Modal Models - State-Space Structural Models

Structures

2.3 State-Space Structural Models

2.3.3 Modal Models

The ith component, or mode, has the state-space representation ( independently obtained from the state equations

, , )

mi mi mi

A B C

, , .

i mi i mi

i mi i

n i i

x A x B u y C x

y y

¦

(2.51)

This decomposition is justified by the block-diagonal form of the matrix and is illustrated in Fig. 2.9 for n= 2. In generic coordinates each state depends on itself (through the gain shown in Fig. 2.9 with a solid line) and on other states (through the gains shown in Fig. 2.9 with a dashed line). In modal coordinates the cross-coupling gains are zero, thus each state is independent and depends only on itself.

m, A Ami

Amij

x

Am21

Am12

³

+ +

+ + +

Am1

Am2

Figure 2.9. Block diagram of the state-space representation of a structure with two modes:

The modal cross coupling (marked with a dashed line) is nonexistent.

We consider three modal representations in this book. The blocks A_mi, B_mi, and of these models are as follows:

Cmi

x Modal model 1:

0 0

, ,

mqi i

mi mi mi mvi

i i i mi i

A B C c

b Z

Z ] Z Z ^c ^;

ª º

ª º ª º

« »

« » « »

¬ ¼ ¬ ¼ ¬ ¼

(2.52)

x Modal model 2:

, 0 , ^mqi

i i i

mi mi mi mvi i mvi

i i i mi i

A B C c c

b ] Z Z

Z ] Z Z ]

ª c º;

ª º ª º

« »

« » « »

¬ ¼ ¬ ¼ ¬ ¼

(2.53)

x Modal model 3:

0 1 0

, ,

mi 2 mi mi mqi mvi

i i i mi

A B C c

Z ] Z b

ª º ª º

c .

ª º

« » « » ¬ ¼

« » ¬ ¼

¬ ¼

(2.54)

Theith state component for the first modal model is as follows:

i mi ,

i mi

x q q

Z ½ ® ¾

¯ ¿ (2.55)

for the second modal model it is

i mi ,

i i i mi mi

x q

q q

Z ] Z

® ¾

¯ ¿ (2.56)

and for the third modal model it is

mi ,

i mi

x q q

½ ® ¾

¯ ¿ (2.57)

where and are the ith modal displacement and velocity, as defined in (2.18). Note that each component consists of modal displacement and velocity which, by (2.18), gives the original (nodal) displacement q and velocity Note also that eigenvalues of are the complex conjugate poles given by (2.32).

qmi q_mi

q. Ami

We obtain the modal models 1, 2, and 3 from the corresponding state-space representations in modal coordinates as in (2.38), (2.42), and (2.45), by simply rearranging the columns of AandCand the rows ofA and B. Consider, for example, the third representation, with the state vector x^T ªq^T_m q^T_mº,

¬ ¼ consisting of modal

displacements followed by modal rates. We transform it to the new state defined as follows:

1 1

1 2

2 ,

m m m m mn n mn

q x

q x q

° °

° ° ½

° ° ° °

® ¾ ® ¾

° ° ° °

° ° °¯ °¿

° °

¯ ¿

(2.58)

where the modal displacement for each mode stays next to its rate. The variable x_i in the above equation is defined by (2.57). Formally, this state-space representation is derived using the transformation matrix R in the form

1 1

1 2

2 2

0 0

0 , where

0 0

i i

i n n

e E

e E e

R e

e E e

ª º

« »

« » ª º

« » « »

E ª º,

« » « » « »

« » « » ¬ ¼

« » « »

« » «¬ »¼

« »

¬ ¼

# # #

(2.59)

while is an n row vector with all elements equal to zero except the ith which is equal to one, and 0 denotes an n row vector of zeros (actually, we simply rearrange the coordinates).

We obtain the modal models 1 and 2 in a similar manner by rearranging the states in the state vectors (2.36) and (2.40). The new state vectors for these representations are as follows:

1 1 1 1

1 1

2 2 2 2

2 2

2 , 2 2 2 .

m m

n n

n mn n mn

mn n n mn mn

q q

q q ₁

2 m

x q x

q q

x x

q q

x x q

x x

q q

Z Z

] Z

Z Z

] Z

Z Z

] Z

½ ½

° ° ° °

° ° ½ ° ° ½

° ° ° ° ° ° ° °

® ¾ ® ¾ ® ¾ ®

° ° ° ° ° ° °

° ° °¯ °¿ ° ° °¯ ¿

° ° ° °

¯ ¿ ¯ ¿

# #

¾°

(2.60)

In the above equation x_i for modal models 1 and 2 are defined by (2.55).

We obtain the state representations from the representations as in (2.38) and (2.42), respectively, by rearranging the columns of AandCand the rows of A and B.

Note that the modal models (2.52), (2.53), and (2.54) are not unique in the sense that for the same matrix A_i, one can obtain different matrices B_i and , as explained in Appendix A.1. In particular, the first component of the input matrix

might not necessarily be zero, unless one finds a unique transformation that preserves the zero entry. For details see Appendix A.1.

In Appendix A the Matlab functions modal1m.m andmodal1n.m determine the modal model 1 using the modal or nodal data; and functions modal2m.m and modal2n.m determine the modal model 2 using the modal or nodal data. Also, the functionsmodal1.mandmodal2.m determine modal model 1 or 2 using an arbitrary state-space representation.

Example 2.9. Obtain modal model 2 from the model in Example 2.8.

Using transformation (2.59) we find

0.0487 3.1210 0 0 0 0

3.1203 0.0487 0 0 0 0

0 0 0.0233 2.1598 0 0

0 0 2.1596 0.0233 0 0 ,

0 0 0 0 0.0030 0.7708

0 0 0 0 0.7708 0.0030

ª º

« »

¬ ¼

1 2 3

0 0.3280 0 0.5910 0 0.7370

m m

B b

ª º

« »

« » ª º

« » « »

« » «¬ »¼

« »

¬ ¼

and

>

1 2 3

@

0.1894 0 0.3412 0 0.4255 0

0.0092 0.5910 0.0080 0.7370 0.0013 0.3280 .

0.0051 0.3280 0.0064 0.5910 0.0028 0.7370

m m m m

C c

ª º

« »

¬ ¼

c c

The transfer function of a structure is defined in (2.5). In modal coordinates it is, of course,

(2.61) ( ) _m( _m) 1 .

G s C sI A B_m

As was said before, the transfer function is invariant under the coordinate transformation; however, its internal structure is different from the generic transfer function (2.6). In modal coordinates the matrix sI A _m is block-diagonal, and it can be decomposed into a sum of transfer functions for each mode, therefore,

Property 2.2. Transfer Function in Modal Coordinates. The structural transfer function is a composition of modal transfer functions:

2 2

1 1

( )

( ) ( ) ,

n n

mqi mvi mi

i i

c j c b

G G

Z Z Z

Z Z ] Z Zi

¦ ¦

^(2.62)

where

2 2

( )

( ) ( ) , 1, , ,

mqi mvi mi

mi mi mi mi

i i i

c j c b

G C j I A B i

Z Z Z

Z Z ] Z Z

!n (2.63)

is the transfer function of the ith mode. The value of the transfer function at the ith resonant frequency is approximately equal to the value of the ith mode transfer function at this frequency:

( )

( ) ( ) .

mqi i mvi mi

i mi i

i i

jc c b

G G Z

Z Z

] Z

# (2.64)

Proof. IntroducingA, B, andC as in (2.45) to the definition of the transfer function we obtain

1 1

( ) ( ) ( ) ( ),

n n

m m m mi mi mi mi

i i

GZ C j I AZ B

¦

C j I AZ B

¦

G ^Z

which proves the first part. The second part follows from the first part by noting that, for flexible structures with distinct natural frequencies and low damping,

2 2

( ) ( )

mj i mi i

G Z G Z for iz j.

Special Models

ª how to describe less-common structures

Do not quench your inspiration and your imagination;

do not become the slave of your model.

—Vincent van Gogh

Models described in the previous chapter include typical structural models, which are continuous-time, stable, and with proportional damping. In this chapter we consider models that are not typical in the above sense but, nevertheless, often used in engineering practice. Thus, we will consider models with rigid-body modes (which are unstable), models with nonproportional damping, discrete-time structural models, models with acceleration measurements, and generalized structural models.

The latter include two kinds of inputs: controlled (or test) inputs and disturbance inputs, and also two kinds of outputs: measured outputs and outputs where the system performance is evaluated.

Dalam dokumen Advanced Structural Dynamics and Active Control of Structures (Halaman 58-64)