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-crossing or curve veering, and a flexible mechanism can be analyzed for each mode efficiently.
Also an efficient stability analysis method for flexible mechanisms is developed by using single-link modal coordinates. Moreover, the relations between the instability conditions and a time-varying natural frequency are derived from the detailed investigation on the monodromy matrix. It is found that the average of a time-varying natural frequency is the key parameter for the stability analysis and unstable regions can be determined through doing the stability analysis only at a few speeds satisfying $Ω=ω_{avg}/m$ or $Ω=ω_{avg}/(m+1/2)$. Also the ranges of unstable regions are determined from the derived relations without doing the secondary analysis in the neighborhood of $Ω=ω_{avg}/m$ or $Ω=ω_{avg}/(m+1/2)$.
From the numerical simulations, the proposed method showed both its numerical efficiency and its accuracy for stability analysis of a flexible mechanism, compared with several existing methods.
The experimental study from a flexible four-bar mechanism was also performed to verify the proposed method. From the numerical and experimental results, it can be concluded that the proposed method shows fairly good prediction of the dynamic responses and critical speeds of flexible mechanisms.
