Model Analysis Theory

Modal Analysis Theory R. Kashani, Ph.D. www.deicon.com Modal Analysis is a computationally elegant technique for modeling structural dynamics. It is based on the use of eigenvalue and Eigenvector information of a system. The elegance and appeal of this technique is mainly due to its decoupling capability. Moreover, it is the basis for understanding modal test methods (known also as experimental modal analysis). Considering the forced response of an underdamped system shown by Equation 1 M x¨ t
 Kx t


f t


(1)

where M and K are n  n positive definite matrices of mass and stiffness, x t
is the vector of displacement, and f t
is the vector of applied force. Equation 1 can be solved usinjg eigenvalue expansion. This is done by first solving the eigenvalue problem for the corresponding homogenous system Aψ  λψ

(2)

where A  M  1 K, λ is the diagonal matirx of eigenvalues, and ψ is the matrix of eigenvectors. The spatial coordiante x can be changed to a new coordinate η using η  ψ 1 x

(3)

where ψ is an invertible matrix. Premultiplying Equation 3 by ψ results in x  ψη

(4)

Substituting for x in Equation 1 from Equation 4 Mψη¨  Kψη  Premultiplying Equation 5 by ψ

f t


(5)

1

ψ  1 Mψη¨  ψ  1 Kψη  ψ 

1

f t


Using similarity transformation (described in the following section) ψ

1

 ψT

Moreover, ψT Mψ and ψT Kψ are diagonal matrices. Mass normalizing the eigenvectors, i.e., ψT Mψ  I 1

(6)

will result in

ψT Kψ  diag ω2n

Thus the use of mass normalized eigenvectors as the columns of the transformation matrix φ, results in a set of n decoupled 2nd order differential equations of η¨ i  ω2i ηi 

fi t

(7)

where fi denotes the i-th element of the vector ψT f . Equation 4 indicates that the physical displacement of the structure, x, is the summation of its modal contributions, i.e., modal displacements ηi ’s scaled by their corresponding eigenvector (mode shape). In other words x t 

n

∑ ψi ηi

i 1

(8)

1 Similarity Transformation Two matrices A and B are similar if they have the same eigenvalues. Their transformation is via a nonsingular matrix P, according to A  P 1 BP When the columns of the similarity transformation P consist of the n eigenvectors of B, the similar matrix of B becomes diagonal, denoted by Λ, i.e., Λ  P 1 BP or This can be rewritten as

B  PΛP

1

BP  PΛ

(9)

The matrix equation 9 is known as eigenvalue problem. If λii denotes the i-th diagonal element of the diagonal matrix Λ, Equation 9 can be rewritten as n seperate equations APi  λii Pi

i  1 2     n

(10)

Equation 10 states that Pi is the i-th eigenvector of the matrix P and λii is the associated eigenvalue.

2 Poles and Zeros Consider the equation of motion in modal domain, i.e., η¨ t  ω2nη t  x 

ψT f t

ψη

Taking the Laplace transform of Equations 11 and 12 results in

2

(11) (12)



s2  ω2n  η  s  x  s  

ψT f  s   ψη  s  ψ s2  ω2n   1 ψT f  s 

(13) (14)

leading to the transfer function matrix mapping the input f to the output x  ψ s2  ω2n   1 ψT f  s  and the frequency response function (FRF) of

 ψ ω2  ω2n   1 ψT   ψ ω2n ω2   1 ψT

α  ω  

(15)

  Note that the term ω2n ω2   1 in Equation 15 is a diagonal matrix. The FRF of Equation 15 can also be written in summation notation by considering the ik-th element of α  w  and partitioning the matrix ψ into columns denoted by ψr . The vectors ψr are the mass normalized modal vectors, i.e., eigenvectors of the matrix K normalized with respect to the mass matrix M. This yields 

n



∑ ω2r

α  ω

r 1

ω2   1 ψr ψTr

(16)

The ik-th element of the α  ω  matrix becomes



n

∑ ω2r

αik  ω



ω2  

r 1

1

ψr ψTr  ik

(17)

 where the matrix elements ψr ψTr  ik is identified as the modal constant or residue for the r-th mode and  T the matrix ψr ψr  is called the residue matrix.

Partial Fraction Expansion The 2nd. order system of Equation 18 can be expressed as the sum of partial fractions shown in Equation 19. H  s 

 λ1   s λ1  c c  1   2  s λ1   s λ1  s



1 M



(18) (19)

 where λ1  σ1  jω1 and λ1  σ1 jω1 are the complex conjugate poles and c1 and c2 defined below are the complex conjugate residues of the rationoal polynomial transfer function describing the 2nd order system. c1 

1 M

c2 

(20)

j2ω1



1 M

j2ω1 3

 c1

(21)

The mathematical representation of a transfer function in terms of a partial fraction expansion in nothing more than a sum of single degree of freedom systems. Therefore the mass of a single dof system, which is by definition the modal mass, canbe related to the residue for a single dof system. Note that the in the jargons of multi-dof system, the transfer function of a single dof system, i.e., Equation 18, is a driving point (collocated) transfer function. Considering that a multi-dof transfer function, or frequency response function, is represented as H  s  11 or H  w  11 , the residue c1 which is the driving point residue will be represented as cqq1 The modal mass for a single dof system is M

1 j2ω1 c1 1 j2ω1 cqq1 j "! 2ω1 cqq1

(22)

Recalling that the residue matrix for a particular pole λr is related mode of an N dof system # c$ r Qr % u & r % u & Tr '( ( u1 u1 u1 u2 * * * u1 um ( u2 u1 u2 u2 Qr .. .. ) . . um u1 um um

to the modal vectors. For an r–th

+, , , (23)

r

where Qr is an arbitrary scaling constant. Now the r-th modal mass of a multi-dof is defined as Mr

1 j2Qr ωr

Example 2.1: A 2-dof system Using the differential Equation 25, . . m1 0 x¨1 c1 c2 ! c2 0 m2 /10 x¨2 243 ! 3 c2 c2 /10

x˙1 x˙2 253

(24)

.

k1 k2 ! k2 ! 3 k2 k2 1 / 0 F 0 0 2

x1 x2 2 (25)

the model of the 2dof undamped system shown in Figure 1 in the absence of damping is given in Equation 26 M x¨  t  Kx  t  u  t  (26) 3 # # where x x1 x2 $ 6 and u f1 f2$ 6 are the vectors of displacements (outputs) of the two masses and forces (inputs) exciting the two masses; see Figure ???. The transfer function matrix mapping the input vector to output vector is evaluated by taking the Laplace transfrom of Equation 26, resulting in # 2 Ms K $ x  s  u  s  (27) 3 4

7

F

c1

c2

8m1

8m2 k2

k1 x1

x2

Figure 1: A 2–DOF discrete system Using the differential Equation 25 x 9 s :;=< Ms2 > K ? @ 1 u 9 s :

> m1 s 2 0 > C k1 k2 F k2 k2 0 m2 s 2 E F k2 @ 1 m s2 > k1 > k2 F k2 C 1 A M2 s2 > K2 E F k2 m s2 > k2 k2 C 2 k2 m1 s2 > k1 > k2 E f A 1 f2 det I J K L G M sN

@

1

EHG

;

f1 f2 G

(29)

G

(30)

;

A

f1 f2 G

;BADC

(28)

where

det ; m1 m2 s4 > 9 m1 k2 > m2 k1 > m2 k2 : s2 > k1 k2 is the determinant of the matrix < Ms2 > K ? . Letting m1 ; 8, m2 ; 15, k1 ; 5, and k2 ; function G 9 s : in Equation 30 becomes G 9 s :;OC

G11 9 s : G12 9 s : G21 9 s : G22 9 s : E 15s2 > 10 10 C 2 10 8s > 15 E 120s4 > 305s2 > 50

10, the transfer

(31)

Looking at the first element of the transfer function matrix 31, G11 9 s :;

15s2 > 10 120s4 > 305s2 > 50 j0 P 0315 > F j0 P 0334 > j0 P 0334 F j0 P 0315 > > > s F j1 P 538 s j1 P 538 s F j0 P 4197 s j0 P 4197

Note that the pairs c111 ; F j0 P 0315, c111 Q ; j0 P 0315 and c112 ; F j0 P 0334, c112 Q ; conjugate residues corresponding to the two complex conjugate pairs of poles. 5

(32)

j0 P 0334 are complex

In a similar fashion the rest of the residues for the remaining transfer functions can be determined. The system transfer function G R s S can now be expressed in terms of partial fractions, shown in Equation 33. 0 X 0315i 0 X 0124i 0 X 0315i 0 X 0124i 0 X 0124i 0 X 0049i Y 0 X 0124i 0 X 0049i Y U W G R s STVU1W Z Z s 1 X 5380i s 1 X 5380i W W Z 0 X 0334i 0 X 0453i 0 X 0334i 0 X 0453i W Z Z 0 X 0453i 0 X 0616i Y 0 X 0453i 0 X 0616i Y Z W U W U Z (33) Z s 0 X 4197i s 0 X 4197i W W Z Note that residues corresponding to a pair W of complex conjugate poles, appear as a complex conjugate pair themselves. Recall that the modal vector associated with each pole is proportional to the residue matrix for that pole. This makes the modal vectors corresponding to a pair of complex conjugate poles complex conjugate of each other, as well. Relating the modal vector to the residue matrix of the first pole Q1 [ u \ 1 [ u \ ]1 T

Q1

U

u1 u1 u1 u2 u2 u1 u2 u2 Y

0 X 0315i 0 X 0124i

T T

U W 1 100

U^W

1

0 X 0124i 0 X 0049i Y

3 X 15i W 1 X 24i 1 X 24i 0 X 49i Y

(34)

W

3 Modal Vector Scaling One of the widely used scaling techniques of modal vectors is “unity modal mass” in which the scaling factor Qr is decided upon based on having unity modal mass Mr , i.e., 1 j2Mr ωr 1 j2ωr j 2ωr

Qr T

T T

(35)

W Using the unity modal mass scaling factor in conjunction with the diagonal elements (corrensponding to driving force measurements) of Equation 23, the scaled modal coefficients can be computed as uqr uqr T

cqqr Qr

(36)

Having the driving point scaled modal coefficients (corresponding to the diagonal elements of coefficient matrix in Equation 23), the other coefficients (off–diagonal elements of the matrix can be calculated. In general, the scaled modal vector is 1 (37) [ u\ r T [ c\ r Qr uqr

6