A parameter optimization framework for determining the pseudo-rigid-body model of cantilever-beams

Analysis of large-displacement compliant mechanisms using an incremental linearization approach

Design of a variable-diameter wheel with a novel distributed compliant mechanism

An intensive study on the dynamic modeling and analysis of compliant mechanisms is presented in this paper based on the pseudo-rigid-body model. The pseudo-rigid-body dynamic model with single degree-of-freedom is proposed at first and the dynamic equation of the 1R pseudo-rigid-body dynamic model for a flexural beam is presented briefly. The pseudo-rigid-body dynamic models with multi-degrees-of-freedom are then derived in detail. The dynamic equations of the 2R pseudo-rigid-body dynamic model and 3R pseudo-rigid-body dynamic model for the flexural beams are obtained using Lagrange equation. Numerical investigations on the natural frequencies and dynamic responses of the three pseudo-rigid-body dynamic models are made. The effectiveness and superiority of the pseudo-rigid-body dynamic model has been shown by comparing with the finite element analysis method. An example of a compliant parallel-guiding mechanism is presented to investigate the dynamic behavior of the mechanism using the 2R pseudo-rigid-body dynamic model.

Compliant mechanisms are single-piece flexible structures that deliver the desired motion by undergoing elastic deformation as opposed to jointed rigid body motions of conventional mechanisms. Compliance in design leads to jointless, no-assembly (Fig. 1), monolithic mechanical devices and is particularly suited for applications with small range of motions. The compliant windshield wiper shown in Fig. 1 illustrates this paradigm of no-assembly. Conventional flexural mechanisms employ flexural joints that connect relatively rigid links as depicted in Fig. 2. Reduced fatigue life, high stress concentration and difficulty in fabrication are some of the drawbacks of flexural joints. Our focus is on designing compliant mechanisms with distributed compliance which employs flexural links (see Fig. 3) and have no joints (neither pin nor flexural joints) for improved reliability, performance, and ease of manufacture. Distributed compliant mechanisms derive their flexibility due to topology and shape of the material continuum rather than concentrated flexion at few regions. This paper focuses on the unique methodology employed to design jointless mechanisms with distributed compliance. The paper also illustrates a compliant stroke amplification mechanism that was recently designed, fabricated and tested for MEMS application.

Dielectric Elastomers (DE) are incompressible dielectrics which can experience deviatoric (isochoric) finite deformations in response to applied large electric fields. Thanks to the strong electro-mechanical coupling, DE intrinsically offer great potentialities for conceiving novel solid-state mechatronic devices, in particular linear actuators, which are more integrated, lightweight, economic, silent, resilient and disposable than equivalent devices based on traditional technologies. Such systems may have a huge impact in applications where the traditional technology does not allow coping with the limits of weight or encumbrance, and with problems involving interaction with humans or unknown environments. Fields such as medicine, domotic, entertainment, aerospace and transportation may profit. For actuation usage, DE are typically shaped in thin films coated with compliant electrodes on both sides and piled one on the other to form a multilayered DE. DE-based Linear Actuators (DELA) are entirely constituted by polymeric materials and their overall performance is highly influenced by several interacting factors; firstly by the electromechanical properties of the film, secondly by the mechanical properties and geometry of the polymeric frame designed to support the film, and finally by the driving circuits and activation strategies. In the last decade, much effort has been focused in the devolvement of analytical and numerical models that could explain and predict the hyperelastic behavior of different types of DE materials. Nevertheless, at present, the use of DELA is limited. The main reasons are 1) the lack of quantitative and qualitative models of the actuator as a whole system 2) the lack of a simple and reliable design methodology. In this thesis, a new point of view in the study of DELA is presented which takes into account the interaction between the DE film and the film supporting frame. Hyperelastic models of the DE film are reported which are capable of modeling the DE and the compliant electrodes. The supporting frames are analyzed and designed as compliant mechanisms using pseudo-rigid body models and subsequent finite element analysis. A new design methodology is reported which optimize the actuator performances allowing to specifically choose its inherent stiffness. As a particular case, the methodology focuses on the design of constant force actuators. This class of actuators are an example of how the force control could be highly simplified. Three new DE actuator concepts are proposed which highlight the goodness of the proposed method.

Compliant mechanisms utilize the elastic deformation of their constituent members to transfer forces or displacements. Compliant mechanisms are advantageous in robotic applications because of their simplified manufacturing, monolithic structure, high precision, low wear and backlash, and flexibility to accommodate a wide range of actuation schemes. Despite these significant advantages, modeling these mechanisms is challenging because of their complex and large nonlinear deformations. Currently, most modeling methods for compliant mechanisms are either kinematic or kinetostatic, such as the pseudo-rigid body method (PRBM), beam constraint model (BCM), and load-flow-based method. These methods are mostly limited to bending deformation and do not consider shear or stretch strain. Furthermore, they are not conducive to modeling compliant mechanisms with embedded actuators in the geometry of the mechanism. To overcome these limitations, here we adopt Cosserat rod modeling, which captures the generalized deformations of the constituent members of compliant mechanisms. Furthermore, it allows the modeling of compliant mechanisms with actuation members. Several compliant mechanism experimental prototypes, including compliant inverters and a compliant gripper, were presented to validate and depict the effectiveness of the Cosserat rod modeling. This modeling approach enables large-scale and efficient modeling of the complaint mechanisms with actuators and sensors.

Abstract. PRBMs (pseudo-rigid-body models) have been becoming important engineering technologies/methods in the field of compliant mechanisms to simplify the design and analysis through the use of the knowledge body of rigid-body mechanisms coupling with springs. This article addresses the PRBMs of spatial multi-beam modules for planar motion, which are composed of three or more symmetrical wire/slender beams parallel to each other where the planar twisting DOF (degree of freedom) is assumed to be very small for specific applications/loading conditions. Simplified PRBMs are firstly proposed through replacing each beam in spatial multi-beam module with a rigid-body link plus two identical spherical joints at its two ends. The characteristics factor, bending stiffness and twisting stiffness for the spherical joint are determined. Load-displacement equations are then derived for a class of spatial multi-beam modules and general spatial multi-beam modules using the virtual work principle and kinematic relationships. Finally, nonlinear FEA (finite element analysis) is employed with comparisons with the PRBMs. The present PRBMs have shown the ability to predict the primary nonlinear constraint characteristics such as load-stiffening effect, cross-axis coupling in the two primary translational directions and buckling load.

Compliant Mechanisms (CMs) are currently employed in several engineering applications requiring high precision and reduced number of parts. For a given mechanism topology, CM analysis and synthesis may be developed resorting to the Pseudo–Rigid Body (PRB) method, in which the behavior of flexible members is approximated via a series of rigid links connected by spring-loaded kinematic pairs. From a CM analysis standpoint, the applicability of a generic PRB model requires the determination of the kinematic pairs’ location and the stiffness of a set of generalized springs. In parallel, from a design standpoint, a PRB model representing the kinetostatic behavior of a flexible system should allow to compute the flexures’ characteristics providing the desired compliance. In light of these considerations, this paper describes a Computer-Aided Design/Engineering (CAD/CAE) framework for the automatic derivation of accurate PRB model parameters, on one hand, and for the shape optimization of complex-shape flexures comprising out-of-plane displacements and distributed compliance. The method leverages on the modelling and simulation capabilities of a parametric CAD (i.e. PTC Creo) seamlessly connected to a CAE tool (i.e. RecurDyn), which provides built-in functions for modelling the motion of flexible members. The method is initially validated on an elementary case study taken from the literature. Then, an industrial case study, which consists of a spatial crank mechanism connected to a fully-compliant four-bar linkage is discussed. At first, an initial sub-optimal design is considered and its PRB representation is automatically determined. Secondly, on the basis of the PRB model, several improved design alternatives are simulated. Finally, the most promising design solution is selected and the dimensions of a flexure with non-trivial shape (i.e. hybrid flexure) is computed. This technique, which combines reliable numerical results to the visual insight of CAD/CAE tools, may be particularly useful for analyzing/designing spatial CMs composed of complex flexure topologies.

Significant reduction in cost and time of bistable mechanism design can be achieved by understanding their bistable behavior. This paper presents bistable compliant mechanisms whose pseudo-rigid-body models (PRBM) are four-bar mechanisms with a torsional spring. Stable and unstable equilibrium positions are calculated for such four-bar mechanisms, defining their bistable behavior for all possible permutations of torsional spring locations. Finite Element Analysis (FEA) and simulation is used to illustrate the bistable behavior of a compliant mechanism with a straight compliant member, using stored energy plots. These results, along with the four-bar and the compliant mechanism information, can then be used to design a bistable compliant mechanism to meet specified requirements.

Compliant mechanisms have been defined as mechanical systems that derive some or all of their mobility from the flexibility of one or more of their members. In general, due to the large-deflection nature of the compliant members, analysis and synthesis of compliant mechanisms have typically required a numerical iterative solution method. Recently, analytical loop-closure synthesis techniques have been extended to the design of compliant mechanisms, by utilizing the concept of an equivalent pseudo-rigid-body model for a compliant mechanism. These developments have greatly simplified the synthesis of compliant mechanisms. These mechanisms have been synthesized for three precision points using analytical and graphical methods. In rigid-link mechanism design, additional precision points may be incorporated by using Burmester theory. With this enhancement as the motivation, this paper presents the first use of Burmester theory to design compliant mechanisms for four and five precision points. This is facilitated by use of the concept of an equivalent pseudo-rigid-body model. Burmester theory yields a linear set of kinematic equations, allowing for their easy utilization herein. Also for some design problems, this theory can provide a graphical representation of all possible solutions for compliant mechanisms. In addition to the specification of precision points for compliant mechanisms, input torque and/or potential energy values may also be prescribed.

Bistable mechanical devices remain stable in two distinct positions without power input. They find application in valves, switches, closures, and clasps. Mechanically bistable behavior results from the storage and release of energy, typically in springs, with stable positions occurring at local minima of stored energy. Compliant mechanisms offer an elegant way to achieve this behavior by incorporating both motion and energy storage into the same flexible element. Interest in compliant bistable mechanisms has also recently increased because of the advantages of bistable behavior in many micro-electro-mechanical systems (MEMS). Design of compliant or rigid-body bistable mechanisms typically requires simultaneous consideration of both energy storage and motion requirements. This paper simplifies this process by developing theory that provides prior knowledge of mechanism configurations that guarantee bistable behavior. Configurations which include one or more translational, or slider, joints are studied in this work. Several different mechanism types are analyzed to determine compliant segment placement that will ensure bistable mechanism operation. Examples demonstrate the power of the theory in design.

This paper proposes a new compliant mechanism for the stroke amplification of piezoelectric actuators. Compared with the existing ones, this compliant stroke amplification mechanism (CSAM) has several advantages including lightweight, less stress concentration, large motion range, as well as ease of manufacture in macro- and micro-scales, mainly due to the completely distributed compliance and largely reduced moving mass. Kinetostatic models of the CSAM, with consideration of the payloads applied on the output stage, are derived for quick and insightful determinate analysis, which enables optimization of the associated parameters to achieve different objectives. The results obtained from the kinetostatic models are in reasonable agreement with the FEA simulation results, with less than 3.8% difference when the input displacement is less than 10 µm. An optimization method, based on the analytical models, is introduced and employed to increase the amplification ratio of the CSAM by up to 240%. The optimized CSAM is an excellent candidate for the stroke amplification of piezoelectric actuators, which is fabricated and experimentally tested. The optimization method is able to be extended to design other compliant mechanisms with optimized size, shape and topology configuration simultaneously.

Compliant mechanisms gain at least some of their mobility from the deflection of flexible members rather than from movable joints only. Dynamic effects are very important to improving the design of compliant mechanisms. An investigation on the dynamics and synthesis of the compliant mechanisms is presented. The dynamic model of compliant mechanisms is developed at first. The natural frequency and sensitivity are then studied based on the dynamic model. Finally, optimal design of compliant mechanism is investigated. The experimental study of natural frequency is performed. The comparison between the experiment results and the theoretical results verifies the validity of the experiment system and theoretical model.

Inspired by nature’s multi-material compliant designs and the capability of modern manufacturing methods to make multi-material components without assembly, and motivated by the versatility and enhanced functionality with multi-material designs, we formalize the notion and the design of heterogeneous compliant mechanisms. In the topology optimization of compliant mechanisms, a fictitious density function is often used for material interpolation to overcome the computational difficulties encountered in the large “0-1” type integer programming problem. In this paper, we illustrate that a gradually formed continuous peak function can be used for material interpolation. One of the advantages of introducing the peak function is that multiple materials can easily be incorporated into the topology optimization without increasing the number of design variables. By using the peak function and the optimality criteria method, we synthesize compliant mechanisms with multiple materials with and without the material resource constraint. The numerical examples include the two, three, and four material designs where void is treated as one material. This new design method enables us to optimally juxtapose stiff and flexible materials in heterogeneous compliant mechanisms that can be built using a variety of modern manufacturing methods.

Compliant mechanisms have shown a great deal of potential, in just a few decades of its development, in providing innovative solutions to design problems. However, their use has been limited due to challenges associated with the materials. With ever increasing focus on the applications of compliant mechanisms, it is necessary to find alternatives to the existing material usage and methods of prototyping. This paper presents a methodology for the design of compliant segments and compliant mechanisms with improved creep resistance and fatigue life properties using the current state-of-the-art materials. The methodology proposes using a stronger material at the core of a softer casing. The paper provides an equivalent pseudo-rigid-body model and a closed-form elliptic integral formulation for a fixed-free compliant segment with an insert. The equivalent pseudo-rigid-body model is verified experimentally for the prediction of beam end point displacements. The paper also presents experimental results that show improvements obtained in the creep recovery properties as expected using the proposed design philosophy.