Efficient Numerical Solution of Fourth-Order Problems in the Modeling of Flexible Structures

Abstract

In the modeling and control of large flexible structures, fourth-order boundary value problems often arise. The present work describes a Sinc-Galerkin method for the solution of fourth-order ordinary differential equations of the form

$$\begin{gathered} u''''\left( x \right) + \upsilon \left( x \right)u''\left( x \right) + \sigma \left( x \right)u\left( x \right) = f\left( x \right), a < x < b \hfill \\ u\left( a \right) = u\left( b \right) = 0 \hfill \\ u'\left( a \right) = u'\left( b \right) = 0. \hfill \\ \end{gathered} $$

(1.1)

Although the procedure to be described is equally applicable to the more general fourth-order problem

$$\begin{gathered} u''''\left( x \right) + \rho \left( x \right)u'''\left( x \right) + \upsilon \left( x \right)u''\left( x \right) \hfill \\ + \tau \left( x \right)u'\left( x \right) + \sigma \left( x \right)u\left( x \right) = f\left( x \right), \hfill \\ \end{gathered} $$

(1.2)

the present focus is toward potential applications in the study of flexible structures. A large class of problems in this area require the efficient and accurate solution of boundary value of the form in (1.1). To illustrate the Sinc-Galerkin method in this setting, the damped beam equation

$$\begin{gathered} y''''\left( x \right) + 2\pi y''\left( x \right) + {\pi ^4}y\left( x \right) = 0 \hfill \\ y\left( 0 \right) = y\left( 1 \right) = 0 \hfill \\ y'\left( 0 \right) = \sqrt 2 \pi , y'\left( 1 \right) = - \sqrt 2 \pi \hfill \\ \end{gathered} $$

(1.3)

will be used as one of the examples. This equation arises when a square root damping mechanism is used to model the transverse vibrations of a uniform “clamped-clamped” beam [6].