DYNAMIC INSTABILITIES IN FLEXIBLE PLANAR LINKAGES (MECHANISM, SLIDER CRANK, COUPLER, CONNECTING ROD)

The exact partial differential equation of the transverse motion of a flexible bar of an elastic linkage is derived. Large deflections are considered and rotating inertia effects are included. The Newton's equations of motion of the bars of the mechanism are employed to determine the dynamic connection forces (reactions) at the ends of each bar. Exact expressions for the kinematics of the linkage and for the inertia forces are used in the Newton's equations, where large deflections are considered.; The terms of the above p.d.e. of a specific bar are expanded in Taylor series with respect to the lateral deflection. The linear part of this expansion, being in agreement to the assumptions of the small deflection theory, appears to be independent of the deflections of the other bars. As a result, the small deformation dynamic instability of elastic linkages can be analyzed by considering flexibility of each link separately.; The Galerkin approximation is applied to both the nonlinear and the linear p.d.e.'s resulting into systems of nonlinear and linear o.d.e.'s, respectively. Under steady state operation of the mechanism, systems of o.d.e.'s with periodic coefficients result. The Floquet theory considers the properties of the systems of linear o.d.e.'s with periodic coefficients, for the study of the stability of the solutions of which, two methods were utilized. In general, the method of solving an initial value problem determines the Floquet multipliers which control stability. In particular, when the coefficients of the differential systems are even functions of time, use is made of the method of determination of T and 2T periodic solutions, which constitute the boundaries between the regions of stability and instability.; Instability diagrams were obtained for the flexible coupler of a four bar mechanism as well as for the flexible coupler of a slider crank. Consideration of only the first two modes of vibration provides sufficient accuracy.; It was shown that the variational equations of a system of nonlinear O.D.E.'s, which control the stability of its solutions, coincide with the corresponding system of linear O.D.E.'s. Therefore, the resulting instability diagrams are exact according to the nonlinear theory, as well.