Page 122 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 4.7 Finite Difference Numerical Computation 109

nonlinear or if the system is excited by a force that cannot be expressed by simple
analytic functions.
In the finite difference method, the continuous variable t is replaced by the
discrete variable and the differential equation is solved progressively in time
increments h = At starting from known initial conditions. The solution is approxi­
mate, but with a sufficiently small time increment, a solution of acceptable
accuracy is obtainable.
Although there are a number of different finite difference procedures avail­
able, in this chapter, we consider only two methods chosen for their simplicity.
Merits of the various methods are associated with the accuracy, stability, and
length of computation, which are discussed in a number of texts on numerical
analysis listed at the end of the chapter.
The differential equation of motion for a dynamical system, which may be
linear or nonlinear, can be expressed in the following general form:

X, = x(0) (4.7-1)
i, = i(0)
where the initial conditions and ij are presumed to be known. (The subscript 1
is chosen to correspond to i = 0 because most computer languages do not allow
subzero.)
In the first method, the second-order equation is integrated without change
in form; in the second method, the second-order equation is reduced to two
first-order equations before integration. The equation then takes the form
X = y
(4.7-2)
y = /(jc, y,t)
Method 1: We first discuss the method of solving the second-order equation
directly. We also limit, at first, the diseussion to the undamped system, whose
equations are
x = f { x , t )
= x(0) (4.7-3)

= i(0)
The following procedure is known as the central difference method, the basis of
which can be developed from the Taylor expansion of and x-_^ about the
pivotal point i.

., ==X, + hx^ + -Tj-x^ -X. + •

(4.7-4)
h^ .. h^...
S - 1 : X, - hx, + -JX, - -g-X; -h