Dynamic Responses - Mechanics of Structures: Variational and Computational Methods - page 627

TABLE 10.2
Procedure for Computer Implementation of the Central Difference Method

Initial Step

1. Select t

2. Calculate the constants related to t

c 1 = 1 / t 2 , c 2 = 1 / t , c 3 = 2 c 1 , and c 4 = c 2 / 2

Initialize V 0 , V 0 , and V 0 3.

4. Calculate V − 1 from Eq. (10.101)

5. Calculate the effective stiffness matrix of Eq. (10.100)

K

= c 1 M + c 4 C

Decompose ∗ (triangularize)
K

6.

= LDL T

At Each Step n:

1. Calcula te the effective load from Eq. (10.100)

P n + 1 = P n − ( K − c 3 M ) V n − ( c 1 M − c 4 C ) V n − 1

Solve KV n + 1 = P n + 1 using K LDL T 2. = (matrix decomposition)

∗

K can be decomposed into LDT T . L is a lower triangular matrix (elements only on the diagonal and below) with
the diagonal elements equal to 1, and D is a diagonal matrix.

solution, the spectral radius, i.e., the maximum eigenvalue of matrix A , has to be smaller
than 1. This can be readily proved. It follows from Eq. (10.102) that

X n = AX n − 1 (10.104)

Substitute Eq. (10.104) into Eq. (10.102)

A 2 X n − 1

X n + 1 = (10.105)

Thus, in general,

A n + 1 X 0 X n + 1 = (10.106)

From the theory of matrices,

Z λ Z − 1 A = (10.107)

where λ are the eigenvalues of A , which are assumed to be distinct, and each column of Z
is a corresponding eigenvector, whose elements are finite.
Also, from the theory of matrix analysis,

A n + 1 Z λ n + 1 Z − 1 = (10.108)

Since the elements of Z are bounded, the elements of A n + 1

will be bounded provided

| λ i |≤ 0 , X n + 1 cannot be very large or approach infinity,

and the integration is stable. The condition of

1 . 0 ,i = 1 , 2 , ... ,n, i.e., if | λ i |≤ 1 .

| λ i |≤ 1 . 0 leads to the critical time step for

stability

2
ω

t ≤ t crit = (10.109)