Substructuring Tutorial Imac2010

8/13/2019 Substructuring Tutorial Imac2010

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IMAC 2010

Dynamic Substructuring Concepts

Tutorial

Daniel RixenDelft University of Technology,

The Netherlands

CU, Boulder


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Substructuring is a way to see things in parts in order

- to simplify the dynamic analysis- to concentrate on specific components

Substructuring ideas are already more then 40 years old but still a research field.

esa


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General concepts in substructuring

1. Reduction of dynamic models2. Substructuring a random walk in history

3. Assembly of substructures

4. Reduction of substructure dynamics

5. Experimental substructuring

Disclaimer:this tutorial is not intending to give a complete overview of all publicationson the subject, but to introduce the main concepts in substructuring


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CU, Boulder

Models very refined

! to represent complex geometry! to compute static stressesBut much too refinedfor representing global dynamics!

1. Reduction of Dynamic Models


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2 options to have a lower order model:

New mesh

q=

fine coarseqR

Reducing the mathematical

space of dofs


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where

" Small system (say 200 dofs)" also called reduced or generalized dofs (q called the physical dofs)" Remember that there is always a related force errorinvolved (residue)" For symmetrical systems, one usually chooses" Once is computed, the physical dofs are found by substitution in


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Which reduction basis R?# sparse# must well approximate the solution


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Idea : cut in piecesand reduce every sub-part

$ Natural approach if model already decomposed$Allows isolating difficult regions (contact, NL )$ Suitable for parallel computing$ Easy re-use of components$Allows combining numerical/experimental submodels

Dynamic substructuring


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4. Numerical reduction



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2. Substructuring A random walk in history

The paradigm of Divide and Conquer

$Already promoted by Julius Caesar himself $ But the first mathematical application of it was by H. Schwarz, in 1890, who wanted toprove the existence and uniqueness of the solution to the Laplace equation in a complexdomain:


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2. Substructuring A random walk in history

$ Later, with the work of Courant (1943), the idea of subdivisionwas used to define Ritz approximation per elements.

Finite Elements were born

$ In the 60s the idea of partitioning a finite element model in substructures was proposed toreduce the complexity of the models

$ In the 80s, in order to speed-up the solution of linear algebraic system, like inKu=F,thecomputational domain was cut in sub-domains in order to share work amongst several CPUs.That was the start of Domain Decomposition.

$ End of the 60s, but especially in the 80s, decomposing a structure in experimentalmechanics was applied in order to simplify the testing


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Representationofsubstructure

(reducedornot)

AssemblyofSubstructure

s

Next

Reduction of Numerical models Experimental models

2. Substructuring General framework


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4. Numerical reduction


Disclaimer:this tutorial is not intending to give a complete overview of all publicationson the subject, but to introduce the main concepts in substructuring


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3. Assembly

Interface (internal) forcesInterface forces

- enforce the compatibility:

- are in equilibrium (action-reaction)


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3. Assembly

In block diagonal matrix form this can be written as

where

Boolean matrix

Signed Boolean matrix


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3. Assembly

Example

Signed Boolean matrix


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3. Assembly

Example


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3. Assembly

Example

Hence one also verifies that

Unique dofsfor total structure

Lis the NULL SPACE ofB

L is clearly the assembly matrix known from F.E.


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3. Assembly

Recall matrix notation (called 3-field formulation)

If one assumes that there is a unique setof dofs on the interface:

( )

where

Primal assembly (see F.E.)


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3. Assembly

Recall back the block 3-field formulation

DUAL assembly

If one assumes that there the interfaceforces are in equilibrium:


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3. General framework Coupling of physical matricesSummary

Interface compatibility

Interface equilibrium

Primal assembly DUAL assembly

3-field formulation

End-result obviously the sameDual assembly works also if non-matching meshes!

0

0

I

0

-I

0 I 0 -I


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More details? .. see then

A General Framework for Dynamic Substructuring-A history, review and classification of DS techniques D. de Klerk, D.J. Rixen and S.N. Voormeeren


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Time for a 10 seconds break

you are doing great, just breath deeply


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4. Reduction of substructure dynamics The Craig- Bampton CMS Variants of the Craig-Bampton method Other ingredients for the CMS: free interface modes The issue of interface reduction



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Super-elements

=

u(1)

u(2)

(1)

(2)

R(1)

R(2)

R(3)

ub ub

(s)

!(s)

(1)

(2)

u(1)b

u(2)b

Can be assembled like normal FE:Super (or macro) elements


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An example of super-element: the Craig-Bampton CMS

Roy Craig is one of the fathers of dynamic substructuring.See also his book and review papers.

This Component Modes Synthesis was published in 1968 and developed while Craig wasa young engineer at Boeing.

The Craig-Bampton method is one of the most used techniques because it yields nicereduced matrices for the super-element.

Many of the CMS methods are variants of the Craig-Bampton method


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Craig-Bampton component mode synthesis (CMS)

q(s)i

Boundary displacements q(s)b

Internal displacements

M(s)ii q(s)i +K(s)ii q(s)i = !K(s)ib q(s)b !M(s)ib q(s)b

In each substructure (s):

computed by modal superposition of fixed interface modes:q(s)i

q(s)i = +"

(s)!(s)! K(s)ibK

(s)!1ii q

(s)b

static solution

[Craig-Bampton]


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q

!!!"###$!!!"

q =%

!!!"

b

q

(1)i...

q(Ns)i

&

###$"

% I 0 0

#(1)

"(1)

0... ...

#(Ns) 0 "(Ns)

&% qb

!(1)...

!(Ns)

&

###$

RCB

Vibration modes (fixed interface)

( k


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KCB = RT CBKRCB =

%

!

!

!!"

Sbb 0%(1)

2

...0 %(Ns)

2

&

#

#

##$

MCB = RT CBMRCB =

%!!!

!

"

M#bb M(1)b" M

(Ns)b"

M(1)"b I 0... ...

M

(Ns)"b

0

I

###

$

SbbM#bb : assembly of statically condensed matrices(Schur complement, Guyan)

" K reduces to a quasi-diagonal matrix" Mass coupling between boundary and internal d.o.f." Good approximation with a small number of modes" Building the reduced system is not very expensive

Craig-Bampton component mode synthesis (CMS)


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Truss frame with beams

690 dof / substructure



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Craig-Bampton

Modes error

4 modes / sub-structure

1 MAC(full,reduced)

Mode number

Craig-Bampton


Craig-Bampton


Craig-Bampton



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How to choose the substructure modes to be included in the reduction ?

" Rule of thumb: include substructure modes having a frequency less than1.8 Xglobal eigenfrequency to ebe computed

" Evaluate how much the internal modes participate to the representation of thesubstructure mass as seen from the interface (effective modal mass)

" Evaluate how much reaction forces an internal mode generates on the interface"A posteriori,it is always possible to check the residual force due to the approximation


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Variants on the Craig-Bampton CMS

q

!!!"###$!!!"

q =%

!!!"

b

q

(1)i...

q(Ns)i

&

###$"

% I 0 0

#

(1)"

(1)0

... ...

#(Ns) 0 "(Ns)

&% qb

!(1)...

!(Ns)

&

###$

TCBAdd high-frequency quasi-static corrections: MTA (Mode Truncation Augmentation)

[Dickens,Rixen]


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Variants on the Craig-Bampton CMS

q

!!!"###$!!!"

q =%

!!!"

b

q

(1)i...

q(Ns)i

&

###$"

% I 0 0

#

(1)"

(1)0

... ...

#(Ns) 0 "(Ns)

&% qb

!(1)...

!(Ns)

&

###$

q

!!!"###$

q =%

!!!"

b

q

(1)i...

q(Ns)i

&

###$"

% I

#

(1)

...

#(Ns)

& qb

Start with a pure Guyan reduction

Compute approximate dynamics of full system

Substitute in original full problem

Compute residual forces per substructure

Compute the static response of

the susbtructures to that force error

!!!"q "

% I#(1)

...

#(Ns)

###$!!!"

0 0X(1) 0

...0 X(Ns)

&% qb!(1)...

!(Ns)

##$

X(s)

Enrich the reduction basis


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Other ingredients for CMS: free interface modes

Primal

=

0 &

0

I

0

-I

0 I 0 -I0

Dual


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4. Other ingredients for CMS: free interface modes

u(s)= +'

(s)!(s)

!K(s)+

B(s)T

&+R(s)(

(s)

Generalized inverse (Factorization + fictitious links)

Interface flexibility modes

'K

(s)!!

(s)2r M

(s)(#(s)r =0

Free interface vibration modes


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u(s)= +'

(s)!(s)

!K(s)+

B(s)T

&+R(s)(

(s)

such that

'(s)TK(s)G(s)res = 0'(s)TM(s)G(s)res = 0

R(s)TM(s)G(s)res = 0

K(s)+

!

q.r=1

)

(s)T

r )

(s)r

!(s)2r

Residual flexibility modes

(attachment modes)

u(s)= B(s)

T & +R(s)((s) +'(s)!

(s)G(s)res!


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u(s)=T(s) =/

G(s)resB(s)

T R(s) '(s)

0%"

&

((s)

!(s)&$

%"

&

((s)

!(s)&$

Mac-Neal, Rubin, Craig-Chang :

eliminate l in term of displacements dof per substructure

Obtain a super-element

Dual Craig-Bampton :

Use dual assembly and reduce the problem


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The issue of interface reduction

If number of substructures increase,

number of increases !

Limitation:

ub

Interface reduction :

Define interface modes :

ub" "b!b

MCB =1"

Tb I

2MCB

"b

I

-

KCB =

1"

Tb I

2 KCB

"b

I-


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4. Reduction of substructure dynamics

5. Experimental substructuringA guitarA car


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Frequency Based Substructuring

Measured FRFs


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Guitar test case

Dynamic combination of guitar body and strings determines the sound quality


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Guitar test case: substructure model

Input on sound table / output on cord (easier to measure)

o

Body (B)

String (A)

c

ci

Assume string and nut clamped (negligible dynamic contribution) Only one string present In-plane vibrations are assumed negligibleAnalytical model for string / experimental model for body Impedance head and laser vibrometer (to minimize additional mass effects)


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Guitar test case: FRF guitar body

Measured by admittance head and laser

o

Body (B)

String (A)

c

ci


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Guitar test case: assembled

Good up to 250 Hz. Bad above. Why ?


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Guitar test caseGood up to 250 Hz. Bad above. Why ?

# Prestress of sound table ?#Vibro-acoustic coupling ?# Out-of-plane motion ?#Validity of substructure model ?

o

Body (B)

String (A)

c

ci

Body (B)

String (A)

c

ci

c

c

o

?


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Guitar test case

Body (B)

String (A)

c

ci

c

c

o

Important dynamics at nut above 250 Hz !

Redo analysis with 2 connection points


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Guitar test case

Very good agreement except around 230 Hz Loose bridge saddle !


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Guitar test case

After re-gluing the bridge saddle:

Even at high frequency !


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Modeling of SubstructuresGear noise propagation path Description

!"#$%#&'" )*+"$",-#' .!/01!"#$%#&'" )*+"$",-#'

234,-,5 .!/0%61

!"#$%#&'" 7#$$*"$ .!/81!"#$%#&'" 7#$$*"$

234,-,5 .!/8%61

!5"#$

93): ;3$$*,5=6*7$3>?3,"=9; 234,-,5=


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Example of results


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A lot of potentialbut remaining challenges:

Model reduction and substructuring

Choose the best substructure reduction automatically A priori error estimates (error propagation) Reducing parametric models (optimization, updating ) Include non-linearities Reducing damping matrices

Experimental substructuring Measuring rotational dofs Obtaining clean FRFs (using a model for instance) What if non-linearities Apply to impact problems Error propagation Substructure decoupling

Substructuring Tutorial Imac2010

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