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MECA029 – Mechanical Vibrations
Gaëtan Kerschen
Space Structures & Systems Lab (S3L)
MECA029 – Mechanical Vibrations
Lecture 6: Reduction and
Substructuring Methods
3
Outline
Preliminary Remarks
4
Quiz
Results and answers…
5
Outline
Previous Lectures
6
Previous Lectures
L1: Analytical dynamics of discrete systems
L2: Undamped vibrations of n-DOF systems
L3: Damped vibrations of n-DOF systems
L4: Continuous systems: bars, beams, plates
L5: Approximation of continuous systems by Rayleigh-Ritz and finite element methods
L6: Reduction and substructuring methods
L7: Direct time-integration methods
L8: Introduction to nonlinear dynamics
7
Previous Lectures
132 Hz
NUMERICAL METHODS OBJECTIVE OF
THIS COURSE
8
Previous Lectures
L1
ODEs:
L2,3
L4
( ) ( ) ( ) ( )t t t t+ + =Mq Cq Kq p&& &0: 0
:
ijjj
i
ji ij
V u Xx
S n tσ
σρ
σ
∂− + =
∂
=
&&
L4LINEARIZATION
L2,3
132 Hz
L5SPATIAL
DISCRETIZATION
PDEs:
L6EFFICIENT
COMPUTATION
ANALYTIC
9
Previous Lectures
Finite element method: a structure is divided into a finite number of elements of simple geometry (bar, beam, membrane, plate, shell)
10
Previous Lectures
1. Kinematic assumptions
2. Write the resulting strain and kinetic energy
3. Determine the interpolation functions
4. Compute the elementary matrices in local axes
5. Coordinate transformation
6. Compute the elementary matrices in structural axes
7. Assembly process
Finite element method:
11
Previous Lectures
Bar element
⇒ Linear interpolation functions inside the element
⇒ Elementary mass and stiffness matrices
2 11 26eL
m ⎡ ⎤= ⎢ ⎥
⎣ ⎦M l 1 1
1 1eLE A −⎡ ⎤
= ⎢ ⎥−⎣ ⎦K
l
12
Previous Lectures
Beam element
⇒ Cubic interpolation functions inside the element
⇒ Elementary mass and stiffness matrices
2 2
2 2
156 22 54 1322 4 13 354 13 156 2242013 3 22 4
eLm
−⎡ ⎤⎢ ⎥−⎢ ⎥=⎢ ⎥−⎢ ⎥− − −⎣ ⎦
M
l l
l l l ll
l l
l l l l
2 2
3
2 2
12 6 12 66 4 6 212 6 12 66 2 6 4
eLE I
−⎡ ⎤⎢ ⎥−⎢ ⎥=⎢ ⎥− − −⎢ ⎥−⎣ ⎦
K
l l
l l l l
l ll
l l l l
13
Previous Lectures
3D Beam element
⇒ Same reasoning as for the previous beam elements
⇒ 6DOFs/node
⇒ 12x12 elementary matrices
14
Previous Lectures
eSeL qRq =
Local ⇒ structural axes
1
X
YZ
O
local axes
3
2y x
z
ψx1
ψy1
ψz1
U1
V1
W1ψx2
U2
V2W2
ψz2
ψy2
, ,eL eL eS eS⇒K M K M
15
Previous Lectures
Assembly process: use localization vectors
, , 0e eS eSΣ ⇒ ⇒ + =K M K M Mq Kq&&
16
Previous Lectures
Convergence to the exact solution when the number of finite elements tends to infinity
Upper bound convergence for a consistent mass matrix (i.e., constructed using the same process as for the stiffness matrix)
Example of a clamped-free bar:
2
2
1 2 1(2 1) 1 ...2 24 2r
EA rrmL N
π πω⎡ ⎤−⎛ ⎞= − + +⎢ ⎥⎜ ⎟
⎝ ⎠⎢ ⎥⎣ ⎦
Finite element method
2(2 1)2
1,...,
kEAkml
k
πω = −
= ∞
Continuous system
17
Previous Lectures
Introduction to reduction methods
⇒ Industrial FE models typically have [105-106] DOFs
⇒ But interest in the lower frequency eigensolutions
)1()()1( ×××
=mmnn
yRx
yMyK 2ω=
Reduced matrices
RMRMRKRK TT == and
m n<?
18
Previous Lectures
Guyan: static condensation
⎭⎬⎫
⎩⎨⎧
⎥⎦
⎤⎢⎣
⎡=
⎭⎬⎫
⎩⎨⎧
⎥⎦
⎤⎢⎣
⎡
C
R
CCCR
RCRR
C
R
CCCR
RCRR
xx
MMMM
xx
KKKK 2ω Retained DOF
Condensed DOF
1CC CR−
⎡ ⎤= ⎢ ⎥−⎣ ⎦
IR
K KRCRCCDS
R
C
R xKKI
xxx
xx
x ⎥⎦
⎤⎢⎣
⎡−
=⎭⎬⎫
⎩⎨⎧
+=
⎭⎬⎫
⎩⎨⎧
= −1
19
Previous Lectures
Guyan: static condensation
CRCCRCRRT
RR KKKKRKRK 1−−==
CRCCCCCCRC
CRCCRCCRCCRCRRT
RR
KKMKK
MKKKKMMRMRM11
11
−−
−−
+
−−==
( )2 0RR RR Rω− =K M x
20
Today
L1: Analytical dynamics of discrete systems
L2: Undamped vibrations of n-DOF systems
L3: Damped vibrations of n-DOF systems
L4: Continuous systems: bars, beams, plates
L5: Approximation of continuous systems by Rayleigh-Ritz and finite element methods
L6: Reduction and substructuring methods
L7: Direct time-integration methods
L8: Introduction to nonlinear dynamics
21
Today
1. Guyan: Illustrative example
2. Craig-Bampton
22
1. Guyan: Illustrative Example
1 1 2l
2 4
w2 Ψ2
l
3l
3
Ψ3
w3
1
1 2
l
2 3
w2 Ψ2
l
3994484537414620214053
1.38039.39560.6720255.50067.50492.5161
32
42
−−
exactelementselementsr
IELm
rω
No reduction (straightforward application of the finite
element method)
23
1. Guyan: Illustrative Example
1 1 2l
2 4
w2 Ψ2
l
3l
3
Ψ3
w3
+ Guyan ???
2 2
2 2
156 22 54 1322 4 13 354 13 156 2242013 3 22 4
eLm
−⎡ ⎤⎢ ⎥−⎢ ⎥=⎢ ⎥−⎢ ⎥− − −⎣ ⎦
M
l l
l l l ll
l l
l l l l
2 2
3
2 2
12 6 12 66 4 6 212 6 12 66 2 6 4
eLE I
−⎡ ⎤⎢ ⎥−⎢ ⎥=⎢ ⎥− − −⎢ ⎥−⎣ ⎦
K
l l
l l l l
l ll
l l l l
24
1. Guyan: Illustrative Example
1 1 2l
2 4
w2 Ψ2
l
3l
3
Ψ3
w3
+ Guyan ???
3994484537414620214053
1.38039.39560.6720255.50067.50492.5161
32
42
−−
exactelementselementsr
IELm
rω
Guyan (4DOFs⇒2DOFs)506.5
4002
25
Today
1. Guyan: Illustrative example
2. Substructuring and component mode
synthesis
26
Substructuring ?
Complex structure: assembly of several subsystems
⇒ Very often, the subsystem are analyzed by different teams
⇒ Necessary to reconstruct the full model from separate analyses (definition of appropriate interfaces)
⇒ Keep the number of DOFs of each subsystem as small as possible
27
Substructuring: Examples
Vehicle body, 1373000 DOFsNatsiavas et al., JCND2, 2007
Model combining the vehicle body, the suspensions, the wheels and 4 passengers; response to road irregularities
Passenger-seat model
28
Substructuring: Examples
Stator (1278000 DOFs)
Objective: analysis of the radial freeplay
Casing (789072 DOFs)
Rotor (113176 DOFs)
29
Concept of Mechanical Impedance
internal DOF q1
(no applied loads)
( ) gqMK =− 2ω
applied force amplitudes
Impedance matrix Z(w2)
boundary DOF q2
The impedance matrix relates the force amplitudes to the displacement amplitudes
It is the inverse of the FRF matrix H(ω)
30
Concept of Mechanical Impedance
( )( ) ( )
∑=
−−−−
−
−
−−−
+−−−
−=
1
1224
212
21)()(212
214
121
11111
1121121
1121121
1121222
121
112122*22
)( ~~
~~~~n
i ii
Ti
Tiii
ωωω
ωωω
ω
MKxxMK
KKMKKMKKKKMM
KKKKZ
Expression of the reduced impedance matrix
2( ),i iω x% % are the eigenfrequencies and eigenmodes of the subsystem
* * 2 * 422 22 22 2 2
i
i i
αω ωω ω
= − +−∑Z K M
%
Statically condensed stiffness
Statically condensed mass
Clamped vibration modes
31
Concept of Mechanical Impedance
Necessary ingredients for appropriate representation of the dynamic behavior of a given subsystem
⇒ Static boundary modes resulting from static condensation
⇒ Subsystem eigenmodes in clamped boundary configuration
32
Two Ingredients
The Craig-Bampton method, also called component mode synthesis (CMS), relies on these two ingredients:
⇒ The static modes resulting from unit forces on the boundary DOFs
⇒ The internal vibration modes of the substructure fixed on its boundary
33
Two Ingredients
Static mode Internal vibration mode
But how do we calculate the reduction matrix R ???
34
Problem Statement
⎭⎬⎫
⎩⎨⎧
=⎭⎬⎫
⎩⎨⎧
⎥⎦
⎤⎢⎣
⎡−
⎭⎬⎫
⎩⎨⎧
⎥⎦
⎤⎢⎣
⎡0g
xx
MMMM
xx
KKKK B
I
B
IIIB
BIBB
I
B
IIIB
BIBB 2ω Boundary DOF
Subsystem DOF
Reaction forces with other parts of the system
2 0RR RC R RR RC R
CR CC C CR CC C
ω⎡ ⎤ ⎧ ⎫ ⎡ ⎤ ⎧ ⎫
− =⎨ ⎬ ⎨ ⎬⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎩ ⎭ ⎣ ⎦ ⎩ ⎭
K K x M M xK K x M M x
Retained DOF
Condensed DOF
Guyan
Craig-Bampton
35
Reduction Matrix
Guyan (static condensation)
Craig-Bampton (static + dynamic information)
1CC CR−
⎡ ⎤= ⎢ ⎥−⎣ ⎦
IR
K K
1
0???II IB
−
⎡ ⎤= ⎢ ⎥−⎣ ⎦
IR
K K
?
36
Reduction Matrix
Guyan (static condensation)
Craig-Bampton (static + dynamic information)
1CC CR−
⎡ ⎤= ⎢ ⎥−⎣ ⎦
IR
K K
1
0
II IB I−
⎡ ⎤= ⎢ ⎥−⎣ ⎦
IR
K K Φ
Internal vibration modes2
II I II Iω=K Φ M Φ
37
Reduction Matrix
Guyan (static condensation, dynamic information neglected)
Craig-Bampton (static + dynamic information, some internal modes neglected)
1CC CR−
⎡ ⎤= ⎢ ⎥−⎣ ⎦
IR
K K
1
0
II IB m−
⎡ ⎤= ⎢ ⎥−⎣ ⎦
IR
K K Φ
Im n<<
38
Reduced Stiffness and Mass Matrices
⎥⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡=
IMMM
MΩ00K
KmB
BmBB
m
BB and2
( ) TBmIBIIIIIB
TmmB
IBIIIIIIBIIBIIBIIBIIBIBBBB
IBIIBIBBBB
MKKMMΦM
KKMKKMKKKKMMM
KKKKK
=−=
+−−=
−=
−
−−−−
−
1
1111
1
with
39
Reduction Performed
Number of DOFs before reduction:
Size of the reduction matrix R:
Number of DOFs after reduction
I Bn n n= +
Internal Boundary
( )Bn n m× +
Bn m+
40
Superelement
The matrices may be interpreted as elementary matrices of a subsystem
The resulting “element” is called a superelement
The superelement of a subsystem is assembled with the rest of the structure like an ordinary finite element
Assumption of linear dynamics for the superelement
andK M
41
Internal Vibration Mode Selection
For simplicity, the choice of the retained modes is usually carried out by considering the first m modes, but this is not optimal !!!
Effective modal masses
⇒ Select modes that contribute significantly to the response of the structure excited at its interface
( )12
1
ni j
T Sj j
m mμ=
− ≈∑Γ e Effective mass of
mode j in direction i
Due to missing mass the equality does not hold
42
Cantilever Beam Example (400 DOFs)
Guyan
Craig-Bampton
xB
xI
1 2 3 199 200
- 1 node at the interface (2DOFs)- We keep the first m modes.
1 2 3 199 200
- Rotational DOFs are condensed- Some translational DOFs are condensed (depending on the reduction).
For example, if the reduction factor is 40, we keep 10 DOFs in the reduced model: DOFs 1, 21, 41, 61, 81, 101, 121, 141, 161, 181
430 5 10 15
0
2000
4000
6000
8000
10000
12000
14000
16000
Mode number
Freq
uenc
y (H
z)
EXACTCRAIG-BAMPTONGUYAN
Cantilever Beam Example (400 DOFs)
NbDOFs_GUYAN=NbDOFs_CB=20
CraigBamptonGuyan
44
Cantilever Beam Example (400 DOFs)
Time series can also be simulated using the superelement
A half-sine pulse is applied to the beam at DOF 1
1 2 3 199 200
0 0.05 0.10
0.5
1
Time
Forc
e
SuperElementTimeSeries
45
Industrial Example (Automotive)
Vehicle body, 1373000 DOFsNatsiavas et al., JCND2, 2007
Model combining the vehicle body, the suspensions, the wheels and 4 passengers; response to road irregularities
Passenger-seat model
46
Industrial Example (Automotive)
Excitation: left suspension / response: vehicle roof
Solid line: exactDashed line: CMS
FRF
Frequency (Hz)
FRF of the vehicle body
47
Industrial Example (Automotive)
Elapsed time (hh:mm:ss) [0-500Hz]
Elapsed time (hh:mm:ss) [0-900Hz]
03:28:27 23:45:31
CMS Full model
09:20:00 >60:00:00
FRF of the vehicle body
48
Industrial Example (Automotive)
Solid line: exactDashed line: CMS
Frequency (Hz)
Number of modes
Modes of the vehicle body
49
Industrial Example (Automotive)
Response of the full, but reduced, model to road irregularities
50
Industrial Example (Space)
Courtesy of EADS Space Transportation, 282000 DOFsAutomated Transfer Vehicle
51
Industrial Example (Space)
All modes up to 100 Hz retained (375 DOFs)
36 interface nodes (216 DOFs)
Total number of DOFs: 591
Reduction: 591/282000=0.21%
52
Industrial Example (Space)
Cumulative effective modal mass
Bas Fransen, 2005
53
Industrial Example (Space)
Example of a dominant lateral mode (high effective modal mass)
54
Model Reduction of Large Molecules
10-monomer amylose chain: 10 substructures
Harmonic excitation (1 Angstrom, 20 GHz)
Amylose structure
55
Model Reduction of Large Molecules
Dowell et al., 2006
Atom displacement
56
Not Discussed in This Course (Chap. 6)
Numerical solution of eigenvalue problems
⇒ Jacobi, Householder, subspace iteration, Lanczosmethods
Sensitivity analysis
⇒ Calculate the sensitivity of eigenfrequencies and eigenmodes to variations of physical parameters (e.g., elasticity modulus, density, thicknesses)
⇒ Useful for structural optimization and model updating