Civil Engineering Reference
In-Depth Information
11
Forced Vibrations
11.1
Introduction
In the previous chapter, programs were described which enable the calculation of the
intrinsic dynamic properties of systems, namely their undamped natural frequencies and
mode shapes. The next stage in a dynamic analysis is usually the calculation of the response
of the system to an imposed time dependent disturbance. This chapter describes seven
programs which enable such calculations to be made.
The types of equations to be solved were derived early in the topic (e.g. 2.13). After
semi-discretisation in space using finite elements, the resulting matrix equations are typified
by (2.17), a set of second order ordinary differential equations in time. On inclusion of
damping, the relevant equations become (3.118) and Section 3.13 describes the principles
behind the various solution procedures used below.
Program 11.1 describes forced vibration analysis of elastic slender beam structures
using direct integration in the “time domain”. Programs 11.2, 11.3, and 11.4 describe
forced vibration analyses of planar 2D elastic solids. Program 11.2 works in the “frequency
domain” using the Modal Superposition Method, and Programs 11.3 and 11.4 work in the
“time domain” utilising two different implicit time-marching algorithms. Program 11.5
illustrates a “mixed” time-marching scheme for 2D analysis in which some parts of the
mesh are integrated “explicitly” and others “implicitly”. Program 11.6 repeats the algo-
rithm of Program 11.3 using a “mesh free” approach with a preconditioned conjugate
gradient solver. Program 11.7 uses a fully explicit time marching scheme to analyse a 2D
elastic-plastic material.
Programs 11.3 and 11.7 have parallel counterparts described in Chapter 12.
