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

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

k 2 c 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

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):

[ ][ ]

Component β

Component α

[ ] [ ] αααω pU2

[ ] [ ] βββω pU2

Uncoupled Systems

Component β

Component α

[ ][ ]

[ ] [ ] 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∆

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

∆∆

[ ] [ ][ ]

[ ] [ ] 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

[ ][ ]

[ ] [ ][ ] [ ]

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

x[ ] [ ] [ ]

[ ] [ ]

[ ] [ ] [ ][ ] [ ]

[ ][ ]

The constraint equation is given by

and the uncoupled equations aregiven as

and the relationship of constraint is

Component β

[ ] [ ][ ] [ ]

ββββ 000

β= pUx cc

[ ][ ]

I00IU0

Substituting and putting into normal form gives

Component β

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

TMT i*Ti*T =

ββ&&

[ ][ ] [ ][ ] [ ]

[ ][ ]

[ ][ ] [ ][ ] [ ]

[ ][ ] 0

ββ &&

Modal to Modal with Physical System Added

Component α and β(modal components):

Component γ(physical component)only connection DOF:

[ ][ ]

2 [ ][ ]

Component α

Component β

Physical [M],[K]

[ ] [ ] [ ][ ] [ ]

K[ ] [ ] [ ][ ] [ ]

Uncoupled system

Constraint relation is

Component α

Component β

Physical [M],[K]

[ ][ ]

[ ]ptpp

For all DOF Component α

Component β

Physical [M],[K]

[ ][ ]

I00IU00U

[ ] [ ][ ]

[ ][ ] *

I00Itt

αβα

This leads to

Component α

Component β

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

TMT c*Tc*T =

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

βββαβ

βαααα

βββαβ

βαααα

tMtMtMttMttMtMTMTM

tKtKtKttKttKtKTKTK

TT*T**

Physical System with Modal Components Added

Component α and β(modal components):

Component γ(physical component)only connection DOF:

[ ][ ]

2 [ ][ ]

Component α

Component β

Physical [M],[K]

[ ] [ ] [ ][ ] [ ]

K[ ] [ ] [ ][ ] [ ]

Uncoupled system

Constraint relation is

Component α

Component β

Physical [M],[K]

[ ] [ ][ ] [ ]

[ ][ ]

[ ] [ ][ ] [ ]

[ ][ ]

[ ] [ ][ ]

αβα

ppUUx ccc

For all DOF Component α

Component β

Physical [M],[K]

[ ] [ ][ ]

[ ][ ]

This then becomes

Component α

Component β

Physical [M],[K]

Physical System Modal Components Added

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

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

βββαββ

βααααα

MUMUUMUMUUMUMUMUMU

UMUMMTMTM

ciccicii*T**

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

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

βββαββ

βααααα

KUKUUKUKUUKUKUKUKU

UKUKKTKTK

ciccicii*T**

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

TKTppx

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

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

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 ≡

ic1ii KK−−

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

where Ψ is the matrix of component modes.

Using the Craig Bampton method, the mass is given by

µµµµ

=µmmmc

( )( ) ccicciicicii

ijiciiTim

mCmmCmC

+++=µ

+=µ=µ

Using the Craig Bampton method, the stiffness is given by

Note that κcc is the Guyan reduced stiffness matrix

κκκκ

=κmmmc

ic1iicicccc

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

[ ]0I0ID −=

[ ]0I0DID

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

In this case

I0000I0I000I

To get synthesized system M and K matrices:

µ=αβ

µµµµ

I0000I0I000I

I0I00I00000I

cmcmcc

M0M0MMMMM

( )( )

βββ

ααα

µ+µ=

cccccc

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

= full matrix of stiffness terms for reduced α and β

ααβ

K000K000K

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 &&

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

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

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

bcbacac fh+fhx =

ab1-aacacb

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

uuq)pj(

uuq)j(H *

jkikkij

−ω+

−ω=ω ∑

ab1aacacb

ccb HHHH

FxH~ −−==

aaababa FHFHx +=

bab1aaa FHHF −−=

bcbacac FHFHx +=

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

[ ] [ ][ ] [ ]

COMPONENT (A)

SYSTEM (S)

COMPONENT (B)

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

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)

cA XXX ==

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)

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

HHHHHHHHH

(4-12)

(4-13)

(4-14)

[ ] [ ] cAcc

A FHFHX +=

[ ] [ ] [ ] bBcb

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

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

aS FHFHFHX ++=

[ ] [ ] cAac

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)

aSX [ ] [ ] cA

aS F~HFHX +=

[ ] [ ] [ ] [ ] [ ] cA

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

S HHHHHH−

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

ijSij HHHHhh

−+−=

COMPONENT A

COMPONENT B

CONNECTION POINTS

FRFsdescribingconnection

points

FRFsdescribinginput force

points

FRFsdescribing

output responsepoints

COMPONENTEVALUATION

Additional relations are dervied as

(4-23)

(4-24)

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

S HHHHH−

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

S HHHHH−

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

S HHHHH−

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

S HHHHH−

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

S HHHHHH−

(4-21)

(4-22)

(4-25)

COMPONENTEVALUATION

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