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CHAPTER 6 6
Direct Numerical
Integration Methods
6.1 INTRODUCTION
The behaviour of many dynamic systems undergoing time-dependant changes (transients) can be described
by ordinary differential equations. When the solution to the differential equation(s) of motion of a dynamic
system cannot be obtained in closed form, a numerical procedure is warranted. Many numerical integration
methods are available for the approximate solution of such equation(s) of motion. All the numerical integration
methods have two basic characteristics. First, they do not satisfy the differential equation(s) at all time t,
but only at discrete time intervals, say ∆ t apart. Secondly, within each time interval ∆ t, a specific type of
variation of the displacement X, velocity X , and acceleration X is assumed. Thus, several numerical
integration schemes are available depending on the type of variation assumed for X, X and X within each
time interval ∆ t.
In this chapter, we discuss several widely used step-by-step numerical integration schemes for solutions of
both single and multi degree of freedom systems. A brief description of these methods is presented for
linear dynamic response analysis and their application is illustrated by several examples.
6.2 SINGLE-DEGREE OF FREEDOM SYSTEM
The general equation of a viscously damped single degree of freedom dynamical system, which is linear,
can be expressed in the following general form:
MX + CX + KX = F () ...(6.1)
t
where M, C and K are the mass, damping and stiffness of the system; F(t) is the applied force; and X, X
and X are the displacement, velocity and acceleration of the system.
6.2.1 Finite Difference Method
If the equilibrium relation (6.1) is regarded as an ordinary differential equation with constant coefficients, it
follows that any convenient finite difference expressions to approximate the velocities and accelerations in
terms of displacements can be used. The central idea in the finite difference method is to use approximations
