csl4 21 j15

ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION

Control Systems

Lecture 4


Modeling different systems: Mechanical Translational system

• Mechanical Translational system (mass-spring-dashpot)

• Mass: The Mass is an inertial element– Force (F) Acceleration Reaction force


• Dashpot (damper)– The reaction damping force F is approximated by

the product of damping f and relative velocity if any.

– F(t) = f (v1-v2) = fv



• Spring – Restoring force Reaction Force on each

end is same– The spring element force equation, in accordance

with Hooke’s Law is given by:

– Fk = K (xc-xd)



Modeling different systems: Mechanical Rotational system

Rotational Systems are similar to Translational Systems except for the difference that the torque equations are written Extending the Newton’s Law of motion

Elements of the Rotational System are;

1.Inertia element

Tj = Jdω/dt =J.d2ϴ/dt2

2.Torsional spring element

Tk = K (ϴ1 - ϴ2) = Kϴ

3.Damper element

TB = B (ω1 – ω2) = Bω


Example 1


1. Decide input and the output2. Draw the free body diagram of the masses (optional)3. Convert to the frequency-domain representations4. Create transfer function

Modeling Steps:

Identify Modeling steps


Step 1: Decide Input and Output

)(tTAny angular position, velocity or acceleration of the rotational masses

Output variable:Input variable:


Step 2: The free body diagram J1 (optional)Step 3: The frequency response representation

111J

1J11D

1K

2k

TKKDJ 211111 T

Terms related to θ1

Step 2: Free Body Diagram


2

22J

2J22D

2K

32D

1K

013222222 KDKDJ


Step 2: The free body diagram J2 (Optional)Step 3: The frequency response representation



3

33J

3J32D

33D

22D 022333233 DDDJ


Step 2: The free body diagram J3 (Optional) Step 3: The frequency response representation



Step 4: Transfer Function

0

0

)(0

0

3

2

1

3232

2

2222

112 T

DDsJssD

sDKsDJsK

KKsDJs

Inertial 1:

Inertial 2:

Inertial 3:

TsKsKssDsJs )()()()( 2111112

0)()()()()( 132222222 sKssDsKssDsJs

0)()()()( 223332332 ssDssDssDsJs

Put the equations in matrix form


Step 4: Transfer Function

If we are interested to control the position of the mass J3, then solve for θ3.Cramer’s rule:

)(0

0

00

0

3232

2

2222

112

2

222

112

3

DDsJssD

sDKsDJsK

KKsDJs

sD

KsDJsK

TKKsDJs

0

0

)(0

0

3

2

1

3232

2

2222

112 T

DDsJssD

sDKsDJsK

KKsDJs


Example 2

Find the transfer function between )( and )( 2 ttT


Example 2

csl4 21 j15

Engineering