Dynamic modeling and stability analysis for an elastic four-bar mechanism

Abstract

As operating speed increases and link weight decreases, inertia forces acting on links become larger and link deformations cannot be neglected. The dyanmic equations of motion for flexible mechanisms result in linear differential equations with periodically time-varying coefficients, due to the change of mechanism configuration. Unlike time-invariant system, the time-varying system have parametric resonant characterisitcs. Even in operating speed ranges lower than the fundamental natural frequency, flexible mechanisms can experience severe vibrations and critical damage to their parts. Therefore, determining critical speeds has drawn a great deal of attention and are still problems of interest in the mechanism design. In this work, a dynamic model is developed for flexible mechanisms by using continuous modeling approach, which give insight the relation between the dynamic characterisitics of a mechanism and single-link parameters. Furthermore an efficient and accurate stability analysis method is presented by using the relation between instability condition and time-varying natural frequency. Flexible links are treated as continuous systems with infinite degree of freedom and axial foreshortening effect as well as joint mass is considered for accurate analysis. The effect of nonlinear terms on dynamic equations is investigated and a linearized model for flexible mechanism is verified through parametric study. The resulting equations of a flexible mechanism show that the dynamic characteristics of a mechanism can be considered the combinations of those of individual links with the inertial coupling due to crank flexibility. So, single-link modes such as coupler mode, follower mode are suggested for mechanism modes. Based on single-link modes the modal properties of a flexible mechanism are obtained from the second order perturbation method. The main advantages of using these single-link modes include that mode shapes are time-invariant, irrespective of mode-c...