Use of generalized mass in the interpretation of dynamic response of bending-torsion coupled beams

The interpretation of mode shapes and dynamic response of bending-torsion coupled beams is assessed by using the concept of generalised mass. In the first part of this investigation, the free vibratory motion of bending-torsion coupled beams is studied in detail. The conventional method of interpreting the normal modes of vibration consisting of bending displacements and torsional rotations is shown to be inadequate and replaced by an alternative method which is focussed on the constituent parts of the generalized mass arising from bending and torsional displacements. Basically, the generalized mass in a particular mode is identified and examined in terms of bending, torsion and bending-torsion coupling effects. It is demonstrated that the contribution of individual components in the expression of the generalized mass of a normal mode is a much better indicator in characterizing a coupled mode. It is also shown that the usually adopted criteria of plotting bending displacement and torsional rotations to describe a coupled mode can be deceptive and misleading. In the second part of the investigation, attention is focussed on the dynamic response characteristics of bending-torsion coupled beams when subjected to random bending or torsional loads. A normal mode approach is used to establish the total response. The input random excitation is assumeed to be stationary and ergodic so that with the linearity assumption, the output spectrum of the response is obtained by using the frequency response function. The contribution of each normal mode to the overall response is isolated. Particular emphasis is placed on bending-induced torsional response and torsion-induced bending response. A number of case studies involving different types of bending-torsion coupled beams with cantilever end conditions are presented. The limitations of existing methods of modal interpretation are highlighted, and an insight into the mode selection for response analysis is provided.