more detail, some small mistakes in formulas by Mohri are corrected and no combination of papers needs to be advised. The element is based on the work of Attard [25], but is extended to comprehend large torsion by not limiting the trigonometric functions. So no assumptions are made on the torsion angle amplitude. Next the shortening effect, pre-buckling deflections and flexural-torsional coupling are naturally included. The element is set up as a 3D beam element with two nodes, where each node contains 7DOFs. The 7th DOF is necessary to deal with warping. In order to obtain the tangent stiffness matrix a classical incremental iterative Newton-Raphson procedure is assumed as solution strategy. The element contains also several assumptions: • There are no shear deformations in the mean surface of the section and the contour of the cross-section is rigid in its own plane. • The model will be applied to slender beams, which means that local and distortional deformations are not included. • Displacements and twist angles can be large, but bending rotations is assumed to be small. • Small deformation are admitted as an elastic material model is used. • Only constant load contribution will be considered and no load eccentricities are present The straight thin-walled beam has a slenderness L, an open cross-section A and a boundary surface ∂A. Master of Science Thesis N.K. Teunisse 68 A-1 Description of the non-linear elastic non-uniform torsional beam element Kinematics A Cartesian coordinate system is chosen for the three-dimensional space. Let us denote by x the initial longitudinal axis and by y and z the first and second principal axes. The origin of these axes is located at the centroid C and the shear center is denoted as S with coordinates ys and zs . Moreover this theory makes use of the sectorial coordinate ω, giving random point M at the section contour the coordinates (y, z, ω).Also the following trigonometric variables are introduced c = cos(θx − 1) (A-1) s = sin(θx ) (A-2) Next the displacements of point M yield uM vM wM = u − y(v 0 + v 0 c + w0 s) − z(w0 + w0 c − v 0 s) − ωθx0 (A-3a) = v − (z − zs )s + (y − ys )c (A-3b) = w + (y − ys )s + (z − zs )c (A-3c) where (·)0 denotes the derivative with respect to x, u is the axial displacement of the centroid and v and w are the displacements of the shear center in y and z direction. The non-linear (Green) strain-displacement relations of the non-vanishing strains are as follows xx = xy = xz = ∂u 1 + ∂x 2 1 ∂u + 2 ∂y 1 ∂u + 2 ∂z ∂u 2 ∂v 2 ∂w 2 + + ∂x ∂x ∂x ∂v ∂u ∂u ∂v ∂v ∂w ∂w + + + ∂x ∂y ∂x ∂y ∂x ∂y ∂x ∂w ∂u ∂u ∂v ∂v ∂w ∂w + + + ∂x ∂z ∂x ∂z ∂x ∂z ∂x " # (A-4a) (A-4b) (A-4c) For thin-walled beams it can be assumed that the axial displacement u is much smaller than v or w so that the products of the derivatives of u can be neglected in the strain-displacement 2 ∂u ∂u relation, meaning that ∂u ∂u ∂x ∂x , ∂y ∂x this assumption to equations A-4(a-c) leads to xx = xy = xz = ∂u 1 + ∂x 2 1 ∂u + 2 ∂y 1 ∂u + 2 ∂z ∂u ∂y + ∂v ∂x and ∂u ∂u ∂z ∂x ∂v 2 ∂w 2 + ∂x ∂x ∂v ∂v ∂v ∂w ∂w + + ∂x ∂y ∂x ∂y ∂x ∂w ∂v ∂v ∂w ∂w + + ∂x ∂z ∂x ∂z ∂x " # ∂u ∂z + ∂w ∂x . Applying (A-5a) (A-5b) (A-5c) Substituting equations A-3(a-c) into equations A-5(a-c) yields the following set of strain resultants 1 xx = − ykz − zky − ωθx00 + R2 θx02 (A-6a) 2 1 ∂ω 0 xy = − z − zs + θx (A-6b) 2 ∂y 1 ∂ω 0 xz = y − ys − θx (A-6c) 2 ∂z N.K. Teunisse Master of Science Thesis A-2 Equilibrium equations 69 The axial strain (eq. A-6a) consists of several terms, namely the first term is the membrane component , the second and third terms are the strains due to bending, the fourth term is caused by warping and the last term is the "Wagner strain" [12]. This term was not included by Vlasov [11] and creates the non-linear effect. Furthermore ky , kz are the beam curvatures and R defines the distance between the point M and the shear center S. These and the membrane component are defined as ky = w00 + w00 c − v 00 s kz = v + v c − w s 00 R 2 00 (A-7) (A-8) 00 2 2 = (y − ys ) + (z − zs ) 1 02 = u0 + v + w02 − ψθx0 2 (A-9) (A-10) The variable ψ associated with the membrane component is defined by ψ = ys (w0 + w0 c − v 0 s) − zs (v 0 + v 0 c + w0 s) (A-11) and includes the flexural-torsional coupling, because (w0 + w0 c − v 0 s) and (v 0 + v 0 c + w0 s) represent the slope between the longitudinal axis in the deformed state and the initial vertical and horizontal planes. A-2 Equilibrium equations To obtain the equilibrium equations the minimum total potential energy principal is used δU − δW = 0 (A-12) where U is the sum of the elastic strain energy and W is the potential energy associated with the applied loads. For simplicity this work only considers tractions and no body forces. The components of the resulting external forces Fxe , Fye and Fze are proportional to the loadfactor λ. The strain energy and the external load variations can be written as δU δW = Z Z (σxx δxx + 2σxy δxy + 2σxz δxz )dA dx LZ AZ = λ L ∂A (A-13) (Fxe δuM + Fye δvM + Fze δwM )ds dx (A-14) where σxx , σxy and σxz are the second Piola-Kirchhoff stress tensor components and δxx , δxy and δxz are Green’s strain tensor variations. We can also write the strain energy variation in terms of stress resultants and virtual strain deformations δU = Z L 1 N δ − My δky − Mz δkz + Msv δθ0 + Bω δθx00 + MR δ(θx0 )2 dx 2 Master of Science Thesis (A-15) N.K. Teunisse 70 Description of the non-linear elastic non-uniform torsional beam element where the stress resultants are defined as follows N Z = ZA My = σxx dA (A-16a) σxx zdA (A-16b) AZ Mz = − Bω = − Msv = Z MR = Z ZA A A A (A-16c) σxx ydA (A-16d) σxx ωdA σxz y − ys − ∂ω ∂z − σxy z − zs + ∂ω dA ∂y (A-16e) σxx R2 dA (A-16f) (A-16g) N is the axial force, My and Mz are the bending moments, Bω is the bimoment acting on the cross-section, Msv is the St-Venant torsion moment and MR is a higher order stress resultant called Wagner’s moment. The contribution of eccentric loads from centroid and shear point to second member and tangent stiffness matrix is not included, so the virtual work by the external loads is represented as δW = λ Z L Fxe δu + Fye δv + Fze δw + Mxe δθx + Mye δw0 + Mze δv 0 + Bωe δθx0 dx (A-17) with Mye and Mze as external bending moments, Mxe as the torque and Bωe as bimoment. Their definitions are Mye = −Fxe z (A-18a) Mze = −Fxe y (A-18b) Mxe = −Fxe ω (A-18c) Using matrix notations we can write the minimum total potential energy (eq. A-12) as Z L T {δγ} {S}dx − λ Z L {δq}T {Fe } + {δθ}t {Me } dx = 0 (A-19) with the following matrices {S}T {γ}T {q}T T {θ} T {Fe } T {Me } = {N = { = {u = {u 0 = {Fxe My Mz − ky v v − kz w 0 w Fye = {0 Mze 0 θx } θx0 Fze Mye Msv Bω θx0 θx00 00 00 v w Mxe } Bωe 0 0 (A-20) MR } 1 02 θ } 2 x (A-21) θx00 θx } (A-23) 0} (A-25) 0 (A-22) (A-24) (A-26) Next we assume small strains and a homogeneous and isotropic material, so we can make use of the Young’s modulus E and shear modulus G. Furthermore the Poisson ratio is considered N.K. Teunisse Master of Science Thesis A-2 Equilibrium equations 71 zero for the thin-walled structures. Consequently, the relationship between the stress vector components in terms of the deformation vector components are as follows N= My = Mz = 1 Exx dA = EA + EAIo θx02 2 ZA Z ZA A Exx zdA = −EIy ((ky − βz θ02 )) (A-27b) Exx ydA = −EIz ((kz − βy θ02 )) (A-27c) Z Msv = 2 A Bω = Z MR = Z A (A-27a) Gxz y − ys − ∂ω ∂z − Gxy z − zs + ∂ω ∂y dA = GJθx0 Exx ωdA = EIω (θx00 − βω θx02 ) (A-27d) (A-27e) 1 Exx R2 dA = EAIo − 2EIz βy kz − 2EIy βz ky − 2EIω βω θx0 + EIR θx02 2 A (A-27f) in the above equation the following geometric constants are used Iy = Iz = Iω = Z ZA ZA A z 2 dA (A-28a) y 2 dA (A-28b) ω 2 dA (A-28c) 1 y(y 2 + z 2 )dA − ys 2Iz A Z 1 z(y 2 + z 2 )dA − zs βz = 2Iy A Z 1 βω = ω(y 2 + z 2 )dA 2Iω A Iy + Iz Io = + ys2 + zs2 A Z Z βy = IR = A (A-28d) (A-28e) (A-28f) (A-28g) ((y − ys )2 + (z − zs )2 )2 dA (A-28h) where Iy , Iz are the second moment of area for the x and y axis, Iω is the warping constant, βy , βz and βω are the Wagner’s coefficients, Io is the polar moment of area about the shear center and IR is the fourth moment of area about the shear center. The resultant force equations (eq. A-27) written in matrix formulation lead to EA 0 0 0 0 EIy 0 0 0 Mz 0 EIz 0 {S} = = Msv 0 0 GJ 0 Bω 0 0 0 0 MR EAIo 2EIy βz 2EIz βy 0 N M y Master of Science Thesis 0 EAIo 0 2EIy βz −k y 0 2EIz βy −k z 0 = [D]{γ} (A-29) 0 0 θx 00 EIω −2EIω βω θ x 1 02 θ −2EIω βω EIR 2 x N.K. Teunisse 72 Description of the non-linear elastic non-uniform torsional beam element with [D] as the material matrix behavior and γ as the strain vector. The latter consists of a linear part and two non-linear parts {γ} = −k y −kz = θx0 00 θ x 1 02 0 u 00 −w 00 −v θx0 00 θ x + 02 + w 02 ) (v 0 0 2 θx 0 0 0 θx02 1 2 + 0 −ψθ x 00 00 −w c + v s 00 00 −v c − w 0 0 0 s (A-30) = {γl } + {γnl (θ)} + {γnlα (θ, α)} with {α} defined as the "rotation" vector {α}T = {c s (A-31) ψ} The three parts of the strain vector can be formulated in term of the deformation gradient vector {θ} 1 0 0 0 0 0 0 0 {γl } = 0 0 0 0 0 0 0 0 0 0 u0 0 v 0 0 w 0 0 0 θx = [H]{θ} v 00 0 00 w 0 00 0 θx (A-32a) u0 0 v 0 0 0 w 0 0 0 0 0 θx 1 = [A(θ)]{θ} 00 2 0 0 v 00 w 0 0 00 θ 0 0 x (A-32b) 0 0 0 0 0 0 −1 0 0 −1 0 0 1 0 0 0 0 0 0 1 0 0 0 0 θx 0 v 0 w0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 θx0 {γnl } = 1 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 θx 0 0 0 0 0 0 0 0 {γnlα } = − 0 0 0 0 0 ψ 0 0 0 0 0 −s c 0 0 0 c s 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 u0 0 0 v w0 0 0 0 θ x = −[Aα (α)]{θ} v 00 0 00 0 w 00 0 θx (A-32c) θx N.K. Teunisse Master of Science Thesis A-3 Finite element discretization 73 Resulting in the following relation between the strain vector {γ} and the deformation gradient vector {θ} 1 {γ} = [H] + [A(θ)] − [Aα (α)] {θ} 2 (A-33) {δγ} = ([H] + [A(θ)] − [Aα (α)] − [Ã(θ, α)]){δθ} (A-34) and its variation yields The matrix [Ã(θ, α)] is explained in further detail by Mohri [9], but it is the product of two matrices [Ã(θ, α)] = [Â(θ)][P (θ, α)] (A-35) which are defined as 0 0 θx0 00 w −v 00 0 00 00 v w 0 [Â(θ)] = 0 0 0 0 0 0 0 0 0 (A-36) 0 0 0 0 0 0 0 −s [P (θ, α)] = 0 0 0 0 0 c + 1 0 0 0 Qc Rc 0 0 0 0 Pc (A-37) The coefficients Pc , Qc and Rc utilized in the matrix [P (θ, α)] hold Pc = −ys (w0 s + v 0 (c + 1)) + zs (v 0 s − w0 (c + 1)) (A-38a) Qc = −ys s − zs (c + 1) (A-38b) Rc = ys (c + 1) − zs s (A-38c) Finally one can substitute the strain vector (eq. A-33) and its variation (eq. A-34) in the equilibrium equation (eq. A-19) which leads to the following Z L {δθ}T ([H] + [A(θ)] − [Aα (α)] − [Ã(θ, α)]){S}dx − λ Z L {δq}T {Fe } + {δθ}t {Me } dx = 0 1 {S} = [D]([H] + [A(θ)] − [Aα (α)]){θ} 2 A-3 (A-39) (A-40) Finite element discretization The element is set up as a 3D beam element with two nodes, where each node contains 7DOFs. The 7th DOF is necessary to deal with warping. For the axial displacements linear shape Master of Science Thesis N.K. Teunisse 74 Description of the non-linear elastic non-uniform torsional beam element functions are used and cubic functions for the other displacements. The vectors {q} and {θ} and their variations {δq} and {δθ} are related to nodal variables {r} and {δr} by {q} = [N ]{r}e (A-41a) {δq} = [N ]{δr}e (A-41b) {δθ} = [G]{δr}e (A-41d) {θ} = [G]{r}e (A-41c) (A-41e) where [N ] is the shape functions matrix and [G] is the strain-displacement matrix. Linear shape functions are assumed for axial displacements and cubic functions for the other displacements. N u(x) 5 v(x) 0 {q} = = w(x) 0 0 θx (x) {r}e = 0 N1 0 0 0 0 N1 0 0 0 0 N1 0 0 0 N2 0 0 0 −N2 0 0 0 N2 N6 0 0 0 0 N3 0 0 0 0 N3 0 0 0 0 N3 0 0 0 N4 0 0 {r}e 0 −N4 0 0 0 N4 (A-42) h 0 0 u1 v1 w1 θx1 v10 w10 θx1 u2 v2 w2 θx2 v20 w20 θx2 iT 1 1 1 N1 = (1 − ξ)2 (2 + ξ), N2 = L(1 − ξ 2 )(1 − ξ), N3 = (1 + ξ)2 (2 − ξ) 4 8 4 1 1 1 1 1 2 N5 = − ξ, N6 = + ξ N4 = L(−1 + ξ )(1 + ξ), 8 2 2 2 2 (A-43) (A-44) (A-45) The dimensionless parameter ξ defines the relative axial position on the beam varying from -1 to 1 and equals ξ = −1 + 2x L . The discretized equilibrium equation can be written as XLZ 1 e 2 −1 {δr}Te {S}dξ −λ XLZ 1 e 2 −1 {δr}Te {f }e dξ = 0 ∀{δr} 1 {S} = [D]([Bl ] + [Bnl (θ)] − [Bnlα (α)]){r}e 2 (A-46) denotes the assembling process over the beam elements. The matrices and vectors used in the formulation of the discretized minimum total potential energy are defined as P e N.K. Teunisse [B(θ, α)] = [Bl ] = [Bnl (θ)] [Bl ] + [Bnl (θ)] − [Bnlα (α)] − [B̃nl (θ, α)] (A-47a) [H][G] (A-47b) = [A(θ)][G] (A-47c) [Bnlα (α)] = [Aα (α)][G] (A-47d) [B̃nl (θ, α)] = [Ã(θ, α)][G] {f }e = [N ] {Fe } + [G] {Me } T (A-47e) t (A-47f) Master of Science Thesis A-3 Finite element discretization A-3-1 75 Solution strategy in nonlinear context This section will be a summary of the equivalent section in Mohri [9], however with a small, ¯ but logical correction of matrix [S̄0 ]. To solve the nonlinear problem (eq. A-46) we adopt a classical incremental-iterative Newton-Raphson procedure. With this aim in view, we have to compute the tangent stiffness matrix. If the unknowns of the nonlinear problem ({U }T = {{r} {S} {α}, λ}) are sought in the form {U } = {U0 } + {∆U } and λ = λ0 + ∆λ (A-48) Given an initial guess of the solution ({U0 }, λ0 ), the increments of the problem ({∆U }, ∆λ), satisfy the following conditions X1Z 1 e 2 −1 ([B(θ0 , α0 )] {∆S})dξ + T X1Z 1 e 2 −1 ([∆B(θ, α)]T {S0 })dξ − ∆λ{f } = {0} 1 1 {∆S} = [D] [Bl ] + [Bnl (θ0 )] − [Bnlα (α0 )] {∆r} + [D] [Bnl (∆θ)] − [Bnlα (∆α)] {r0 } 2 2 (A-49) The first term in equation A-49 leads to the geometric stiffness matrix X1Z 1 e 2 −1 ([B(θ0 , α0 )]T {∆S})dξ = X1Z 1 e 2 −1 ([B(θ0 , α0 )]T [D][B(θ0 , α0 )])dξ{∆r} = [Kg ]{∆r} (A-50) and the second term in equation A-49 concerns the formulation of the initial stiffness matrix X1Z 1 e 2 −1 ([∆B(θ, α)]T {S0 })dξ = [G]T ([S̄0 ] − [S̄¯0 ][P (θ0 , α0 )] Master of Science Thesis ¯ − [P (θ0 , α0 )]T [S̄¯0 ]T − [S̄0 ][P̄ (θ0 , α0 )])[G]{∆r} = [KS0 ]{∆r} (A-51) N.K. Teunisse 76 Description of the non-linear elastic non-uniform torsional beam element ¯ The matrices [S̄0 ], [S̄¯0 ], [S̄0 ] are function on the initial stresses and are as follows 0 0 0 0 [S̄0 ] = 0 0 0 0 0 N0 0 0 0 0 0 0 0 0 N0 0 0 0 0 0 0 0 0 MR 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 [S̄¯0 ] = Mz 0 My0 0 0 0 0 −zs N0 θx0 0 ys N0 θx0 0 ¯ [S̄0 ] = 0 0 00 −My0 v0 Mz0 w000 0 0 0 0 0 0 0 0 0 0 0 0 −My0 Mz 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (A-52a) 0 0 0 N0 0 0 0 0 (A-52b) 0 0 0 0 0 0 0 0 (A-52c) 0 0 −ys N0 θx0 0 −zs N0 θx0 0 0 0 −My0 w000 −Mz0 v000 The matrices [P (θ0 , α0 )], [P̄ (θ0 , α0 )] include the flexural-torsional coupling. [P (θ0 , α0 )] is defined in equation A-37 and [P̄ (θ0 , α0 )] is defined as [P̄ (θ0 , α0 )] = 0 0 0 P̄0 = (A-53b) Q̄0 = ys (v00 s0 − w00 (1 + c0 )) + zs (v00 (1 + c0 ) + w00 s0 ) R̄0 = −ys s0 − zs (c0 + 1) (A-53d) 0 0 Q̄0 0 0 R̄0 0 0 0 0 0 0 −ys (c0 + 1) + zs s0 0 0 0 0 0 0 −s0 c0 + 1 P̄0 (A-53a) (A-53c) The integration is performed using the Gaussian quadrature rule and the number of points adapts to keep the results equal to the exact solution. Finally the incremental problem (eq. A-49) can be written as [Kt ]{∆r} − ∆λ = {0} N.K. Teunisse (A-54) Master of Science Thesis A-3 Finite element discretization 77 where the tangent stiffness matrix [Kt ] is defined as [Kt ] = [Kg ] + [KS0 ] A-3-2 (A-55) Mass matrix including rotary inertia Normally a mass matrix is constructed as follow Z L [M ] = 0 [N ]T [N ]ρA dx (A-56) However by doing this the rotary inertia is omitted which is essential for studying rotations. Therefore, the total mass matrix (Mtot ) is constructed as the sum of the translational (Mtrans ) and rotational (Mrot ) mass matrices [Mtot ] [Mtrans ] [Mrot ] = [Mtrans ] + [Mrot ] (A-57) = Z L [Ntrans ]T [Ntrans ]ρAdx (A-58) = Z L [Nrot ]T [Nrot ]IR dx (A-59) 0 0 with Ntrans the shape functions for the translations degrees of freedom (1-3 rows of [N ]), Nrot the shape functions of the rotational degrees of freedom (4th row of [N ]), ρ density of the material, A the cross-sectional area and IR as the moment of inertia per unit length. The shape functions are the same as used for the stiffness matrix, see eq.A-42. For homogeneous beams with a constant cross-section the moment of inertia IR per unit length can be calculated easily IR = Z A ρ(r)r2 dA = ρ Z A r2 dA = ρ Z A (y 2 + z 2 )dA =ρ Z A y 2 dA + Z A z 2 dA = ρ(Iyy + Izz ) (A-60) where r is the radius of a point on the cross-section to the shear center and Iyy , Izz are respectively the second moment of area of the second and third principal axes. Note that at the end the moment of inertia of the whole body is taken as IR is placed in the integral over the length of the beam (eq. A-59). Master of Science Thesis N.K. Teunisse 78 N.K. Teunisse Description of the non-linear elastic non-uniform torsional beam element Master of Science Thesis B Approximation of a triangular function The objective of the optimization is to get a more triangular response instead of a solely sinusoidal response. A pretty good approximation of a triangular function in time can be composed out of just three cosines: A cos 1 A 3 A 5 π(4f t − 1) + cos π(4f t − 1) + cos π(4f t − 1) 2 9 2 25 2 (B-1) with A as amplitude and f as frequency. The result of eq.B-1 is shown in fig.B-1 for the first, the first two and all three cosine terms. So the optimization has to optimize to a situation that the response holds only 3 cosines with the second cosine having a 3 times higher frequency and 9 times lower amplitude than the first cosine. The third cosines needs to have a 5 times higher frequency and 25 times smaller amplitude. Master of Science Thesis N.K. Teunisse 80 Approximation of a triangular function Figure B-1: Approximation of a triangular function in time using one, two and three cosines. N.K. Teunisse Master of Science Thesis C Higher-order approximations of the analytical solution This appendix describes the work that is performed in obtaining the analytic higher-order approximations of the governing equation describing a beam excited in its torsional natural frequency. The higher-order approximations are necessary to capture the amount of triangularity of the dynamic response of the tip of the beam which is used for optimization purposes. Unfortunately the author never obtained the full higher-order approximations, but a good amount of work is made towards that direction. To help possible projects in the future that continue on this subject, the already constructed calculations are provided in this appendix. C-1 Solution of order 3 In section 3-2-2 we derived the third order problem, but the third order problem was not solved since only the secular terms were eliminated. This led to the first order approximation of θ. To capture the third order approximation the third order problem needs to be solved, scale it by an unknown complex function and determine this function by letting the secular terms vanish in the fifth order problem. The third order problem with homogeneous boundary conditions is defined in eq.3-95, but repeated here for convenience D02 θ̃ˆ3 + ηw θ̃ˆ30000 − θ̃ˆ300 = [γ1 (ωn2 x3 − 3ωn2 x2 + 6x − 6) + β1 ]eiωn t0 + [γ2 (−9ωn2 x3 + 27ωn2 x2 − 6x + 6) + β2 ]e3iωn t0 + cc (C-1) subject to θ̃ˆ3 = θ̃ˆ30 = 0 at x = 0 θ̃ˆ30 − ηw θ̃ˆ3000 = 0 and θ̃ˆ300 = 0 at x = 1 Master of Science Thesis (C-2) (C-3) N.K. Teunisse 82 Higher-order approximations of the analytical solution with −2iωn ζAn Θn − 2iωn D2 An Θn + 3ηn A2n Ān (Θ0n )2 Θ00n β1 (x, t2 ) = ηn A3n (Θ0n )2 Θ00n β2 (x, t2 ) = (C-4) (C-5) We can rewrite eq.C-1 as D02 θ̃ˆ3 + ηw θ̃ˆ30000 − θ̃ˆ300 = F1 eiωn t0 + F2 e3iωn t0 + cc (C-6) with F1 and F3 as coefficients of the forcing terms. Thanks to the homogeneous boundary conditions we can write both θ̃ˆ3 as the forcing terms as an infinite summation of the modes multiplied by time-dependent coefficients. So let us rewrite F1 as a summation of Θm and f (t̃). t̃ represents all the time scales except t0 , because t0 is already explicitly mentioned in the equation and holds for all modes. Rewriting yields F1 = ∞ X fm (t̃)Θm = m=1 n−1 X ∞ X fm Θm + fn Θn + m=1 fm Θm (C-7) m=n+1 To make sure the third order solution doesR not contain secular terms we forced F1 in eq.3-102 to be orthogonal to the n-th mode with 01 F1 Θn dx = 0 resulting in fn = 0. Consequently, F1 turns into F̃1 F̃1 = n−1 X ∞ X fm Θm + m=n+1 m=1 fm Θm = F1 − fn Θn (C-8) Finally the third order partial differential equation becomes D02 θ̃ˆ3 + ηw θ̃ˆ30000 − θ̃ˆ300 = (F1 − fn Θn )eiωn t0 + F2 e3iωn t0 + cc (C-9) The new forcing term The term fn in eq.C-9 is already defined in eq.3-104. Multiplied with Θn and rearranged the equation now looks like − 2iωn ζAn Θn + −2iωn D2 An Θn = − κ2 cx1 M̄ Θn iσt2 e −6 − 12ηw ηn cx1 (Θ0n (1))3 Θn 2 An Ān − 3ηn cx2 Θn A2n Ān (C-10) + −3 − 6ηw Substituting eq.C-10 into β1 of eq.C-1 yields D02 θ̃ˆ3 + ηw θ̃ˆ30000 − θ̃ˆ300 = [γ1 (ωn2 x3 − 3ωn2 x2 + 6x − 6) + β3 ]eiωn t0 + [γ2 (−9ωn2 x3 + 27ωn2 x2 − 6x + 6) + β2 ]e3iωn t0 + cc (C-11) with the new variable β3 (x, t2 ) defined as β3 (x, t2 ) = 3ηn A2n Ān (Θ0n )2 Θ00n − N.K. Teunisse κ2 cx1 M̄ Θn iσt2 e −6 − 12ηw ηn cx1 (Θ0n (1))3 Θn 2 + An Ān − 3ηn cx2 Θn A2n Ān (C-12) −3 − 6ηw Master of Science Thesis C-1 Third order solution 83 Note that Ān is not an unknown function anymore as we constructed it in the derivation of the first order approximation. Let us make an estimate of θ̃ˆ3 (1) (3) θ̃ˆ3 = Θ̃3 (x, t2 )eiωn t0 + Θ̃3 (x, t2 )e3iωn t0 (C-13) Filling this into eq.C-11 results in a set of differential equations (1) (1)0000 −ωn2 Θ̃3 + ηw Θ̃3 (3) −9ωn2 Θ̃3 + (3)0000 ηw Θ̃3 (1) − (1)00 − Θ̃3 (3)00 Θ̃3 = γ1 (ωn2 x3 − 3ωn2 x2 + 6x − 6) + β3 = γ2 (−9ωn2 x3 + 27ωn2 x2 − 6x + 6) + β2 (1) (3) (C-14) (C-15) (3) ∞ 2 When we rewrite Θ̃3 = ∞ m=1 hm (t2 )Θm (x) and Θ̃3 = m=1 hm (t2 )Θm (x), use −ωn Θn + 00 ηw Θ0000 n − Θn = 0(eq.3-71) and the mode orthogonality, then it follows that P h(1) m h(3) m = R1 R1 0 P [γ1 (ωn2 x3 − 3ωn2 x2 + 6x − 6) + β3 ]Θm dx 2 − ω2 ωm n (C-16) [γ2 (−9ωn2 x3 + 27ωn2 x2 − 6x + 6) + β2 ]Θm dx 2 − 9ω 2 ωm n (C-17) [γ1 (ωn2 x3 − 3ωn2 x2 + 6x − 6) + 3ηn A2n Ān (Θ0n )2 Θ00n ]Θm dx 2 − ω2 ωm n (C-18) = 0 Again the problem can be simplified using orthogonality of the modes when we consider β3 . So the coefficients are now defined as h(1) m h(3) m = = R1 0 R1 0 [γ2 (−9ωn2 x3 + 27ωn2 x2 − 6x + 6) + ηn A3n (Θ0n )2 Θ00n ]Θm dx 2 − 9ω 2 ωm n (C-19) Note that the terms would go to infinity if the n-th mode was included. Furthermore one (1) can notice that the magnitude of hm is high when the frequency of the m-th mode is close to the frequency of the n-th mode or when the forcing term greatly overlaps with the m-th (3) mode. This also holds for hm , however now the frequency of the m-th mode should be close to three times the frequency of the n-th mode. So we can truncate a lot of modes, without a noticeable difference in the coefficients of the m-th mode. Results in Maple showed that it is justified to only include the n + 1-th mode for our problem. When the results aren’t accurate enough, more modes can be added without problem. The solution Finally after all these calculations we can write θ̃ˆ3 (1) (3) 3iωn t0 θ̃ˆ3 = hn+1 Θn+1 eiωn t0 + h(3) + hn+1 Θn+1 e3iωn t0 + cc n Θn e (C-20) With the help of eq.3-87 θ̂3 = −3a3 x2 + a3 x3 + θ̃ˆ3 and eq.3-92 a3 = γ1 (t2 )eiωn t0 − γ2 (t2 )e3iωn t0 + cc Master of Science Thesis N.K. Teunisse 84 Higher-order approximations of the analytical solution one can write θˆ3 as (1) θ3 = Bn1 (t4 , t6 )[(−3x2 + x3 )γ1 + hn+1 Θn+1 ]eiωn t0 (3) 3iωn t0 + Bn2 (t4 , t6 )[(3x2 − x3 )γ2 + h(3) + cc n Θn + hn+1 Θn+1 ]e (C-21) where Bn1 and Bn2 are complex functions of the slower time scales. C-2 The higher-order problems Altogether we need least 3 cosine to make a rough approximation of a triangular function. This leads to the following expansion of θ̂(x, t) θ̂(x, t) = θ̂1 (x, t0 , t2 , t4 , t6 ) + 3 θ̂3 (x, t0 , t2 , t4 , t6 ) + 5 θ̂5 (x, t0 , t2 , t4 , t6 ) + 7 θ̂7 (x, t0 , t2 , t4 , t6 ) + . . . (C-22) where t0 = t is the fast scale and t2 = 2 t, t4 = 4 t, t6 = 6 t are the slow time scales. The time derivatives are defined by 2 2 d /dt = D02 2 4 d/dt = D0 + 2 D2 + 4 D4 + 6 D6 + . . . + 2 D0 D2 + (2D0 D4 + D22 ) 6 + 2 (D2 D4 + D0 D6 ) + . . . (C-23) (C-24) We rescale the damping factor ζ as 2 ζ and the forcing moment amplitude M̄ as 3 M̄ . Thus, substituting the assumed expansion for θ̂, the rescaled damping and forcing and the time derivatives yields Order D02 θ̂1 + ηw θ̂10000 − θ̂100 = 0 (C-25) subject to the boundary conditions θ̂1 = θ̂10 = 0 at x = 0 θ̂10 Order 3 − ηw θ̂1000 = 0 and θ̂100 (C-26) = 0 at x = 1 (C-27) D02 θ̂3 + ηw θ̂30000 − θ̂300 = −2D0 D2 θ̂1 − 2ζD0 θ̂1 + ηn (θ̂10 )2 θ̂100 (C-28) subject to the boundary conditions 1 (θ̂30 − ηw θ̂3000 ) = κ2 M̄ cos Ωt0 − ηn (θ̂10 )3 3 θ̂3 = θ̂30 = 0 at x = 0 and θ̂300 = 0 at x = 1 (C-29) (C-30) Order 5 D02 θ̂5 + ηw θ̂50000 − θ̂500 = −D22 θ̂1 + 2ζD2 θ̂1 − 2D0 D4 θ̂1 2 θ̂300 − 2D0 D2 θ̂3 − 2ζD0 θ̂3 + 2ηn θ̂10 θ̂100 θ̂30 + ηn θ̂10 N.K. Teunisse (C-31) Master of Science Thesis C-3 Finding slow time scale coefficients for the third order solution 85 subject to the boundary conditions (θ̂50 − ηw θ̂5000 ) = −ηn (θ̂10 )2 θ̂30 θ̂5 = θ̂50 = 0 at x = 0 and θ̂500 (C-32) = 0 at x = 1 (C-33) Order 7 D02 θ̂7 + ηw θ̂70000 − θ̂700 = −2D0 D6 θ̂1 − 2D2 D4 θ̂1 − 2ζD4 θ̂1 + 2ηn θ̂10 θ̂100 θ̂50 + ηn θ̂100 (θ̂30 )2 + 2ηn θ̂10 θ̂30 θ̂300 + ηn (θ̂10 )2 θ̂500 − D22 θ̂3 − 2D0 D4 θ̂3 − 2ζD2 θ̂3 − 2D0 D2 θ̂5 − 2ζD0 θ̂5 (C-34) subject to the boundary conditions (θ̂70 C-3 − ηw θ̂7000 ) = −ηn θ̂10 (θ̂30 )2 − ηn (θ̂10 )2 θ̂50 θ̂7 = θ̂70 = 0 at x = 0 and θ̂700 = 0 at x = 1 (C-35) (C-36) Finding slow time scale coefficients for the third order solution Previously the complete solution of the first order problem was found θ̂1 = 2a cos((σ2 + ωn )t0 − ν)Θn (x) = 2a cos(Ωt0 − ν)Θn (x) (C-37) and a part of the solution of the third order problem (1) θ̂3 = Bn (t4 ) (−3x2 + x3 )γ1 (t2 ) + hn+1 (t2 )Θn+1 (x) eiωn t0 h i (3) 3iωn t0 + Bn (t4 ) (3x2 − x3 )γ2 (t2 ) + h(3) + cc (C-38) n (t2 )Θn (x) + hn+1 (t2 )Θn+1 (x) e h i where Bn and Bn are the still unknown complex functions of the slow time scales. To determine these, one should focus on the fifth order problem: D02 θ̂5 + ηw θ̂50000 − θ̂500 = −D22 θ̂1 + 2ζD2 θ̂1 − 2D0 D4 θ̂1 2 + 2ηn θ̂10 θ̂100 θ̂30 + ηn θ̂10 θ̂300 − 2D0 D2 θ̂3 − 2ζD0 θ̂3 (C-39) subject to the boundary conditions (θ̂50 − ηw θ̂5000 ) = −ηn (θ̂10 )2 θ̂30 θ̂5 = θ̂50 = 0 at x = 0 and θ̂500 = 0 at x = 1 (C-40) (C-41) Unfortunately due to time constraints the author was unable to find the slow time scale coefficients for the third order problem. Master of Science Thesis N.K. Teunisse 86 N.K. Teunisse Higher-order approximations of the analytical solution Master of Science Thesis D Analytical solutions of the cross-sectional properties The analytical solutions of the cross-sectional properties I˜n , I˜pp , I˜p , I˜y , I˜z and C̃s for an I-shaped cross-section are provided in this appendix. The cross-sectional features are width b̃, height h̃, flange thickness t̃f and wall thickness t̃w . The tilde represent the division of the cross-sectional feature by the constant length of the beam. 2 1 1 1 2 I˜y = b̃h̃2 t̃f − b̃h̃t̃2f + t̃3f b̃ + t̃w h̃3 − t̃w h̃2 t̃f + t̃w h̃t̃2f − t̃w t̃3f 2 3 12 2 3 1 1 3 3 I˜z = t̃w (h̃ − 2t̃f ) + b̃ t̃f 12 6 ˜ ˜ ˜ Ip = Iy + Iz (D-1) (D-2) (D-3) (D-4) 1 1 1 1 1 I˜pp = t̃3w h̃t̃2f + b̃h̃2 t̃3f − b̃h̃t̃4f − t̃w h̃4 t̃f + t̃w h̃3 t̃2f − b̃3 h̃t̃2f + b̃3 h̃2 t̃f 6 8 2 6 12 1 3 2 1 4 1 3 2 2 5 1 2 3 4 − t̃w h̃ t̃f + t̃w h̃t̃f + b̃h̃ t̃f − t̃w h̃ t̃f − b̃h̃ t̃f + b̃t̃f + t̃5w h̃ 12 8 2 5 80 1 3 3 1 3 3 1 1 5 2 5 1 3 3 1 5 − t̃w t̃f + t̃w h̃ − t̃w t̃f + t̃w h̃ − t̃w t̃f + b̃ t̃f + t̃f b̃5 (D-5) 40 72 9 80 5 9 40 I˜n = I˜pp − I˜p2 (D-6) Ã The warping constant C̃s is analytically obtained for an I-shaped cross-section using Vlasov’s theory (eq.2-2). Vlasov’s theory is described in chapter 2 and holds as long as the beams remains thin-walled. The analytical solution of the warping constant C̃s for an I-shaped cross-section holds 1 C̃s = t̃f b̃3 6 Master of Science Thesis h̃ − t̃f 2 !2 (D-7) N.K. Teunisse 88 N.K. Teunisse Analytical solutions of the cross-sectional properties Master of Science Thesis E Optimization algorithms The choice of an optimization algorithm totally depends on the problem one faces. The optimization problem in this work yields a non-linear objective and constraint. Section E-1 shows that the only solver in MATLAB that can handle both a non-linear constraint as objective is the solver fmincon. Originally it was assumed that the torsion constant and warping constant need to be obtained from ANSYS. This also means that only 0th order information is available, because ANSYS does not provide the derivatives of those constants and hence the derivative of the objective function is unknown. The solver fmincon has different algorithms that can be used as described in section E-1. The best algorithm for the optimization using ANSYS is the Sequential Quadratic Programming (SQP) algorithm, because it doens’t need a Hessian supplied by the user and less steps are expected in comparison with the Interior-Point algorithm. Namely, the latter makes use of a linearized Lagrangian causing more ANSYS calls, while the SQP algorithm creates a Quadratic Programming-subproblem which is more expensive to calculate, but less steps and ANSYS calls are expected to reach the optimum. Furthermore the SQP algorithm respects the constraints even for every finite step which is important, because steps outside the feasible domain may cause ANSYS to crash. An example is a wall thickness that is larger than the profile width. At the end ANSYS wasn’t necessary and an approximate model, see section 6-2-1, is used. If the approximate model was computationally more expensive, the Trust region reflective algorithm would be the best optimization algorithm, because it employs the Hessian to ensure fast convergence and due to the trust-region the global optimum is likely to be found. The Hessian is can be computed since the approximate model is analytical and continuous. E-1 Different optimization solvers The best solver of an optimization problem depends on the type of the objective function and the constraint functions. MATLAB has a lot of solvers available to solve all kind of problems. These problems differ in various ways, for example by boundedness of the problem, the type of boundaries, the order of the objective function and more. In this work the objective function can be characterized as a non-linear single-objective function and the equality constraint is also non-linear. In table E-1 the solvers of MATLAB are listed and explained why they can or Master of Science Thesis N.K. Teunisse 90 Optimization algorithms can not be used for the problem in this work. The most appropriate solver for our constrained problem is the constrained non-linear minimization (fmincon). This solver supports multiple algorithms: • Trust region reflective • Interior point • Active set • Sequential Quadratic Programming (SQP) In the next sections every algorithm will be discussed briefly, based on [26]. E-1-1 Trust region reflective algorithm The trust region reflective algorithm is an approximation based method. It constructs an approximation of the objective function q near the current point x. The validity of the approximation is not true for the full space of the model because of the non-linear functions. The subspace wherefore the original problem is approximated is called the trust region N . Generally, if the computations indicate the approximate model fit the original problem quite well, the trust region can be enlarged. Otherwise when the approximate model seems to be not good enough the trust region will be reduced. In the trust region a minimum is calculated using the approximation of the objective function q: min{q(s), s ∈ N }. (E-1) s where s is the trial step. The trial step is the step taken from the current point to the approximated minimum of the trust region. The current point is updated to x + s if the objective value is indeed lower at x + s. If this is not true, the approximation of the objective function seems to be bad and an approximation is made for a smaller region. So the trust region N is shrunk and the trial step computation is performed again [26]. This process is executed iteratively to find the minimum of the objective function. The approximation of the model is calculated by taking the first terms of the Taylor expansion, usually the first and second term. So it is logical that the approximation is only reliable close to the point where the Taylor expansion was evaluated. So the mathematically subproblem in the trust-region typically states: 1 T~ ~ min s Hs + sT ∇f such that ||Ds|| ≤∆ 2 (E-2) ~ is the Hessian matrix, D ~ is a diagonal where ∇f is the gradient of f at the current point ~x, H scaling matrix which scales the step size, ∆ is the trust region radius, which is updated at each step. A lot of algorithms are provided to solve eq.E-2, however they require time proportional to ~ To speed up the process the trust-region is restricted to a twoseveral factorizations of H. ~ dimensional subspace S. This makes the computation of the solution of eq.E-2 easy. So the hard part has become to determine the appropriate subspace. N.K. Teunisse Master of Science Thesis E-1 Different optimization solvers 91 Solver Applicable to our problem Motivation bintprog No The solutions generated by this solver are binary integers, which is not applicable for cross-section dimensions. Next to that it does not support nonlinear constraints. fgoalattain No This solver is intended for multi-objective functions, but the objective function used in this work is a single-objective function. fminbnd No No constraints are supported in this function, which makes is unusable. Additionally it can not handle vector inputs, which makes it unsuitable even if no constraints are present. fmincon Yes This solver is suited for the optimization problem. It supports non-linear constraints, it can handle vector inputs and works with decimal numbers. fminimax No The minimization of the maximum in a function does not correspond with our goal, which is the minimization of the objective function. fminsearch No It does not support constraints. fminunc No It does not support constraints. The difference with fminsearch is that this method uses (generated) derivative information. fseminf No This method is used when the constraints do depend on other variables then used in the objective function. linprog No Linear Programming can not handle nonlinear objective functions and non-linear constraints. quadprog No Quadratic Programming can not handle nonlinear objective functions and non-linear constraints. Table E-1: Solvers available in MATLAB. Master of Science Thesis N.K. Teunisse 92 Optimization algorithms The subspace is calculated by a preconditioned conjugate gradient process ensures global convergence and when possible fast local convergence, via the Newton search direction. E-1-2 Interior point algorithm The interior point algorithm make use of slack variables s to transform all the inequality constraints to equality constraints, which are easier to solve. Next a barrier function is added to the objective function min fµ (x, s) = min f (x) − µ x,s x,s s.t. ln(si ) (E-3) h(x) = 0 (E-4) g(x) + s = 0 (E-5) X i Both µ and s are always positive. Note that if µ decreases to zero, the minimum of fµ approaches the minimum of f . To solve the eq.E-3 the algorithm generates a linearized Lagrangian at (x,s) to try to solve the Karush–Kuhn–Tucker (KKT) conditions. It does this by LDL factorization of the matrix. If the Hessian is not positive definite, it will use the conjugate gradient method with a trust region. E-1-3 Active Set algorithm The Active Set Algorithm makes use of the KKT conditions to find the optimal solution for the problem. The Langrange multipliers in the KKT conditions are iteratively calculated by SQP. The SQP method makes every major iteration an approximation of the Hessian matrix of the Langrangian function using the Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm. This approximated Hessian in combination with the linearized nonlinear constraints is subsequently used to generate a QP subproblem whose solution is used as search direction in a line search. Next a line search is performed to find a new starting point with a lower objective value. From here the process will repeat itself. The algorithm consists of two phases. The first phase is the calculation of a feasible point. Subsequently feasible points are generated that converge to the solution. During the iterations an active set is maintained, which is an estimate of the active constraints at the solution point. This active set makes sure the generated results are always feasible. E-1-4 SQP algorithm The SQP algorithm is similar to the active set algorithm. However there are some small changes [26] • The bounds are not strict • It can handle a non-double result of an objective or constraint function • Different routines are used to solve the quadratic programming subproblem • Two new approaches can be used for the quadratic programming subproblem when the constraints are not satisfied. N.K. Teunisse Master of Science Thesis Bibliography [1] C. Bolsman, Flapping wing actuation using resonant compliant mechanisms. PhD thesis, 2010. [2] A. Hughes, D. Iles, and A. Malik, Design of steel beams in torsion. SCI, 2011. [3] M. Shama, “Torsion and Shear Stresses in Ships,” pp. 91–104, 2011. [4] E. J. Sapountzakis, “Bars under Torsional Loading: A Generalized Beam Theory Approach,” ISRN Civil Engineering, vol. 2013, pp. 1–39, 2013. [5] K. Y. Ma, P. Chirarattananon, S. B. Fuller, and R. J. Wood, “Controlled flight of a biologically inspired, insect-scale robot.,” Science (New York, N.Y.), vol. 340, no. 6132, pp. 603–7, 2013. [6] F. Bos, D. Lentink, B. V. Oudheusden, and H. Bijl, “Influence of wing kinematics on performance in hovering insect flight,” Journal of Biomechanics, vol. 39, no. November 2015, p. S358, 2006. [7] C. Felippa, “Nonlinear Finite Element Methods,” tech. rep., University of Colorado at Boulder, Boulder, 2014. [8] W. Pilkey, Analysis and Design of Elastic Beams: Computational Methods, vol. 56. New York :: Wiley„ 2003. [9] F. Mohri, N. Damil, and M. P. Ferry, “Large torsion finite element model for thin-walled beams,” Computers & Structures, vol. 86, pp. 671–683, Apr. 2008. [10] N. W. Murray, Introduction to the theory of thin-walled structures. The Oxford engineering science series 13, Oxford: Clarendon, 1984. [11] V. Z. Vlasov, Thin-walled elastic beams. Jerusalem: Monson, 1961. [12] N. Trahair, “Nonlinear elastic nonuniform torsion,” Journal of structural engineering, vol. 131, no. 7, pp. 1135–1142, 2005. [13] A. Ghobarah and W. Tso, “A non-linear thin-walled beam theory,” International Journal of Mechanical Sciences, vol. 13, pp. 1025–1038, Dec. 1971. Master of Science Thesis N.K. Teunisse 94 Bibliography [14] N.-I. Kim and M.-Y. Kim, “Exact dynamic/static stiffness matrices of non-symmetric thin-walled beams considering coupled shear deformation effects,” Thin-Walled Structures, vol. 43, pp. 701–734, May 2005. [15] E. Sapountzakis and V. Tsipiras, “Nonlinear nonuniform torsional vibrations of bars by the boundary element method,” Journal of Sound and Vibration, vol. 329, pp. 1853–1874, May 2010. [16] S. a. Sina, H. Haddadpour, and H. M. Navazi, “Nonlinear free vibrations of thin-walled beams in torsion,” Acta Mechanica, vol. 223, pp. 2135–2151, June 2012. [17] E. Sapountzakis and V. Tsipiras, “Non-linear elastic non-uniform torsion of bars of arbitrary cross-section by BEM,” International Journal of Non-Linear Mechanics, vol. 45, pp. 63–74, Jan. 2010. [18] A. H. Nayfeh, Introduction to Perturbation Techniques. 1993. [19] S. Howison, Practical applied mathematics; modelling, analysis, approximation. Cambridge texts in applied mathematics, Cambridge, UK: Cambridge University Press, 2005. [20] R. D. Cook, D. S. Malkus, M. E. Plesha, and R. J. Witt, Concepts And Applications of Finite Element Analysis. Wiley, fourth ed., 2002. [21] E. Sapountzakis and V. Mokos, “Nonuniform torsion of bars of variable cross section,” Computers & Structures, vol. 82, pp. 703–715, Apr. 2004. [22] P. Tiso, Finite element based reduction methods for static and dynamic analysis of thinwalled structures. Dissertation, Delft University of Technology, 2006. [23] T. D. Canonsburg, “ANSYS Mechanical APDL Structural Analysis Guide,” vol. 3304, no. November, pp. 724–746, 2013. [24] ANSYS Inc., “ANSYS Mechanical APDL Element Reference,” ANSYS Manual, vol. 15317, no. October, pp. 1377–1390, 2012. [25] M. M. 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