A study of the motion of a flexible rod in a quick return mechanism.

Abstract

This study deals with the analysis of a quick return mechanism, with the crank regarded as rigid and the rod treated as flexible, modelled by Euler-Bernoulli beam theory. Several tasks are addressed, the first being the development of independently arrived at solutions so that accuracy can be assessed. The second is an investigation of the applicability, in view of its use in commercial codes, of the free-free mode approach for this class of mechanism. Finally, a study is undertaken to see if currently available techniques for analyzing stability are applicable and to determine whether stability is a problem in practical mechanisms. The partial differential equations of motion are developed from Hamilton's Principle. The study naturally falls into two parts depending on whether the the crank length is small. For small cranks, Galerkin's method, together with time dependent pinned-pinned overhanging beam modes and Taylor series expansions in crank length, is used to generate ordinary differential equations of motion. Response is obtained from them using numerical integration. Hsu's technique is employed to obtain regions of parametric resonance, and these are compared to the results of a numerical method using the monodromy matrix. Forced motion instability is investigated using an extension of Hsu's technique. For the large crank, two different approaches are employed. The first involves the use of a set of time dependent polynominal modes. The results obtained using these modes are shown to be accurate by comparison with the small crank results. The modes lead to linear equations for addressing stability. It is shown that instabilities are not a concern at practical operating speeds. In the other approach, free-free modes with respect to a suitable floating frame are employed. A major finding of the work is this latter approach gives accurate response results for both large and small cranks and is the most computationally efficient. Stability investigations using this approach require the development of suitable linearization procedures. This still remains an open question, but several conjectures are set forth in this thesis.