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Mathematical Modelling of Mechanical Systems

Vidhya Ragesh06-02-2014

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Outline of this Lecture

•Part-I: Laplace Transform

•Part-II: Transfer Function

•Part-III: Translational Mechanical System

Overview

• The transfer function of a process shows the characteristics of its dynamic behavior assuming a linear representation of the process.

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What is a Transfer Function?

• Given a linear system, then the transfer function, G(s), of the system is the ratio of the transform of the output to the transform of the input.

Examine the diagram

• In this system, we have the following:– The system is linear. – The input is c(t), and the transform of the input is C(s).– The output is y(t), and the transform of the output is Y(s).

• Then, the transfer function is given by: G(s) = Y(s)/C(s)

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Basic Types of Mechanical Systems

• Translational– Linear Motion

• Rotational– Rotational Motion

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TRANSLATIONAL MECHANICAL SYSTEMSPart-I

Basic Elements of Translational Mechanical Systems

Translational Spring

i)

Translational Massii)

Translational Damperiii)

Translational Spring

i)

Circuit Symbols

Translational Spring• A translational spring is a mechanical element that

can be deformed by an external force such that the deformation is directly proportional to the force applied to it.

Translational Spring

Translational Spring• If F is the applied force

• Then is the deformation if

• Or is the deformation.

• The equation of motion is given as

• Where is stiffness of spring expressed in N/m

2x1x

02 x1x

)( 21 xx

)( 21 xxkF

k

F

F

Translational Mass

Translational Massii)

• Translational Mass is an inertia element.

• A mechanical system without mass does not exist.

• If a force F is applied to a mass and it is displaced to x meters then the relation b/w force and displacements is given by Newton’s law.

M)(tF

)(tx

xMF

Translational Damper

Translational Damperiii)

• When the viscosity or drag is not negligible in a system, we often model them with the damping force.

• All the materials exhibit the property of damping to some extent.

• If damping in the system is not enough then extra elements (e.g. Dashpot) are added to increase damping.

Common Uses of DashpotsDoor Stoppers

Vehicle Suspension

Bridge SuspensionFlyover Suspension

Translational Damper

xCF

• Where C is damping coefficient (N/ms-1).

)( 21 xxCF

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Modelling a simple Translational System• Example-1: Consider a simple horizontal spring-mass system on a

frictionless surface, as shown in figure below.

or kxxm

0 kxxm

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Example-2• Consider the following system (friction is negligible)

• Free Body Diagram

MF

kfMf

k

F

xM

• Where and are force applied by the spring and inertial force respectively.

kf Mf

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Example-2

• Then the differential equation of the system is:

kxxMF

• Taking the Laplace Transform of both sides and ignoring initial conditions we get

MF

kfMf

Mk ffF

)()()( skXsXMssF 2

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)()()( skXsXMssF 2

• The transfer function of the system is

kMssF

sX

2

1

)()(

• if

12000

1000

Nmk

kgM

2

00102

ssF

sX .)()(

Example-2

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• The pole-zero map of the system is

2

00102

ssF

sX .)()(

Example-2

-1 -0.5 0 0.5 1-40

-30

-20

-10

0

10

20

30

40Pole-Zero Map

Real Axis

Imagin

ary

Axis

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Example-3• Consider the following system

• Free Body Diagram

k

F

xM

C

MF

kf

Mf

Cf

CMk fffF

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Example-3

Differential equation of the system is:

kxxCxMF

Taking the Laplace Transform of both sides and ignoring Initial conditions we get

)()()()( skXsCsXsXMssF 2

kCsMssF

sX

2

1

)()(

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Example-3

kCsMssF

sX

2

1

)()(

• if

1

1

1000

2000

1000

msNC

Nmk

kgM

/

1000

00102

sssF

sX .)()(

-1 -0.5 0 0.5 1-2

-1.5

-1

-0.5

0

0.5

1

1.5

2Pole-Zero Map

Real Axis

Imag

inar

y A

xis

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Example-4• Consider the following system

• Free Body Diagram (same as example-3)

MF

kf

Mf

Bf

BMk fffF kBsMssF

sX

2

1

)()(

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Example-5• Consider the following system

• Mechanical Network

k

F

2x

M1x B

↑ M

k

BF

1x 2x

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Example-5

• Mechanical Network

↑ M

k

BF

1x 2x

)( 21 xxkF

At node 1x

At node 2x

22120 xBxMxxk )(

Example-6• Find the transfer function X2(s)/F(s) of the following system.

1M 2M

k

B

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Example-7

k

)(tf

2x

1M4B3B

2M

1x

1B 2B

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Example-8• Find the transfer function of the mechanical translational

system given in Figure-1.

Free Body Diagram

Figure-1

M

)(tf

kf

Mf

Bf

BMk ffftf )(kBsMssF

sX

2

1

)()(

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Example-9• Restaurant plate dispenser

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Example-10• Find the transfer function X2(s)/F(s) of the following system.

Free Body Diagram

M1

1kf

1Mf

Bf

M2

)(tF

1kf

2Mf

Bf2kf

2k

BMkk fffftF 221

)(

BMk fff 11

0

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Example-11

1k

)(tu

3x

1M

4B3B

2M

2x

2B 5B

2k 3k

1x

1B

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Example-12: Automobile Suspension

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Automobile Suspension

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Automobile Suspension

).()()( 10 eq ioioo xxkxxbxm

2 eq. iiooo kxxbkxxbxm

Taking Laplace Transform of the equation (2)

)()()()()( skXsbsXskXsbsXsXms iiooo 2

kbsms

kbs

sX

sX

i

o

2)(

)(

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Example-13: Train Suspension

Car BodyBogie-2

Bogie

Frame

Bogie-1

WheelsetsPrimary

Suspension

Secondary

Suspension

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Example: Train Suspension