Introduction to Aircraft Aeroelasticity and Loads

4

Vibration of Continuous Systems – Discretization Approach

In Chapter 3 the vibration of continuous systems was considered using a summation of assumed shapes to describe the motion. The approximate analysis led to differential equations of motion expressed in terms of the unknown coefficients (or generalized coordinates) that multiply each assumed shape. Standard MDoF analysis approaches as described in Chapter 2 could then be employed to solve for natural frequencies, normal modes and the response to various forms of excitation.

In this chapter, the vibration of continuous systems will be approached using a physical discretization of the system, i.e. the structure is divided into finite width strips (or elements) and the motion of the structure is described via the displacements and rotations of the strips. An early approach to this discretization was using the flexibility influence coefficients (Rao, 1995), but this was superceded by the finite element (FE) approach (NAFEMS, 1987; Cook et al, 1989; Rao, 1995). In this chapter, the FE method will be introduced, where the deformation within each strip (or so-called ‘finite element’) is approximated using a polynomial representation and the distributed stiffness and mass behaviour is represented by stiffness and mass matrices for each element. The FE approach will be illustrated upon simple slender ‘beam-like’ members (e.g. representing a wing or fuselage by a ‘stick-like’ model). Such approaches have been traditionally employed ...