The stability and response of a flexible rod in a quick return mechanism

Abstract

A quick return mechanism is analyzed for deflection and stability when the rod is considered an Euler-Bernoulli beam. The crank is assumed rigid and to be rotating at a constant angular velocity. The equations of motion and natural boundary conditions are obtained using Hamilton's principle. Spatial dependence is suppressed using Galerkin's method with the time dependent pinned-pinned overhanging beam modes. Using a small crank length approximation, zones of parametric resonance are found using Hsu's method. The accuracy of these is verified using a monodromy matrix technique. The technique is also used to explore the possibility of resonances not covered by Hsu's (first order) method. A particular solution instability is found to exist using Hsu's method and is verified by direct numerical integration. For a somewhat flexible configuration, all instabilities were found to lie outside the range of normal operating speeds. For stiffer configurations likely to be found in practice, this conclusion can be asserted even stronger, at least for small cranks.