LECT05 - MDOF Part 1 [Compatibility Mode]

8/10/2019 LECT05 - MDOF Part 1 [Compatibility Mode]

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MULTI DEGREE OFFREEDOM (M-DOF)


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Outline of the Chapter

1 Introduction

2 Potential and Kinetic EnergyExpressions in Matrix Form

3 Generalized Coordinates andGeneralized Forces

4 Using Lagranges Equations to derive

Equations of Motion


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1. INTRODUCTION

For simplicity continuous systems are

approximated as multidegree of freedom(MOF) systems.

Equations of motion of MOF systems can be

obtained either from Newtons 2nd law ofmotion or from Lagranges equations.

Analysis of MOF systems can be simplifiedusing the orthogonality property of the modeshapes of the systems natural frequencies.


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Modeling of Continuous systems as MOF

systems

Method 1: Lumped-mass system

Replace distributed mass of the system byfinite number of lumped masses

Lumped masses are connected by masslesselastic and damping members

E.g. Model the 3-storey

building as a 3 lumpedmass system


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Method 2: Finite Element Method (FEM)

Replace geometry of the system by largenumber of small elements

Principles of compatibility and equilibrium areused to find an approx. solution to the originalsystem


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Using Newtons 2nd

Law to deriveEquations of MotionStep 1: Set up suitable coordinates to describe

positions of the various masses in the system.Step 2: Measure displacements of the massesfrom their static equilibrium positions

Step 3: Draw free body diagram and indicateforces acting on each mass

Step 4: Apply Newtons 2nd

Law to each mass


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Example 1

Derive the equations of motion of the spring-

mass damper system shown below.


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Solution

Free-body diagram is as shown:

Applying Newtons 2nd Law gives:

Set i=1 with x0=0and i=nwith xn+1=0:

( ) ( ) ( ) ( ) 1,...,3,2,111111

=+++=++++

niFxxcxxcxxkxxkxm iiiiiiiiiiiiiii &&&&&&

( ) ( )( ) ( ) nnnnnnnnnnnnn Fxkxkkxcxccxm

Fxkxkkxcxccxm

=++++

=++++

++ 1111

1221212212111

&&&&

&&&&


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Solution

Notes:

The equations of motion can be expressed inmatrix form:

[m] is the mass matrix

[c] is the damping matrix

[k] is the stiffness matrix

[ ] [ ] [ ] Fxkxcxmrr&r&&r =++

[ ]

=

nm

m

m

m

0...0

0

00

0...0

2

1

OM

M

[ ] [ ]

( )

( )

( )

( )

( )

( )

=

=

+

+

+

+

=

+

+

+

=

+

+

tF

tF

tF

F

tx

tx

tx

x

kkk

kkk

kkkkkkk

k

ccc

cccc

ccc

c

nnn

nnn

3

2

1

3

2

1

1

433

3322

221

1

3322

221

,,

000

0

00

0000

,000

00

rr

OO

L

MM

M

M


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Potential and Kinetic Energy

Elastic potential energyof the ith spring, Vi=0.5Fixi Total potential energy

Since

Matrix form:

= =

==

n

i

n

i

iii xFVV1 12

1

,1

=

=

n

j

jiji xkF

= == ==

=

n

ij

n

jiij

n

ii

n

jjij xxkxxkV 1 11 1 2

1

2

1

[ ]xkxV T rr

2

1=


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Potential and Kinetic Energyexpressions in Matrix form Kinetic energyof mass mi:

Total KE of system:

Matrix form

2

2

1iii xmT &=

==

==

n

i

ii

n

i

i xmTT1

2

1 2

1&

[ ]

==

n

T

q

q

q

qqmqT

&

M

&

&

&r&r&r 2

1

where2

1


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Generalized coordinates andGeneralized forces

n independent

coordinates areneeded to describethe motion of a n

DOF system.

E.g. Consider the

triple pendulum asshown.


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Generalized coordinates andGeneralized forces

(xj,yj) are constrained by the following:

(xj,yj) are not independent, thus they cannot

be called generalized coordinates. If angular displacements j are used to

specify the locations of the masses mj, there

will be no constraints on j. Thus they form a set of generalized

coordinates and are denoted by qj= j,j=1,2,3

( ) ( ) ( ) ( ) 232

23

2

23

2

2

2

12

2

12

2

1

2

1

2

1 ,, lyyxxlyyxxlyx =+=+=+


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Generalized coordinates andGeneralized forces When external forces act on the system, the

new system configuration is obtained bychanging qjby qj, j=1,2,,n

The corresponding generalized force

Qj=Uj/qj, where Uj is the work done inchanging qj by qj.

Qj will be a moment when qj is an angular

displacement.


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Using Lagranges Equations to deriveEquations of Motion

Lagranges Equations:

= qj/ t is the generalized velocity Qj(n) is the non-conservative generalized force

corresponding to qj.

Qj(n) can be computed as follows:

Fxk, Fyk and Fzk are the external forces acting on the

kth mass in the x, y and z directions xk ,yk and zk are the displacements of the kth mass

in the x, y and z directions

njQq

V

q

T

q

T

dt

d nj

jjj

,...,2,1,)(

==

+

&

jq&

j

kzk

j

kyk

k j

kxk

n

jq

zF

q

yF

q

xFQ

+

+

= )(


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Using Lagranges Equations to deriveEquations of Motion

For a torsional system: Fxk is replaced by Mxk, the moment acting about

the x axis.

xk is replaced by xk, the angular displacement

about the x axis. For conservative system, Qj(n)=0

Thus Lagranges equations reduce to

This is a set of n differential equations, onecorresponding to each of the n generalized

coordinates

njq

V

q

T

q

T

dt

d

jjj,...,2,1,0 ==

+

&


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Example 2

Derive the equations of motion of the systemshown using Lagranges Equationby treating

i as generalized coordinates.

Ji is the mass moments of inertial

Mti are the external moments acting on the

components

kti are the torsional spring constants of the shaft

between the components.


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Solution

q1=1, q2=2, q3=3

Kinetic energy: (E1)

Potential energy (PE):

Total PE of system: (E2)

Generalized force:

233

222

211

21

21

21 &&& JJJT ++=

==

0

2

2

1)( tt kdkV

2

233

2

122

2

11 )(2

1)(

2

1

2

1 ++= ttt kkkV

= =

=

=3

1

3

1

)(

k k j

ktk

j

ktk

n

j Mq

MQ


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Solution

Therefore

(E3)

1

1

33

1

22

1

11

)(

1 tttt

n MMMMQ =

+

+

=

2

2

3

3

2

22

2

11

)(

2 tttt

n MMMMQ =

+

+

=

3

3

3

3

3

22

3

11

)(

3 tttt

nMMMMQ =

+

+

=


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Solution

Substituting E1, E2 and E3 into Lagranges

equations, we get

12212111 )( tttt MkkkJ =++

&&

2331223222 )( ttttt MkkkkJ =++ &&

3233333 ttt MkkJ =+ &&


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LECT05 - MDOF Part 1 [Compatibility Mode]

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