lesson 11: solving for mechanical dynamics using laplace ... notes... · write a differential...
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Lesson 11: Solving Mechanical System
Dynamics Using Laplace Transforms
ET 438a Automatic Control Systems Technology
1
lesson11et438a.pptx
Learning Objectives
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After this presentation you will be able to: Draw a free body diagram of a translational mechanical
system, Write a differential equation that describes the position of a
mechanical system as it varies in time. Use Laplace transforms to convert differential equations into
algebraic equations. Take the Inverse Laplace transform and find the time
response of a mechanical system. Examine the impact of increased and decreased damping on
a mechanical system.
Using Laplace Transforms to Solve
Mechanical Systems
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Example 11-1: Write the differential equation for the system shown with respect to position and solve it using Laplace transform methods. Assume f(t) = 50∙us(t) N, M= 1 Kg, K=2.5 N/m and B=0.5 N-s/m. The mass slides on a frictionless surface. x(0)=0.
Draw a free body diagram and label the forces
+
M= 1 Kg
f(t) - fI(t) f (t) fI(t)
fB(t)
- fB(t)
fK(t)
- fK(t) = 0
f(t)=fI(t)+fB(t)+fK(t)
Laplace Solution to Mechanical
Systems (1)
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)t(u50)t(f s2
2
2
2
Idt
)t(xd1
dt
)t(xdM)t(f
dt
)t(dx5.0
dt
)t(dxB)t(fB
)t(x5.2)t(xK)t(fK
Solve for X(s)
Laplace Solution to Mechanical
Systems (2)
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Solve using inverse Laplace transform and partial fractions expansion
Use Quadratic formula to find factors in denominator
Laplace Solution to Mechanical
Systems (3)
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Factored form
s2 s1
Partial Fractions Expansion- Find A
Laplace Solution to Mechanical Systems (7)
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Reverse angle rotation
Expanded Laplace relationship
Remember
Time Plot of System Position
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0 5 10 15 200
10
20
30
40
Time (Seconds)
Blo
ck P
osi
tio
n (
m)
20x t( )
t
Damping of system is B=0.5
Effects of Increased Damping
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0 2 4 6 8 10 120
10
20
30
40
Position B=1.0Position B=2.0
Time (Seconds)
Blo
ck P
osi
tio
n (
m)
40
9.188 103
x 1 t( )
x 2 t( )
120 t
Large damping value reduces oscillations and overshoot but slows response
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