system modeling concepts 071206 -...

1 Dr. Peter AvitabileModal Analysis & Controls LaboratorySystem Modeling Concepts

System Modeling Concepts

Peter AvitabileModal Analysis & Controls LaboratoryMechanical 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

m

k

k

c

c

1

1

2

2

1

2

4 . 0 0 . 1 1 . 0

L i n e a r F r e q u e n c y ( H z ) 3 . 0

2 . 0 5 . 0

1.0 E+1

1.0 E+2

m

k 2 c 2

2

4 . 0 0 . 1 1 . 0

L i n e a r F r e q u e n c y ( H z ) 3 . 0

2 . 0 5 . 0

1.0 E+3

1.0 E+4

m

k

1

1 c 1



System models are generated from component models for a variety of applications

Modal MethodsComponent Mode SynthesisFrequency Based Substructuring


Modal to Modal with Tie Matrix

For component α(modal component):

For component β(modal component):

[ ][ ]

α

α

αω

p

U

2

[ ][ ]

β

β

βω

p

U

2

Component β

Component α

[ ] [ ] αααω pU2

[ ] [ ] βββω pU2



Uncoupled Systems

Component β

Component α

[ ][ ]

[ ][ ]

=

+

β

α

β

α

β

α

β

α

00

pp

0

0pp

I00I

2

2

Ω

Ω&&

&&

[ ] [ ] αααω pU2

[ ] [ ] βββω pU2


The two systems are connected with tie matrices

The tie matrix can be projected from physical to modal space using

withComponent β

Component α

[ ]TM∆ [ ]TK∆

[ ]TM∆ [ ]TK∆

[ ] [ ] [ ][ ][ ] [ ] [ ][ ]UKUK

UMUM

TT

T

TT

T

∆∆

∆∆

=

=

[ ] [ ][ ]

= β

α

U00UU


[ ] [ ] αααω pU2

[ ] [ ] βββω pU2


Adding this to the uncoupled equations gives

The eigensolution of this gives the new system frequencies and mode shapes

Component β

Component α

[ ][ ]

[ ] [ ][ ] [ ]

=

+

+

+

β

α

β

α

β

α

β

α

00

ppK

0

0ppM

I00I

T2

2

T ∆Ω

Ω∆&&

&&



Component α (physical) can be partitioned into connection DOF (c)and other or interior DOF (i)

Component β (modal) only considers connection DOF

Component β

Component αModal to Physical with Constraint

=i

c

xx

x[ ] [ ] [ ]

[ ] [ ]

[ ] [ ] [ ][ ] [ ]

=

=

α

α

iiic

cicc

iiic

cicc

KKKK

K

MMMM

M

[ ][ ]

β

β

βω

p

U

2


The constraint equation is given by

and the uncoupled equations aregiven as

and the relationship of constraint is

Component β


[ ] [ ][ ] [ ]

[ ]

[ ] [ ][ ] [ ]

[ ]

=

+

ββββ 000

pxx

KKKKK

pxx

MMMMM

i

c

iiic

cicc

i

c

iiic

cicc

&&

&&

&&

β= pUx cc

[ ][ ]

[ ][ ]

=

=

β px

Tpx

I00IU0

pxx

iic

i

c


Substituting and putting into normal form gives

or

Component β


[ ] [ ] [ ] [ ] [ ] [ ] 0px

TKTpx

TMT i*Ti*T =

+

ββ&&

&&

[ ][ ] [ ][ ] [ ]

[ ][ ]

[ ][ ] [ ][ ] [ ]

[ ][ ] 0

px

TK

KKKK

T

px

TM

MMMM

T

iiiic

ciccT

iiiic

ciccT

=

+

ββ

ββ &&

&&


Modal to Modal with Physical System Added

Component α and β(modal components):

Component γ(physical component)only connection DOF:

[ ][ ]

α

α

αω

p

U

2 [ ][ ]

β

β

βω

p

U

2

Component α

Component β

Physical [M],[K]

=i

c

xx

x

[ ] [ ] [ ][ ] [ ]

=

iiic

cicc

KKKK

K[ ] [ ] [ ][ ] [ ]

=

iiic

cicc

MMMM

M



Uncoupled system

Constraint relation is

Component α

Component β

Physical [M],[K]

[ ][ ]

[ ][ ]

[ ][ ]

0ppx

KK

Kppx

MM

M

c

c

=

+

β

α

β

α

β

α

β

α

&&

&&

&&

[ ]ptpp

UU

xx

c

c

c

c =

=

β

α

β

α

β

α



For all DOF Component α

Component β

Physical [M],[K]

[ ][ ]

[ ][ ]

[ ]

==

=

β

αβ

αβ

α

β

α

β

α

pppT

pp

I00IU00U

ppxx

c

c

c

c

[ ] [ ][ ]

[ ][ ] *

c

pTpp

I00Itt

ppx

z =

=

= β

αβα

β

α



This leads to

where

Component α

Component β

Physical [M],[K][ ] [ ] [ ] [ ] [ ] [ ] 0px

TKTpx

TMT c*Tc*T =

+

&&

&&

[ ] [ ] [ ] [ ] [ ] [ ] [ ][ ] [ ] [ ][ ][ ] [ ][ ] [ ] [ ] [ ][ ]

[ ] [ ] [ ] [ ] [ ] [ ] [ ][ ] [ ] [ ][ ][ ] [ ][ ] [ ] [ ] [ ][ ]

++==

++==

βββαβ

βαααα

βββαβ

βαααα

tMtMtMttMttMtMTMTM

tKtKtKttKttKtKTKTK

TT

TT*T**

TT

TT*T**


Physical System with Modal Components Added

Component α and β(modal components):

Component γ(physical component)only connection DOF:

[ ][ ]

α

α

αω

p

U

2 [ ][ ]

β

β

βω

p

U

2

Component α

Component β

Physical [M],[K]

=i

c

xx

x

[ ] [ ] [ ][ ] [ ]

=

iiic

cicc

KKKK

K[ ] [ ] [ ][ ] [ ]

=

iiic

cicc

MMMM

M


Uncoupled system

Constraint relation is

Component α

Component β

Physical [M],[K]


[ ] [ ][ ] [ ]

[ ][ ]

[ ] [ ][ ] [ ]

[ ][ ]

+

β

α

β

α

β

α

β

α

ppxx

KK

KKKK

ppxx

MM

MMMM

i

c

iiic

cicc

i

c

iiic

cicc

&&

&&

&&

&&

[ ] [ ][ ]

= β

αβα

ppUUx ccc


For all DOF Component α

Component β

Physical [M],[K]


[ ] [ ][ ]

[ ][ ]

=

β

α

βα

β

α

ppx

II

IUU0

ppxx

icc

i

c


This then becomes

where

Component α

Component β

Physical [M],[K]

Physical System Modal Components Added

[ ] [ ] [ ] [ ][ ] [ ][ ] [ ][ ]

[ ] [ ] [ ] [ ][ ] [ ] [ ] [ ][ ][ ] [ ] [ ] [ ][ ] [ ] [ ][ ] [ ]

++==

βββαββ

βααααα

βα

MUMUUMUMUUMUMUMUMU

UMUMMTMTM

cccT

ccccT

cciT

c

cccT

ccccT

cciT

c

ciccicii*T**

[ ] [ ] [ ] [ ][ ] [ ][ ] [ ][ ]

[ ] [ ] [ ] [ ][ ] [ ] [ ] [ ][ ][ ] [ ] [ ] [ ][ ] [ ] [ ][ ] [ ]

++==

βββαββ

βααααα

βα

KUKUUKUKUUKUKUKUKU

UKUKKTKTK

cccT

ccccT

cciT

c

cccT

ccccT

cciT

c

ciccicii*T**

[ ] [ ] [ ] [ ] [ ] [ ] 0ppx

TKTppx

TMTi

*Ti

*T =

+

β

α

β

α

&&

&&

&&


Component Mode Synthesis

Two components: α and β

For each component, the equation of motion can be written in partitioned form, with

xc = juncture coordinatesxi = interior coordinates

=

+

i

c

i

c

iiic

cicc

i

c

iiic

cicc

ff

xx

KKKK

xx

MMMM

&&

&&



We can represent the physical coordinates x in terms of component generalized coordinates p:

where Ψ = a matrix of component modes.For the Craig-Bampton method, constraint modes and fixed-interface normal modes are used as component modes.

px Ψ=



Fixed-Interface normal modes Um (where “m” refers to kept modes)All juncture coordinates are constrainedObtain the normal modes by solving the eigenvalueproblem:

It is assumed that the modes are scaled to unit modal mass.

( ) 0xmk 2 =ω−



Constraint modesPartition physical coordinates x into two sets: c, the constrained coordinates—the coordinates relative to which the constraint modes will be defined; and i, the remaining (interior) coordinatesStatically impose a unit displacement on one constrained coordinate, and maintain a zero displacement on the other constrained coordinates. The remaining i coordinates are free to deform.



Constraint modesThe constraint mode matrix is therefore

Where Tc = constraint mode transformation

−

=

= −

ic1ii

cc

ic

cccKK

ICI

T



Applying the Craig Bampton method:Let for each component, whereUm = kept fixed-interface normal modes, andC = constraint modes.

cmm CppUx +=



Since the normal modes are fixed-interface modes, and , this can be written in partitioned form aswhereIcc = identity associated with unit displacement at connection DOF (for constraint modes)Cic = Resulting constraint modes, equal to Uim = Normal modes of system with c constraints applied (fixed-interface normal modes)

Note: n – c = i n = all points (as usual)c = constrained points i = interior points

cc xp ≡

=

m

c

imic

cc

i

c

pp

UC0I

xx

ic1ii KK−−



In general, for each component, the mass and stiffness matrices are

where Ψ is the matrix of component modes.

ΨΨ

ΨΨ

K

MT

T

=κ

=µ



Using the Craig Bampton method, the mass is given by

where

µµµµ

=µmmmc

cmcc

( )( ) ccicciicicii

Ticcc

ijiciiTim

Tcmmc

mmmm

mCmmCmC

mCMU

I

+++=µ

+=µ=µ

=µ



Using the Craig Bampton method, the stiffness is given by

Where

Note that κcc is the Guyan reduced stiffness matrix

κκκκ

=κmmmc

cmcc

ic1iicicccc

cmTmc

2mmmm

KKKK

0−−=κ

=κ=κ

==κ ΩΛ



To assemble system matrices, first write the equation of interface displacement compatibility (with pc

α being the dependent coordinate):

Written in the form therefore

this is

D = dependent

I = linearly independent

0pp cc =− αβ

0Dp =

[ ] 0

pppp

0I0I

m

c

m

c

=

−

β

β

α

α

[ ]0I0ID −=

[ ]0I0DID

DI

DD

=−=



Then define S so , wherep = original generalized coordinates, andq = new generalized coordinates.

In this case

Sqp =

=

β

α

β

β

β

α

α

m

m

c

m

c

m

c

ppp

I0000I0I000I

pppp



To get synthesized system M and K matrices:

SSK

SSMT

T

κ=

µ=αβ

αβ

µµµµ

µµµµ

=

ββ

ββ

αα

αα

αβ

I0000I0I000I

0000

0000

I0I00I00000I

M

mmmc

cmcc

mmmc

cmcc



Then

where

=ββ

αα

βα

αβ

mmmc

mmmc

cmcmcc

M0M0MMMMM

M

( )( )

βα

βββ

ααα

ββ

αα

µ+µ=

µ==

µ==

=

=

cccccc

mcT

cmmc

mcT

cmmc

mmmm

mmmm

M

MM

MM

IM

IM



And

Where= diagonal matrix, modal stiffness of α= diagonal matrix, modal stiffness of β

= full matrix of stiffness terms for reduced α and β

=

β

ααβ

kk

mm

cc

K000K000K

K

αα = mmmmK Λββ = mmmmK Λ

βα κ+κ= ccccccK



Then solve the equation of motion for the assembled system:

Transform from q to p coordinates using

and then from p to u (physical coordinates) using

0qKqM =+ αβαβ &&

Sqp =

=

m

c

imic

cc

i

c

pp

UC0I

xx


Impedance Modeling Techniques

Consider a cantilever beam. It is desired to estimate the FRF between point c and b when the tip of the beam is pinned to ground.In particular, the FRF hcb when xa = 0

abc



The response at "a" is related to the force at "a" and "b" through

where xa is the vertical translation at the tip of the beam. With the constraint xa = 0, the force at point "a" becomes

abc

aaababa fhfhx +=

bab1-aaa fhhf −=



The response at "c" due to an excitation at “a” and "b" is

In order to include the effects of the constraint at “a” , the force at point "a" with the constraint xa = 0, changes this equation to

which are obtained from the unconstrained system

abc

bcbacac fh+fhx =

ab1-aacacb

b

ccb~

hhh-h=fxh =


Summary of Impedance Modeling

Frequency Response Functions can also be used to investigate structural modifications. The FRF can be written as

Using force balance and compatibility equations, the effects of a modification can be written in terms of the unmodified system as

)pj(

uuq)pj(

uuq)j(H *

k

*jk

*ikk

m

1k k

jkikkij

−ω+

−ω=ω ∑

=

ab1aacacb

b

ccb HHHH

FxH~ −−==

aaababa FHFHx +=

bab1aaa FHHF −−=

bcbacac FHFHx +=

c b a


Frequency Based System Modeling Techniques

Consider combining two systems together

COMPONENT (A)

a-DOFs b-DOFsc-DOFs c-DOFs

SYSTEM (S)

COMPONENT (B)


Frequency Based System Modeling Techniques

The equation of motionfor each component is

where n = a + c for component (A) n = b + c for component (B).

Note that the number of "c" coordinates are the same on component (A) and (B)

COMPONENT (A)


SYSTEM (S)

COMPONENT (B)

[ ] nnnn FHX =


FBS Modeling Techniques

Component A can be partitioned as

Component B can be partitioned as

[ ] [ ][ ] [ ]

ncAa

A

nnccA

caA

acA

aaA

ncAa

A

FF

HHHH

XX

=

[ ] [ ][ ] [ ]

nbBc

B

nnbbB

bcB

cbB

ccB

nbBc

B

FF

HHHH

XX

=

COMPONENT (A)


SYSTEM (S)

COMPONENT (B)

(4-8)(4-8)(4-8)

(4-8)

(4-9)



When rigidly connecting Component A to Component B,compatibility implies that

and equilibrium at the "c" DOFs requires that

where "S" superscript is used to represent system comprised of Component A rigidly coupled to Component B at the connection DOFs "c"

COMPONENT (A)


SYSTEM (S)

COMPONENT (B)

cScB

cA XXX ==

cScB

cA FFF =+

(4-10)

(4-11)



The FRFs of the uncoupled system can be defined as

From the partitioned equations for Component A and Component B, the connection DOF are

COMPONENT (A)


SYSTEM (S)

COMPONENT (B)

[ ] [ ] [ ][ ] [ ] [ ][ ] [ ] [ ]

nbSc

Sa

S

nnbbS

bcS

baS

cbS

ccS

caS

abS

acS

aaS

nbSc

Sa

S

FFF

HHHHHHHHH

XXX

=

(4-12)

(4-13)

(4-14)

[ ] [ ] cAcc

Aa

Aca

Ac

A FHFHX +=

[ ] [ ] [ ] bBcb

Bc

Bcc

Bc

B FHFHX +=



These two equations can be equated and used to solve for the connection force as

From these equations derived above, the coupled system FRFs can be determined in terms of the uncoupled FRFs of the individual components. Equations (4-8), (4-9), (4-12) and (4-15) are used in the development of the coupled system.

COMPONENT (A)


SYSTEM (S)

COMPONENT (B)

(4-15)

[ ] [ ][ ] [ ] [ ] [ ] [ ]cSccB

aA

caA

bB

cbB1

ccB

ccA

cA FHFHFHHHF~ +−+=

−



As an example,will be derived The first equation of (4-12) of the coupled system is

and the first equation of (4-8) of the uncoupled system is

COMPONENT (A)


SYSTEM (S)

COMPONENT (B)

(4-16)

(4-17)

[ ]aaSH

[ ] [ ] [ ] bSabS

cS

acS

aS

aaS

aS FHFHFHX ++=

[ ] [ ] cAac

Aa

Aaa

Aa

A FHFHX +=



When the systems are coupled, the force on "A" from "B" is given by from (4-15) and the corresponding response associated withthat coupling force is which then becomes

then (4-16) and (4-18) combine to give

COMPONENT (A)


SYSTEM (S)

COMPONENT (B)

(4-18)

(4-19)

cAF~

aSX [ ] [ ] cA

acA

aA

aaA

aS F~HFHX +=

[ ] [ ] [ ] [ ] [ ] cA

acA

aA

aaA

bS

abS

cS

acS

aS

aaS

F~HFH

FHFHFH

+=

++



The FRF, is developed realizing that are zero

Substituting (4-15) into (4-19) and simplifying, allows for the calculation of in terms of the uncoupled FRF matrices as

COMPONENT (A)


SYSTEM (S)

COMPONENT (B)

(4-20)

[ ]aaSH

cSF bSF bBF

[ ]aaSH

[ ] [ ] [ ] [ ] [ ][ ] [ ]caA1cc

Bcc

Aac

Aaa

Aaa

S HHHHHH−

+−=



Schematically this is shown as [ ] [ ][ ] cjA1

ccB

ccA

icAA

ijSij HHHHhh

−+−=

COMPONENT A

COMPONENT B

CONNECTION POINTS

ji

FRFsdescribingconnection

points

FRFsdescribinginput force

points

FRFsdescribing

output responsepoints

COMPONENTEVALUATION



Additional relations are dervied as

(4-23)

(4-24)

[ ] [ ] [ ] [ ][ ] [ ]ccB1cc

Bcc

Aac

Aac

S HHHHH−

+=

[ ] [ ] [ ] [ ][ ] [ ]cbA1cc

Bcc

Aac

Aab

S HHHHH−

+=

[ ] [ ] [ ] [ ][ ] [ ]ccB1cc

Bcc

Acc

Acc

S HHHHH−

+=

[ ] [ ] [ ] [ ][ ] [ ]cbB1cc

Bcc

Acc

Acb

S HHHHH−

+=

[ ] [ ] [ ] [ ] [ ][ ] [ ]cbB1cc

Bcc

Abc

Bbb

Bbb

S HHHHHH−

+−=

(4-21)

(4-22)

(4-25)

COMPONENTEVALUATION

system modeling concepts 071206 -...

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