MEG6007: Advanced Dynamics -Principles

and Computational Methods (Fall, 2017)

Lecture 13: Reduced-Order Modelingof Structural Dynamics Systems

This lecture covers:

• Variational formulation of partitioned equations of motionfor structures.

•Model reduction of large-order partitioned FEM equations

• The Craig-Bampton method as a component mode syn-thesis technique

13.1 Introduction

The previous chapter has devoted to the reduced-order model-ing of a single vibrating substructure. Once all of the substruc-tures in a total system are approximated by their correspondingreduced-order models, the next task is to assemble the reduced-order substructural models. Third, the assembled reduced-ordertotal system is either used for performance evaluation and/ordesign improvements.

In practice, the size of the assembled total structural model thatconsists of reduced-order substructural models is often consid-ered too large. For such a case, it is customary to carry outadditional reduction via total system modal analysis.

Figure 1 illustrates the sequence of model development in large-scale vibrating structural systems.

In the following, the variational formulation of partitioned equa-

1

Fig. 1. Sequence of Reduced-Order Modeling and Ap-plications

tions of motion will be discussed first. In particular, two treat-ment of interface constraints will be discussed: classical (or global)λ-method (which reads as Lagrange multiplier method), and lo-calized λ-method. The partitioned equations of motion employ-ing the two λ-methods are then derived. we will then focus onone of the most widely used component mode synthesis method,the Craig-Bampton method. Finally, a component mode synthe-sis technique based on the localized λ-method will be described.

13.2 Variational Formulation of Partitioned StructuralSystems

Consider a structure that consists of two substructures as shownin Fig. 21.2. When the structure is partitioned into two struc-tures, Ω(1) and Ω(2), interactions forces, λ(1) and λ(2) (or λ(12)),are developed along the interface boundaries of Γ(1) and Γ(1). Inaddition, the displacement for substructure 1 consists of the in-terior ones u

(1)I and along the partition boundary u

(1)Γ . Similarly,

for substructure 2 we have u(2)I and u

(2)Γ . These can be expressed

as

u(1) =

u(1)I

u(1)Γ

, u(2) =

u(2)I

u(2)Γ

(13.1)

Using these notations, the energy functionals for substructures

2

Fig. 2. Partitioning of a Structure into Two Substruc-tures

1 and 2 may be written as

Substructure 1:

δΠ(1) = (δu(1))TK(1) u(1) − (f (1) −M(1) u(1))

Substructure 2:

δΠ(2) = (δu(2))TK(2) u(2) − (f (2) −M(2) u(2))(13.2)

where M and K are mass and stiffness matrix, respectively, fora substructure, and the superscripts, (1, 2), denote substructure.

While the virtual energy is completely contained in the preced-ing energy expressions, the interface conditions between thesetwo substructures are needed for partitioning as well as assem-bly. The kinematic interface compatibility condition may be de-scribed in one of the two possible ways:

Classical (or Global) form:

u(1)Γ − u

(2)Γ = 0

Localized form: u

(1)Γ

u(2)Γ

−[

L(1)f

L(2)f

]uf = 0

(13.3)

3

which states that the interface displacement along substructure1, u

(1)Γ , must be equal to that of substructure 2, u

(2)Γ ; and, Lf is

the interface displacement displacement operator.

The constraint functional that incorporates the above constraintscan be expressed as

Classical (or Global) form:

πclassical = (λ(12))T (u(1)Γ − u

(2)Γ )

Localized form:

πlocalized =λ(1)

λ(2)

T

u(1)Γ

u(2)Γ

−[

L(1)f

L(2)f

]uf

(13.4)

Finally, the total energy functional is simply the sum of twosubstructural energy expressions, (13.2), plus one of the the con-straint functionals, (13.4):

Classical interface form:

δΠsystem = δΠ(1) + δΠ(2) + δπclassicalLocalized interface form:

δΠsystem = δΠ(1) + δΠ(2) + δπlocalized

(13.5)

We now derive the partitioned equations of motion for the twointerface treatment cases.

13.3 Partitioned Equations of Motion Employing Clas-sical λ-Method

The total energy of the system for this case which is the one givenby the first of (13.5), consists of the two substructural energyexpressions (13.2) plus the interface constraint functional, viz.,

4

the first expression in (13.4), as

δΠtotal = (δu(1))TK(1) u(1) − (f (1) −M(1) u(1))+ (δu(2))TK(2) u(2) − (f (2) −M(2) u(2))+ (δλ(12))T (u

(1)Γ − u

(2)Γ ) + (δu

(1)Γ − δu

(2)Γ )T λ(12)

= (δu(1))TK(1) u(1) − (f (1) −M(1) u(1)) + (B(1))T u(1)+ (δu(2))TK(2) u(2) − (f (2) −M(2) u(2))− (B(2))T u(2)+ (δλ(12))T (B(1) u(1) −B(2) u(2))

u(1)Γ = B(1) u(1), u

(2)Γ = B(2) u(2)

(13.6)

where B(k) is the Boolean matrix that extracts the interfacedegrees of freedom at the interface of substructure k .

The stationarity of the above variational equation, viz., δΠtotal =0, yields the following partitioned equations of motion:M(1) 0 0

0 M(2) 00 0 0

u(1)

u(2)

λ(12)

+

K(1) 0 (B(1))T

0 K(2) −(B(2))T

B(1) −B(2) 0

u(1)

u(2)

λ(12)

=

f (1)

f (2)

0

⇓ ⇓M 0

0 0

u

λ(12)

+

K BTcl

Bcl 0

u

λ(12)

=

f

0

(13.7)

In the above equation set, the first row is the equations of motionfor substructure 1, the second row for substructure 2, and thethird is the interface constraint equation. To illustrate the com-positions of the partitioned equation further, we express each ofthe three equations in terms of the interior degrees of freedom,uI , and the interface degrees of freedom, uΓ as follows:

For substructure 1:MII MIΓ

MΓI MΓΓ

(1)

uI

uΓ

(1)

+

KII KIΓ

KIΓ KΓΓ

(1)

uI

uΓ

(1)

=

f

(1)I

f(1)Γ − λ(12)

(13.8)

5

Fig. 3. Partitioning of a Structure into Two Substruc-turesFor substructure 2:MII MIΓ

MΓI MΓΓ

(2)

uI

uΓ

(2)

+

KII KIΓ

KIΓ KΓΓ

(2)

uI

uΓ

(1)

=

f

(2)I

f(2)Γ + λ(12)

(13.9)

If one is interested in constructing the equations of motion forthe entire system, then all one has to do is to assemble thecoefficient matrices that are associated with u

(1)Γ and u

(2)Γ into

the same rows and the columns. This corresponds to an explicitenforcement of the second of the above constraint condition, e.g.,uΓ = u

(1)Γ = u

(1)Γ . The assembled equations of motion therefore

becomesM

(1)II M

(1)IΓ 0

M (1)ΓI M

(1)ΓΓ +M

(2)ΓΓ M

(2)IΓ

0 M(2)ΓI M

(2)II

u(1)I

uΓ

u(2)I

+

K

(1)II K

(1)IΓ 0

K(1)ΓI K

(1)ΓΓ +K

(2)ΓΓ K

(2)IΓ

0 K(2)ΓI K

(2)II

u(1)I

uΓ

u(2)I

=

f(1)I

fΓ

f(2)I

(13.10)

This assembly process is precisely the assembly procedure of atypical finite element software system, except it is repeated sev-eral hundreds or thousands times. The modes and mode shapesof the total system can, in principle, be extracted from the eigen-problem associated with the above assembled equations of mo-tion.

6

13.4 Partitioned Equations of Motion Employing Lo-calized λ-Method

The total energy of the system for this case which is the one givenby the second of (13.5), consists of the two substructural energyexpressions (13.2) plus the interface constraint functional, viz.,the second expression in (13.4), as

δΠtotal = (δu(1))TK(1) u(1) − (f (1) −M(1) u(1))+ (δu(2))TK(2) u(2) − (f (2) −M(2) u(2))

+δλ(1)

δλ(2)

T

u(1)Γ

u(2)Γ

−[

L(1)f

L(2)f

]uf

+ δ

u(1)Γ

u(2)Γ

−[

L(1)f

L(2)f

]δufT

λ(1)

λ(2)

= (δu(1))TK(1) u(1) − (f (1) −M(1) u(1)) + (B(1))Tλ(1)+ (δu(2))TK(2) u(2) − (f (2) −M(2) u(2)) + (B(2))Tλ(2)

+δλ(1)

δλ(2)

T

B(1)u(1)

B(2)u(2)

−[

L(1)f

L(2)f

]uf

− δuTf

[L

(1)f

L(2)f

]T λ(1)

λ(2)

u

(1)Γ = B(1) u(1), u

(2)Γ = B(2) u(2)

(13.11)

7

The stationarity of the above variational equation, viz., δΠtotal =0, yields the following partitioned equations of motion:

M(1) 0 0 0 0

0 M(2) 0 0 00 0 0 0 00 0 0 0 00 0 0 0 0

u(1)

u(2)

λ(1)

λ(2)

uf

+

K(1) 0 (B(1))T 0 00 K(2) 0 (B(2))T 0

B(1) 0 0 0 −L(1)f

0 B(2) 0 0 −L(2)f

0 0 −(L(1)f )T −(L

(2)f )T 0

u(1)

u(2)

λ(1)

λ(2)

uf

=

f (1)

f (2)

000

⇓ ⇓

M 0 00 0 00 0 0

uλ`uf

+

K BT` 0

B` 0 −Lf

0 −LTf 0

uλ`uf

=

f00

(13.12)

13.5 Eigenvalue Problem Using Reduced-Order Parti-tioned Equations of Motion

Model reduction of a substructure has been presented in theprevious chapter. In this section we will use the reduction pro-cedure based on the constrained interface modes for assemblingthe substructures into a total system. To this end, let’s expressthe reduction form for substructure k as

u(k) = Ψ(k) q(k)

(13.13)

where u(k) is approximated as:

u(k) ≈

uI

uΓ

=

ΨI −K−1II KIΓ

0 IΓΓ

qI

uΓ

(13.14)

8

Substituting the above reduction formula into (13.7), one ob-tainsM

(1) 0 00 M(2) 00 0 0

q(1)

q(2)

λ(12)

+

K(1) 0 (B(1))T

0 K(2) −(B(2))T

B(1) −B(2) 0

q(1)

q(2)

λ(12)

=

p(1)

p(2)

0

M(1) = (Ψ(1))T M(1) Ψ(1), M(2) = (Ψ(2))T M(2) Ψ(1)

K(1) = (Ψ(1))T K(1) Ψ(1), K(2) = (Ψ(2))T K(2) Ψ(1)

B(1) = (Ψ(1))T B(1), B(2) = (Ψ(2))T B(2)

p(1) = (Ψ(1))T f (1), p(2) = (Ψ(2))T f (2)

(13.15)

Vibration analysis of the total system based on the above reduced-order model is carried using the following equation:

Ktotal Ψtotal = Mtotal Ψtotal Λtotal

Ktotal =

M(1) 0 0

0 M(2) 00 0 0

Mtotal =

K(1) 0 (B(1))T

0 K(2) −(B(2))T

B(1) −B(2) 0

(13.16)

It is noted thatΦtotal is not the eigenvectors of the assembledmodel. The correct eigenvectors (mode shapes) of the assembledmodel are obtained by

u(1)

u(2)

λ12

=

Ψ(1) 0 00 Ψ(2) 00 0 I

q(1)

q(2)

λ12

=

Ψ(1) 0 00 Ψ(2) 00 0 I

Ψtotal qtotal

⇓⇓

Ψtotal =

Ψ(1) 0 00 Ψ(2) 00 0 I

Ψtotal

(13.17)

which includes not only the modes that span the substructuresbut also the interface modes pertaining to the interface forceλ(12). However, the eigenvalues Λtotal represent the assembled

9

structural system. In other words, (Ψtotal,Λtotal) constitute themode shapes and modes of the assembled system even though wehave obtained them from the partitioned equations of motion.

13.6 Eigenvalue Problem Using Reduced-Order Assem-bled Equations of Motion

In the preceding section the reduced-order partitioned equationsof motion has been directly utilized for the formulation of eigen-value problem. While computationally equivalent, the resultingeigenvalue problem given by (13.16) involves non-definite ma-trices, thus requiring a special care. One way to circumvent thenon-definite matrices is to assemble the partitioned equations ofmotion into the assembled form akin to the equation given in(13.10). This can be accomplished in the following way.

First, we note that the constraint condition(13.3) implies thatthe interface Boolean matrices B(1) and B(2) yield the followingrelation:

B(1)u(1) =[0 I

(1)Γ

] u(1)I

u(1)Γ

B(2)u(2) =[0 I

(2)Γ

] u(2)I

u(2)Γ

(13.18)

This means that the assembled degrees of freedom, (q(1)I ,uΓ,q

(2)I ),

can be related to the partitioned degrees of freedom, (q(1)I ,uΓ,q

(2)I ,u

(2)Γ ),

according to

q

(1)I

uΓ

q(2)I

u(2)Γ

=

I

(1)I 0 0

0 0 I(1)Γ

0 I(2)I 0

0 0 I(2)Γ

q(1)I

q(2)I

uΓ

⇒ qpart = La qa

(13.19)

where the superscripts, (part, a), denote the partitioned andassembled degrees of freedom, and La is an assembly Booleanmatrix. Therefore, teh complete transformation relation can be

10

expressed as

q(1)I

uΓ

q(2)I

u(2)Γ

λ(12)

=

qpart

λ(12)

= Ttotal

qa

λ(12)

, Ttotal =

[La 00 I

(12)Γ

]

(13.20)

Substituting the above assembly transformation into (13.16) andafter some simplifications, one arrives at the following reduced-order assembled equations of motion:

M ¨q +K q = p

M = (La)T[M(1) 0

0 M(2)

]La

K = (La)T[K(1) 0

0 K(2)

]La

p = (La)T

p(1)

p(2)

q = (La)T

q(1)

q(2)

(13.21)

As one can see, the above reduced-order assembled equations ofmotion is difficult to follow through. We will examine a step-by-step derivation of the above equation below, which is known inthe literature as the Craig-Bampton component mode synthesis(CMS) or substructuring method method.

13.7 The Craig-Bampton Method

Equation(13.21) may be considered a generic component modesynthesis as it can accommodate several possible substructuralreduction methods. One of its specializations was proposed byR.R Craig M.C.C. Bampton in 1968. As the Craig-Bamptoncomponent mode synthesis technique is perhaps the most widelyused substructuring method, we present a step-by-step formula-tion of their method below.

11

13.7.1 Step 1: Approximate the substructural displacements

It approximates the displacement of each substructure by a setof fixed-interface normal modes plus a set of constraint modes.Specifically, for substructure 1, u(1) is approximated by

u(1) ≈

u

(1)I

u(1)Γ

=

φ(1)I ψ

(1)IΓ

0 I(1)ΓΓ

q(1)I

u(1)Γ

, ψ(1)IΓ = −(K

(1)II )−1 K

(1)IΓ

(13.22)

Similarly, the substructural displacement u(2) for substructureis approximated by

u(2) ≈

u

(2)I

u(2)Γ

=

φ(2)I ψ

(2)IΓ

0 I(2)ΓΓ

q(2)I

u(2)Γ

, ψ(2)IΓ = −(K

(2)II )−1 K

(2)IΓ

(13.23)

13.7.2 Step 2: Obtain the approximate substructural kineticand strain energy

The approximate substructural strain energy and kinetic energyare restated below:

U (1) = 12(u(1))T K(1) u(1)

≈

q

(1)I

u(1)Γ

T Ψ

(1)I Ψ

(1)IΓ

0 I(1)ΓΓ

T K

(1)II K

(1)IΓ

K(1)ΓI K

(1)ΓΓ

Ψ

(1)I Ψ

(1)IΓ

0 I(1)ΓΓ

q(1)I

u(1)Γ

T (1) = 1

2u(1) M(1) u(1)

≈

q

(1)I

u(1)Γ

T Ψ

(1)I Ψ

(1)IΓ

0 I(1)ΓΓ

T M

(1)II M

(1)IΓ

M(1)ΓI M

(1)ΓΓ

Ψ

(1)I Ψ

(1)IΓ

0 I(1)ΓΓ

q(1)I

u(1)Γ

(13.24)

12

The reduced-order mass for substructure 1, M(1), is thus givenby

M(1) =

Ψ(1)I

T0

Ψ(1)IΓ

TI

(1)ΓΓ

M

(1)II M

(1)IΓ

M(1)ΓI M

(1)ΓΓ

Ψ

(1)I Ψ

(1)IΓ

0 I(1)ΓΓ

=

M(1)II M(1)

IΓ

M(1)ΓI M(1)

ΓΓ

(13.25)

where

M(1)II = Ψ

(1)I

TM

(1)II Φ

(1)I = III (due to mass normalization)

M(1)IΓ = Ψ

(1)I

T(M

(1)IΓ + M

(1)II Ψ

(1)IΓ )

M(1)ΓΓ = Ψ

(1)IΓ

T(M

(1)II Ψ

(1)IΓ + M

(1)IΓ ) + M

(1)ΓI Ψ

(1)IΓ + M

(1)ΓΓ

(13.26)

The reduced-order stiffness for substructure 1, KA, is thus givenby

K(1) =

K(1)II K(1)

IΓ

K(1)ΓI K(1)

ΓΓ

(13.27)

where

K(1)II = Ψ

(1)I

TK

(1)II Ψ

(1)I = Λ

(1)II =

ω21

. . .

ω2I

(1)

K(1)IΓ = Ψ

(1)I

T(K

(1)IΓ + K

(1)II Ψ

(1)IΓ ) = Ψ

(1)I

T(K

(1)IΓ −K

(1)II (K

(1)II )−1 K

(1)IΓ ) = 0

K(1)ΓΓ = K

(1)ΓΓ + Ψ

(1)IΓ

T(K

(1)II Ψ

(1)IΓ + K

(1)IΓ ) + K

(1)ΓI Ψ

(1)IΓ

= K(1)ΓΓ −K

(1)ΓI (K

(1)II )−1 K

(1)IΓ

(13.28)

Observe that K(1) can be written as

K(1) =

Λ(1)II 0

0 K(1)ΓΓ

(13.29)

which consists of the diagonal interior substructural modes andthe Guyan-reduced matrix KΓΓ.

13

For substructure 2, a similar procedure employed for substruc-ture A can be repeated to yield:

M(2) =

M(2)II M(2)

IΓ

M(2)ΓI M(2)

ΓΓ

K(2) =

Λ(2)II 0

0 K(2)ΓΓ

(13.30)

where it is understood that the substructural displacement u(2)

is approximated by the fixed-interface interior modes plus theconstrained modes.

13.7.3 Step 3: Sum up the substructural kinetic and strainenergy expressions

The strain energy and the kinetic energy of the total structurecan be approximated by

T = T (1) + T (2)

U = U (1) + U (2)

U (1) ≈ 12

q

(1)I

u(1)Γ

T Λ

(1)II 0

0 K(1)ΓΓ

q(1)I

u(1)Γ

T (1) ≈ 1

2

q

(1)I

u(1)Γ

T M

(1)II M(1)

IΓ

M(1)ΓI M(1)

ΓΓ

q(1)I

u(1)Γ

(13.31)

U (2) ≈ 12

q

(2)I

u(2)Γ

T Λ

(2)II 0

0 K(2)ΓΓ

q(2)I

u(2)Γ

T (2) ≈ 1

2

q

(2)I

u(2)Γ

T M

(2)II M(2)

IΓ

M(2)ΓI M(2)

ΓΓ

q(1)I

u(2)Γ

(13.32)

14

13.7.4 Step 4: Derive the reduced equations of motion for thetotal system

The Lagrangian of the total system is given by

L = T − U + (λ(12))T (u(1)Γ − u

(2)Γ ) (13.33)

where the last term involving the Lagrange multiplier λ(12) isintroduced to enforce the interface displacement compatibilityconstraint

u(1)Γ − u

(2)Γ = 0 (13.34)

The equations of motion for free vibration (f (1) = 0, f (2) = 0)can be derived from (13.33) as

[M(1) 00 M(2)

] q(1)

qB

+[K(1) 0

0 K(2)

] q(1)

q(2)

= CT λ(12)

q(1) =

q

(1)I

u(1)Γ

, q(2) =

q

(2)I

u(2)Γ

CT = 〈0 I 0 −I 〉

(13.35)

If desired, the interface force λ(12) can be eliminated by express-ing the interface displacement u

(1)Γ in terms of u

(2)Γ or vice versa.

This can be accomplished by the reduction

q

(1)I

u(1)Γ

q(2)I

u(2)Γ

= La

q

(1)I

q(2)I

uΓ

, La =

I 0 00 0 I0 I 00 0 I

, uΓ = u(1)Γ = u

(2)Γ

(13.36)

Substituting (13.36) into (13.35) and premultiplying the result-ing equation by (La)T we obtain the following reduced-order free

15

vibration equation:

M q +K q = 0 , q =

q

(1)I

q(2)I

uΓ

M =

M(1)

II 0 M(1)IΓ

0 M(2)II M(2)

IΓ

M(1)ΓI M(2)

ΓI MΓΓ

, MΓΓ =M(1)ΓΓ +M(2)

ΓΓ

K =

Λ

(1)II 0 0

0 Λ(2)II 0

0 0 KΓΓ

, KΓΓ = K(1)ΓΓ +K(2)

ΓΓ

(13.37)

13.7.5 Step 5: Perform eigenanalysis of the total system

First, we perform an eigenanalysis of (13.37):

K Φ =M Φ Λg, Φ =

Φ(1)I

Φ(2)I

ΦΓ

(13.38)

Second, once (Φ, Λg) are obtained, the global eigenvector Φg isobtained by the following expression

Φg =

φ

(1)I 0 Ψ

(1)IΓ

0 Ψ(2)I Ψ

(2)IΓ

0 0 IΓ

Φ(1)I

Φ(2)I

ΦΓ

(13.39)

Observe that the eigenvalues are preserved under a similaritytransformation. Thus, the global eigenvalues and eigenvectorpairs are given by (Φg, λg ).

The component mode synthesis for other techniques due to Ben-field and Hruda, Hurty, MacNeal, Rubin, Hintz, Dowell and

16

Klein, and Craig and Chang may be similarly constructed.

Remark 1: Note that from (13.37) equation, in carrying out thecomponent mode synthesis by the Craig-Bampton method, viz.,

ω2M Φ = K Φ (13.40)

the mass matrixM becomes dense even if the original substructural-level mass matrices are diagonal.

Remakr 2: The stiffness matrix at the interface is given by

KΓΓ = K(1)ΓΓ +K(2)

ΓΓ

K(1)ΓΓ = K

(1)ΓΓ −K

(1)ΓI (K

(1)II )−1 K

(1)IΓ

K(2)ΓΓ = K

(2)ΓΓ −K

(2)ΓI (K

(2)II )−1 K

(2)IΓ

(13.41)

Note that both K(1)ΓΓ and K(2)

ΓΓ are the Schur complements (orin structural mechanics known as Guyan-reduced matrices) thatpreserve the strain energy content of each substructure. Hence,no approximation is introduced at the interface strain energycontents.

On the other hand, the same cannot be said regarding the kineticenergy. This can be seen by examing the interface mass traixMΓΓ:

MΓΓ =M(1)ΓΓ +M(2)

ΓΓ

M(1)ΓΓ = Ψ

(1)IΓ

T(M

(1)II Ψ

(1)IΓ + M

(1)IΓ ) + M

(1)ΓI Ψ

(1)IΓ + M

(1)ΓΓ

M(2)ΓΓ = Ψ

(2)IΓ

T(M

(2)II Ψ

(2)IΓ + M

(2)IΓ ) + M

(2)ΓI Ψ

(2)IΓ + M

(2)ΓΓ

(13.42)

In other words, the constraint modes ΨIΓ play the role ofaugmenting the interface kinetic energy by infusing the interiormasses MII unto the interface nodes.

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