# InvestIgatIon of the DynamIc ProPertIes of a mIllIng tool holDer

Investigation of the Dynamic Properties of a Milling Tool Holder Investigation of the Dynamic Properties of a Milling Tool Holder Henrik Åkesson, Tatiana Smirnova, Lars Håkansson, Thomas Lagö, Ingvar Claesson Henrik Åkesson, Tatiana Smirnova, Lars Håkansson, Thomas Lagö, Ingvar Claesson Blekinge Institute of Technology Research report No. 2009:07 Copyright © 2009 by individual authors. All rights reserved. Printed by Printfabriken, Karlskrona 2009. ISSN 1103-1581 ISRN BTH-RES–07/09–SE Investigation of the Dynamic Properties of a Milling Tool Holder H. Åkesson, T. Smirnova, L. Håkansson, T. Lagö and I. Claesson Blekinge Institute of Technology Research Report No 2009:07 ISSN: 1103-1581, November, 2009. Abstract Vibration problems during metal cutting occur frequently in the manufacturing industry. The vibration level depends on many different parameters such as the material type, the dimensions of the workpiece, the rigidity of tooling structure, the cutting data, and the operation mode. In milling, the cutting process subjects the tool to vibrations, and having a milling tool holder with a long overhang will most likely result in high vibration levels. As a consequence of these vibrations, the tool life is reduced, the surface finishing becomes poor, and disturbing sound appears. In this report, an investigation of the dynamic properties of a milling tool holder with moderate overhang has been carried out by means of experimental modal analysis and vibration analysis during the operating mode. Both the angular vibrations of the rotating tool and the vibrations of the machine tool structure were examined during milling. Also, vibration of the workpiece and the milling machine was examined during cutting. This report focuses on identifying the source/sources of the dominant milling vibration components and on determining which of these vibrations that are related to the structural dynamic properties of the milling tool holder. Investigation of the Dynamic Properties of a Milling Tool Holder 3 Contents 1 Introduction 1.1 Literature Review . . . . . . . . . . . . . . 1.1.1 Chatter Theory . . . . . . . . . . . . 1.1.2 Force Models . . . . . . . . . . . . . 1.1.3 Stability . . . . . . . . . . . . . . . . 1.1.4 Vibration Control . . . . . . . . . . 1.1.5 Motivation . . . . . . . . . . . . . . 1.2 Basic Concepts of Metal Cutting in Milling 1.3 Measurement of Forces and Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 4 4 5 5 5 6 6 9 2 Materials and Methods 2.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Tool Holder and Tool . . . . . . . . . . . . . . . . . . . . 2.1.2 Cutting Data . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Measurement Equipment and Setup . . . . . . . . . . . . 2.1.4 Spatial Measurements of the Acceleration During Milling 2.1.5 Modal Analysis Setup . . . . . . . . . . . . . . . . . . . . 2.1.6 Excitation Signal for the Experimental Modal Analysis . 2.2 Spectral Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Operating Deflection Shape Analysis . . . . . . . . . . . . . . . . 2.4 Experimental Modal Analysis . . . . . . . . . . . . . . . . . . . . 2.5 Modal Parameter Estimation . . . . . . . . . . . . . . . . . . . . 2.5.1 Spectral Estimation Parameters . . . . . . . . . . . . . . . 2.6 Distributed Parameter Model of the Milling Tool Holder . . . . . 2.6.1 A Geometrical Model of the Clamped Milling Tool Holder 2.6.2 A Model of Transverse Vibrations . . . . . . . . . . . . . 2.6.3 A Model of Torsional Vibrations . . . . . . . . . . . . . . 2.7 A Finite Element Model of the Milling Tool Holder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 9 10 10 12 13 13 14 14 16 16 16 17 19 19 20 21 21 . . . . . . . 21 22 24 27 31 31 34 37 3 Results 3.1 Spatial Measurements of Vibration . . 3.1.1 Operating Deflection Shapes . 3.1.2 Angular Vibrations . . . . . . . 3.2 Modal Analysis Results . . . . . . . . 3.2.1 Mode Shapes . . . . . . . . . . 3.3 Distributed Parameter Model Results 3.4 Finite Element Model . . . . . . . . . 4 Summary and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4 1 H. Åkesson, T. Smirnova, L. Håkansson, T. Lagö and I. Claesson Introduction Metal cutting is generally used in the manufacturing industry to machine, e.g., workpieces to desired geometries with certain tolerances. During the machining process, a number of different machining operations may be involved. There are several different machining operations including turning, milling, drilling, boring, threading, etc. [1]. Today, there are many advanced machines that have several axes and that can perform complex milling and turning operations about non-fixed axes [1] by, for example, rotating or leaning the axis of the spindle. Another example of the type of advanced operation that modern machines are capable of is the production of an oval or ellipsoidal cross-section of a workpiece by controlling the tool motion in the radial direction of the workpiece during turning. The metal cutting operation may sometimes produce high server vibration levels. The cause of these vibrations can be attributed to many different factors such as the cutting parameters, the workpiece material and shape, the tooling structure, the insert, and the stability of the machine [2]. Thus, there are many different parameters that influence the stability of the cutting process in milling operations, and there has been a lot of research done in this area. 1.1 Literature Review Turning has been one of the most studied metal cutting processes due to the fact that it is comparatively easy to monitor the forces applied to the tools under controlled conditions [3]. Turning has also been used to imitate the periodic excitation present in the milling operation by rotating a non-continuously shaped workpiece that produces an intermittent tooth pass excitation [4]. Many similarities between the cutting processes in turning operations and in milling operations may be observed, and chatter theory developed for turning operations is also used in milling theory to produce rough estimates of stability limits [5, 6]. 1.1.1 Chatter Theory Some of the earliest studies on the principles of chatter in simple machine tool systems were produced by Arnold [7] in 1946. In 1965, Tobias [8] presented an extensive summary of results from a large number of research studies concerning the dynamic behavior of the lathe application and the chatter theory, and he further developed the research into the chatter phenomena in consideration of the chip thickness variation and the phase lag of the undulation of the surface. The same year, Merritt [9] discussed the stability of structures with n-degrees of freedom, assuming the absence of dynamics in the cutting process. He also proposed a simple stability criterion. Together with Tobias and Merritt, Koenigsberger and Tlusty are also considered to be the pioneers of chip regeneration formulation for basic chatter theory [10]. Furthermore, Pandit et al. [11] developed a procedure for modeling chatter from time series by including unknown factors of random disturbances present in the cutting process. They formulated self-excited random vibrations with white noise as a forcing function. Finally, Kato et al. [12] investigated regenerative chatter vibration caused by the deflection of the workpiece and introduced a differential equation describing chatter vibration based on experimental data. Investigation of the Dynamic Properties of a Milling Tool Holder 1.1.2 5 Force Models There are many force models of various complexity and properties that describe the cutting process [13–17]. Tlusty [13] presented the relationship between maximum depth of cut, stiffness of the structure, and a specific cutting coefficient of the cutting process for turning, where the maximum width of cut is proportional to the static stiffness and the damping ratio at the cutting tool point of the machine tool. Later, in 1991, Smith and Tlusty [18] summarized the force models and simulation methods of the milling process currently used so far. In 1996, Altintas presented a force model focusing on the helical end mill geometry [14]. In addition to this, Tlusty et al. presented numerical simulations of the milling dynamics, including saturations such as the tool jumping out of cut [19]. Also, Engin and Altintas [20] presented a generalized mathematical model of inserted cutters for the purpose of predicting cutting forces. The model is capable of considering various insert geometries, angles, and positions relative to the cutter body. 1.1.3 Stability In 1981, Tlusty and Ismail [19] studied the basic non-linearity of the cutting processes by analyzing the vibrations that occur when the tool leaves the workpiece for a part of the cycle. This was done for both turning and milling and took into consideration the mode coupling self-excitation. No further conclusions with respect to turning were made, however, they found that stability boundaries related to the milling were calculated erroneously by a factor of two to three by the methods known at the time, thus opening up possibilities for the improvement of stability methods. Furthermore, an improved method for obtaining stability lobes was presented in 1983 by Tlusty et al. [21]. In the beginning of the 1990s, the milling operation received more consideration, which resulted in more accurate stability lobes/diagrams based on various additional properties related to the milling operation [5, 22–27]. A practical example of how to increase cutting performance by considering stability lobes was presented by Tlusty et al. [28]. They studied the performance of a long and slender tool in high speed milling and increased the metal removal rate by choosing the appropriate tool length with respect to stability lobes that allowed maximum spindle speed in high speed milling. However, more contributions can be done to this field, since the variation of machines, tool configuration, inserts, workpiece material, etc., seem to be virtually unlimited. 1.1.4 Vibration Control The common methods used to control vibrations in milling systems utilize the control mechanisms of the cutting parameters related to the machine. Other methods are either based on passive vibration absorption or active control that applies a secondary control force. Several methods have been developed to find cutting parameters that can be used to avoid instabilities [29–32]. The most commonly used method is to change some of the parameters during cutting, i.e., changing the spindle speed or the feed rate of the workpiece. The goal to be achieved by changing one of the cutting parameters is to reduce the dynamic feedback into the system and thus to avoid instability. The most common parameter to change is the spindle speed. An example of a passive solution was put forth in 2008 by Rashid et al. [33]. They presented the development and testing of a tuned viscoelastic dampers in a milling 6 H. Åkesson, T. Smirnova, L. Håkansson, T. Lagö and I. Claesson operation that were able to attenuate vibrations. Active solutions have also been proposed, for example, by J.L. Dohner et al. [34] who developed an active structural control system able to alter the dynamics of the system. Furthermore, active solution based on embedded piezoelectric actuators in a palletized workpiece holding system for milling was presented by Rashid et al. [35] in 2006. Another active solution focusing on the spindle was implemented by Denkena et al. who used a contactless magnetic guide in a milling machine prototype to sense and actuate harmonic disturbances [36]. 1.1.5 Motivation A major part of the research within metal cutting technology concerns methods for improving the cutting performance and increasing the tool life [3]. Cutting performance may be defined in terms, of e.g., the material removal rate, the surface roughness, and the forces the cutting process applies to the different machine parts [3]. Usually, work concerning methods for improving cutting performance focuses on maximizing the material removal rate, while keeping the surface finish below a certain roughness limit [3]. Methods that focus on tool life basically aim at the development of technology able to increase the time the tools continue to have the desired cutting performance during machining [3]. Thus, it is important to maintain the required chip formation/chip breakage and cutting forces, etc., for as long as possible [1]. From the literature review, it can be seen that much work has been done on both turning and milling theory. Many force models and methods for producing stability diagrams have been developed. Also, different methods for handling vibration problems have been presented in various articles, see for example section 1.1.4 concerning the review on vibration control. However, due to the complexity of different cutting operations, there is still much work to be carried out in order to identify and understand the causes of the problems that arise during machining. It appears that little experimental work has been done on the identification of the dynamic properties of milling vibration and spatial dynamic properties of the milling tool holder and the spindle house. Dynamic modeling of the cutting dynamics is an important research area for the manufacturing industry. Developments in this area are dependent on, among other things, knowledge of the dynamic properties of milling vibration during cutting and the spatial dynamic properties of the milling tool holder and the spindle house. In order to gain further understanding of the dynamic behavior of the milling tool holder and the spindle house in the metal cutting process, both analytical and experimental methods may be utilized. This paper investigates the dynamic properties of the milling vibration and the spatial dynamic behavior of the milling tool holder and the spindle house during milling and mode shapes and corresponding resonance frequencies for the first two modes of a milling tool holder clamped to milling machines. For the purpose of the investigation, spectrum analysis, operating deflection shapes analysis (ODS), experimental modal analysis (EMA), FE-modeling, and distributed-parameter system modeling have been utilized. 1.2 Basic Concepts of Metal Cutting in Milling A large number of different types of milling cutters, designed for different milling operations, are available today. Some of the most common types of cutters are end Investigation of the Dynamic Properties of a Milling Tool Holder 7 mill cutters, ball nosed cutter slot drills, side and face cutters, gear cutters, and hobbing cutters, see Fig. 1. a) b) c) d) Figure 1: a) Side and face cutter b) end cutters, c) slot mill cutter, and d) hobbing cutter. Since there are many different types of milling cutters, the understanding of the cutting parameters and their influence on the machining process is important in order to be able to use them properly [3]. The cutting parameters control the basic properties of the cutting process where chip formation is one of the crucial parts. As the milling cutter rotates, the material to be cut is fed into the cutter at a certain speed denoted as the feed rate, and each tooth of the cutter cuts away small chips of workpiece material. During the machining of a workpiece, the chip formation process and chip breakage are of vital importance for maintaining an efficient cutting process. The size and shape of the chip depend on many parameters. The most significant ones are: depth of cut, the feed rate, the cutting speed, the number of teeth, insert/tooth geometry, and the workpiece material [1]. A simplified drawing of the material removal process and the cutting setup in milling is presented in Fig. 2. Figure 2: A simplified sketch of the material-removal process and the cutting configuration in milling where one tooth forms a chip and removes it from a workpiece. ae is the width of cut in the radial direction (mm), ap is the cutting depth (mm), Dc is the diameter of the tool (mm), fz is the feed per tooth (mm/tooth), vC is the cutting speed (m/min), vf is the feed speed (mm/min), and h1 (α) is the un-deformed chip thickness (mm) at the angle α (rad). The cutting speed vC (m/min), is related to the spindle speed n (r.p.m) according 8 H. Åkesson, T. Smirnova, L. Håkansson, T. Lagö and I. Claesson to vc = nπDc 1000 (1) where Dc is the diameter of the cutter or tool (mm). The relation between the feed speed vf (mm/min), and the feed per tooth fz (mm/tooth) is vf = nzn fz (2) where zn is the number of efficient teeth used in the cutter during machining. One of the parameters which is usually considered in the overall process for optimal efficiency of the production line is the rate at which material is being removed. The material removal rate MRR (mm3 /min) depends on the three main cutting parameters: the feed rate vf , the depth of cut ap (mm), and the width of cut ae (mm). It may be expressed as MRR = ae ap vf . (3) Another important configuration of the cutting setup involves the entrance and exit phases of the tool to and from the workpiece. The configurations used are usually referred to as conventional milling, slot milling and climb milling. These configurations are illustrated in Fig. 3. Figure 3: Three different cutting configurations. To the left is the climb configuration, in the center the slot configuration and to the right is the conventional configuration presented. The thin arrows represent the counterclockwise rotation of the milling tool seen from the under side of the tool while the thicker strait hollow arrows represent the feed direction of each cutter. Conventional milling starts with a thin chip thickness at the entrance phase and ends with a larger chip thickness at the exit phase. In order for the insert to start to cut, a sufficient chip thickness must be built up and before the actual cutting starts workpiece material will slide along the surface [1]. This may result in a deformation hardening of the surface and also poor surface finish. At the exit phase, the insert will be exposed to severe tensile stress and the workpiece material might also remain on the edge of the insert. By contrast, climb milling starts with a large chip thickness and exits with a thin chip thickness. The insert does not slide or rub the material, which allows for longer tool life and better surface finish when compared to the conventional setup. However, climb milling usually expose the machine to larger loads compared to conventional milling [2]. Investigation of the Dynamic Properties of a Milling Tool Holder 1.3 9 Measurement of Forces and Vibrations The most common method of analyzing the properties and the performance of milling tools are done by measuring a number of different forces during cutting operations with the help of dynamometers. The measurement of the forces is carried out by either using a table-mounted dynamometer or using a spindle-mounted dynamometer. The table-mounted dynamometer is mounted on the table of the milling machine and any component to be milled can be fixed over the dynamometer. Forces in x, y and z directions may be measured and the coordinate systems of the measured signals stay fixed relative to the milling table. There are also tables/fixtures that measure the ”feed force”, the ”deflection force” and the moment applied to the table. The spindle-mounted dynamometer, which is mounted between the spindle and milling tool, usually measures the cutting forces in the x, y and z directions and moment applied to the spindle, but in this case the x-y coordinate system is rotating relative to the table. In other words, the x-y coordinates rotate with the milling tool. Examples of these types of dynamometers are presented in Fig. 4. a) b) c) Figure 4: Three types of dynamometers; in a) a table-mounted dynamometer measuring forces in the x-y-z directions is presented, b) shows a table-mounted dynamometer measuring torque and c) is a spindle-mounted dynamometer. When measuring the vibrations of a milling tool holder or a milling tool, laser vibrometers are usually used [37]. This method requires a line of sight and may limit the conditions for the machining. For example, the use of cooling liquids may not be possible. Also, the chips removed during the cutting process might interfere with the measurement. Other types of vibration sensors that may be utilized for the measurement of milling tool vibration are the strain gauge and the piezo film. Such sensors usually require amplifiers mounted on the tool holder and wireless communication such as telemetric equipment to transfer the sensor signals to data acquisition systems. For non-rotating parts, accelerometers are commonly used for vibration measurements. 2 2.1 Materials and Methods Experimental Setup The first milling machine used in the experiments was a Hurco BMC-50 vertical CNC machining centre. The spindle was of the ATC type, which means that the spindle 10 H. Åkesson, T. Smirnova, L. Håkansson, T. Lagö and I. Claesson speed can be varied between 10-3000 rpm in steps of 20 rpm, and the maximum torque was 428 Nm, see Fig. 1. The second machine used in the experiments was a DMU 80FD Duoblock which is a 5-axis milling machine, see Fig. 6. In addition to boring and milling operations, this machine can also carry out turning operations in a single machine setup. This is possible because it has a rotary table which can rotate with up to 800 rpm. It has a maximal a torque of 2050 Nm and a holding torque of 3000 Nm. The spindle has a maximal rotation speed of 8000 rpm and a maximal torque of 727 Nm. Figure 5: The Hurco BMC-50 milling machine. 2.1.1 Tool Holder and Tool The milling tool holder is the interface between the spindle and the tool which holds all the inserts. The milling tool holder used in the experiments was of the type E3471 5525 22160 which has an overhang of 140mm and a diameter of 48mm, see Fig. 7. Mounted on the tool holder was the tool R220.69-0050-12-7A presented to the left in Fig. 7. This tool has a cutting diameter of Dc = 50mm and a seventeeth,zn = 7 (inserts) configuration. The insert used in the tool configuration was XOMX120408TR-M12 T250M. The material type of the milling tool holder is SS-2511 (EN-16NiCrS4) and the material composition and properties are presented in Table 1. 2.1.2 Cutting Data Three cutting parameters were considered in the experiments: cutting depth, spindle speed and table feed rate. While two of the cutting parameters were kept constant, the third was changed in five small steps. This was done for each parameter. In Table 2, Investigation of the Dynamic Properties of a Milling Tool Holder 11 Figure 6: The DMU 80FD Duoblock milling machine. Figure 7: The tool holder E3471 5525 22160 is illustrated with the tool R220.690050-12-7A mounted. The tool is configured with seven inserts of the type XOMX120408TR-M12 T250M. H. Åkesson, T. Smirnova, L. Håkansson, T. Lagö and I. Claesson 12 Material composition besides Fe, [%] C Si Mn P S Cr Ni 0.13-0.18 0.15-0.40 0.7-1.1 0.035 0.050 0.60-1.00 0.80-1.20 Material properties Young’s Modulus Poisson’s ratio Mass density Tensile strength 210 GPa 0.3 7850 kg/m3 207 MPa Table 1: Composition and properties of the material EN-16NiCrS4. the cutting data used in the experiments are given. In the table, it is also observable that the width of the cut ap (how much of the workpiece is removed in the y-direction per tool pass) varied slightly, see Fig. 8. These variations were, however, inevitable due to the settings of the cutting data used in the experiments. The influence of these small changes is likely to be insignificant in respect to the degree of forces expected from the overall setup. Setup 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Spindle speed n [rev/min] 1401 1401 1401 1401 1401 1401 1465 1528 1592 1656 1401 1401 1401 1401 1401 Table feed vf [mm/min] 1401 1401 1401 1401 1401 1401 1401 1401 1401 1401 1401 1501 1601 1701 1801 Cutting depth ap [mm] 1 2 3 4 5 2 2 2 2 2 2 2 2 2 2 Width of cut ae [mm] 26.0 25.6 25.2 24.4 24.0 26.0 25.8 25.4 25.2 25.0 24.6 24.2 23.8 23.4 23.0 Table 2: The cutting data used during the milling measurements. 2.1.3 • • • • • • • • Measurement Equipment and Setup 12 PCB Piezotronics, Inc. 333A32 accelerometers. 2 Brüel & Kjǽr 8001 impedance head. 1 Brüel & Kjǽr NEXUS 2 channel conditioning amplifier 2692. OSC audio power amplifier, USA 850. Ling dynamic systems shaker v201. Gearing & Watson electronics shaker v4. Hewlett Packard VXI mainframe E8408A. Hewlett Packard E1432A 4-16 channel 51.2 kSa/s digitizer. Investigation of the Dynamic Properties of a Milling Tool Holder 13 Figure 8: The side milling configuration of the cutting setup during the milling measurements. • • • • 2.1.4 PC with I-DEAS 10 NX Series. Custom designed slit disk for measuring angular frequency. Rotec 5.3. Autodesk Inventor. Spatial Measurements of the Acceleration During Milling To examine the spatial dynamic behavior of milling machine components during the milling process, the acceleration at a number of different spatial locations on the structure was measured simultaneously. The accelerometers had to be positioned on non-rotating parts. Thus six accelerometers were positioned on the spindle frame and three accelerometers on the workpiece, see Fig. 8 (for the setup on the Hurco milling machine). The sensor setup on the DMU 80FD Duoblock milling machine was almost the same. The only difference was that instead of using nine accelerometers for the four positions defined in Fig. 9, twelve accelerometers were used. Thus, all directions were measured in the four nodes in the DMU 80FD Duoblock milling machine setup. Furthermore, the angular velocity along the tool holder was also measured at three positions, see Fig. 9. Two disks with 500 uniformly distributed gaps on the tool holder and a reflector tape on the spindle, together with optical scanning, was used to handle the measurements of angular velocities. All data from the milling measurements were collected with a sampling frequency of fs = 51200 Hz, using a VXI Mainframe, Matlab and VibraTools SuiteTM. The workpiece material was carbon steel SS1312 (EN 10 025) and the different workpieces used in the experiments had approximately the dimensions of 70x60x530 mm (y, z, x), see Fig. 9. The workpiece was clamped to the milling table which moved in the x-direction, resulting in a continuous cutting process along the workpiece, see Fig. 8. 2.1.5 Modal Analysis Setup The next step was to examine the dynamic properties of the milling tool holder mounted in the spindle. This was done using two shakers that excited the tool holder close to the tool in two orthogonal directions, see Fig. 10 a). Each shaker excited H. Åkesson, T. Smirnova, L. Håkansson, T. Lagö and I. Claesson 14 2 () () 2 1() 2 () S p in d le fra m e W o rk p ie c e 1 T o o l 3() () 3 T o o l h o ld e r () 2 () 3 () M illin g ta b le 4 () 4 () 4 () A c c e le ro m e te rs o n w o rk p ie c e Figure 9: The sensor configuration during the milling measurements conducted on the Hurco BMC-50 milling machine. Six accelerometers where positioned on the spindle frame and three on the workpiece. Three sensors for the velocity measurements of the spindle and the tool holder were positioned according to the illustration. the tool holder via a stinger rod connected to an impedance head, thus measuring the driving point in the respective direction, see Fig. 10 b). At the same time, the acceleration at 11 other locations along the tool holder and spindle frame was measured, see Fig. 10 where the modal analysis setup on the Hurco BMC-50 milling machine is presented. The modal analysis setup in the DMU 80FD Duoblock milling machine was almost identical to the setup in the Hurco BMC-50 milling machine. The differences between the sensor setups concerns the number of sensors and positions are presented in Fig. 11. 2.1.6 Excitation Signal for the Experimental Modal Analysis All the measurements that were performed as a basis for the experimental modal analysis were done using the excitation signal burst random, 80% noise and 20% silent. 2.2 Spectral Properties Non-parametric spectrum estimation may be utilized to produce non-parametric linear least-squares estimates of dynamic systems [38]. By using the Welch spectrum estimator [39], the cross-power spectral density P̂yx (fk ) between the input signal x(n) and the output signal y(n), and the power spectral density P̂xx (fk ) for the input k F s is the discrete frequency, signal x(n), may be produced [38, 40], where fk = N k = 0, . . . , N − 1, where N is the length of the data segments used to produce the periodograms and fs is the sampling frequency. Investigation of the Dynamic Properties of a Milling Tool Holder 15 a) b) Figure 10: The experimental modal analysis setup in the Hurco BMC-50 milling machine. a) b) Figure 11: Cross-section view in the y-z plane of the spindle house and the tool holder illustrating the positions of the sensors measuring acceleration and force in the y direction for the EMA setup in the two milling machines, the sensor configurations are identical for the x-z plane. The black squares represents the accelerometers and the black rectangular represents the impedance heads. In a) the EMA setup in the Hurco milling machine is presented where the positions of the sensor one to six are placed along the z-axis according to {260, 180, 140, 100, 60, 20} mm from the tool tip and in b) the EMA setup for the DMU 80 milling machine is shown and the positions of the sensor one to five are placed along the z-axis according to {560, 370, 100, 60, 20} mm from the tool tip. 16 H. Åkesson, T. Smirnova, L. Håkansson, T. Lagö and I. Claesson In the case of a multiple-input-multiple-output (MIMO) system with P number of responses and Q number of references, an estimate of the cross-spectrum matrix [P̂xx (fk )] between all the inputs is produced, where the diagonal elements are power spectral densities (PSDs) for the respective input signal and the of-diagonal consists of cross-spectral densities. Also, a cross-spectrum matrix [P̂yx (fk )] between all the inputs and outputs may be estimated in the same way. The least-square estimate for a MIMO system may be written as [38], −1 . (4) Ĥ(fk ) = P̂yx (fk ) P̂xx (fk ) In the case of multiple inputs, the multiple coherence is of interest as a measure of the quality of the MIMO system’s estimates [38]. 2.3 Operating Deflection Shape Analysis The spatial motion of a machine or a structure during real operating conditions may be investigated using operating deflection shapes analysis (ODS). By simultaneous measurements of N responses at discrete points on a structure the forced spatial motion of the machine or structure, either at a moment in time, or at a specific frequency may be estimated [40]. Thus, by considering the phase and amplitude of the response signals from e.g. N accelerometers distributed on an operating structure, it is possible to produce estimates of operating deflection shapes for the operating structure. The amplitude is measured by either power spectrum or power spectral density estimates depending whether the signal is tonal or random [38, 41]. And the phase between each spatial position is estimated from cross-power spectra or cross-power spectral densities [38, 41]. An estimate of a frequency domain operating deflection shape may be constructed as follows [40]: {ODS(f )}RM S = P̂11 (f ) P̂22 (f )e j θ̂21 (f ) ··· T j θ̂N 1 (f ) e . P̂N N (f ) (5) Where P̂nn (f ) are e.g. estimated power spectra and ej θ̂n1 (f ) are phase functions of cross-power spectra P̂n1 (f ), n ∈ {2, · · · , N }. 2.4 Experimental Modal Analysis The primary goal of experimental modal analysis is to identify the dynamic properties of the system under examination or the modal parameters. In other words, the purpose is to determine the natural frequencies, mode shapes and damping ratios from experimental vibration measurements. The procedure of modal analysis may be divided into two parts: the acquisition of data followed by the parameter estimation or parameter identification that can be determined with these data, a process also known as curve fitting [42]. Acquiring good data and performing accurate parameter identification is an iterative process, based on various assumptions along the way [42]. 2.5 Modal Parameter Estimation There are several different methods for the identification of the modal parameters [42, 43]. There are two basic curve fitting methods. One consists of curve fitting in the frequency domain using measured frequency response function (FRF) data and a Investigation of the Dynamic Properties of a Milling Tool Holder 17 parametric model of the FRF. The other method employs curve fitting toward the measured impulse response function (IRF) data using a parametrical model of the IRF [42]. Many methods use both domains, depending on which parameter that estimated [42]. A parametric model of the FRF matrix, Ĥ(f ) , expressed as the receptance between the reference points, or the input signals, and the responses or the output signals, may be written as [42], N Qr {ψ}r {ψ}Tr Q∗ {ψ}∗r {ψ}H r + r Ĥ(f ) = ∗ j2πf − λr j2πf − λr r=1 (6) where r is the mode number, N is the number of modes used in the model, Qr is the scaling factor of mode r, {ψ}r is the mode shape vector of mode r, and λr is the pole belonging to mode r. Because two sources (references) were used during data acquisition, a method capable of handling multi-references is required. One such method is the polyreference least square complex exponential method developed by Vold [44, 45]. This method is defined for identification of MIMO-systems with the purpose of obtaining a global least-square estimate of the modal parameters. While this method was used in this work, the mode shapes were estimated using the frequency domain polyreference method [46]. The modal scaling method used was unity modal mass [43]. To assess the quality of the estimated parameters, the FRF’s were synthesized using the estimated parameters and overlayed with the estimated FRF’s. Furthermore, the Modal Assurance Criterion (MAC) [42] defined by 2 {ψ}H {ψ} k l (7) M ACkl = H {ψ} {ψ}H {ψ} {ψ} k l k l was used as a measure of correlation between the mode shape {ψ}k belonging to mode k, and the mode shape {ψ}l belonging to mode l, where H is the Hermitian transpose operator. 2.5.1 Spectral Estimation Parameters The estimation parameters used for the spectral density estimates, frequency response functions and operating deflection shapes are presented in Table 3, Table 4 and Table 5 respectively. Parameter Excitation signal Sampling frequency f s Block length N Frequency resolution ∆f Number of averages L Burst length Window Overlap Value Cutting process 51200 Hz 40960 1.25 Hz 20 Hanning 50% Table 3: Spectral density estimation parameters used in the production of the milling tool holder spectra during continuous machining. 18 H. Åkesson, T. Smirnova, L. Håkansson, T. Lagö and I. Claesson Parameter Excitation signal Sampling frequency f s Block length N Frequency resolution ∆f Number of averages L Burst length Window Overlap Value Burst random 51200 Hz 40960 1.25 Hz 200 80% Rectangular 0% Table 4: Spectral density estimation parameters used in the production of the frequency response functions for the modal analysis. Parameter Excitation signal Sampling frequency f s Block length N Frequency resolution ∆f Number of averages L Burst length Window Overlap Value Cutting process 51200 Hz 40960 1.25 Hz 6 Hanning 50% Table 5: Spectral density estimation parameters used in the production of the operating deflection shapes. Investigation of the Dynamic Properties of a Milling Tool Holder 2.6 19 Distributed Parameter Model of the Milling Tool Holder The milling tool holder may be considered to be a beam with the cross section A(z) and the length l. The Euler-Bernoulli beam theory may be utilized to approximately model a milling tool holder’s lower order bending modes [40,47]. The Euler-Bernoulli beam theory is generally considered for slender beams that have a diameter to length ratio exceeding 10 as this ratio allows the effects of shear deformation and rotary inertia to be ignored [48]. As a result, this theory tends to slightly overestimate the eigenfrequencies. This problem increases when dealing with the eigenfrequencies of higher modes [48]. 2.6.1 A Geometrical Model of the Clamped Milling Tool Holder The milling tool holder has a complex structure and a cone interface is used for the particular clamping mechanism that attaches the tool holder to the spindle. Furthermore, the tool holder consists of a cylindrical shaft with a lip towards the spindle and in the center of the tool holder is cooling channel. In the model, spindle and the tool holder are assumed to be clamped rigidly. The geometry of the tool holder has also been simplified into a pipe in the analytical model. The cross-section of the milling tool holder and the corresponding analytical model are illustrated in Fig. 12 together with the assumed clamping. Figure 12: a) The cross-section of the milling tool holder and in b) the simplified analytical model, where l=140.90 mm is the length of the overhang, Ro = 24.00 mm is the radius of the tool holder and Ri = 10.25 mm is the radius of the coolant channel. The cross-sectional properties of the simplified model are presented in Table 6. Variable A I J Value 1.4795 · 10−3 2.5191 · 10−7 4.0838 · 10−7 Unit m2 m4 m4 Table 6: The cross-sectional properties of the milling tool, where A is the area, I is the moment of inertia and J is the polar moment of inertia. H. Åkesson, T. Smirnova, L. Håkansson, T. Lagö and I. Claesson 20 2.6.2 A Model of Transverse Vibrations The Euler-Bernoulli differential equation describing the transversal motion of the milling tool holder in the y-direction may be written as [48] ∂2 ∂ 2 u(z, t) ∂ 2 u(z, t) = f (z, t) (8) + 2 EIx (z) ρA(z)dz ∂t2 ∂z ∂z 2 where A(z) is the milling tool holder’s cross-sectional area, E is Young’s elastic modulus for the tool holder, I(z) is the cross-sectional area moment of inertia about the ”x axis”, ρ is the density, t is the time, u(z, t)is the deflection in the y-direction and f (z, t) is the external force per unit length. It is assumed that both the cross-sectional area A(z) and the flexural stiffness EI(z) are constant along the milling tool holder. Eq.8 is often referred to as the Euler-Bernoulli beam equation. The model assumes that the following assumptions regarding the beam and its plane are true: • The beam is uniform along its span-, or length-, and slender (diameter to length ratio¿10). • The beam is composed of a linear, homogenous, isotropic elastic material without axial loads. • The plane section remains plane. • The plane of symmetry of the beam is also the plane of vibration so that rotation and translation are decoupled. • Rotary inertia and shear deformation can be neglected. To model the milling tool holder, a Fixed-Free Euler-Bernoulli beam model was applied. The beam has four boundary conditions, two at each end. One end is clamped and the other is free, see Fig. 13. Figure 13: Model of a Fixed - Free beam, where ρ is the density, E is the elasticity modulus (Young’s coefficient), G is the shear modulus, A is the cross-sectional area, I is the moment of inertia, J is the polar moment of inertia and the length of the beam l = 140.9 mm. The clamped side of the beam will be fixated. Thus the displacement and the slope of the displacement in this point z = 0 will equal zero and the two first boundary conditions become ∂u(z, t) =0 (9) u(z, t)|z=0 = 0, ∂z z=0 The other end is free, so that no bending moment or shear force constrains the beam at the coordinate z = l when the beam vibrates. This yields two other boundary conditions that can be described as ∂ 3 u(z, t) ∂ 2 u(z, t) = 0, EI = 0. (10) EI ∂z 2 z=l ∂z 3 z=l Investigation of the Dynamic Properties of a Milling Tool Holder 2.6.3 21 A Model of Torsional Vibrations In the same way as for the transverse vibration model, a vibration model for the torsional vibrations may be derived by considering the equation of motion for an infinitesimal element of the beam [48]. The differential equation describing the torsional motion for the milling tool holder around and along the z axis may be written as [48] 2 G ∂ θ(z, t) ∂ 2 θ(z, t) − = τ (z, t) (11) 2 ∂t ρ ∂z 2 where θ(z, t) is the angular deflection, G is the shear modulus, ρ is the density and τ (z, t) is the externally applied torque load per unit length. The clamped boundary condition is applied to where the milling tool holder is connected to the spindle and yields zero deflection. At the other end there is no torque in the case of the free vibration model. Thus, the boundary conditions for a milling tool holder with a coolant channel modeled as a hollowed shaft becomes θz (z, t)|z=0 = 0 π 4 Ro − Ri4 Gθz (z, t)|z=l = 0 2 (12) (13) where Ro is the outer radius of the milling tool holder and Ri is the radius of the coolant channel. The relation between the shear modulus G and the elasticity modulus E is given by [49] G= E 2(1 + ν) (14) where ν is Poisson’s ratio. 2.7 A Finite Element Model of the Milling Tool Holder The milling tool holder was modeled in a CAD program and a finite element analysis was conducted to estimate the natural frequencies and mode shapes of the tool holder. The finite element mesh of the milling tool holder, consisted of 73470 nodes and 42728 elements, is presented in Fig. 14 where the white lines represent the borders of the elements connected at the nodes. In the FE analysis the clamping surface on the back of the holder, i.e. behind the lip to the left in Fig. 14, was constrained to be fixed for all degrees of freedoms. 3 Results This experimental investigation resulted in a large amount of vibration data that was collected from both experimental setups from both milling machines. However, the results presented in this report only constitute a small part of the investigation, but they represent the essence of the results. The results from the experimental examination are presented in terms of measured acceleration signals as a function of time and power spectral densities of the acceleration signals. Operating deflection shapes were estimated for one of the milling machines and are presented. Also, results from experimental modal analysis of the tool holder mounted in the milling machine are given. Finally, resonance frequencies and mode shapes calculated based H. Åkesson, T. Smirnova, L. Håkansson, T. Lagö and I. Claesson 22 Figure 14: The finite element mesh of the milling tool holder consisting of 73470 nodes and 42728 elements. on distributed-parameter system models of the milling tool holder were generated as well as the corresponding results from the finite element analysis of the milling tool holder. 3.1 Spatial Measurements of Vibration In order to get an overview of the measured acceleration signals during machining, the acceleration of the workpiece moving in the feed direction (+x4 ) on the DMU 80FD Duoblock milling machine during machining is presented in the time domain in Fig. 15. 80 Acceleration [m/s2 ] Acceleration [m/s2 ] 100 60 40 20 0 −20 −40 −60 −80 −100 0 60 40 20 0 −20 −40 −60 2 4 6 8 10 Time t [sec] a) 12 14 16 1.8 1.82 1.84 1.86 Time t [sec] b) 1.88 1.9 Figure 15: a) Accelereation of the workpiece in the feed direction (+x4 ) during a milling operation performed in the DMU 80FD Duoblock milling machine. The radial depth ae was 23 mm, the axial depth ap was 2 mm, the feed speed vf was 1401 mm/min and the spindle speed n was 1401 rev/min b) and the corresponding acceleration record zoomed in. Results in terms power spectral density estimates of workpiece vibration in the feed direction (+x4 ) are presented in Fig. 16 a) and b) for five different axial depths (ap = 1, 2, 3, 4, 5 mm) in the DMU 80FD Duoblock milling machine. The radial Investigation of the Dynamic Properties of a Milling Tool Holder 23 40 Depth=1mm Depth=5mm 30 20 10 0 −10 −20 −30 −40 −50 −60 400 600 800 1000 1200 1400 Frequency [Hz] a) 1600 1800 PSD [dB rel 1 ((m/s2 )2 /Hz)] PSD [dB rel 1 ((m/s2 )2 /Hz)] depth was ae = 26.0, 25.6, 25.2, 24.2, 24.0 mm, the feed speed vf was mm/min and the spindle speed n was 1401 r.p.m. The periodic components found in the power spectral density estimates in Fig. 16 a) are related to the spindle speed n. Furthermore, a broadband response to an underlying structure may be observed in Fig. 16 a). Power spectral density estimates workpiece vibration, zoomed in frequency to the interval of the dominating resonance peak at approx. 770 Hz for, for the five different axial depths are presented in Fig. 16 b). 40 30 20 10 0 −10 −20 Depth=1mm Depth=2mm Depth=3mm Depth=4mm Depth=5mm −30 −40 −50 −60 700 750 800 850 Frequency [Hz] b) 900 Figure 16: Power spectral densities of workpiece vibration in the feed direction (+x4 ) during milling for different axial depths ap in the DMU 80FD Duoblock milling machine. In a) (for the gray solid line ap = 1 mm and for the black solid line ap = 5 mm) and the radial depth ae was 26 mm and 24 mm respectively, the feed speed vf was mm/min and the spindle speed n was 1401 r.p.m. b) Zoomed in frequency to the interval of the dominating resonance peak for the five different axial depths (ap = 1, 2, 3, 4, 5 mm) and the corresponding radial depth was ae = 26.0, 25.6, 25.2, 24.2, 24.0 mm. Results in terms power spectral density estimates of workpiece vibration in the feed direction (+x4 ) when changing the feed speed vf are presented in Fig. 17 for five different feed speeds (vf = 1401, 1501, 1601, 1701, 1801 mm/min) in the DMU 80FD Duoblock milling machine. The radial depth was ae = 24.6, 24.2, 23.8, 23.4, 23.0 mm, the axial depth ap was 2 mm and the spindle speed n was 1401 r.p.m. In Fig. 17 a) no particular changes can be observed, but when zooming in on the peaks as illustrated in Fig. 17 b) a small difference in magnitude is observable. Results in terms power spectral density estimates of workpiece vibration in the feed direction (+x4 ) when changing the spindle speed n are presented in Fig. 18 a), for five different spindle speeds (n = 1401, 1465, 1528, 1592, 1656 r.p.m.) in the DMU 80FD Duoblock milling machine. The radial depth was ae = 26.0, 25.8, 25.4, 25.2, 25.0 mm, the axial depth ap was 2 mm and the feed speed vf was 1401 mm/min. In Fig. 18 a), it is observable how the frequency of the harmonics changes with the change of spindle speed. In Fig. 18 b), typical power spectral density estimates of workpiece vibration in the feed direction (+x4 ) during machining conducted in the Hurco BMC50 milling machine is presented. The radial depth ae was 26 mm, the axial depth ap was 1 mm, the feed speed vf was 1401 mm/min and the spindle speed n was 1401 r.p.m. Also, when carrying out the machining in the Hurco BMC-50 milling machine both a large number of narrow-banded peaks and a broadband response of PSD [dB rel 1 ((m/s2 )2 /Hz)] 10 5 0 −5 −10 −15 −20 1401 mm/min 1501 mm/min 1601 mm/min 1701 mm/min 1801 mm/min −25 −30 −35 −40 750 775 800 825 Frequency [Hz] a) 850 PSD [dB rel 1 ((m/s2 )2 /Hz)] H. Åkesson, T. Smirnova, L. Håkansson, T. Lagö and I. Claesson 24 4 2 0 −2 −4 1401 mm/min 1501 mm/min 1601 mm/min 1701 mm/min 1801 mm/min −6 −8 814 815 816 817 818 Frequency [Hz] 819 820 b) Figure 17: a) Power spectral densities of workpiece vibration in the feed direction (+x4 ) during milling for five different feed speeds (vf = 1401, 1501, 1601, 1701, 1801 mm/min) in the DMU 80FD Duoblock milling machine. The radial depth was ae = 24.6, 24.2, 23.8, 23.4, 23.0 mm, the axial depth ap was 2 mm and the spindle speed n was 1401 r.p.m. b) Corresponding spectra zoomed in at one of the harmonics. an underlying structure may be observed in the spectra, see Fig. 18 b). 3.1.1 Operating Deflection Shapes To obtain information on how the spindle frame vibrates relative to the workpiece, spatial measurements of the acceleration of these structural parts were carried out. Accelerometer positions and measurement directions on the spindle frame and the workpiece are illustrated in Fig. 19 a). To facilitate illustration of the operating deflection shapes, the spindle frame and the workpiece are simplified into a skeleton structure where the measurement positions are illustrated by black circles, defined as nodes, as also shown in Fig. 19 a). In Fig. 19 b) the simplified skeleton structure of the spindle frame and the workpiece is shown in the y-z plane and in Fig. 19 c) it is shown in the x-z plane. Observe that the fourth node in the skeleton structure is fixed on the workpiece and thus moving away from the other nodes, along the x-axis, as the tool is cutting the workpiece. The spatial motion of this structure has a complex behavior and changes with time. However, an operating deflection shape at one of the dominant peeks in the spectral density previously presented (see Fig. 16), i.e. at 780 Hz, was estimated during a short time sequence and is presented in Fig. 20 a) and b). The deformation shape is presented with arrows in the figure. Observe that the size of the arrows does not represent the absolute magnitudes of the four positions deflection; their magnitudes are displayed in an enlarged scale to make them observable. In order to show the complex spatial behavior of the measurement positions on the spindle frame and the workpiece, a trajectory for node two is presented during a time sequence of 15.625 ms, see Fig. 21 a) and b). The trajectory was produced by filtering the acceleration signals with a band-pass filter having a center frequency at 780 Hz. The frequency response function for the band-pass filter is presented in Fig. 22. By combining the trajectory plots for each of the four measurement positions 10 5 0 −5 −10 −15 −20 1400 rev/min 1464 rev/min 1528 rev/min 1591 rev/min 1655 rev/min −25 −30 −35 −40 750 770 790 810 Frequency [Hz] a) 830 850 PSD [dB rel 1 ((m/s2 )2 /Hz)] PSD [dB rel 1 ((m/s2 )2 /Hz)] Investigation of the Dynamic Properties of a Milling Tool Holder 25 10 5 0 −5 −10 −15 −20 −25 −30 −35 −40 400 600 800 1000 1200 1400 Frequency [Hz] b) 1600 1800 Figure 18: a) Power spectral densities of workpiece vibration in the feed direction (+x4 ) during milling for five different spindle speeds (n = 1401, 1465, 1528, 1592, 1656 r.p.m.) in the DMU 80FD Duoblock milling machine. The radial depth was ae = 26.0, 25.8, 25.4, 25.2, 25.0 mm, the axial depth ap was 2 mm and the feed speed vf was 1401 mm/min. b) Power spectral density of workpiece vibration in the feed direction (+x4 ) performed in the Hurco BMC-50 milling machine. The radial depth ae was 26 mm, the axial depth ap was 1 mm, the feed speed vf was 1401 mm/min and the spindle speed n was 1401 r.p.m. a) b) c) Figure 19: a) presents a 3d-view of the spindle frame, tool holder and the milling table with the workpiece. The measurement positions shown as black circles connected by straight black lines forming a skeleton structure, b) present the skeleton structure in the y-z plane and c) presents the skeleton structure in the x-z plane. Node position on the z-axis [cm] 100 Node 2 Node position on the z-axis [cm] H. Åkesson, T. Smirnova, L. Håkansson, T. Lagö and I. Claesson 26 Node 1 80 60 Node 3 40 20 Node 4 0 −60 −40 −20 0 20 40 60 100 Node 2 80 60 Node 3 40 20 Node 4 0 −40 Node position on the y-axis [cm] a) Node 1 −20 0 20 40 60 80 Node position on the x-axis [cm] b) z-axis z-axis Figure 20: Operating deflection shape for the spindle frame and the milling table with the workpiece at the frequency 780 Hz, estimated during machining. The radial depth ae was 24.4 mm, the axial depth ap was 4 mm, the feed speed vf was 1401 mm/min and the spindle speed n was 1401 r.p.m. In a) the shape is presented in the y-z plane and in b) the shape is presented in the x-z plane. y-axis a) x-axis y-axis x-axis b) Figure 21: Trajectory plot of the measured acceleration signals in node two at 780 Hz during machining. The radial depth ae was 24.4 mm, the axial depth ap was 4 mm, the feed speed vf was 1401 mm/min and the spindle speed n was 1401 rev/min. a) and b) viewed from two different perspectives perspective. Investigation of the Dynamic Properties of a Milling Tool Holder 0 −10 Phase [degree] Magnitude [dB] [cm] 0 −20 −30 −40 −50 −1000 −2000 −3000 −4000 −5000 −60 −70 0 27 400 800 1200 Frequency [Hz] a) 1600 2000 −6000 0 400 800 1200 Frequency [Hz] b) 1600 2000 Figure 22: The frequency response function for the filter used in the production of the time domain ODS, a) is the magnitude and b) is the phase. on the spindle frame and workpiece and plotting them together with the skeleton structure its spatial motion for a short time interval may be illustrated as in Fig. 23. Observe that the ellipses are displayed in a enlarged scale to make them observable. 3.1.2 Angular Vibrations The angular vibrations of the milling tool holder were measured at three different positions, two on the tool holder and one at the spindle close to the clamping of the tool holder. In Fig. 24, the angular vibrations of the three positions versus the number of revolutions of the tool holder are shown in the same diagram. In this figure, the angular vibrations during approximately the first 35 revolutions are measured prior to engagement of the tool in the workpiece. Note how, during the first revolutions, when no machining is carried out, the angular vibration of the tool holder and the spindle are still observable. All three sensors show a good agreement on the angular vibrations when there is no cutting, see Fig. 25 a). However, during machining a discrepancy between the angular vibrations measured by the sensor on the spindle and the angular vibrations measured by the sensors on the tool holder is observable, see Fig. 25 b). By plotting a waterfall diagram of the order spectra of the angular tool holder vibrations closest to the tool, it is obvious that the main angular vibration is directly related to the first order, see Fig. 26. The radial depth ae was 24.0 mm, the axial depth ap was 5 mm, the feed speed vf was 1401 mm/min and the spindle speed n was 1401 rev/min. To facility observability of the peaks of the higher orders in the order spectra the first order was excluded from them and they were again plotted in a waterfall diagram as illustrated in Fig. 26 b). Also, in this figure an underlying broadband dynamic angular response of the tool holder may be observed. The seventh, 14:th and 21:th order of the spindle speed are slight higher than the direct neighboring orders, see Fig. 26 b). These orders are also the first, second and third order of the tooth-passing frequency. Node position on the z-axis [cm] 100 Node 2 Node position on the z-axis [cm] H. Åkesson, T. Smirnova, L. Håkansson, T. Lagö and I. Claesson 28 Node 1 80 60 Node 3 40 20 Node 4 0 −60 −40 −20 0 20 40 60 Node position on the y-axis [cm] a) 100 Node 2 Node 1 80 60 Node 3 40 20 Node 4 0 −40 −20 0 20 40 60 80 Node position on the x-axis [cm] b) Vibration angle [degree] Figure 23: Spatial motion of the spindle frame and the workpiece for a short time interval, based on band pass filtered acceleration signal measured during cutting. The radial depth ae was 24.4 mm, the axial depth ap was 4 mm, the feed speed vf was 1401 mm/min and the spindle speed n was 1401 r.p.m. The ellipses represent the motion of the measured nodes for the frequency 780 Hz. The circle on each ellipse represents a synchronization point for all the nodes (measurement positions) at a certain time instant and is followed by a solid ellipse line which indicates the direction of the motion. In a) the motion is presented in the y-z plane and in b) the motion is presented in the x-z plane. 0.4 0.3 0.2 Upper disc Middle disc Lower disc 0.1 0 −0.1 −0.2 −0.3 −0.4 15 20 25 30 35 40 Revolutions 45 50 55 60 Figure 24: Angular vibrations of tool holder and the spindle, measured at two positions on the tool holder and at one position on the spindle. The first 35 revolutions are measured prior to engagement of the tool in the workpiece directly followed by the engagement phase. After approx. 40 revolutions the material removal process is carried out according to the selected cutting data. The radial depth ae was 24.2 mm, the axial depth ap was 2 mm, the feed speed vf was 1501 mm/min and the spindle speed n was 1401 rev/min. Vibration angle [degree] Investigation of the Dynamic Properties of a Milling Tool Holder 0.1 29 Upper disc Middle disc Lower disc 0.05 0 −0.05 −0.1 21 22 23 24 25 Revolutions 26 27 28 29 Vibration angle [degree] a) 0.6 Upper disc Middle disc Lower disc 0.4 0.2 0 −0.2 −0.4 401.5 402 402.5 403 403.5 Revolutions 404 404.5 405 405.5 b) Figure 25: The angular vibrations of the milling tool holder and the spindle during machining when the radial depth ae was 24.2 mm, the axial depth ap was 2 mm, the feed speed vf was 1501 mm/min and the spindle speed n was 1401 rev/min. a) Shows the angular vibrations prior to cutting and b) shows the angular vibrations during cutting. H. Åkesson, T. Smirnova, L. Håkansson, T. Lagö and I. Claesson 30 a) b) Figure 26: Waterfall plot of the order spectra of the angular vibration during 30 s of machining. The radial depth ae was 24.0 mm, the axial depth ap was 5 mm, the feed speed vf was 1401 mm/min and the spindle speed n was 1401 rev/min. a) presents the order spectra with the first order included in the plot and b) presents the order spectra when the first order has been removed. Investigation of the Dynamic Properties of a Milling Tool Holder 3.2 31 Modal Analysis Results 20 10 0 −10 −20 −5X:+1X −5X:+3X −5X:+4X −5X:+5X −5X:+6X −30 −40 400 600 800 1000 1200 1400 Frequency [Hz] a) 1600 1800 Accelerance [dB rel 1(m/s2 )/N] Accelerance [dB rel 1(m/s2 )/N] Results in terms of accelerance function estimates and coherence function estimates from the experimental modal analysis (EMA) carried out on two milling machines are presented first in this section next to each other. In other words, the accelerance function estimates from the Hurco BMC-50 milling machine are presented in Fig. 27 a), Fig. 28 a) and Fig. 29 a) and the accelerance function estimates from the DMU 80FD Duoblock milling machine are presented in Fig. 27 b), Fig. 28 b) and Fig. 29 b). The two EMA setups differed slightly between the two machines, see section 2.1.5. A significant peak is noticeable around 650 Hz in the accelerance functions produced from the EMA carried out in Hurco BMC-50 milling machine and a peak around 750 Hz is noticeable in the accelerance functions produced from the EMA carried out in the DMU 80FD Duoblock milling machine. 20 10 0 −10 −20 −5X:+1X −5X:+2X −5X:+4X −5X:+5X −5X:+6X −30 −40 400 600 800 1000 1200 1400 Frequency [Hz] b) 1600 1800 Figure 27: Accelerance magnitude function estimates between the force input in xdirection and the acceleration responses in x-direction based on the experimental modal analysis measurements, a) in the Hurco BMC-50 milling machine and b) in the DMU 80FD Duoblock milling machine. Typical multiple coherence function estimates obtained during the experimental modal analysis are illustrated in Fig. 30. The coherence function estimates presented in Fig. 30 shows values above 0.9 for most frequencies between 350 Hz and 1800 Hz. This indicates that the level of forces and accelerations was fairly good in the region of interest, that is, between 350 Hz up to 1400 Hz. Some dips may be observed around 800 Hz in estimates done from both EMA setups and one larger around 1500 Hz in the coherence function estimate from the EMA carried out in the DMU 80FD Duoblock milling machine. 3.2.1 Mode Shapes Based on the accelerance functions estimated from the modal analysis setups carried out in the Hurco BMC-50 milling machine and in the DMU 80FD Duoblock milling machine, a number of resonance frequencies were estimated. The estimated resonance frequencies and their relative damping are presented in Table 7. Furthermore, for each resonance frequency a corresponding mode shape of the spindle house - tool holder system was estimated. A figure (Fig. 31) defining the positions of the sensors together with two tables ( Table 8 and Table 9) presents Accelerance [dB rel 1(m/s2 )/N] 20 10 0 −10 −5Y:+1Y −5Y:+3Y −5Y:+4Y −5Y:+5Y −5Y:+6Y −20 −30 −40 400 600 800 1000 1200 1400 Frequency [Hz] a) 1600 1800 Accelerance [dB rel 1(m/s2 )/N] H. Åkesson, T. Smirnova, L. Håkansson, T. Lagö and I. Claesson 32 20 10 0 −10 +5Y:−1Y +5Y:−2Y +5Y:−4Y +5Y:−5Y +5Y:−6Y −20 −30 −40 400 600 800 1000 1200 1400 Frequency [Hz] b) 1600 1800 20 10 0 −10 −20 −5Y:+1X −5Y:+3X −5Y:+4X −5Y:+5X −5Y:+6X −30 −40 400 600 800 1000 1200 1400 Frequency [Hz] a) 1600 1800 Accelerance [dB rel 1(m/s2 )/N] Accelerance [dB rel 1(m/s2 )/N] Figure 28: Accelerance magnitude function estimates between the force input in ydirection and the acceleration responses in y-direction based on the experimental modal analysis measurements, a) in the Hurco BMC-50 milling machine and b) in the DMU 80FD Duoblock milling machine. 20 10 0 −10 −20 +5Y:+1X +5Y:+2X +5Y:+4X +5Y:+5X +5Y:+6X −30 −40 400 600 800 1000 1200 1400 Frequency [Hz] b) 1600 1800 Figure 29: Accelerance magnitude function estimates between the force input in ydirection and the acceleration responses in x-direction based on the experimental modal analysis measurements, a) in the Hurco BMC-50 milling machine and b) in the DMU 80FD Duoblock milling machine. 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 *:+6Y *:+6X 0.1 0 400 600 800 1000 1200 1400 Frequency [Hz] a) 1600 1800 Multiple Coherence γ 2 Multiple Coherence γ 2 Investigation of the Dynamic Properties of a Milling Tool Holder 33 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 *:−5Y *:+5X 0.1 0 400 600 800 1000 1200 1400 Frequency [Hz] b) 1600 1800 Figure 30: Typical coherence functions between the force inputs (the asterisk denotes all inputs) and the acceleration responses +6Y,-5Y (dashed line) and +6X,+5X (solid line) from the experimental modal analysis measurements, a) in the Hurco BMC-50 milling machine and b) in the DMU 80FD Duoblock milling machine. Mode 1 2 3 4 5 6 7 8 9 10 Hurco milling machine Freq. [Hz] Damping [%] 615.4 3.91 650.4 2.58 783.0 0.60 920.6 1.49 985.5 1.23 1141.1 1.75 1305.3 1.96 1349.9 1.68 1518.0 4.11 1611.3 0.38 DMU 80FD milling machine Mode Freq. [Hz] Damping [%] 1 744.1 2.68 2 755.4 2.61 3 809.6 2.38 4 925.8 1.24 5 992.8 2.03 6 1064.0 1.34 7 1128.2 1.50 8 1509.5 1.17 9 1692.8 1.04 Table 7: Estimated resonance frequencies and their relative damping coefficients from the modal analysis setups carried out in the Hurco BMC-50 milling machine and in the DMU 80FD Duoblock milling machine. 34 H. Åkesson, T. Smirnova, L. Håkansson, T. Lagö and I. Claesson the two first mode shapes for the respective milling machine setup in the form of magnitude and angle. a) b) Figure 31: Cross-section view of the spindle house and the tool holder illustrating the mode shape for the first bending mode. In a) for the Hurco milling machine and in b) for the DMU 80 milling machine. Hurco milling machine mode at 615.4 Hz Position Norm. mag Angle [◦ ] 1 0.05 65.4 2 3 0.36 50.4 4 0.58 47.9 5 0.75 44.7 6 1.00 45.3 DMU 80FD milling machine mode at 744.1 Hz Position Norm. mag Angle [◦ ] 1 0.02 47.5 2 0.01 48.3 3 0.28 54.2 4 0.69 56.2 5 1.00 57.1 - Table 8: Mode shape table presenting magnitude values for each measurement position and the angle relative to the x-axis, for the first mode estimated for respective milling machine setup. In order to be able to evaluate the quality of the estimated modal parameters, a synthesized accelerance function are produced and overlaid on top of the corresponding estimated accelerance function. In Fig. 32 and Fig. 33, the driving point accelerance functions and the transfer accelerance functions between x and y direction in the driving point position, are presented together with their synthesized functions. The synthesized functions show good agreement with the estimated accelerance functions. In order to check the quality of the estimated mode shapes, a MAC matrix was produced. These matrixes are presented in Fig. 30, and as can be seen the orthogonality between the two first modes are excellent, while a high correlation exists for higher order modes. 3.3 Distributed Parameter Model Results The results from the distributed parameter models of a clamped milling toolholder, in terms of natural frequency estimates, for both bending and torsional modes, are presented in Table 10. The mode shapes, based on the distributed parameter models of a clamped milling tool holder, for the first and second bending mode as well as for the first torsional mode is presented in Fig. 35. Note that the first and second bending mode shapes are Investigation of the Dynamic Properties of a Milling Tool Holder Hurco milling machine mode at 650.4 Hz Position Norm. mag Angle [◦ ] 1 0.03 -24.2 2 3 0.35 -39.3 4 0.55 -42.1 5 0.72 -45.1 6 1.00 -44.5 35 DMU 80FD milling machine mode at 755.4 Hz Position Norm. mag Angle [◦ ] 1 0.01 -42.2 2 0.01 -41.8 3 0.22 -35.6 4 0.68 -33.9 5 1.00 -32.8 - 15 −5X:+5X, Estimated −5X:+5X, Synthesized 10 5 0 −5 −10 −15 −20 400 600 800 1000 1200 1400 Frequency [Hz] 1600 1800 Accelerance [dB rel 1(m/s2 )/N] Accelerance [dB rel 1(m/s2 )/N] Table 9: Mode shape table presenting the magnitude values for each measurement position and angle relative to the x-axis, for the second mode estimated for respective milling machine setup. 15 −5Y:+5X, Estimated −5Y:+5X, Synthesized 10 5 0 −5 −10 −15 −20 400 600 800 1000 1200 1400 Frequency [Hz] 1600 1800 20 10 0 −10 −20 −30 −40 −50 −4X :+4X, Estimated −4X :+4X, Synthesized 400 600 800 1000 1200 1400 Frequency [Hz] 1600 1800 Accelerance [dB rel 1(m/s2 )/N] Accelerance [dB rel 1(m/s2 )/N] Figure 32: The magnitude of the synthesized and the measured accelerance functions for the milling tool holder when clamped in the Hurco BMC-50 milling machine, a) between the force signal and the acceleration signal from location 5X and b) between the force signal at location 5Y and the acceleration signal from location 5X. 20 10 0 −10 −20 −30 −40 −50 −4X:−4Y, Estimated −4X:−4Y, Synthesized 400 600 800 1000 1200 1400 Frequency [Hz] 1600 1800 Figure 33: The magnitude of the synthesized and the measured accelerance functions for the milling tool holder when clamped in the DMU 80FD Duoblock milling machine, a) between the force signal and the acceleration signal from location 4X and b) between the force signal at location 4Y and the acceleration signal from location 4X. H. Åkesson, T. Smirnova, L. Håkansson, T. Lagö and I. Claesson 36 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 615 650 783 921 986 1141 1305 1350 1518 l 1611 M od eΨ [H z] 1611 1518 1350 1305 1141 986 921 783 k 650 615 z] de Mo Ψ [H 0 744 755 810 926 993 1064 1128 1510 l 1693 M od eΨ [H z] a) 1693 1510 1128 1064 993 926 810 k 755 744 ] de Mo Ψ [Hz b) Figure 34: a) MAC matrix presenting the correlation between the estimated mode shapes. In a) the matrix based on the mode shapes estimated from the Hurco BMC-50 milling machine setup, while in b) the matrix is based on the mode shapes estimated from the DMU 80FD Duoblock milling machine setup. Mode 1 2 3 Frequency [Hz] 1902.34 11921.74 6321.53 Type of mode First bending Second bending First torsional Table 10: Natural frequency estimates based on the distributed parameter model of a clamped milling tool holder. Investigation of the Dynamic Properties of a Milling Tool Holder 37 in the transverse direction, while the first torsional mode shape represents a rotation deformation of the structure around and along its own centerline. Normalized mode shape 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 0 First bending mode Second bending mode First torsional mode 0.02 0.04 0.06 0.08 0.1 0.12 Distance from clamping [m] 0.14 Figure 35: The first and second bending modes together with the first torsional mode. Calculated based on the distributed-parameter system models of a clamped milling tool holder. 3.4 Finite Element Model The first six natural frequencies estimated based on the milling tool holder FE-model are presented in Table 11. The corresponding mode shapes are illustrated in Fig. 36to Fig. 38. Mode 1 2 3 6 7 8 Frequency [Hz] 1437.7 1440.4 5417.6 6728.7 6744.2 7789.2 Type of mode First bending First bending First torsional Second bending Second bending First longitudinal Table 11: Natural frequency estimates based on FE analysis of a clamped milling tool holder. 4 Summary and Conclusion A number of different machining measurements have been conducted which show good agreement with the expectations grounded in cutting theory. A pronounced periodicity is present in acceleration measurements carried out during milling operations. This periodicity is directly related to the spindle speed n and also to the harmonics observable in Fig. 16 a) and b). Furthermore, when increasing the cutting depth ap the vibration level increases as can be seen in Fig. 16 b). This also agrees with the theory [1], showing that increasing cutting depth results in increasing cutting forces, and thus also in greater excitation levels. When changing the feed speed (feed rate), no significant changes are observed although a small increase in vibration level with 38 H. Åkesson, T. Smirnova, L. Håkansson, T. Lagö and I. Claesson a) b) Figure 36: a) The milling toolholder mode shape belonging to the natural frequency at 1437.7 Hz, and b) the mode shape belonging to the natural frequency at 1440.4 Hz, estimated with a FE model. The solid part represents the shape of the mode and the transparent part drawn in thin lines is the un-deformed part. a) b) Figure 37: a) The milling toolholder mode shape belonging to the natural frequency at 5417.6 Hz, and b) the mode shape belonging to the natural frequency at 6728.7 Hz, estimated with a FE model. The solid part represents the shape of the mode and the transparent part drawn in thin lines is the un-deformed part. a) b) Figure 38: a) The milling toolholder mode shape belonging to the natural frequency at 6744.2 Hz, and b) the mode shape belonging to the natural frequency at 7789.2 Hz, estimated with a FE model. The solid part represents the shape of the mode and the transparent part drawn in thin lines is the un-deformed part. Investigation of the Dynamic Properties of a Milling Tool Holder 39 increasing speed may be noticed, see Fig. 17. The dependence of the harmonics on the spindle speed is observable in Fig. 18 where the spindle speed was changed. In all the acceleration records, an underlying dynamic response may be observed. This is confirmed from the results obtained from the modal analysis carried out in both milling machines; compare the acceleration spectra in Fig. 18 b) with the accelerance function in Fig. 27 a) where a peak near 620 Hz is present in both figures. The results from the modal analysis, presented in Table 8 and Table 9, suggest that this is the first bending mode of the tool holder when the tool holder is clamped in the Hurco BMC-50 milling machine. This can also be seen when comparing the acceleration spectra in Fig. 16 with the accelerance function in Fig. 28 b), where a peak near 790 Hz is present in both figures. The results from the modal analysis, presented in Table 8 and Table 9, from the setup in the DMU 80FD Duoblock milling machine suggest this is the first bending mode of the tool holder. The difference in frequency of the first bending modes of the tool holder in the different machines indicates a more rigid clamping in the DMU 80FD Duoblock milling machine. Also the fact that the natural resonance frequencies estimated for both EMA setups in the same frequency range, presented in Table 7, are in general higher in the case of the DMU 80 Duoblock milling machine compare to the case of the Hurco BMC-50 milling machine, supports this conclusion. The operating deflection shape analysis of the spindle frame - workpiece of the milling machine provided information concerning their spatial motion during machining. The deflection shape of the spindle frame-workpiece, can also be connected to the first bending mode, and the complex behavior of the shape may be explained by the fact that the milling tool holder is rotating while it is assumed to have a motion that is in itself related to a first bending mode. This assumption is however, not confirmed since no accelerometer measured the bending vibration of the milling tool holder during cutting. To move on, angular vibration measurements of the milling tool holder showed the significance of the various orders. For example, the first order of the spindle speed had a major impact on the angular vibration level which suggests a significant unbalance of spindle - tool holder system. An unbalance will generally introduce transverse vibration directly related to the first order of the spindle speed. However, due to the sensor setup, the discs mounted on the milling tool holder, will because of the unbalance, be displaced from the rotation center and thus influence the angular velocity measured by the sensors. The first order of the spindle speed was already present in the angular vibration even before cutting took place, see the angular vibration before cutting in Fig. 25 a). It may also be noticed that the largest vibrations for all measurements in all nodes occur at a frequency, where one of the orders of the spindle speed coincides with a fundamental bending mode of the milling tool holder. The main purpose with using rotation sensors was to discover any angular motion relating to torsional modes of the milling tool holder. One important conclusion from the measurement was that no such dynamic behavior could be observed or demonstrated. The analytical models together with the finite element model provided rough estimates of where in frequency the natural frequencies of the various modes may be expected to be found and how the mode shapes will look like. This is important when setting up the measurement and selecting sensor configuration. Both the analytical model and the finite element model were configured in order to overestimate the natural frequencies. In the real setups, the milling tool holder was configured with a tool that made the structure longer and at the same time adding mass to the end of the 40 H. Åkesson, T. Smirnova, L. Håkansson, T. Lagö and I. Claesson structure. The clamping of the milling tool holder was furthermore assumed to be infinitely rigid for the model, which is not the case in reality. One of the most interesting results, can be found in the accelerance function in Fig. 29, where significant peaks near 800, 1000 and 1600 Hz were found in the HurcoBMC-50 milling machine setup, while peaks near 925 was found in the DMU 80FD Duoblock milling machine setup. These peaks were not found in the accelerance functions in Fig. 27 and Fig. 28. The presence of the peaks in the accelerance function in Fig. 29 suggests that the structure is sensitive to forces applied in the orthogonal direction with respect to the direction of the response. 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Investigation of the Dynamic Properties of a Milling Tool Holder Investigation of the Dynamic Properties of a Milling Tool Holder Henrik Åkesson, Tatiana Smirnova, Lars Håkansson, Thomas Lagö, Ingvar Claesson Henrik Åkesson, Tatiana Smirnova, Lars Håkansson, Thomas Lagö, Ingvar Claesson Blekinge Institute of Technology Research report No. 2009:07 Copyright © 2009 by individual authors. All rights reserved. Printed by Printfabriken, Karlskrona 2009. ISSN 1103-1581 ISRN BTH-RES–07/09–SE

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