1 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 – Review MDOF Theory
Review MDOF Theory
Peter AvitabileMechanical Engineering DepartmentUniversity of Massachusetts Lowell
[ K ]n
[ M ]n [ M ]a[ K ]a [ E ]a
[ ω ]2
Structural Dynamic Modeling Techniques & Modal Analysis Methods
m
k
1
1 c1
mp
3
3
c3
m
k
2
2 c2
f3
k 3
p 2 f2f1p 1
MODE 1 MODE 2 MODE 3
2 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 – Review MDOF Theory
Multiple Degree of Freedom Systems
• Referred to as a Multiple Degree of Freedom • An NDOF system has ‘N’ independent degrees of freedom to describe the system
• There is one natural frequency for every DOF in the system description
Systems with more than one DOF:
3 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 – Review MDOF Theory
Multiple Degree of Freedom Systems
• Each natural frequency has a displacement configuration referred to as a ‘normal mode’
• Mathematical quantities referred to as ‘eigenvalues’ and ‘eigenvectors’ are used to describe the system characteristics
• While the resulting motion appears more complicated, the system set of equations can always be decomposed into a set of equivalent SDOF systems for each mode of the system.
Properties of a MDOF system:
4 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 – Review MDOF Theory
Multiple Degree of Freedom Systems
• lumped mass• stiffness proportional to displacement
• damping proportional to velocity
• linear time invariant• 2nd order differential equations
Assumptions
m
m
k
k
c
c
1
1
2
2
1
2
f 1
f 2
x1
x2
5 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 – Review MDOF Theory
Multiple Degree of Freedom Systems
Free body diagram
f 1
f 2
(k 2 x1x2 )- (c 2 x1x2 )-
x1
x2
k 1x1 c1x1
m2
m1
6 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 – Review MDOF Theory
Multiple Degree of Freedom Systems
Newton’s Second Law
Rearrange terms
( ) ( ) ( )( ) ( ) ( )122122222
1221112211111
xxkxxctfxmxxkxkxxcxctfxm
−−−−=−+−−+−=
&&&&
&&&&&
( ) ( ) ( )( )tfxkxkxcxcxm
tfxkxkkxcxccxm
22212221222
1221212212111
=+−+−=−++−++
&&&&
&&&&
f 1
f 2
(k 2 x1x2 )- (c 2 x1x2 )-
x1
x2
k 1x1 c1x1
m2
m1
7 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 – Review MDOF Theory
MDOF Equations of Motion
Equation of Motion for 2 DOF
can be written in compact matrix form as
( )
( )
=
−
−++
−
−++
)t(f)t(f
xx
kkkkkxx
ccccc
xx
mm
2
1
2
1
22
221
2
1
22
221
2
1
2
1
&
&
&&
&&
[ ] [ ] [ ] FxKxCxM =++ &&&
Matrices and Linear Algebra are important !!!
8 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 – Review MDOF Theory
MDOF Equations of Motion
This coupled set of equations can be uncoupled by performing an eigensolution to obtain ‘eigenpairs’for each mode of the system, that is
‘eigenvalues’ and ‘eigenvectors’
or
‘frequencies’ (poles) and ‘mode shapes’
9 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 – Review MDOF Theory
MDOF Equations of Motion
Equation of Motion[ ] [ ] [ ] )t(FxKxCxM =++ &&&
Eigensolution
Frequencies (eigenvalues) and Mode Shapes (eigenvectors)
[ ] [ ][ ] 0xMK =λ−
[ ] [ ]L2122
21
2 uuUand\\
\=
ω
ω=
Ω
10 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 – Review MDOF Theory
MDOF - Orthogonality of Eigenvectors
Normal modes are ‘orthogonal’ with respect to the system mass and stiffness matrices.The eigen problem can be written as
For the ith mode of the system, the eigenproblemcan be written as
[ ][ ] [ ][ ][ ]2UMUK Ω=
[ ] [ ] iii uMuK λ=
11 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 – Review MDOF Theory
MDOF - Orthogonality of Eigenvectors
Premultiply that equation by the transpose of a different vector uj
and write a second equation for uj and premultiplyit by the transpose of vector ui
[ ] [ ] iTjii
Tj uMuuKu λ=
[ ] [ ] jTijj
Ti uMuuKu λ=
12 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 – Review MDOF Theory
MDOF - Orthogonality of Eigenvectors
Subtracting the previous two equations, yields
Since the eigenvalues for each mode are different,
When i=j, then the values are not zero
( ) [ ] 0uMu jT
iji =λ−λ
[ ] [ ]
jikuKumuMu
iiiT
i
iiiT
i =
==
[ ] ji0uMu jT
i ≠=
13 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 – Review MDOF Theory
Modal Matrix and Modal Space Transformation
Define the modal matrix as the collection of modal vectors for each mode organized in column fashion in the modal matrix
This modal matrix is then used to define the modal transformation equation with a new coordinate with ‘p’ as the principal coordinate
[ ] [ ]L321 uuuU =
[ ] [ ]
==
M
L 2
1
21 pp
uupUx
14 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 – Review MDOF Theory
Modal Space Transformation
Substitute the modal transformation into the equation of motion
In order to put the equations in normal form, this equation must be premultiplied by the transpose of the projection operator to give
[ ] [ ][ ] [ ] [ ][ ] [ ] [ ][ ] [ ] FUpUKUpUCUpUMU TTTT =++ &&&
[ ][ ] [ ][ ] [ ][ ] FpUKpUCpUM =++ &&&
15 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 – Review MDOF Theory
Modal Space Transformation
The first term of the modal acceleration can be expanded as
[ ] [ ][ ]
[ ] ( ) [ ] ( ) [ ] ( ) [ ] ( ) [ ] ( ) [ ] ( ) [ ] ( ) [ ] ( ) [ ] ( )
=
OMMM
L
L
L
3T
3T
31T
3
3T
22T
21T
2
3T
12T
11T
1
T
uMu2uMuuMuuMuuMuuMuuMuuMuuMu
UMU
16 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 – Review MDOF Theory
Modal Space Transformation
Recall from orthogonality that
[ ] ji0uMu jT
i ≠=
so that
[ ] [ ][ ]
[ ] ( ) [ ] ( )
[ ] ( )
=
OMMM
L
L
L
3T
3
2T
2
1T
1
T
uMu000uMu000uMu
UMU
17 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 – Review MDOF Theory
Modal Mass, Modal Damping, Modal Stiffness
The mass becomes
The stiffness becomes
The damping becomes (under special conditions)
[ ] [ ][ ]
=
OMMM
L
L
L
33
22
11
T
m000m000m
UMU
[ ] [ ][ ]
=
OMMM
L
L
L
33
22
11
T
k000k000k
UKU
[ ] [ ][ ]
=
OMMM
L
L
L
33
22
11
T
c000c000c
UCU
18 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 – Review MDOF Theory
Unit Modal Mass Scaling
The mass orthogonality results in an arbitrary mass called the modal mass
It may be desirable to scale the mode shapes so that the modal mass is 1.0The mode shape must be scaled by the inverse of the square root of the modal mass
[ ] iiiT
i muMu =
[ ] iiiT
i muMu =
19 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 – Review MDOF Theory
Unit Modal Mass Scaling
Each mode can be conveniently scaled as follows
[ ] [ ]
=
OMMM
L
L
L
33
22
11
n
m100
0m10
00m1
UU
20 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 – Review MDOF Theory
Scaled Unit Modal Mass, Damping, and Stiffness
Modal Mass
Modal Damping
Modal Stiffness
[ ] [ ][ ]
=
OMMM
L
L
L
100010001
UMU nTn
[ ] [ ][ ]
ωω
ω
=
OMMM
L
L
L
23
22
21
nTn
000000
UKU
[ ] [ ][ ]
ωζωζ
ωζ
=
OMMM
L
L
L
33
22
11
nTn200020002
UCU
21 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 – Review MDOF Theory
Modal Space Transformation
The physical set of highly coupled equations are transformed to modal space through the modal transformation equation to yield a set of uncoupled equations
Modal equations (uncoupled)
=
+
+
MMM
&
&
M
&&
&&
FuFu
pp
\k
kpp
\c
cpp
\m
mT
2
T1
2
1
2
1
2
1
2
1
2
1
2
1
22 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 – Review MDOF Theory
Modal Space Transformation
Diagonal Matrices -Modal Mass Modal Damping Modal Stiffness
Highly coupled system
transformed into simple system
[ ] FUp\
K\
p\
C\
p\
M\
T=
+
+
&&&
m
k
1
1 c1
mp
3
3
c3
m
k
2
2 c2
f3
k 3
p 2 f2f1p 1
MODE 1 MODE 2 MODE 3
23 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 – Review MDOF Theory
Modal Space Transformation
It is very clear to see that these modal space equations result in a set of independent SDOF systemsThe modal transformation equation uncouples the highly coupled set of equationsThe modal transformation appropriates the force to each modal oscillator in modal spaceThe modal transformation equation combines the response of all the independent SDOF systems to identify the total physical response
24 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 – Review MDOF Theory
Modal Space Response Analysis
Since the MDOF system is reduced to equivalent SDOF systems with appropriate force, the response of each SDOF system can be determined using SDOF approaches discussed thus far.The total response due to each of the SDOF systems can be determined using the modal transformation equation
25 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 – Review MDOF Theory
Modal Space Transformation
[ M ]p + [ C ]p + [ K ]p = [U] F(t)T
m
k
1
1 c1
f1p1
MODE 1
m
k
2
2 c2
p2 f2
MODE 2
mp
3
3
c3
f3
k3
MODE 3
MODAL
PHYSICAL MODEL[M]x + [C]x + [K]x = F(t)
x = [U]p = [
SPACE
x = [U]p = ++ u p 33u p 22u p 11
u 3u 2u 1 ]p
.. .
.. .
+
+
Σ=
26 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 – Review MDOF Theory
Numerical Response - Mode Superposition
Using the modal space formulation, a simple integration on the SDOF system can be performed
=
+
+
MMM
&
&
M
&&
&&
FuFu
pp
\k
kpp
\c
cpp
\m
mT
2
T1
2
1
2
1
2
1
2
1
2
1
2
1
m
k
1
1 c 1
f 1 p 1
MODE 1
m
k
2
2 c 2
p 2 f 2
MODE 2
m
p
3
3
c 3
f 3
k 3
MODE 3
x =
x =
x =
u p 1 1
u p 2 2
u p 3 3
+
x = [U]p = + + u p 3 3
u p 2 2
u p 1 1
+ Σ =
27 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 – Review MDOF Theory
Modal Space - Modal Matrices
[ ] [ ][ ]
=
\M
\UMU 111
T1
[ ] [ ][ ]
=
\C
\UCU 111
T1
[ ] [ ][ ]
=
\K
\UKU 111
T1Modal Stiffness
Modal Damping
Modal Mass
The mode shapes are linearly independent and orthogonal w.r.t the mass and stiffness matrices
TRUE !!!
???????
TRUE !!!
28 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 – Review MDOF Theory
Proportional Damping
[ ] [ ] [ ][ ][ ]
β+α=β+α
\KM
\UKMU 1
T1
The damping matrix is only uncoupled for a special case where the damping is assumed to be proportional to the mass and/or stiffness matrices
Many times proportional damping is assumed since we do not know what the actual damping isThis assumption began back when computational power was limited and matrix size was of critical concern. But even today we still struggle with the damping matrix !!!
29 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 – Review MDOF Theory
Non-Proportional Damping
If the damping is not proportional then a state space solution is required (beyond our scope here)
[ ] [ ][ ] [ ]
[ ] [ ][ ] [ ]
=
−
−
F0
xx
K00M
xx
CMM0
1
1
11
1 &
&
&&
[ ] [ ] QYAYB 11 =−&
[ ] [ ] [ ][ ] [ ]
=
11
11 CM
M0B [ ] [ ] [ ]
[ ] [ ]
−
=1
11 K0
0MA
[ ] [ ][ ] 0YBA 11 =λ− [ ] 11 pY Φ=
The equation of motion can be recast as
The eigensolution and modal transformation is then