lect05 - mdof part 1 [compatibility mode]
TRANSCRIPT
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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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