7.1 Introduction

The usual first step toward a solution of the linear, small deflection matrix equation of structural motion, [ M]{ } + [ C]{ } + [ K]{ q} = { Q} (called the m, c , k, Q equation), is to obtain the system natural frequencies and mode shapes. As discussed in the previous chapter , the natural frequencies and mode shapes are eigenpair solutions to the undamped, homogeneous ( m, k) form of this same matrix differential equation. The focus of this chapter is on the usual final steps in the solution process. The usual step after acquiring the modal frequencies and mode shapes is to use that data to isolate the effect of each mode on the system response. The final step is to solve each of the selected modal differential equations and sum those results. For the sake of instruction and intuition building, this chapter and the next focus only on obtaining analytical solutions to those modal differential equations. To further clear the underbrush, the example problems will omit damping. Chapter 9 discusses the far more prevalent approach of numerical integration of the modal differential equations. The available commercial software packages, of course, provide numerical solutions rather than analytical solutions. Numerical solutions are almost always the only option when the original matrix differential equation is complicated by material or geometric nonlinearities.

Before introducing the specifics of the two final steps to a complete solution...