Random vibration of MDOF systems -1
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Deterministic Excitations - Stochastic Structural Dynamics - Lecture Slides, Slides of Structural Analysis
Main points of this lecture are: Deterministic Excitations, Random Vibration, Equations of Motion, Input-Output Relations, Frequency Domain, Modal Expansion, Forced Vibration Analysis, Static Coupling, Inertial Coupling, Coordinate System
Typology: Slides
2012/2013
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Random vibration of MDOF systems - 1
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PreliminariesDiscrete MDOF systems under deterministic excitations Nature of equations of motion Input - output relations in time domain Input - output relations in frequency domain Forced vibration analysis using modal expansion
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^
^
^
^
^
^
1
1
2
2
2
2
2
1
1
1
1
2
1 1^
2 2 2
2
1 1^
2 2^
1 1^
2 2
my^^0
k^
y^
l^
k^
y^
l
I^
k^
y^
l^
l^
k^
y^
l^
l
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k l^
k l
m^
y^
y
k l^
k l
k l^
k l
I
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^
^
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^
^
^
^
^
^
^
^
^
^
^ is diagonal and
is non-diagonal
M^
K
Static coupling
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1
2
2
2
1 1
2 2 0
0
0 k^
k
m^
me
z^
z
k L
k L
me
m
^
^
^
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^
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^
^
is non-diagonal and
is diagonal
M^
K
Inertial coupling
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Remarks • Equations of motion for MDOF systems are generallycoupled • Coupling between co-ordinates is manifest in the form ofstructural matrices being nondiagonal • Coupling is not an intrinsic property of a vibrating system.It is dependent upon the choice of the coordinate system.This choice itself is arbitrary. • Equations of motion are not unique.They depend upon the choice of coordinate system.
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Remarks (continued) • The best choice of coordinate system is the onein which the coupling is absent. That is, the structuralmatrices are all diagonal. • These coordinates are called the natural coordinatesfor the system. Determination of these coordinates fora given system constitutes a major theme in structuraldynamics. Theory of ODEs and linear algebra help us.
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1
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2 3
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x^ t^ ^
y t^ ^
t
O
O
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3 4
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3 4 3 3 2 1 12
;
12
h
I E
k
h
I E
k k k^
s s
a a^
^
^
^
2
2
(^22) (^22)
(^22) (^21) (^21) (^22) (^21) (^21)
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2
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2 2 2 2 3 2 ) ( 2
2
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columns slab
r hA
r hA
y hx A y x y x y
hx A
d y b x dtb
d b dtb I
hA hA
dtb
m m m
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^
^
^
^
0
0
0
;^
0
,^
and
, in general, are non-diagonal
Equations are coupledSuppose we introduce a new set of dependentvariables
( ) using the transformation
( ) where
is a
tra
MX
CX
KX
F t
X^
X^
X^
X
M^
C^
K Z t
X t
TZ tT^
n^
n
^
^
^
^
nsformaiton matrix, to be
How to uncouple equations of motion? selected.
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^
^
^
^
^
0
0
0
;^
0
( )
( )
( )
( )
,^
, &
structural matrices in the new coordinate system.
( )
force vector in the new coord t^
t^
t^
t
MX
CX
KX
F t
X^
X^
X^
X
X t
TZ t MTZ t
CTZ t
KTZ t
F t
T MTZ t
T CTZ t
T KTZ t
T F t
MZ t
CZ t
KZ t
F t
M C
K
F t
^
^
^
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^
^
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^
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inate system
Can we select
such that
,^
, &
are all
?
If yes, equation for
( ) would then represent a set of uncoupled
equations and hence can be solved easily.
T^
M C
K
Z t
Question
DIAGONAL
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