Vibration of Continuous Systems – Assumed Shapes Approach

In Chapters 1 and 2, some basic concepts were introduced for single and multiple DoF ‘discrete parameter’ systems, where motion was defined via displacement or rotation coordinates. However, for most problems encountered in aircraft aeroelasticity and loads, the systems are ‘continuous’, involving mass and stiffness properties distributed spatially over the system. An aircraft wing or fuselage may be considered as elastic continuum components able to bend and twist, but these require a different analysis approach.

There are several ways of modelling ‘continuous’ systems, namely:

(a) an exact approach using the partial differential equations of the system to achieve exact modes,

(b) an approximate approach using a series of assumed shapes to represent the deformation or

(c) an approximate approach using some form of spatial ‘discretization’.

The exact approach is satisfactory for simple systems such as slender members under bending, torsional or axial deformation, but is impractical for ‘real’ systems with complex geometry. (The term ‘slender member’ is used here in place of the many terms often used to describe what are essentially similar members that experience different types of loading, namely beams, shafts, bars and rods. The term ‘slender’ implies that their length is significantly greater than their cross-section dimensions.)

In this chapter, the Rayleigh–Ritz approach for modelling a system using a series of ...

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