1. CHAPTER 3 MESB 374 System Modeling and Analysis
ME375 Handouts - Spring 2002
CHAPTER 3 MESB 374 System Modeling and Analysis
Rotational Mechanical Systems

2. Rotational Mechanical Systems
ME375 Handouts - Spring 2002
Rotational Mechanical Systems
Variables
Basic Modeling Elements
Interconnection Laws
Derive Equation of Motion (EOM)

3. Variables q : angular displacement [rad]
ME375 Handouts - Spring 2002
Variables q
t (t) J K B
q : angular displacement [rad]
w : angular velocity [rad/sec]
a : angular acceleration [rad/sec2]
t : torque [Nm]
p : power [Nm/sec]
w : work ( energy ) [Nm]
1 [Nm] = 1 [J] (Joule)

4. Basic Rotational Modeling Elements
ME375 Handouts - Spring 2002
Basic Rotational Modeling Elements
Spring
Stiffness Element
Analogous to Translational Spring.
Stores Potential Energy.
E.g. shafts
Damper
Friction Element
Analogous to Translational Damper.
Dissipate Energy.
E.g. bearings, bushings, ... tD tD tS B tS q1 q2 t = B q
&
& t = K q - q - q = B w - w D 2 1 2 1 S 2 1

5. Basic Rotational Modeling Elements
ME375 Handouts - Spring 2002
Basic Rotational Modeling Elements
Moment of Inertia
Inertia Element
Analogous to Mass in Translational Motion.
Stores Kinetic Energy.
Parallel Axis Theorem
Ex: d

6. Massless or small acceleration
ME375 Handouts - Spring 2002
Interconnection Laws
Newton’s Second Law
Newton’s Third Law
Action & Reaction Torque
Lever
(Motion Transformer Element) d J w = J q
&
& å = t dt 1 2 3
EXTi i
Angular
Momentum
Massless or small acceleration

7. Motion Transformer Elements
ME375 Handouts - Spring 2002
Motion Transformer Elements
Rack and Pinion
Gears
Ideal case
Gear 1
Gear 2

8. Example Derive a model (EOM) for the following system:
ME375 Handouts - Spring 2002
Example
Derive a model (EOM) for the following system:
FBD: (Use Right-Hand Rule
to determine direction) q
t (t) J K B J K B

9. Energy Distribution J q & & + B q & + K q = t ( t ) J q & & × q & + B
ME375 Handouts - Spring 2002
Energy Distribution
EOM of a simple Mass-Spring-Damper System
We want to look at the energy distribution of the system. How should we start ?
Multiply the above equation by angular velocity term w : Ü What have we done ?
Integrate the second equation w.r.t. time: Ü What are we doing now ? J q
&
& + B q
& + K q = t ( t )
Contributi on
Contributi on
Contributi on
Total
of Inertia
of the Dam
per
of the Spr
ing
Applied To
rque J q
&
& × q
& + B q
& × q
& + K q × q
& = t t × w

10. ME375 Handouts - Spring 2002
Energy Method
Change of energy in system equals to net work done by external inputs
Change of energy in system per unit time equals to net power exerted by external inputs q
t (t) J K B
EOM of a simple Mass-Spring-Damper System

11. Example Rolling without slipping FBD: Elemental Laws:
ME375 Handouts - Spring 2002
Example
Rolling without slipping
FBD:
Elemental Laws:
Coefficient
of friction m x q K R
f(t) B
J, M
Kinematic Constraints: (no slipping)
f(t) fs fd ff Mg N q x

12. ME375 Handouts - Spring 2002
Example (cont.)
Q: How would you decide whether or not the disk will slip?
I/O Model (Input: f, Output: x): If
Slipping happens.
Q: How will the model be different if the disk rolls and slips ?
does not hold any more

13. Example (Energy Method)
ME375 Handouts - Spring 2002
Example (Energy Method)
Rolling without slipping
Power exerted by non-conservative inputs
Coefficient
of friction m x q K R
f(t) B
Energy equation
J, M
Kinetic Energy
Kinematic Constraints: (no slipping)
Potential Energy
EOM

14. Example (coupled translation-rotation)
ME375 Handouts - Spring 2002
Example (coupled translation-rotation)
Real world
Think about performance of lever and gear train in real life