\n Flexure-based compliant mechanisms are becoming increasingly promising in precision engineering, robotics, and other applications due to the excellent advantages of no friction, no backlash, no wear, and minimal requirement of assembly. Because compliant mechanisms have inherent coupling of kinematic-mechanical behaviors with large deflections and/or complex serial-parallel configurations, the kinetostatic and dynamic analyses are challenging in comparison to their rigid-body counterparts. To address these challenges, a variety of techniques have been reported in a growing stream of publications. This paper surveys and compares the conceptual ideas, key advances, and applicable scopes, and open problems of the state-of-the-art kinetostatic and dynamic modeling methods for compliant mechanisms in terms of small and large deflections. Future challenges are discussed and new opportunities for extended study are highlighted as well. The presented review provides a guide on how to select suitable modeling approaches for those engaged in the field of compliant mechanisms.

/pdf/kinetostatic-and-dynamic-modeling-of-flexure-based-compliant-1slw5ldbnk.pdf

Kinetostatic and Dynamic Modeling of Flexure-Based Compliant Mechanisms: A Survey

This paper studies analytically and numerically the spring softening and hardening phenomena that occur in electrostatically actuated microelectromechanical systems comb drive resonators utilizing folded suspension beams. An analytical expression for the electrostatic force generated between the combs of the rotor and the stator is derived and takes into account both the transverse and longitudinal capacitances present. After formulating the problem, the resulting stiff differential equations are solved analytically using the method of multiple scales, and a closed-form solution is obtained. Furthermore, the nonlinear boundary value problem that describes the dynamics of inextensional spring beams is solved using straightforward perturbation to obtain the linear and nonlinear spring constants of the beam. The analytical solution is verified numerically using a Matlab/Simulink environment, and the results from both analyses exhibit excellent agreement. Stability analysis based on phase plane trajectory is also presented and fully explains previously reported empirical results that lacked sufficient theoretical description. Finally, the proposed solutions are, once again, verified with previously published measurement results. The closed-form solutions provided are easy to apply and enable predicting the actual behavior of resonators and gyroscopes with similar structures.

Nonlinear Dynamics of Spring Softening and Hardening in Folded-MEMS Comb Drive Resonators

In the past two decades, a large number of robot fishes have been built, though their performances are still far inferior compared with that of the real fish. One of the key reasons is the design. This paper presents a novel robot fish. The key innovation is the combination of an active wire-driven body with a soft compliant tail to accomplish undulatory swimming (i.e., flapping in “S” motion). First, the design is presented. Second, the design and analysis model is given. In particular, it is shown that the compliant tail can be modeled through the proposed multipseudolink model. Then, a prototype and experimental results are presented. The experiment results demonstrate that the kinematics of the robot fish can be predicted by the presented model with reasonable errors. Moreover, it has several advantages; simple in structure, easy to control, capable of high-speed swimming (the maximum speed is 2.15 body lengths/s), and good maneuverability (the average turning speed is 63°/s). Most importantly, it is highly efficient; the swimming performance of the robot fish is close to that of a real fish.

A Novel Robot Fish With Wire-Driven Active Body and Compliant Tail

The platform based vibratory bowl feeders are essential elements in automatic assembly. Taking the bowl feeder as a typical three-legged rigidly connected compliant platform device, this paper applies von Mises' compliance matrix to each of the leaf-spring legs, establishes screw systems of the legs and develops the Jacobian of the platform using the adjoint transformation. Based on the force equilibrium between the supporting and external wrenches and the twist deflection, a platform compliance matrix is proposed as a congruence transformation of the legs' compliance matrices. The matrix is then decomposed into a central compliance matrix and an adjoint transformation, leading to the decomposition of the legs' parameter effect from the platform assembly influence. The analysis presents the necessary and sufficient condition for the existence of the twist deflection that is equivalent to the characteristics equation of the compliant platform. Further based on the eigencompliances and eigentwist decomposition, the legs' parameter effect and the platform assembly parameter influence are identified. This reveals the compliance characteristics of this type of devices and the parameters' effect on the compliance and presents a suitable parameter range for design of the compliant platform device.

Compliance Analysis of a Three-Legged Rigidly-Connected Platform Device

Compliant mechanisms have been used in many engineering areas where high precision and sensitivity are required. One of the major challenges of designing compliant mechanisms lies in understanding and analyzing the nonlinear deflections of flexible members. The pseudo-rigid-body model (PRBM) method, which simplifies the modeling of the nonlinear deflection by approximating it as motion of rigid links, has been accepted as one of the most important tools for synthesis and analysis of compliant mechanisms. In this paper, a review of various PRBMs is presented. The 3R PRBM whose characteristic parameters are independent of external loads is discussed in detail. For the purpose of finding the optimal set of the characteristic parameters for the 3R PRBM, a six-dimensional objective function is formulated by combining the approximation errors of both tip point and tip slope for the two extreme load cases, i.e., pure moment load and pure vertical force load. A particle swarm optimizer was employed to conduct a continuous search on the objective function. The resulting 3R PRBM with the optimized characteristic parameters shows better performance in predicting large deflections of cantilever beams over the original 3R PRBM.

Finding the optimal characteristic parameters for 3R pseudo-rigid-body model using an improved particle swarm optimizer

Based on the principle of dynamic equivalence, a new dynamic model of compliant mechanisms is developed using the pseudo-rigid-body model. The dynamic equation of general planar compliant mechanisms is derived. The natural frequency of a compliant mechanism is obtained in the example of a planar compliant parallel-guiding mechanism. The numerical results show the effectiveness and advantage of the proposed method compared with the methods of FEA and flexible mechanisms.

Yue-Qing Yu

Papers

Dynamic Modeling of Compliant Mechanisms Based on the Pseudo-Rigid-Body Model