Non-linear and parametric vibration of thin-walled beams of equal angle-section

Abstract

The linear coupling of the various types of motion of thin-walled beams of equal angle-section was discussed in reference [1]. Here their interaction due to non-linear effects is studied. In addition to the usual displacements in the axial direction, a second-order effect is introduced. The mutual interaction of two types of motion at a time is studied. This is done by retaining quadratic non-linearities while ignoring cubic non-linearities in the derived differential equations of motion, so obtaining pairs of (quadratically) coupled non-linear partial differential equations in the relevant dependent variables. Application of Galerkin's approximation gives a set of ordinary differential equations with periodic coefficients. The regions of instability are evaluated, and are expressed in non-dimensional terms. Experimental tests on a series of cantilever beams show that torsional and cross-sectional modes are parametrically excited by support motion in either the axial or the transverse direction. When the forcing frequency approaches one of the natural frequencies (longitudinal or transverse), it is shown that the parametric instability region becomes substantially wider. The effect on the instability region due to changes in the basic dimensions of the beam is also discussed.