70811_28.pdf

CHAPTER 28, PART I

MATRIX METHODS

OF ANALYSIS

Stephen H. Crandall

Robert B. McCalley, Jr.

INTRODUCTION

The mathematical language which is most convenient for analyzing multiple degree-

of-freedom vibratory systems is that of matrices. Matrix notation simplifies the pre-

liminary analytical study, and in situations where particular numerical answers are

required, matrices provide a standardized format for organizing the data and the

computations. Computations with matrices can be carried out by hand or by digital

computers. The availability of programs such as MATLAB makes the solution of

many complex problems in vibration analysis a matter of routine.

This chapter describes how matrices are used in vibration analysis. It begins with

definitions and rules for operating with matrices. The formulation of vibration prob-

lems in matrix notation then is treated. This is followed by general matrix solutions

of several important types of vibration problems, including free and forced vibra-

tions of both undamped and damped linear multiple degree-of-freedom systems.

Part II of this chapter considers finite element models.

MATRICES

Matrices are mathematical entities which facilitate the handling of simultaneous equa-

tions. They are applied to the differential equations of a vibratory system as follows:

A single degree-of-freedom system of the type in Fig. 28.1 has the differential

equation

where m is the mass, c is the damping coefficient, k is the stiffness, F is the applied

force, x is the displacement coordinate, and dots denote time derivatives. In Fig. 28.2

a similar three degree-of-freedom system is shown. The equations of motion may be

obtained by applying Newton’s second law to each mass in turn:

mx 1

cx 1

5 kx 1 −

2 kx 2

F 1

2 mx 2

+ 2 cx 2 − 2 cx 3 − 2 kx 1 + 3 kx 2 − kx 3 = F 2

(28.1)

3 m¨ 3

−

2 cx 2 +

2 cx 3

−

kx 2 +

kx 3 =

F 3

28.1

28.2

CHAPTER TWENTY-EIGHT, PART I

FIGURE 28.1

Single degree-of-freedom sys-

FIGURE 28.2

Three degree-of-freedom sys-

tem.

The accelerations, velocities, displacements, and forces may be organized into

columns, denoted by single boldface symbols:

x 1

F 1

x =

x 2

x =

x 2

x =

x 2

f =

F 2

(28.2)

x 3 x 3 x 3 F 3

The inertia, damping, and stiffness coefficients may be organized into square

arrays:

−2 k

M =

02 m

C =

0 c

−2 c

K =

−2 k

3 k

− k

(28.3)

00 3 m

−2 c

2 c

− kk

By using these symbols, it is shown below that it is possible to represent the three

equations of Eq. (28.1) by the following single equation:

Mx + Cx + Kx = f

(28.4)

Note that this has the same form as the differential equation for the single degree-of-

freedom system of Fig. 28.1. The notation of Eq. (28.4) has the advantage that in sys-

tems of many degrees-of-freedom it clearly states the physical principle that at every

coordinate the external force is the sum of the inertia, damping, and stiffness forces.

Equation (28.4) is an abbreviation for Eq. (28.1). It is necessary to develop the rules

of operation with symbols such as those in Eqs. (28.2) and (28.3) to ensure that no

ambiguity is involved. The algebra of matrices is devised to facilitate manipulations of

simultaneous equations such as Eq. (28.1). Matrix algebra does not in any way sim-

plify individual operations such as multiplication or addition of numbers, but it is an

organizational tool which permits one to keep track of a complicated sequence of

operations in an optimum manner. Matrices are essential elements of linear algebra, 1

and are widely employed in structural analysis 2 and vibration analysis. 3

DEFINITIONS

A matrix is an array of elements arranged systematically in rows and columns. For

example, a rectangular matrix A , of elements a jk , which has m rows and n columns is

a 11

a 12

... a 1 n

a 21

a 22

... a 2 n

A = [ a jk ] =

... ... ... ...

a m 1 a m 2

... a mn

28.3

MATRIX METHODS OF ANALYSIS

The elements a jk are usually numbers or functions, but, in principle, they may be any

well-defined quantities. The first subscript j on the element refers to the row number

while the second subscript k refers to the column number. The array is denoted by

the single symbol A , which can be used as such during operational manipulations in

which it is not necessary to specify continually all the elements a jk . When a numeri-

cal calculation is finally required, it is necessary to refer back to the explicit specifi-

cations of the elements a jk .

A rectangular matrix with m rows and n columns is said to be of order ( m,n ). A

matrix of order ( n,n ) is a square matrix and is said to be simply a square matrix of

order n. A matrix of order ( n, 1) is a column matrix and is said to be simply a column

matrix of order n. A column matrix is sometimes referred to as a column vector. Simi-

larly, a matrix of order (1, n ) is a row matrix or a row vector. Boldface capital letters are

used here to represent square matrices and lower-case boldface letters to represent

column matrices or vectors. For example, the matrices in Eq. (28.2) are column matri-

ces of order three and the matrices in Eq. (28.3) are square matrices of order three.

Some special types of matrices are:

1. A diagonal matrix is a square matrix A whose elements a jk are zero when j

≠

The only nonzero elements are those on the main diagonal, where j

k. In order to

emphasize that a matrix is diagonal, it is often written with small ticks in the direc-

tion of the main diagonal:

A =

a jj

2. A unit matrix or identity matrix is a diagonal matrix whose main diagonal elements

are each equal to unity. The symbol I is used to denote a unit matrix. Examples are

100

1 0

0 1 0

01 001

3. A null matrix or zero matrix has all its elements equal to zero and is simply

written as zero.

4. The transpose A T of a matrix A is a matrix having the same elements but with

rows and columns interchanged. Thus, if the original matrix is

[ a jk ]

the transpose matrix is

A T

[ a jk ] T

[ a kj ]

For example:

−14

3−1

A =

A T

The transpose of a square matrix may be visualized as the matrix obtained by rotat-

ing the given matrix about its main diagonal as an axis.

The transpose of a column matrix is a row matrix. For example,

x =

− 4

x T

= [3

−2]

−2

28.4

CHAPTER TWENTY-EIGHT, PART I

Throughout this chapter a row matrix is referred to as the transpose of the corre-

sponding column matrix.

5. A symmetric matrix is a square matrix whose off-diagonal elements are sym-

metric with respect to the main diagonal. A square matrix A is symmetric if, for all j

and k,

a jk = a kj

A symmetric matrix is equal to its transpose. For example, all three of the matrices

in Eq. (28.3) are symmetric. In addition, the matrix M is a diagonal matrix.

MATRIX OPERATIONS

Equality of Matrices. Two matrices of the same order are equal if their corre-

sponding elements are equal. Thus two matrices A and B are equal if, for every j

and k,

a jk = b jk

Matrix Addition and Subtraction. Addition or subtraction of matrices of the

same order is performed by adding or subtracting corresponding elements. Thus,

A + B = C if for every j and k,

a jk +

b jk =

c jk

For example, if

−14

−1

A =

B =

then

−6 −2

4 0

A + B =

A − B =

Multiplication of a Matrix by a Scalar. Multiplication of a matrix by a scalar c

multiplies each element of the matrix by c. Thus

c A

c [ a jk ]

[ ca jk ]

In particular, the negative of a matrix has the sign of every element changed.

Matrix Multiplication. If A is a matrix of order ( m,n ) and B is a matrix of order

( n,p ), then their matrix product AB = C is defined to be a matrix C of order ( m,p )

where, for every j and k,

c jk =

a jr b rk

(28.5)

r = 1

The product of two matrices can be obtained only if they are conformable, i.e., if the

number of columns in A is equal to the number of rows in B . The symbolic equation

( m,n ) × ( n,p ) = ( m,p )

28.5

MATRIX METHODS OF ANALYSIS

indicates the orders of the matrices involved in a matrix product. Matrix products

are not commutative, i.e., in general,

AB ≠ BA

The matrix products which appear in this chapter are of the following types:

Square matrix

square matrix

Square matrix

column vector

Row vector

square matrix

row vector

Row vector

column vector

scalar

Column vector

row vector

square matrix

In all cases, the matrices must be conformable. Numerical examples are given below.

−1

−(3 × 1) + (2 × 5)

(3 × 2) + (2 × 6)

−

(1 × 1) + (4 × 5)

−(1 × 2) + (4 × 6)

= −(1 × 5) + (4 × 3)

−1

Ax =

−14

y T A = [−21]

= [−(2 × 3) − (1 × 1) − (2 × 2) + (1 × 4)] = [−70]

= (−10 + 3) =−7

y T x = [−2

−(3 × 2) (3 × 1)

−

−6

xy T

[−2

1] =

The last product always results in a matrix with proportional rows and columns.

The operation of matrix multiplication is particularly suited for representing sys-

tems of simultaneous linear equations in a compact form in which the coefficients

are gathered into square matrices and the unknowns are placed in column matrices.

For example, it is the operation of matrix multiplication which gives unambiguous

meaning to the matrix abbreviation in Eq. (28.4) for the three simultaneous differ-

ential equations of Eq. (28.1). The two sides of Eq. (28.4) are column matrices of

order three whose corresponding elements must be equal. On the right, these ele-

ments are simply the external forces at the three masses. On the left, Eq. (28.4) states

that the resulting column is the sum of three column matrices, each of which results

from the matrix multiplication of a square matrix of coefficients defined in Eq.

(28.3) into a column matrix defined in Eq. (28.2). The rules of matrix operation just

given ensure that Eq. (28.4) is exactly equivalent to Eq. (28.1).

Premultiplication or postmultiplication of a square matrix by the identity matrix

leaves the original matrix unchanged; i.e.,

Two symmetrical matrices multiplied together are generally not symmetric. The

product of a matrix and its transpose is symmetric.

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