Dynamic Responses - Mechanics of Structures: Variational and Computational Methods - page 638

The numerical determination of Ritz vectors takes less computational work than needed
to calculate mode shapes. This reduces the numerical effort needed in time-domain solu-
tions of complex structural problems. In addition, the method possesses the advantages
of static condensation, Guyan reduction, and higher mode truncation [Wilson, 1985]. The
examples studied in Wilson, et al. (1982) have shown that a dynamic analysis based on Ritz
vectors gives more accurate results, with fewer vectors, than when the usual mode shapes
are employed.
Suppose the applied force P in the governing equations of Eq. (10.97) can be expressed
as a product of a vector of spatial distributions of loading and a function of time. That is,

C V

M V + + KV = P = F ( s ) g ( t ) (10.130)

The time-independent vector F ( s ) represents the spatial distribution of the external force

and g is a function of time only. Equation (10.130) is reduced in size by using a set of

global Ritz vectors x 1 , x 2 ,

( t )

... . ×

n
matrix whose i th column is x i . The displacement vector V is written in terms of the Ritz
vectors as

, x n Let the number of entries of V be n d and let X be the n d

V = Xy (10.131)

where y is a vector with n elements. When Eq. (10.131) is substituted into Eq. (10.130) and
the results premultiplied by X T , the following set of differential equations is obtained

M 1 y + C 1 y ˙ + K 1 y = F 1 g ( t ) (10.132)

The number of scalar differential equations in Eq. (10.132) is equal to n , the number of Ritz
vectors chosen in the expansion of Eq. (10.131). In Eq. (10.132)

X T MX

M 1 = (10.133)

X T KX

=

K 1 (10.134)

X T CX

C 1 = (10.135)

X T F F 1 = (10.136)

The condensed relationship of Eq. (10.132) can be solved by one of the direct step-by-step
integration methods discussed in Section 10.5. To complete the derivation of Eq. (10.132),
Ritz vectors will now be explicitly defined. The first Ritz vector x 1

is taken as the displace-

ment vector obtained from a static analysis with F ( s ) as the external force and is found by

solving the linear algebraic equation

Kx 1 = F ( s ) (10.137)

for x 1 . The Ritz vector x 1 , which is the first column of the matrix X , is found from x 1

by

normalizing x 1

with respect to the mass matrix M

x 1

x 1 = x ∗ T

1

(10.138)

Mx 1

The normalized vector x 1 has the property

x 1 Mx 1

=

1