This document outlines examples for modeling mechanical systems using idealized modeling elements like springs, dampers, masses, and Lagrange's equations. It provides 10 examples of mechanical systems that involve automotive shock absorbers, multimass systems, electric motors, copying machines, and double pendulums. Differential equations of motion are derived for each example by drawing free body diagrams and applying Newton's second law. Pair-sharing exercises are also included for students to work through modeling problems together.

1. Modeling Mechanical Systems
Dr. Nhut Ho
ME584
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2. Agenda
• Idealized Modeling Elements
• Modeling Method and Examples
• Lagrange’s Equation
• Case study: Feasibility Study of a
Mobile Robot Design
• Matlab Simulation Example
• Active learning: Pair-share exercises, case study
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3. Idealized Modeling Elements
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4. Inductive storage
Electrical inductance
Translational spring
Rotational spring
Fluid inertia
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5. Capacitive Storage
Electrical capacitance
Translational mass
Rotational mass
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6. Energy dissipators
Electrical resistance
Translational damper
Rotational damper
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7. Springs
- Stiffness Element
- Stores potential energy
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8. Actual Spring Behavior
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9. Spring Connections
• Spring in series: KEQ=K1K2/(K1+K2)
• Spring in parallel: KEQ=K1+K2
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10. Dampers and Mass
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11. Dampers Connections
• Dampers in series: BEQ=B1B2/(B1+B2)
• Dampers in parallel: BEQ=B1+B2
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12. Modeling Mechanical Systems
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13. Modeling Methods
• State assumptions and their rationales
• Establish inertial coordinate system
• Identify and isolate discrete system elements (springs,
dampers, masses)
• Determine the minimum number of variables needed to
uniquely define the configuration of system (subtract
constraints from number of equations)
• Free body diagram for each element
• Write equations relating loading to deformation in system
elements
• Apply Newton’s 2nd Law:
– F = ma for translation motion
– T = Iα for rotational motion
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14. Example 1: Automobile Shock Absorber
Spring-mass-damper Free-body diagram
)()(
)()(
2
2
trtky
dt
tdy
b
dt
tyd
M 
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15. Example 2: Mechanical System
• Draw a free body diagram, showing all
forces and their directions
• Write equation of motion and derive
transfer function of response x to input u
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16. Example 2: Mechanical System
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17. Example 3: Two-Mass System
• Derive the equation of motion for x2 as a
function of Fa. The indicated damping is
viscous.
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18. Example 3: Two-Mass System
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19. Example 4: Three-Mass System
• Draw the free-body-diagram for each
mass and write the differential equations
describing the system
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20. Example 4: Three-Mass System
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21. Example 5: Pair-Share Exercise
• All springs are identical with
constant K
• Spring forces are zero when
x1=x2=x3=0
• Draw FBDs and write equations
of motion
• Determine the constant
elongation of each spring caused
by gravitational forces when the
masses are stationary in a
position of static equilibrium and
when fa(t) = 0.
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22. Example 5: Pair-Share Exercise:
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23. Example 5: Pair-Share Exercise:
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24. Example 6: Pair-Share Exercise
• Assume that the pulley is
ideal
– No mass and no friction
– No slippage between
cable and surface of
cylinder (i.e., both move
with same velocity)
– Cable is in tension but
does not stretch
• Draw FBDs and write
equations of motion
• If pulley is not ideal,
discuss modeling
modifications
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25. Example 6: Pair-Share Exercise
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• Pulley is not ideal
– Add rotation mass and friction
– Model the slippage behaviors
– Add spring to model cable
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26. Example 7: Electric Motor
• An electric motor is attached to a load inertia through a
flexible shaft as shown. Develop a model and
associated differential equations (in classical and state
space forms) describing the motion of the two disks J1
and J2.
• Torsional stiffness is given in Appendix B
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27. Example 7: Electric Motor
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28. Example 7: Electric Motor
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29. Example 7: Electric Motor
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30. Example 8: Pair-Share Exercise: Copy Machine
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• The device from a copying machine is shown. It moves in a
horizontal plane. Develop the dynamic model, assuming that mass
of bar is negligible compared to attached mass m2 and angular
motions are small. The mass is subjected to a step input F, find an
expression for the displacement of point B after the transient
motions have died out.
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31. Example 8: Pair-Share Exercise: Copy Machine
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32. Example 9: Mass-Pulley System
• A mechanical system with a rotating
wheel of mass mw (uniform mass
distribution). Springs and dampers are
connected to wheel using a flexible cable
without skip on wheel.
• Write all the modeling equations for
translational and rotational motion, and
derive the translational motion of x as a
function of input motion u
• Find expression for natural frequency
and damping ratio
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33. Example 9: Mass-Pulley System
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34. Example 9: Mass-Pulley System
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35. Example 9: Mass-Pulley System
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36. Example 10: Pair-Share Exercise:
Double Pendulum
• The disk shown in the figure
rolls without slipping on a
horizontal plane. Attached to
the disk through a frictionless
hinge is a massless pendulum
of length L that carries another
disk. The disk at the bottom of
the pendulum cannot rotation
relative to the pendulum arm.
• Draw free-body diagrams and
derive equations of motion for
this system.
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37. Example 10: Pair-Share Exercise:
Double Pendulum
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38. Example 10: Pair-Share Exercise:
Double Pendulum
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39. Example 10: Pair-Share Exercise:
Double Pendulum
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