Physical Laws for Mechanical

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Physical Laws for Mechanical
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Example car suspension
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Car 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)
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• Newtons Law
• or
• num
• den
• T.F.H(s)
• or

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State 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

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Quarter car suspension
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u
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Electromechanical systems

• Motors
• DC motors
• Induction motors
• Variable reluctance motors
• Generators
• Angular position sensors
• Encoders
• Tachometers

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For field control with constant armature current
For armature control with constant field current
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Armature controlled motor in feedback
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Get TF from wd to w and Td to w.
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DC Motor Driving an Inertial Load
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• 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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State Space model
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Matlab
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)
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Matlab 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
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SS 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)
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• 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.

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4 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)
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4 ways to enter system model
sys_zpk zpk(,-9.996 -4.004,
1.5) Zero/pole/gain
1.5 ------------------------- (s9.996) (s4.004)
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Modeling

• Types of systems electric
• mechanical
• 
  electromechanical
• fluid systems
• thermal systems
• Types of models I/O o.d.e. models
• Transfer Function
• state space models

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• I/O o.d.e. model o.d.e. involving input/output
  only.
• linear
• where u input
• y output

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• 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

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• Other types of models
• Transfer function model (This is I/O model) from
  I/O o.d.e. model, take Laplace transform

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• Then I/O ODE model in L.T. domain becomes
• or

denote
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ODE or TF to SS
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• 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
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• 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

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Example
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gtgt 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