Lesson 11: Solving Mechanical System

Dynamics Using Laplace Transforms

ET 438a Automatic Control Systems Technology

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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 (4)

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Laplace Solution to Mechanical Systems (5)

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Find B

Laplace Solution to Mechanical Systems (6)

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Find C

0 0

Laplace Solution to Mechanical Systems (7)

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Reverse angle rotation

Expanded Laplace relationship

Remember

Laplace Solution to Mechanical Systems (8)

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From Laplace table

Where

Ans

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

ET 438a Automatic Control Systems Technology

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