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CHAPTER 3 MESB 374 System Modeling and Analysis. Rotational Mechanical Systems. Rotational Mechanical Systems. Variables Basic Modeling Elements Interconnection Laws Derive Equation of Motion (EOM). J. q. K. t ( t ). B. Variables. q :angular displacement [rad]

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CHAPTER 3 MESB 374 System Modeling and Analysis

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Chapter 3 mesb 374 system modeling and analysis

CHAPTER 3MESB 374System Modeling and Analysis

Rotational Mechanical Systems


Rotational mechanical systems

Rotational Mechanical Systems

  • Variables

  • Basic Modeling Elements

  • Interconnection Laws

  • Derive Equation of Motion (EOM)


Variables

J

q

K

t (t)

B

Variables

  • 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)


Basic rotational modeling elements

Basic Rotational Modeling Elements

  • Damper

    • Friction Element

    • Analogous to Translational Damper.

    • Dissipate Energy.

    • E.g. bearings, bushings, ...

  • Spring

    • Stiffness Element

    • Analogous to Translational Spring.

    • Stores Potential Energy.

    • E.g. shafts

tD

tD

tS

B

tS

q1

q2

&

&

t

=

q

-

q

=

w

-

w

B

B

t

=

q

-

q

K

D

2

1

2

1

S

2

1


Basic rotational modeling elements1

d

Basic Rotational Modeling Elements

  • Parallel Axis Theorem

  • Ex:

  • Moment of Inertia

    • Inertia Element

    • Analogous to Mass in Translational Motion.

    • Stores Kinetic Energy.


Chapter 3 mesb 374 system modeling and analysis

Interconnection Laws

  • Newton’s Second Law

  • Newton’s Third Law

    • Action & Reaction Torque

  • Lever

    (Motion Transformer Element)

d

å

&

&

w

=

q

=

t

J

J

EXTi

1

2

3

dt

i

Angular

Momentum

Massless or small acceleration


Motion transformer elements

Motion Transformer Elements

  • Rack and Pinion

  • Gears

Ideal case

Gear 1

Gear 2


Example

J

q

K

t (t)

B

Example

Derive a model (EOM) for the following system:

FBD: (Use Right-Hand Rule

to determine direction)

J

K

B


Energy distribution

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 termw : Ü What have we done ?

  • Integrate the second equation w.r.t. time:Ü What are we doing now ?

&

&

&

q

+

q

+

q

=

t

J

B

K

(

t

)

Total

Contributi

on

Contributi

on

Contributi

on

Applied To

rque

of Inertia

of the Dam

per

of the Spr

ing

&

&

&

&

&

&

q

×

q

+

q

×

q

+

q

×

q

=

t

×

w

J

B

K

t


Energy method

J

q

K

t (t)

B

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

  • EOM of a simple Mass-Spring-Damper System


Example1

Rolling without slipping

FBD:

Elemental Laws:

x

x

q

q

fs

f(t)

fd

Mg

ff

N

Example

Coefficient

of friction m

K

R

B

f(t)

J, M

Kinematic Constraints: (no slipping)


Example cont

Example (cont.)

I/O Model (Input: f, Output: x):

Q:How would you decide whether or not the disk will slip?

If

Slipping happens.

Q:How will the model be different if the disk rolls and slips ?

does not hold any more


Example energy method

Rolling without slipping

Kinetic Energy

x

q

Example (Energy Method)

Power exerted by non-conservative inputs

Coefficient

of friction m

K

R

B

f(t)

Energy equation

J, M

Kinematic Constraints: (no slipping)

Potential Energy

EOM


Example coupled translation rotation

Example (coupled translation-rotation)

Real world

Think about performance of lever and gear train in real life


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