Title: Physical Laws for Mechanical
1Physical Laws for Mechanical
2Example car suspension
3Car suspension simplified
- Ignore tire deformation.
- Suppose y1(t) is measured
- from equilibrium position
- when gravity has set in.
- So gravity is canceled by
- spring force at eq. pos.
- ?There are two forces on m
y1(t)
x(t)
4- Newtons Law
- or
- num
- den
- T.F.H(s)
- or
5State Space Model
- For linear motion
- Define two state variables for each mass
- x1position, x2 velocity x1-dot x2
- x2-dot is acc and solve for it from Newtons
- For angular motion
- Define two state variables for each rotating
inertia - x1 angle, x2 angular velocity x1-dot x2
- x2-dot is angular acc and solve for it from
Eulers law
6Quarter car suspension
7u
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12Electromechanical systems
- Motors
- DC motors
- Induction motors
- Variable reluctance motors
- Generators
- Angular position sensors
- Encoders
- Tachometers
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14For field control with constant armature current
For armature control with constant field current
15Armature controlled motor in feedback
16Get TF from wd to w and Td to w.
17DC Motor Driving an Inertial Load
18- w(t) angular rate of the load, output
- vapp(t) applied voltage, the input
- i(t) armature current
- vemf(t) back emf voltage generated by the motor
rotation - vemf(t) constant motor velocity
- t(t) mechanical torque generated by the motor
- t(t) constant armature current
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20State Space model
21Matlab
R 2.0 Ohms L 0.5 Henrys Km .015
torque constant Kb .015 emf constant Kf
0.2 Nms J 0.02 kg.m2 A -R/L -Kb/L
Km/J -Kf/J B 1/L 0 C 0 1 D
0 sys_dc ss(A,B,C,D)
22Matlab output
a x1 x2
x1 -4 -0.03 x2
0.75 -10 b u1
x1 2 x2 0 c
x1 x2 y1
0 1 d u1
y1 0
23SS to TF or ZPK representation
gtgt sys_tf tf(sys_dc) Transfer function
1.5 ------------------------ s2 14 s
40.02 gtgt sys_zpk zpk(sys_dc) Zero/pole/gain
1.5 ------------------------- (s4.004)
(s9.996)
24- Note The state-space representation is best
suited for numerical computations. For highest
accuracy, convert to state space prior to
combining models and avoid the transfer function
and zero/pole/gain representations, except for
model specification and inspection.
254 ways to enter system model
sys tf(num,den) Transfer function sys
zpk(z,p,k) Zero/pole/gain sys ss(a,b,c,d)
State-space sys frd(response,frequencies)
Frequency response data s tf('s') sys_tf
1.5/(s214s40.02) Transfer function
1.5 ------------------------ s2 14 s
40.02 sys_tf tf(1.5,1 14 40.02)
264 ways to enter system model
sys_zpk zpk(,-9.996 -4.004,
1.5) Zero/pole/gain
1.5 ------------------------- (s9.996) (s4.004)
27Modeling
- Types of systems electric
- mechanical
-
electromechanical - fluid systems
- thermal systems
- Types of models I/O o.d.e. models
- Transfer Function
- state space models
28- I/O o.d.e. model o.d.e. involving input/output
only. - linear
- where u input
- y output
29- State space model
- linear
- or in some text
- where u input
- y output
- x state vector
- A,B,C,D, or F,G,H,J are const matrices
30- Other types of models
- Transfer function model (This is I/O model) from
I/O o.d.e. model, take Laplace transform
31- Then I/O ODE model in L.T. domain becomes
- or
denote
32ODE or TF to SS
33- State space model to T.F. / block diagram
- s.s.
- Take L.T.
- From sX(s)-AX(s)BU(s)
- sIX(s)-AX(s)BU(s)
- (sI-A)X(s)BU(s)
- X(s)(sI-A)-1BU(s)
1
2
1
34- into Y(s)C(sI-A)-1BU(s)DU(s)
- Y(s)C(sI-A)-1BD U(s)
- H(s) DC(sI-A)-1B
- is the T.F. from u to y
- from
2
1
35Example
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37gtgt n1 2 3d1 4 5 6 gtgt A,B,C,Dtf2ss(n,d)
A -4 -5 -6 1 0 0 0
1 0 B 1 0 0 C 1
2 3 D 0 gtgt tf(n,d) Transfer
function s2 2 s 3 ---------------------
s3 4 s2 5 s 6
- In Matlab
- gtgt A0 1-2 -3
- gtgt B01
- gtgt C1 3
- gtgt D0
- gtgt n,dss2tf(A,B,C,D)
- n
- 0 3.0000 1.0000
- d
- 1 3 2