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# CHAPTER 3 MESB 374 System Modeling and Analysis - PowerPoint PPT Presentation

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 3MESB 374 System Modeling and Analysis

Rotational Mechanical Systems

• Variables

• Basic Modeling Elements

• Interconnection Laws

• Derive Equation of Motion (EOM)

q

K

t (t)

B

Variables

• q : angular displacement [rad]

• t: torque [Nm]

• p : power [Nm/sec]

• w : work ( energy ) [Nm]

1 [Nm] = 1 [J] (Joule)

• 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 Elements

• Parallel Axis Theorem

• Ex:

• Moment of Inertia

• Inertia Element

• Analogous to Mass in Translational Motion.

• Stores Kinetic Energy.

• 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

• Rack and Pinion

• Gears

Ideal case

Gear 1

Gear 2

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

• 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

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

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.)

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

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

Real world

Think about performance of lever and gear train in real life