Modeling Of Compliant Mechanisms Research Articles

Graphic Transfer Matrix Method for Kinetostatic and Dynamic Analyses of Compliant Mechanisms

Abstract Kinetostatic and dynamic analyses of compliant mechanisms with complex configurations continue to be an attractive issue for obtaining a process-concise and result-accurate solution. In this paper, the transfer matrix method (TMM) is improved for a unified linear kinetostatics and dynamic modeling of compliant mechanisms with complex serial-parallel configurations in an oriented graphic way. In detail, the transfer matrices of typical building blocks commonly used in compliant mechanisms are summarized and derived. Then, a graphic transfer matrix modeling procedure capturing both the kinetostatics and dynamics of general compliant mechanisms is introduced. The displacement amplification ratio, input/output stiffness, parasitic error, natural frequencies, and frequency response of a typical compliant microgripper and a planar parallel three-degrees-of-freedom (3DOF) nanopositioner are calculated with such a graphic transfer matrix method. The advantages of the proposed modeling method lie in its convenience and uniformity in formulating both the kinetostatic and dynamic behaviors of a class of compliant mechanisms with distributed and lumped compliances in a transfer matrix manner, which has minimal DOF and is easily programmed.

A Substructure Condensed Approach for Kinetostatic Modeling of Compliant Mechanisms with Complex Topology.

Compliant mechanisms with complex topology have previously been employed in various precision devices due to the superiorities of high precision and compact size. In this paper, a substructure condensed approach for kinetostatic analysis of complex compliant mechanisms is proposed to provide concise solutions. In detail, the explicit relationships between the theoretical stiffness matrix, element stiffness matrix, and element transfer matrix for the common flexible beam element are first derived based on the energy conservation law. The transfer matrices for three types of serial–parallel substructures are then developed by combining the equilibrium equations of nodal forces with the transfer matrix approach, so that each branch chain can be condensed into an equivalent beam element. Based on the derived three types of transfer matrices, a kinetostatic model describing only the force-displacement relationship of the input/output nodes is established. Finally, two typical precision positioning platforms with complex topology are employed to demonstrate the conciseness and efficiency of this modeling approach. The superiority of this modeling approach is that the input/output stiffness, coupling stiffness, and input/output displacement relations of compliant mechanisms with multiple actuation forces and complex substructures can be simultaneously obtained in concise and explicit matrix forms, which is distinct from the traditional compliance matrix approach.

Dynamics of Compliant Mechanisms Using Transfer Matrix Method

In this paper, a new closed-form dynamic model describing vibration characteristics of compliant mechanisms is proposed by using transfer matrix method. The advantage of the method is that it can simply and accurately describe the dynamic performance without the need for overall dynamic equation of the system, which is different from previous the dynamic modeling methods of compliant mechanisms. The vibration mechanical model of compliant mechanisms is established at first and the transfer matrices of the compliant mechanisms with serial and parallel configuration are derived. The transfer matrices of planar elliptic flexure hinges and vibration rigid bodies are deduced in detail according to the basic idea of the free vibration characteristics of elements. Finally, the dynamic performances of compliant mechanisms, such as the natural frequency, vibration model, and frequency response, are analyzed utilizing the established model. A comparison of the method with the finite element method for three typical compliant mechanisms discovers a maximum error of the natural frequency is less than 4.65%, which verifies the feasibility of the presented approach.

Viscoelastic Snapping Metamaterials

Abstract Mechanical metamaterials are artificial composites with tunable advanced mechanical properties. Particularly, interesting types of mechanical metamaterials are flexible metamaterials, which harness internal rotations and instabilities to exhibit programable deformations. However, to date, such materials have mostly been considered using nearly purely elastic constituents such as neo-Hookean rubbers. Here, we experimentally explore the mechanical snap-through response of metamaterials that are made of constituents that exhibit large viscoelastic relaxation effects, encountered in the vast majority of rubbers, in particular, in 3D printed rubbers. We show that they exhibit a very strong sensitivity to the loading rate. In particular, the mechanical instability is strongly affected beyond a certain loading rate. We rationalize our findings with a compliant mechanism model augmented with viscoelastic interactions, which qualitatively captures well the reported behavior, suggesting that the sensitivity to the loading rate stems from the nonlinear and inhomogeneous deformation rate, provoked by internal rotations. Our findings bring a novel understanding of metamaterials in the dynamical regime and open up avenues for the use of metamaterials for dynamical shape-changing as well as vibration and impact damping applications.

Topology optimization of a 6-DOF spatial compliant mechanism based on Stewart propotype platform

To improve the global stiffness and conveniently build a model of a compliant mechanism with spatial multiple degrees of freedom (DOF), the topology optimization method, combined with the isomorphic mapping matrix, is proposed in this paper for structure synthesis of a 6-DOF spatial compliant mechanism. By using the differential approximation method, the Jacobian matrix of the Stewart prototype platform is calculated as the isomorphic mapping matrix, and its eigenvalues and eigenvectors are considered. Combining the isomorphic mapping matrix with the solid isotropic material with the penalization topology optimization method, the topological model of the 6-DOF spatial compliant mechanism is constructed, and a topological structure of the 6-DOF spatial compliant mechanism is derived which has the same differential kinematic characteristics as the Gough–Stewart prototype platform. Piezoelectric actuators are mounted inside the topological structure during the three-dimensional printing manufacturing process, and its driver directions are in accordance with the driver configuration directions of the Gough–Stewart prototype platform. The effectiveness of the proposed method for topological structure synthesis of the 6-DOF spatial compliant mechanism is demonstrated through several numerical examples and experimental studies.

Topology design and analysis of compliant mechanisms with composite laminated plates

To improve the small deformation and high stress level in hinge zones of compliant mechanisms with isotropic material, a topology optimization method of compliant mechanisms with composite laminated plates was proposed. Based on the anisotropy and designability of composite laminated plates, a topology optimization model of compliant mechanisms with composite laminated plates was built to maximize the deformable capability. Numerical examples of designing compliant inverters and grippers were investigated to demonstrate the effectiveness of the proposed method. The influence mechanism of layer sequences on topologic shapes, deformation and loading capability were also discussed. The results showed that the deformable capability and stress levels of compliant mechanisms with composite laminated plates were further improved by a reasonable configuration of layer sequences.

A Pseudo-Static Model for Dynamic Analysis on Frequency Domain of Distributed Compliant Mechanisms

This paper presents a pseudo-static modeling methodology for dynamic analysis of distributed compliant mechanisms to provide accurate and efficient solutions. First, a dynamic stiffness matrix of the flexible beam is deduced, which has the same definition and a similar form as the traditional static compliance/stiffness matrix but is frequency dependent. Second, the pseudo-static modeling procedure for the dynamic analysis is implemented in a statics-similar way based on D'alembert's principle. Then, all the kinematic, static and dynamic performances of compliant mechanisms can be analyzed based on the pseudo-static model. The superiority of the proposed method is that when it is used for the dynamic modeling of compliant mechanisms, the traditional dynamic modeling procedures, such as calculation of the elastic and kinetic energies as well as using Lagrange's equation, are avoided and the dynamic modeling is converted to a statics-similar problem. Comparison of the proposed method with an elastic-beam-based model in previous literature and finite element analysis for an exemplary XY precision positioning stage reveals its high accuracy and easy operation.

The Honeybee's Protrusible Glossa is a Compliant Mechanism

AbstractMany biological structures can perform highly-dexterous actions by using dynamic surfaces. To deal with the contradictive demands of high feeding efficiency and low energy expenditure during nectar feeding, the glossal surface of a honeybee undergoes shape changes, in which glossal hairs erect together with segment elongation in a drinking cycle. In this paper, we extracted a transmission link embedded in the glossa from postmortem examination and found that the compliance of the intersegmental membranes provides more possibilities for this highly kinematic synchronicity. According to the morphing phenomena of honeybee's glossa, we proposed a compliant mechanism model to predict the deformation behavior of honeybee considering elastic properties of the glossal intersegmental membranes. The increase of membrane stiffness may improve the capacity of elastic potential energy transfer, but will still result in the increase of mass. An index is introduced to evaluate the contradiction for optimizing structural parameters. This work may arouse new prospects for conceptual design of micro-mechanical systems equipped with bio-inspired compliant mechanisms.

Stress‐based multi‐material topology optimization of compliant mechanisms

AbstractIn this paper, a level‐set‐based method is presented to deal with the multi‐material topology optimization of compliant mechanisms with stress constraints. A novel stress‐based multi‐material topology optimization model of compliant mechanisms is proposed. In this model, the multi‐material level set topology description model and the separable stress interpolation scheme are adopted. The weighted sum method is used to deal with the multi‐objective optimization of the output displacement and compliance of compliant mechanisms. The penalty of stresses is also considered in the objective function to control the local stress level in different materials. To solve the optimization problem, the parametric level set method is employed, and the sensitivity analysis is conducted. Application of the method is demonstrated by 2 numerical examples. Results show that the multi‐material structures without undesirable de facto hinges can be obtained. The output displacement and compliance of the compliant mechanisms are optimized, and stress constraints in different materials are simultaneously satisfied.

Modeling and optimization of magnetostrictive actuator amplified by compliant mechanism

Magnetostrictive actuators are commonly used in precision engineering with the advantages of high resolution and fast response. Their limited strokes are always amplified by compliant mechanisms without wear and backlash. This paper proposes a hybrid model for the actuation system considering the coupling of the actuator and the amplifier. The magnetostrictive model, based on the Jiles–Atherton model, is related to the input stiffness of the amplifier when quantifying the magneto-mechanical effects, including stress-dependent magnetization, stress-dependent magnetostriction and ΔE effect. The compliant mechanism model aims at constructing the flexibility matrix with the amplification ratio and input stiffness related to the spring factor of the load. The deformation and structural stress of the amplifier are also dependent on the output strain of magnetostrictive material. Experiments under both free load and spring load conditions have been done to verify the effectiveness of the hybrid model. The proposed model is suitable for parameter optimization and the performance indicators can be precisely quantified. Optimization based on hybrid model is more preferred than optimizing the actuator and amplifier independently for maximum output displacement. Furthermore, ‘stiffness match principle’ is no longer applicable when considering ΔE effect, and the optimal external stiffness problem can be numerically solved by the hybrid model for maximum output energy of magnetostrictive material.

Analysis and design of an underactuated compliant five-bar mechanism

Generally, the pseudo-rigid-body model of compliant mechanisms is single degree-of-freedom in the literature. In order to obtain more flexibility, compliant mechanisms can also be synthesized from multi degree-of-freedom pseudo-rigid-body models. In this study, analysis and design procedures for such a compliant five-bar mechanism are presented with underactuation. Analysis is performed with different loading scenarios. An approximate method for dead center estimation is introduced. Some results are verified by using finite element analysis. A generalized design procedure for the mechanism is proposed. An advantage of underactuation over constrained mechanisms is presented with an example.

Flexure Hinge Mechanisms Modeled by Nonlinear Euler‐Bernoulli‐Beams

AbstractA flexure hinge is an innovative engineering solution for providing relative motion between two adjacent stiff members by the elastic deformation of an arbitrary shaped flexible connector. In the literature, modeling of compliant mechanisms incorporating flexure hinges is mainly focused on linear methods. However, geometrically nonlinear effects cannot be ignored generally. This study presents a nonlinear modeling technique for flexure hinges based on the Euler‐Bernoulli beam theory, in contrast to the predominant linear modeling approaches. Higher order beam elements of variable cross‐section are employed to model the flexure hinge region. A Newton‐Raphson scheme is applied to solve the resulting nonlinear system equations. The proposed approach reduces the overall degrees of freedom and is computationally efficient compared to commonly applied 3D finite element methods. A compliant displacement amplification mechanism is studied by means of the proposed method, where an excellent agreement with results of a reference solution is achieved. The modeling approach is suitable for the structural optimization of compliant mechanisms towards a less intuitive design process. (© 2015 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Dynamic modeling and model order reduction of compliant mechanisms

In this work, a novel approach towards statical and dynamical modeling of compliant mechanisms is presented which serves as an origin for model order reduction procedures to provide small, efficient and accurate approximations. As 3D finite element modeling of compliant mechanisms results in very large-scale systems, both model reduction and real-time controlling of the mechanism are not possible. This and the fact that in compliant mechanisms mostly the flexure hinges contribute to the overall performance motivates a procedure which is based on partitioning the structure into elastically deformable hinges and rigid linkages. Unlike common modeling techniques, the reproduction of the non-negligible nonlinear behavior is assured and contributes to precise approximation as well as the fact that not only the flexure itself deforms but also adjacent structures. Furthermore, the proposed methodology is applicable to all kinds of flexure hinges with concentrated compliances and compliant mechanisms of complex geometric shape as well as spatial loading cases.First, an analysis of the inserted flexure hinges yields the significantly deformed region of which corresponding master models with connecting nodes are created. With their geometric properties the mechanism is further divided into remaining stiff sections. Their spatial centroid location and moments of inertia about mass centroid are calculated and according point masses are generated. A finite element model of the mechanism is then developed by rigidly linking the master models with the point masses. Therewith, an accurate 3D model with appreciable less degrees of freedom arises, named significant region model. The system matrices K, M in combination with the input matrix B and the output matrix C yield a closed-form description of the mechanism.In a second step, a considerable decrease of system size is performed by applying modern model order reduction techniques, namely Krylov subspace reduction. A comparison of the new two step approach with full 3D finite element modeling reveals only marginal deviations of less than 6% regarding the static displacements and less than 0.0001% relating to the frequency response of an exemplary mechanism with a substantial lower number of degrees of freedom.

Nonlinear modeling of compliant mechanisms incorporating circular flexure hinges with finite beam elements

Modeling of compliant mechanisms incorporating flexure hinges is mainly focused on linear methods. However, geometrically nonlinear effects cannot be ignored generally. This work shows that nonlinear behavior plays an important role in the deformation and stress analysis, which consequently impacts the design of compliant mechanisms. In this study a nonlinear higher order finite beam element based modeling approach is presented strongly reducing the computation time of nonlinear models. Planar deformation and mechanical stress of a single circular flexure hinge under a wide range of loads is modeled and computed with the proposed approach. A comparison with a 3D-nonlinear finite element model shows very good agreement and validates the beam model. It is shown that the linear and nonlinear deformation behavior of a single flexure hinge deviate marginally so that linear modeling approaches are sufficient. Furthermore a planar displacement amplification mechanism incorporating circular flexure hinges is studied by means of the same method highlighting the distinct deviation of the behavior of the geometrically nonlinear model from its linear prediction. In conclusion the nonlinear behavior at the system level can not longer be neglected. Finally, a study shows that different designs of the displacement amplification mechanism are achieved when linear or nonlinear modeling approaches are applied.

A comparative analysis of joint clearance effects on articulated and partly compliant mechanisms

Clearance is inevitable in the articulated mechanisms due primarily to the design, manufacturing and assembly processes or a wear effect. This phenomenon affects the kinematic and dynamic performances of mechanism negatively. Compliant mechanism, which consists of at least one flexible member along with the conventional rigid links, becomes a favorable choice to decrease the number of movable joints and also their clearance effects. In this study, conventional and compliant slider–crank mechanisms having joints with clearance are used to investigate and compare the effects of joint clearance. Pseudo-rigid-body model of compliant mechanism is constituted. For the case of different clearance sizes and running speeds, kinematic and dynamic performances of mechanisms are compared to each other. The results show that the joint clearance leads to chaotic behavior on kinematic and dynamic outputs of mechanism. The flexibility of small-length flexural pivot, that is, pseudo-joint, has clear suspension effects to decrease the undesired reflections of joints clearance on the system outputs. Also, this pseudo-joint constitutes a force-closed kinematic pair behavior between journal and bearing in joint having clearance. This leads to continuous contact mode by preventing the separation of journal and bearing parts.

Reliability-Based Topology Optimization of Compliant Mechanisms with Geometrically Nonlinearity

A new reliability-based topology optimization method for compliant mechanisms with geometrical nonlinearity is presented. The aim of this paper is to integrate reliability and geometrical nonlinear analysis into the topology optimization problems. Firstly, geometrical nonlinear response analysis method of the compliant mechanisms is developed based on the Total-Lagrange finite element formulation, the incremental scheme and the Newton-Raphson iteration method. Secondly, a multi-objective topology optimal model of compliant mechanisms considering the uncertainties of the applied loads and the geometry descriptions is established. The objective function is defined by minimum the compliance and maximum the geometric advantage to meet both the stiffness and the flexibility requirements, and the reliabilities of the compliant mechanisms are evaluated by using the first order reliability method. Thirdly, the computation of the sensitivities is developed with the adjoint method and the optimization problem is solved by using the Method of Moving Asymptotes. Finally, through numerical calculations, reliability-based topology designs with geometric nonlinearity of a typical compliant micro-gripper and a multi-input and multi-output compliant sage are obtained. The importance of considering uncertainties and geometric nonlinearity is then demonstrated by comparing the results obtained by the proposed method with deterministic optimal designs, which shows that the reliability-based topology optimization yields mechanisms that are more reliable than those produced by deterministic topology optimization.

A numerical method for static analysis of pseudo-rigid-body model of compliant mechanisms

This paper presents a numerical method for analyzing the pseudo-rigid-body model of compliant mechanisms based on finite elements and the principle of minimum potential energy. The proposed method represents the links of pseudo-rigid-body model with truss elements. As a result, the pseudo-rigid-body model is modeled into a compliant system that consists of finite elements and springs. The static equilibrium position of the pseudo-rigid-body model can be obtained by minimizing the potential energy function of this compliant system. A comparison between the proposed method and the MinPE method is presented. Lastly, a case study is provided to demonstrate the application of this method in the automated analysis of pseudo-rigid-body models. This numerical method paves the way for introducing the topology optimization techniques into the synthesis of flexure-based compliant mechanisms.

New PR Pseudo-rigid-body Model of Compliant Mechanisms Subject to Combined Loads

Based on the pseudo-rigid-body model(PRBM),a new 2-Degree of freedom pseudo-rigid-body model with a prismatic pair is proposed.The axial displacement and lateral displacement is presented by the prismatic pair and revolute pair,respectively,and so the deflection of a compliant link subject to a combined force and moment loads can be simulated.The PR pseudo-rigid-body model of planar compliant mechanism is developed.A three-dimensional optimization for the characteristic radius factors and a linear regression method for the spring stiffness coefficients are presented.The advantage of the PR pseudo-rigid-body model is illustrated numerically through the comparison with the 1R,2R PRBM models and compliant link.

Design of computer experiments applied to modeling of compliant mechanisms for real-time control

This article discusses the use of design of computer experiments (DOCE) (i.e., experiments run with a computer model to find how a set of inputs affects a set of outputs) to obtain a force–displacement meta-model (i.e., a mathematical equation that summarizes and aids in analyzing the input–output data of a DOCE) of compliant mechanisms (CMs). The procedure discussed produces a force–displacement meta-model, or closed analytic vector function, that aims to control CMs in real-time. In our work, the factorial and space-filling DOCE meta-model of CMs is supported by finite element analysis (FEA). The protocol discussed is used to model the HexFlex mechanism functioning under quasi-static conditions. The HexFlex is a parallel CM for nano-manipulation that allows six degrees of freedom (x, y, z, θ x , θ y , θ z ) of its moving platform. In the multi-linear model fit of the HexFlex, the products or interactions proved to be negligible, yielding a linear model (i.e., linear in the inputs) for the operating range. The accuracy of the meta-model was calculated by conducting a set of computer experiments with random uniform distribution of the input forces. Three error criteria were recorded comparing the meta-model prediction with respect to the results of the FEA experiments by determining: (1) maximum of the absolute value of the error, (2) relative error, and (3) root mean square error. The maximum errors of our model are lower than high-precision manufacturing tolerances and are also lower than those reported by other researchers who have tried to fit meta-models to the HexFlex mechanism.

Dynamic Analysis and Design of Compliant Mechanisms Using the Finite Element Method

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.