Introduction to Nonlinear Aeroelasticity

3Time Integration

3.1 Introduction

In the previous chapter, we introduced a number of interesting characteristics of the response of nonlinear systems but we did not actually solve the nonlinear equations of motion. The problem is that we cannot solve nonlinear ODEs, except from some very special cases. In contrast, we can solve most nonlinear algebraic equations, either analytically or numerically. Therefore, nearly all treatments of nonlinear systems do not solve the differential equations themselves but similar nonlinear algebraic equations. The basis of these treatments lies in the careful selection of the algebraic equations that approximate accurately the response of the original differential equations.

In this chapter, we will introduce different strategies for the integration of nonlinear ODEs in time. These procedures are known as time integration because they aim to solve nonlinear equations of motion of the form

(3.1)

from time and with initial conditions up to the final time instance of interest, . Note that the nonlinear function is written as a function of both ...