Lecture 1 I. Introduction and Dynamics Fundamentals A. Welcome B

Lecture 1 I. Introduction and Dynamics Fundamentals A. Welcome B
Lecture 1
I. Introduction and Dynamics Fundamentals
A. Welcome
B. What we will do today
1. Introduction: Syllabus
2. Talk about design project
3. Review of dynamics fundamentals you should already know
C. Go over syllabus
D. Design Project
1. To be Determined
E. Assignment
1. Problem 10-1
2. Also, if the mallet is spun at 200 rpm about the tip of the handle, how much force
is exerted by the handle to keep the mallet head on?
F. Dynamics Fundamentals
a. Kinematics
b. Kinetics, or dynamic force analysis
10.1 Newton's Laws of Motion
a. Who was Newton?
b. Laws
c. Forward dynamics problem
to be found
d. Inverse dynamics (kinetostatics) problem
to be found
e. In the design process, the two cannot be
completely separated
Possible design process for rotating machines
10.2 Dynamic Models
a. Simplified models used to describe motion of real parts
b. Three rules for such models
i. Equal _________
ii. Equal _____________________
iii. Same ___________________________
10.3 Mass
a. Not weight--what's the difference?
b. What are the units?
c. f = m a
d. m = rho V
10.4 Center of Gravity (Center of Mass)
a. First moment of mass about an axis is equal to
___________ at the center of mass
b. Formula for the center of mass
c. Center of mass of a composite body
10.5 Mass Moment of Inertia
a. Rotational version of Newton's second law
b. Formula
Tabulated values for mass moment of inertia of common shapes:
Appendix B, pp. 677-678
d. Kinetic energy
KE =
KE =
10.6 Parallel Axis Theorem
a. I_zz = I_gg + m d^2
b. Example
Lecture 2
II. Fundamentals of Dynamics, Part II
A. Assignment. Empirically find the hoe’s MOI and center of rotation corresponding to
a center of percussion at the edge of the hoe blade. You may work together to
measure hoe properties, but each student must complete the problem himself. Above
and beyond: design an attachment to relocate the center of rotation to a more useful
10.7 Radius of Gyration
a. Definition
The radius at which the entire ______ of the body could be
concentrated such that the resulting model will [sic] have the same
_______________________ as the original body.
b. Formula
c. Depends upon the axis about which body is being rotated.
10.8 Center of percussion
a. Definition
_________________________ is a point on a body which, when
struck with a force, will have associated with it another point called
the _________________________ at which there will be a zero
reaction force.
b. Hockey stick example
lp =
The acceleration at R due to rotation about the CG is equal and
opposite to the _______________ at R due to
c. Interpretation of formula
i. Negative sign?
ii. x small?
iii. x large?
iv. Symmetry?
v. What if the radius of gyration is small (mass is highly
vi. What if the radius of gyration is large (mass is highly
10.11 Solution Methods
a. Superposition
i. Description
ii. Quick example
b. Matrix
i. Description
ii. Quick example
10.12 The Principle of D'Alemert
a. Quasi-static formula
b. System looks like a statics problem
i. Used in graphical solutions to achieve
ii. Useful intuitive concept, especially for
iii. Example: free-body diagram and kinematic diagram combined
into a quasi-static free-body diagram
c. Centrifugal force
d. Centripetal acceleration
10.13 Energy methods
a. Convert to power
b. Equate power dissipation with power input
c. Collect on left-->virtual work equation (really a power equation)
d. Only unknown is the ________________. This is the only thing you can
solve for unless a special method is used. Good method if this is all
you want, or you are solving the forward dynamics problem.
e. Can be given in terms of generalized displacments, velocities--Kane's
f. One step further--LaGrange's equation
Lecture 3
III. Newtonian Solution Method; 3-bar Linkage; Intro to Project
A. Assignment: Fill in the details for Example 11-2 in the text.
1. Draw the free-body diagrams.
2. Show your work in finding the magnitude and direction for each of the position vectors.
3. Show your work in obtaining the magnitude and direction a_{G2}.
4. What section in the text shows how to get a_{G3}?
5. Write out by hand each of the equilibrium equations which go into the matrix system.
6. Solve the matrix equation using TK Solver, MATRIX, MATLAB, or other software. Provide
hard copy of the set-up and the solution.
11.1 Newtonian Solution Method
a. Vector form of Newton's laws
b. Component form of Newton's laws
11.2 Single Link in Pure Rotation
a. Kinematic diagram
b. Free-body diagram
c. Force and torque equilibrium equations
i. Equations
ii. Unknowns
d. Matrix formulation
e. Example
11.3 Force Analysis of a Three-bar Crank-Slider
a. Kinematic diagram
i. How are they getting the accelerations?
b. Free-body diagrams
c. Equations of equilibrium
d. Matrices
e. Example
B. Introduction to Design Project
Lecture 4
IV. Force Analysis of Fourbar Linkages
A. Assignment (Lectures 4 and 5)
Create a TK or Matlab program to solve kinematics and dynamic forces of a 4-bar mechanism.
Solve 11-5b. Then create list variables 2, FPx, and FPy to solve the forces in the rotating
mechanism in all positions.
11.4 Force Analysis of a Fourbar Linkage
a. Previous kinematic analysis
b. Three moving links
c. _______ equations, _________ unknowns
d. Free-body diagram
e. Unknowns: pin forces and torque
f. Equations
g. Matrix form: Where are the knowns?
Where are the unknowns?
h. Example solution for one position (Example 12-3)
i. Masses
ii. FBD's
iii. Position vectors
iv. Accelerations
v. Forces (known)
vi. Assemble matrix
vii. Solve
viii. Convert to polar
ix. Coordinate systems
Intra-link locations:_________________
Placing values in matrix:_________________
11.5 Force Analysis of a Fourbar Slider-Crank Linkage
a. Applicable to slider-cranks within internal combustion engines
b. Equations for bars ___ and ___: same as before
c. Equations for link ____.
i. Angular and y accelerations are zero
ii. Coulomb friction between links 3 and 4
iii. F_p acts on link 4
d. Eight remaining equations assembled into a matrix
C. Whence the information about displacement, velocity, and acceleration coming?
Lecture 5
V. Inverted Slider-Crank and Linkages with More Than Four Bars
A. Assignment
See Lecture 4. Especially, the list-solve part.
B. Book
11.6 Force Analysis of Inverted Slider-Crank
a. Coriolis acceleration
b. Free-body diagram (what's wrong with this picture?)
c. Kinematic diagram
d. How could one find the moment between links 3 and 4?
11.7 Force Analysis of Linkages with More Than Four Bars
a. Generalizations
b. Is there a limitation concerning the number of degrees of freedom for the
c. What must be known about the mechanism?
Lecture 6
VI. Shaking and Dynafour Program
A. Assignment: Problem 11-3, sentence 4, parts a and b (without Dynafour). Which mechanism has
larger shaking force? Why? Using dynafour, reposition the centers of mass until you have
reduced the shaking force by 50%. State the new position of the mass centers. Print out the
forces for the original and new design, as well as the positions of the balance masses.
B. Book
11.8 Shaking Forces and Shaking Torque
a. Shaking force
i. Definition: _______ of all forces acting on the ___________ ____________.
ii. Equation
iii. Example: Slider-crank, p. 469, Fig. 12-4 b.
iv. Can you give an example of shaking force?
b. Shaking torque
i. Definition: __________ torque felt by the ground plane.
ii. Equation
iii. Example, p. 469. How well does this fit our definition?
iv. Engine tends to "shake" in rotation
c. Methods of limiting shaking
i. Balancing (mass distribution)
ii. Flywheels
iii. Vibration-isolating engine mounts
11.9 Program Dynafour
a. Menu system and capabilities
b. Demonstration
Lecture 7
VII. Energy Methods and Flywheels
A. Assignment: Problem 11-4a.
B. Book
11.10 Linkage Force Analysis by Energy Methods
a. Can solve for individual forces without Newtonian matrix approach
b. Fourbar Linkage
i. Diagram
ii. Virtual work equation
iii. Vector form
iv. Expand to create scalar equation
v. Additional information that must be provided: _______________ (on p. 547).
(Newtonian method reuqired only _________________.
vi. Solve one equation for T12
vii. Only works for input forces or torque, not for internal forces
11.11 Flywheels
a. Reason for flywheels
i. Torque fluctuations at the crank occur due to inertia distribution of mechanism
(diagram: Figure 11-18, p. 548)
ii. The mechanism stores and delivers kinetic energy each cycle
iii. Maximum torque can be much larger than the __________ torque
For most motors, an increased torque results in a ______________
_______________, but we want speed to be constant
Speed and torque throughout a cycle can be smoothed out by
vi. Example: old JD tractor
a) Slow _________
b) ___________ cylinders
c) Large, visible flywheel keeps mechanism going, even through the
_____________ portion of the cycle
b. Energy equations for a flywheel system.
i. Assume a _____________ torque, Tavg for now
ii. FB Diagram and equation of equilibrium
iii. Look at speed as a function of _____________
iv. Resulting nifty integral
v. Interpretation of integral: the minimum and maximum speeds of the crank
throughout the cycle are related to the area under the ________________
between the orientations of maximum and minimum ______________.
vi. The energy the flywheel must store between its high-energy point and lowenergy point is given by the difference in squared
c. Example 11-5
i. Torque profile: Figure 11-11
a) Great variation in torque. Some torque is even negative, indicating
b) How do we figure out how much energy is needed to be stored?
ii. _________________ to find area under T-theta curve: units of ?
iii. Why is minimum omega at B?
iv. Why is omega largest at C?
v. Summation of areas under torque curve
Lecture 8
VIII. Flywheels; Class Design Project
A. Assignment: Specify a flywheel I for the mechanism in Problem P11-3a so that the coefficient of
fluctuation k is 0.05. Use your four-bar solver to produce the torque profile and confirm the
effectiveness of your design, but compute the flywheel size by hand and show your work.
Assume input speed to be the average. Show your work and record the computer program output
to justify your answer.
B. Book
11.11 Sizing the flywheel (continued)
a. From last time: energy equation for flywheel
b. One can integrate area on the torque diagram ofr a constant-speed mechanism to
arrive at the energy which must be stored by the flywheel (Example 11-5).
c. Once the energy is known,
d. Coefficient of fluctuation is defined as the ratio of _________ fluctuation to average
_______________ _______________
omegaave =
e. Energy in terms of design parameters
Is =
f. Is is the moment of inertia of the _________ ____________. The flywheel would be
just part of this.
g. Is small: ________________ fluctuation, ____________ k
h. Is large: ________________ fluctuation, ____________ k
i. Benefit of flywheel, Figure 11-12. Much smoother ___________ output. (However,
in the calculation of this curve, constant speed was still used as an assumption!)
j. Procedure:
Put on flywheel:
Re-analyze torques
11.13 Practical Considerations
a. Typical design procedure
i. First-pass geometric design
ii-iii. Figure mass, I, CG's
iv. Dynamic force analysis
v. Redesign parts to achieve satisfactory geometry, kinematics, and dynamics
b. By now, you see that this design is an iterative process!
c. Repeated design iterations--number crunching--is best done in the computer: TK
solver, etc. Try to minimize "human handling" of the numbers: avoid errors due
to pushing buttons repeatedly.
d. Which leads us to a discussion of the Class design project:
C. Design Problem
Lecture 9
IX. Static Balance, Quiz 2
A. Assignment
1. Problem 12-1 a,b.
2. Problem 12-2 (suggestion: use a circular shape with one edge at the pin)
12.0 Introduction
a. Balancing modifying the mass distribution of a linkage to eliminate shaking forces
and torques
b. Accepted design practice is to balance all parts
c. Static vs. dynamic balancing
d. Balancing often must be done after manufacture
12.1 Static Balance
a. Definition: Static balance is achieved when the sum of all the shaking forces
(including D'Alembert) in the plane is equal to zero.
b. Also known as single-plane balance
i. Part must be relatively short in the axial direction
ii. Examples: propellor, bike tire
c. Vee-shaped link rotating at T
i. Dynamic model: replace legs with concentrated masses at the center of mass
of each leg
a) Critical thought question: can you do this?
b) Implications on centripetal accelerations of masses
Since forces are a linear function of R, all mass may be placed at
the "average" location.
c) Implications on angular acceleration of the link
d) The model is valid for balancing, but not for the calculation of
___________________ ________________!
ii. Equation
a) Critical thought question: how does this differ from simply placing the
center of mass at the center of rotation of the link?
b) Is there only one position at which the mass may be placed?
iii. Conversion to polar coordinates
a) Why leave in the negative signs?
b) The atan2 function: quadrant-specific. If
tan(2) = y/x, then
atan2(y,x)= 2 or sometimes
atan2(x,y) = 2
Rb =
Rb mb =
d. Example (if time allows)
C. Quiz 2 (30 min)
Lecture 10
X. Dynamic Balancing and Balancing of Fourbar Mechanisms
A. Assignment
1. Problem 12-5 a,b.
2. Extra credit problem. Starting from equations (12.8a), develop a method for perfectly
balancing a slider-crank mechanism by specifying the mass center of the crank and of the
connecting rod. Clue: for a slider-crank, l4 = b4 = INF , and N4 = 0.
12.2 Dynamic Balance
a. Using static balance, you can balance in a plane, but what if you have a long shaft
with unbalanced masses in several planes?
b. Can we avoid balancing each unbalanced plane separately?
c. Process known as dynamic balancing (also known as ________-___________
d. Dynamic balance: all reaction forces and moments must be zero
(applies to moments in any plan which includes the shaft)
e. Dynamic balancing required in rotating parts with relatively long axial dimensions
f. Balance by design first, then after manufacture, on a balancing machine
g. ______________ balance individual components before assembly, then dynamically
balance the assembly (e.g. __________ assembly)
h. Derivation
Two equations, four pairs of unknowns:
This gives force balance in the x,y directions; now we need moment balance.
Two more equations:
i. Note: you would use the equations from _________ first, since they allow design of
mbRb. Then you would use that value in the force equilibrium equations to find
j. Example: add masses in planes A and B to dynamically balance a long shaft with one
unbalanced mass
12.3 Static Balancing the Fourbar Linkage
a. Static balancing means making the center of mass stationary
b. This is easy enough for a single link, but what about a fourbar linkage? The couple
does not rotate about a fixed point.
c. However, there is a method to force the overall mass center of the linkage to be
d. Berkof-Lowen method
e. Diagram of a fourbar
f. Mass center
g. Complex vector notation for convenience
h. Loop equation
i. Author gets to Equation (12.8b)
j. Position of CG3 (N3, b3) may be determined at will by the designer. Equations 12.8c
and 12.8d specify the position-mass combination for the ________ (link 2) and
the ______________ (link 4)
k. Euler's identity
l. Equations (12.8) could be rearranged so that the position of the CG of link 2 or 4
could be specified at will. Then the equations would specify the position-mass
combinations of the other two. This must be done for the extra credit problem.
m. One link is free to be designed at will. Balancing can then be done using the other
two links.
Lecture 11
XI. Balancing with Dynafour; Correcting Imbalance; Engine Design
A. Assignment: Problem 12-6
12.4 Effect of Balancing on Input Torque
a. Balancing the fourbar can ___________ the individual links and ___________ their
individual moment of inertia
b. This can increase the magnitude of the fluctuations in the _____ _______ to a
c. It can also increase the _________ _____________
d. The next section gives and example of this
12.5 Balancing with Dyanfour
a. You have already done one HW problem in which you used Dynafour for balancing
a fourbar. Section 12.5 gives you another example to look at.
b. Overhead of animation of the example linkage (Figure 12-5)
c. Overhead of the force balance menu
i. First: Balance, specify radii and masses on links 2 and 4
ii. Then: 10 Point Forces, 8 Shaking Force on the Ground Plane directly after
performing the balance (do not recalculate)
iv. Explanation
d. Unbalanced Forces (Figure 12-7)
e. Force on ground--balanced
i. Magnitude change______________________
f. Forces on ground plane points are ___________ images (Figure 12-9, 12-10)
Maximum torque and torque variation (shaking torque) at the crank both
_________________ for this balanced linkage.
h. You can print out the balance locations-masses for complete balance
i. Print, forces, shaking forces
13.6 Measuring and Correcting Imbalance
Balanced design will not be built perfectly
_______________________ _________________.
b. Example: tire will have imbalance due to imperfect shape
c. Static balance of a tire
i. Tire spun horizontally on cone
ii. Weights placed on machine until tire spins level
iii. Weights then attached to rim
iv. Not best--tire has relatively long axial dimension
d. Dynamic balance of tire
i. Figure 12-12
ii. Force transducer measures forces relative to the angle measured by the
_________ __________ at a given speed
iii. Computer determines necessary locations and masses
iv. Planes and radius correspond to the inner and outer ______ ____________.
13.0 Introduction to Engine Design
a. Figure 13-1. Discuss the parts of an engine
13.1 Engine Design
a. Multi-cylinder, vee engines very popular
b. Especially v6, v8
_______________ of the slider-crank mechanism greatly influence these
d. First we will look at single-cylinder engines
e. Figure 13-3
i. Reverse of pumps made out the same mechanism type
ii. TDC= ____________________
ii. BDC= ____________________
f. Four-stroke cycle (Figure 13-4)
g. Two-stroke engine (Figure 13-5) (Lawnboy)
i. Not as efficient
ii. Higher pollution (some unburnt fuel escapes)
iii. Simpler
h. Program Engine has a built-in gas force curve
C. Don't forget: quiz next time, over Chapter 12
Lecture 12
XII. Slider-Crank Kinematics and Quiz 3
A. Assignment: Problem 13-1.
13.2 Slider-Crank Kinematics
(Subtitled: Every mathematical technique you never thought you'd see again integrated
into one problem)
(Sub-subtitled: Mental calisthenics to warm up for the quiz)
a. Overview
i. Get x, v, a of the slider (piston) as functions of time for a constant speed
ii. Diagram of slider-crank mechanism
iii. You can reason that the slider motion would be near sinusoidal, something
iv. It turns out that its motion can be estimated even more closely by a Fourier
series of the form
v. We will try to dust off as many rusty mathematical techniques as possible in
order to derive this
b. Slider motion as a function of time
i. Law of cosines!
ii. Quadratic formula!!
We could differentiate this with respect to time, but this doesn't give
comprehensible interpretation.
iii. Binomial theorem!!!
remember . . .
in general,
If a = 1 and n = 1/2,
iv. Beloved trigonometric identities!!!!
v. This is our punchline, a truncated Fourier series!!!!!
General form:
can represent any periodic wave form exactly.
vi. Now that we have the displacement in terms of Tt, we can differentiate quite
Note that the 2T terms are more significant in the velocity and
acceleration equations.
vii. Interpretation: Figure 13-8
c. Preview of next time: Superposition
i. Constant T assumed
ii. Analyze forces based on gas forces alone
iii. Analyze forces based on inertia forces and torques
iv. Sum the effects of ii and iii
C. Quiz (50 pt, about 20 minutes)
Lecture 13
XIII. Gas Forces; Equivalent Masses
A. Assignment: Problems 13.3, 13.7
B. Overview of the next two sessions
1. Force and torque analysis by superposition on slider-crank
a. Gas forces only
b. Inertia forces only
c. Summation
2. Gas force and torque--13.3--today
3. Equivalent lumped-mass models which greatly simplify computation of inertia forces--13.4
4. Inertia torques, total torques (13.5, 13.6)--next time
13.3 Gas Force and Gas Torque
a. First part of super position--consider as if the linkage is sitting still
b. Goal: get the driving torque to the crank as a function of the gas force on the piston
c. First: some definitions
i. ______ _________: Force on the piston due to the exploding fuel-air mixture
ii. Gas torque (driving torque Tg21): Torque of the crank on the ground due only
to the gas pressure
iii. Gas reaction torque Tg12 (in this case): the reaction torque felt by the crank
due to torque exerted on ground. Causes the engine to "rock" when
d. Free-body diagrams of links
e. Gas torque in terms of N, Tt
f. Simplify to terms of Tt only
g. Approximation
h. Gas Force curve in Figure 13-9
13.4 Equivalent Masses
a. How this fits into big picture
i. Superposition requires gas and inertia forces to be calculated separately
ii. We want to calculate inertia forces
iii. We want to do this using an extremely simple lumped-mass model for the
initial design stages
b. Our goal in this section
i. Nearly-dynamically equivalent model
ii. Lumped (point) masses
iii. Masses concentrated at the pin joints
iv. We will talk about the computation of the inertia forces next time
c. Model of the conrod (connecting rod)
i. Complex motion--must be dynamically equivalent
*equal _____________________________
*equal _____________________________
*equal _____________________________
ii. Diagram of Conrod (13-10a)
iii. Diagram of the two-mass model
iv. Three equations for equivalence
v. For simplicity of analysis, we specify that one of the masses is at the
__________ pin
vi. Because of this, the masses must be:
Location of second
viii. Typical conrod
* CG toward crank
* Center of percussion very close to the crank pin
* Approximate model--just place 2nd mass at crank pin
* Design stage--CG not fully known anyway
d. Model of Crank
i. Simple rotary motion--statically equivalent model
ii. Only mass and center of gravity must be equal. We are not worried about
angular acceleration.
iii. This is called statically equivalent
iv. Equations
v. One mass at stationary center (inconsequential), and one at the crank pin
e. Resulting model
i. Diagram
ii. Very simple: two lumped masses
iii. Allows simple, approximate analysis for design calculations
iv. Drawback: may have locked in an assumption concerning the shape and
mass distribution of your conrod
Lecture 14
XIV. Shaking; Total Torque; Flywheels
A. Assignment: Problems 13-9, 13-11
B. Overview
1. Use the simplified, lumped mass model from last time to figure the inertia forced and torques
2. Use _________________ to combine gas and inertia forces and torques to get their totals
3. Look at some results of total torque from program ENGINE
4. Tie the results into flywheel design
13.5 Inertia and Shaking Forces
a. Diagram of slider-crank
b. Definition of total inertia force:
c. Total inertia force
d. Components of the total inertia force:
e. Definition of shaking force:
f. Components of the shaking force:
g. Program ENGINE computes shaking force (diagram)
13.6 Inertia and Shaking Forces
a. Definition of inertia torque
(I think that Norton has not followed his sign convention in this section, but I
think that he gets the correct equation for shaking torque in the end)
b. Easiest way to get it (remember from Lecture 13?)
c. Expanded (Equation 13.15b)
d. Can you believe this trig identity?
e. Inertia torque as a function of frequency
Note that it has a third harmonic
f. Shaking torque
Compare this with Equation 12.15c, p. 474
He has corrected the problem with the sign convention by this
13.7 Total Engine Torque
a. Superposition
b. Program ENGINE computes this sum: example, Figure 13-15
i. Low vs. high speed, and increased effects of inertia torque
ii. Negative, as well as positive torques at high speed
iii. What would be the result of this?
13.8 Flywheels
a. Flywheel (or inertia on crankshaft) is mandatory on a ________________________
b. Procedure for designing a flywheel is the same as that of a fourbar linkage
c. Average crank speed must be specified. One must consider the "worst" case, which
occurs at _____ speed
Program engine will specify flywheel I for a user-supplied coefficient of
_________________________, k.
Lecture 15
XV. Pin Forces and Balancing
A. Assignments (last assignments before the Chapter 13 Quiz two class sessions from today)
1. Problem 13-19
2. Build your own one-cylinder, four-stroke engine using the ENGINE software.
a. Use the English system of units
b. Specify your choice of bore, stroke, crank-to connecting rod ratio, and other
parameters requested as you input information from the keyboard. Print out your
engine information.
c. Plot and print the shaking forces, torque, and crankpin forces at redline (5000 rpm).
d. Perfectly balance the crank, and plot and print the shaking forces.
e. Now overbalance the crank so that the maximum shaking force is as small as you can
get it. Plot and print the shaking forces again.
f. Use the software to help you size a flywheel such that the maximum torque of the
motor is 1.5 times its average torque. Plot and print the torque curve. What
moment of inertia, I, is required? What is the coefficient of fluctuation?
B. Overview
1. We have used ______________ to combine gas and inertia torques and forces.
2. Using ________________ models for the crank and the connecting rod
3. We now need to solve for the pin forces using superposition
4. We will still use the lumped-mass model, but will have to separate the mass which belongs
to the __________ from that which belongs to the ____________________________
5. We will look at the effect of each mass individually
6. Balancing the lumped-mass slider-crank model
13.9 Pin Forces in the Single-Cylinder Engine
a. Will look at inertia forces in pins due to each of our modeled lumped masses
b. Nomenclature conventions
i. First subscript
* g: due to gas forces
* i: due to inertia forces
Second subscript
* p: due to piston
* w: due to conrod mass at wristpin
* c: due to conrod mass at ___________
* r: due to __________ mass at the crankpin
c. Goal: get pin forces F41, F34, F32, and F21
d. Gas forces were determined in Section 13.3
e. Forces due to piston mass alone
i. Free-body diagrams
ii. Forces are determined from right to left, since the crank speed is "set in stone"
iii. Summation of forces in the x direction on body 4
iv. Body 4, y direction
v. Force 3-4
vi. Link 3 is a two-force member
vii. Two forces on link 2
f. Forces due to conrod mass at the wrist pin
i. Free-body diagrams
ii. Similarly, the equations of equilibrium are found
where do each of the equations come from?
g. Forces due to the conrod mass at the crank pin
i. Free-body diagram
ii. D'Alembert force
Above, body 3 is acting on body 2
h. Forces due to the crank mass at the crank pin
j. Summation (superposition) of forces
i. Force of piston on ground (what does each component come from?
ii. Force on wristpin (what does each component come from?
iii. Force on crankpin (what does each component come from?
iv. Force on main pin (what does each component come from?
k. Program ENGINE uses these equations to get the pin forces
i. Experiment around with it
ii. Wristpin forces at medium speed can be smaller than at either low or high
13.10 Balancing the Single-Cylinder Engine
a. You can perfectly balance the crank (for the lumped masses at the crank pin
b. Equation
c. Resulting shaking forces are all in the x direction (Figure 13-24)
d. One can overbalance the crank to lessen the maximum shaking force.
i. Some shaking force is now in the y direction
D. If time permits, go to lab and play with program ENGINE
Lecture 16
XVI. Design Trade-offs; Discuss Design Project; Multicylinder Engines
A. Assignment: Continue to work on problems assigned last time, and get ready for the quiz next
13.11 Design Trade-offs
a. Conrod-crank ratio (l/r)
i. Discuss
ii. High: smooth-running (piston acceleration near ____________)
iii. Low
* More compact
* Higher pin forces (cylinder walls, etc.)
b. Bore-stroke ratio (B/S)
i. Definitions
* Bore: __________ __________ of cylinder
* Stroke: _________ to ________, or 2r
ii. Trade-off
iii. Designs
* Square: B/S = 1
* Oversquare
- B/S ____ 1
- pancake
- High gas forces==> high pin forces
* Undersquare
- B/S ______1
- pencil
- larger stroke, inertia forces
iv. Production engines: .75 < B/S < 1.5
c. Materials
Al alloy
Cast or forged steel;
Al in small engines
Engine blocks
Cast iron;
C. Class Design Project
1. Go over revised problem statement
2. Report format
3. Help on equations for analyzing problem
4. Questions
14.1 Multicylinder Engine Designs
a. Figure 14-1
b. Terms
i. Crankshaft
ii. Crank throw: each cylinder's ____________ on the crank ____________
iii. Phase angle on crank
iv. Banks
E. Review questions from class, if time permits
1. Piston position as function of Tt
2. Gas force and torque
3. Equivalent lumped masses
4. Inertia and shaking forces due to individual masses
5. Superposition of 2 and 4
6. Design trade-offs
F. Go look at an actual engine, if time permits
Lecture 17
XVII. Multicylinder Engines; Crank Phase; Quiz 4
A. Assignment: 14-1a, first sentence
14.1 Multicylinder Engines
a. Figure 14-2 Inline 4
b. Figure 14-4 BMW V 12
i. Note vee angle: ___
ii. Vee angle in V-8: 90o
c. Radial engines
i. WWII airplanes--Wright-Patterson
d. What is your idea? Why not ___________________?
14.2 Crank Phase Diagram
a .
D e l t a
p h a s e
a n g l e
b. Rules
i. Diagram of four-bars on common crank
i s
c. Crank phase diagram
i. Sine wave
ii. Square representation
C. Quiz 4. 100 pt. Over Chapter 13.
Lecture 18
XVIII. Forces, Torques, and Moments in In-line Engines
A. Assignment: Problem 14-1
B. Overview
1. We have computed shaking (inertia) forces and torques in single-cylinder engines
2. Used lumped-mass approach
3. Now we will sum inertia forces and torques for inline multicylinder engines, accounting for
crank phase angles
4. Since the crankshaft has length, we also will look at shaking moments (in a plane which
includes the shaft)
5. We will come up with rules for eliminating shaking at fundamental frequency or at harmonics
6. Not concerned about gas forces today, just inertia shaking effects
14.3 Shaking Forces in Inline Engines
a. Equation for shaking force in one crank-slider, with the crank balanced for mA
b. Cranks are oriented with phase angle Ni
i. Ni is a lag
ii. For crank i, the force in the reference direction (i-hat) is determined by
iii. Diagram
c. Shaking force in reference direction
i. Note all forces in the ____-direction--the crank has been balanced for mA
ii. Trig identity
d. Collecting summations
i. Same trig identity can be used on the fourth and sixth harmonics, which were
not written
f. This simple rule illustrates the value of using lumped masses and Fourier series to
approximate slider-crank dynamics (Ch. 13)
i. But what have we lost by making this assumption?
14.4 Inertia torque in engines
a. Review: what is the definition of inertia torque?
b. Inertia torque was given in equation 13.15e
i. Which harmonic is predominant?
c. Including all cylinders
d. Similar trigonometric identity used
e. Similar rules for canceling inertia torque for harmonics
f. Note the presence of the third harmonic
14.5 Shaking Moments in Inline Engines
a. What is meant by shaking moments?
b. Single cylinder engines had no shaking moments--all masses were modeled in one
c. Like the long shaft which needed __________ balancing, we now must deal with
shaking moments in a plane which includes the shaft
d. Diagram
e. Summation of moments in the xz plane
f. Modify the equation from shaking forces to get one for shaking moments
i. How is this different from the equation for shaking force?
g. Conditions for zero shaking moment
h. Can be extended to 4th and 6th harmonics as well
D. Thought questions
1. What will each of these types of shaking look like in an automobile engine?
2. Which do you think is easiest to balance out? Hardest?
E. Example
1. Figure 14.7
2. Table 14-1 (also balances torques)
3. Table 14-2
Lecture 19
XIX. Even Firing
A. Assignment: Problem 14-2a, (there are six possible firing orders). If none of these gives even firing,
also specify a 4-cylinder engine design which does.
B. Overview
1. Last time: how crank phase angles influence inertia shaking forces and torques
2. This time: how firing order and phase angles influence gas forces and torques
14.6 Even Firing
a. Last time we talked about crank phase for minimal inertia effects
b. This time, we will use the presupposition that crank angles should be oriented such
that firing is ________________ over a ______________
i. Even firing
ii. Two-stroke engine (360o cycle)
iii. Four-stroke engine (720o cycle)
iv. Example: 4-cylinder, 4-stroke engine
)N required for inertia effects to cancel in a 4-stroke engine conflicts
with the )N required for even firing
We will analyze how to achieve even firing and cancel as many inertia
effects as possible
c. Delta power stroke angle )R
i. The actual phase between firings
ii. Ideal 2-stroke engine
iii. Ideal 4-stroke engine
d. Example
i. Four stroke 0o, 90o, 180o, 270o
ii. Diagram of the shaft
iii. Crank phase diagram
iv. Possible crank power stroke angles
v. Note: a two-stroke engine can achieve even firing for this crank configuration
e. Second example
i. We would rather sacrifice some balance (inertia effects) to achieve even firing
ii. Crank phases: 0o, 180o, 360o, 540o, 720o
iii. )N = 180o, not the best to cancel inertia
iv. Diagram of shaft
iv. Crank phase diagram
vi. Power strokes for even firing
vii. Trade-off: Table 14-3
* Force:
* Moments:
f. Primary moments canceled by mirror symmetry
i. Schematic of engine, Figure 14-16
ii. Crank phase diagram, Figure 14-17
iii. Force and moment constants, Table 14-4
iv. Demo unit
Lecture 20
XX. Vee and Opposed Engines
A. Determine a feasible crank phase diagram and firing orders for a V-6 engine. Will shaking forces for
frequencies T and 2T be cancelled?
B. Overview
1. What was the conflict in the design of the inline 4 cylinder engine?
2. Today, we will see how the vee engine can overcome this conflict!
14.7 Vee Engine Configurations
a. Vee engine schematic, Figure 14-20
i. Vee angle 2(
ii. Unit vectors l^ and r^
iii. Shaking forces in these two directions
iv. Still using assumption of balanced crank
b. Shaking forces
i. Forces in the directions of the two banks, Equation 14.10d
1 measured from the x-axis (halfway between banks)
ii. Second trigonometric identity leads to Equation 14.10h
iii. Criteria for zero shaking forces are the same as for inline engines
iv. Resolve shaking forces in the x,y directions (Equation 14.10)
c. Shaking moments
i. Directions of the m^, n^ unit vectors
ii. Equations 14.11 a,b
iii. Similar criteria for zero shaking moments as in the inline engine
iv. Resolve into x, y directions
d. Inertia torques
i. Equation, Eq. 14.12a,b
ii. Sufficient conditions for zero inertia torque
iii. Implication: this can be achieved by other means
e. Gas torque
i. Equation 14.13
f. Conrod length
i. Two conrods per throw
ii. Vee engine shorter than inline
iii. Vibrational modes of the crankshaft--best if it is short and stiff, in order to drive
these frequencies up
g. Example engine: V8
i. Why not use 0, 180, 180, 0o for each bank?
ii. Crank phase angle and power phase angles for V8
iii. We can get both the balancing benefits of the 90o crank phases and even firing
with a V8 engine
iv. Crank phase angles of Ni = 0, 90, 180, 270o do not give mirror symmetry
v. Ni = _____________________________ does allow mirror symmetry, cancelling
out inertia moments, except for the primary (T). We can cancel that another
vi. Crank phase diagram (Figure 14-23)
vii. Note the vee angle 2( =
In general, 2( =
where m is an integer, for even firing
viii. Note both _______ firing and cancelled ___________ effects (desirable
crankphase angles) are possible with this engine
ix. Firing usually alternates between left and right banks--why?
x. Force and moment balance state of example engine. Table 14-5.
14.8 Opposed Engine Configurations
a. What would be the ideal vee angle of a V4?
b. Possible crank phase diagram of a V4
c. Can you say, "v6"?
Lecture 21
XXI. Balancing Multicylinder Engines; Program Engine
A. Assignment:
1. Make a chart comparing the advantages and disadvantages of the following engines: V8, V6,
Inline 4, Inline 6
2. Work on the class design project
B. Overview
1. Cover various topics in balancing multicylinder engines
2. Go to lab and use program engine
15.9 Balancing Multicylinder Engines
a. General Statements
i. Two-stroke engines with even firing are balanced for shaking forces except for
frequencies nω
Four-stroke with even firing balanced for shaking forces except for what
iii. Same shaking moments are cancelled if ________ _______ is used
b. Characteristics of specific engines
i. Inline engine needs to be 6-cylinder in order to be balanced through the fourth
ii. Inline 6 (0, 240, 120, 120, 240, 0o) has zero shaking forces and moments through
the 4th harmonic and through the 3rd harmonic for the inertia _________
(there are reasons why popular engines are popular)
iii. V12 is two inline 6's on a single crank. Has these same balance properties
iv. V8 has balanced primary and secondary forces, torques, and moments, except
for the primary _________ (Table 14-5)
v. V6 has unbalanced primary and secondary moments
Vee angle should be m120o for even firing, where m is an integer
Sometimes 60o or 90o to reduce width and to reuse _________ made for
Runs rough unless connecting rod splayed by 30o on crank pin
c. Flywheels
i. Used to smooth out unbalanced inertia torque
ii. Total torque curve (integral) must be used to design the flywheel
d. Balance shafts
i. Unbalanced forces and moments can be balanced using pairs of ___________________ balance shafts (Figures 14-26,27)
ii. One pair must spin at each harmonic for each bank of cylinders--why?
iii. A pair of equivalent eccentricities on the two shafts results in an oscillating
force in a single direction which is used to cancel shaking force
iv. If the pair of equivalent eccentricities is offset, a moment may be created which
cancels the engine's shaking moment
v. At ω, one of the balance shafts may be the ___________ itself
vi. Special case:
V8 has only unbalanced moments in the first harmonic (ω) (Table 14-5)
(until the fourth harmonic)
This primary moment rotates at a constant amplitude
Therefore only one balance shaft is needed, and the __________ may be
How could this be achieved using the smallest amount of mass?
16 Program Engine
a. We will go to the computer lab and work on the engine design exercise provided
Lecture 22
XXII. Program Engine Mini-lab and Quiz
A. Program engine minilab
1. See handout.
B. Quiz
Lecture 23
XXIII. Lumped-Parameter Models
A. Overview
1. We wish to use lumped-parameter models to describe the dynamics of cam-follower systems
2. Figure 15-6
3. Reduce multi-degree-of-freedom model to an approximate single-degree-of-freedom model
4. We will use second-order linear differential equations for the most part. We will analyze the
response of the sdof system next time
15.1 Lumped-Parameter Dynamic Models
a. Simple lumped-parameter dynamic model of a follower (Figure 15-1)
b. Free-body diagram
c. Equation
d. Convert to terms of modeled parameters
e. Particular solution
f. Various types of damping
i. Coulomb
ii. Viscous (linear)
iii. Quadratic
iv. Linear approximation
15.2 Equivalent systems
a. Talk about Figure 15.6
i. Components are not rigid
ii. A model may be built out of springs, masses, and dampers.
iii. Levers
iv. How do we combine these components?
b. Types of variables
i. Through: force. What would be an electrical analogy?
ii. Across: voltage. What would be the mechanical analogy?
c. Types of connections
i. Series: components have the same force (____________ variable)
ii. Parallel: components have the same across variable (displacement, velocity)
d. Combining springs
i. Series
ii. Parallel
e. Combining dampers
i. Series
ii. Parallel
f. Combining masses
i. If they have the same displacement, they are parallel
ii. Summation
g. Levers
i. Goal: eliminate lever to get an equivalent sdof model
ii. Keep spring--replace mass
iii. Keep mass--replace spring
iv. Dampers are similar to springs in this
h. Example 15-1
i. Equivalent stiffnesses come from stiffness equations of components
ii. Equivalent masses come from the masses of components. Sometimes the mass
of a flexible member is shared between nodes
Masses are combined (to get the model to a sdof one).
approximation; why?
This is an
iv. Springs, dampers, and finally the lever are reduced to the sdof equivalents
j. We will look at the dynamic response of this single-dof system next time. It will be the
solution to a 2nd order ________________________
k. Alternate way: we do not have to make the approximation of reducing the system to a
single degree of freedom
i. There is a dof for each mass node
ii. Displacements xi of each node can be put into a vector
iii. Matrix equation
iv. Another form (state-space)
A system of several differential equations including the same number of
variables intermingled
vi. Can solve with TK solver, matlab, simulink, etc.
Lecture 24
XXIV. Dynamic Force Analysis of the Force-Closed Cam Follower
A. Assignment: work on class design project
B. Overview
1. We have learned how the lumped-parameter model of a cam-follower may be reduced to a
single-dof spring-mass-damper system
2. Now we will analyze the response of that system to a forcing input
3. This leads to limitations as to how fast this system can run
15.3 Dynamic Force Analysis of the Force-Closed Cam Follower
a. Free-body diagram
b. Equation of motion
c. Restated
d. Homogeneous solution (Fc = 0)
i. Overdamped case: (. > 1)
s1, s2 are real
Superposed exponential decays
ii. Critically damped (. = 1)
s1, s2 are real and equal
Response looks similar to overdamped case
iii. Underdamped case (. < 1)
Many physical systems, including most followers fit this description
e. Particular solution
i. Forcing function
ii. Assume the solution
iii. Amplitude and phase
iv. Plot of the particular response
f. Complete response = homogeneous + particular
Lecture 25
XXV. Resonance; Force-Closed Cam Follower
A. Assignment: work on class design project
B. Overview: How does the forced response of the one-dof cam-follower model affect its design?
15.4 Resonance
a. Look at plot of forced response amplitude (Figure 15-6)
b. Response amplitude is a function of frequency ratio and damping
c. Resonance occurs at ωf = ωn
d. Rule of thumb: Keep ωf < ωn/10 to avoid large response amplitudes. What would a
large amplitude response look like?
e. Second rule (non-sinusoidal motion)
i. Sketch
ii. Rise-time ratio
Keep τ below 0.5
f. Small τ is equivalent to large ___________. This is good.
g. Interpretation
Rise time must be at least twice as large as the part of the free response cycle
which represents one radian
h. Design ramification: design hardware so that ωn is much higher than ωf
j. Aside: some machines run above the resonance speed. They break through the
resonance when the speed of the motor ramps up. Example?
15.5 Kinetostatic Force Analysis of the Force-Closed Cam Follower
a. Kinetostatics (inverse dynamics problem)
===> ?
b. Inverse dynamical equation with a spring preload
c. Fpl must be large enough to maintain cantact with the cam at all times. Keep Fc
d. Fc goes negative--______________ __________
i. Large impact forces
ii. Criterion for redline maximum RPM
e. Example 15-2
i. Cam displacement program discussed in Chapter 9
ii. Dynamic force Fc for two preloads. Figure 15-10
iii. One results in follower jump, one does not
iv. Higher speed--higher preload needed
v. Smoothness of cam profiles will affect spring preload needed and the redline
speed of the engine
vi. Fundamental law of cam design: ________ _______!
Lecture 26
XVI. Form-Closed Cam Follower; Camshaft Torque
A. Assignment: Continue working of class design project
B. Overview
1. Last time we talked about a spring-preloaded (force-closed) cam-follower, and criteria for
avoiding follower jump
2. This time we will talk about a follower design which eliminates the problem of follower
jump: the form-closed cam-follower
3. Also calculating the camshaft torque from the cam force diagram!
15.6 Kinetostatic Force Analysis of the Form-Closed Cam Follower
a. Diagram of the form-closed cam follower
i. Grooved or track cam
ii. Conjugate cams on a common shaft
b. Another name for the form-closed cam system (almost sure to be on the quiz):
c. Equation for cam force
i. What's missing?
ii. Why is this significant?
d. Example with same cam profile as last time (Example 15-11)
i. Eight segments
ii. Four rises and falls
iii. Dynamic force: Figure 15-12
iv. How does the maximum dynamic force compare to that of the force-closed
cam follower without follower jump?
e. Advantages of the desmodromic cam-follower
i. No resonance (no spring, except the compliance of the follower itself)
ii. Can be operated at high speeds without fear of follower jump (used in race
car engines)
iii. Lower cam force and camshaft torque
f. Disadvantages of the desmodromic cam-follower
i. Must be clearance in cam track
ii. Inner and outer surface must be precisely ground (not necessary in forceclosed cam followers)
iii. When cam force changes sign, follower roller goes to the other side of the
groove. Its direction of rotation must reverse, and the impact on the
opposite wall is called _____________ __________.
15.7 Camshaft Torque
a. We have discussed camshaft torque as a function of angle
b. We can use this to compute camshaft torque
c. Power in = power out
d. Comparison of camshaft torque for force-closed and desmodromic cam-followers
(Figure 15-13)
e. Variations in torque cause small changes in ω which can be smoothed out with a
flywheel on the camshaft (Program DYNACAM)
Lecture 27
XVII. Polydyne Cams; Dynamic Force Measurement
A. What is left in the class
1. Class design project
2. Quiz next time
3. Study for final
B. Problem associated with valve trains
1. Follower ________
2. Why?
a. Equation 15.6b
b. Figure 15–6
3. Measurements of engine valves have shown that the motion of the valve may differ greatly
from that of the cam profile
4. Is there a way one could account for the dynamic response when designing the cam profile?
15.8 Polydyne cams
a. Definition
i. Polynomial + dynamic
ii. Polynomial motion profiles
iii. The dynamics of the follower are considered in the design of the cam profile
b. Procedure
i. Start with a desired valve motion function
ii. Work backward through the differential equation for the response of the
valve train
iii. Come up with the cam profile to make the desired valve motion happen
iv. Polynomial motion functions work well because they are easy to manipulate
c. Example from Section 15.7
i. Polynomial rise and fall had the lowest dynamic force
ii. Figure 15-14
d. A simpler solution to the valve train problem stated above
i. Overhead cam has a very stiff follower system which has a higher-frequency
ii. Motion of the follower deviates less from that of the cam profile
iii. Examples, pp. 642-3. What type of follower?
BMW V-12 _______________
Opposed 6 _______________
e. Now let's look at the type of measurements which show the need for the polydyne or
overhead cam
15.9 Dynamic Force Measurement
a. Frequency response function
Input_______________ ==> Output__________________
b. Frequency response
i. Figure 17-19, 1st ed. only
ii. Resonances of the follower are apparent
c. Response curves as functions of time
i. Figure 15-16
ii. Higher-order follower dynamics show up in actual response curves
17.10 Practical Considerations (good questions for a quiz; answer why for each one)
a. Keep follower lift to a minimum
b. Arrange follower spring to preload all pivots in a consistent direction
c. Keep the duration of the rises and falls as long as possible
d. Keep follower train stiffness low and mass high
e. Keep lever ratios close to 1
f. Keep camshaft itself stiff in torsion and bending
g. Keep the cam pitch circle to a maximum
h. Use low backlash gears in the camshaft drivetrain
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