Compliant Mechanism Design Using a Strain Based Topology Optimization Method

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

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.

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.

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.

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.

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.

The design of compliant mechanisms has been aided by the development of pseudo-rigid-body models to predict the motion of flexible members undergoing large displacements. Many of these models are based on the fact that the end of a cantilever beam follows a near-circular path when planar loads are applied. This paper shows that the application of 3-dimensional end-loading causes a beam to follow a near-spherical path, even for beams with non-circular cross-sections. A 3D pseudo-rigid-body model is presented that allows the motion of an end-loaded rectangular beam to be predicted using a rigid link and a spherical joint. Two sets of deflection limits for 0.5% error are presented and shown to be dependent upon the aspect ratio of the cross-section of the beam. The model has the potential for aiding in the design of spatial compliant mechanisms and analysis of planar compliant mechanisms undergoing large out-of-plane motions.

The formulation of design procedures for rigid-body mechanisms has benefited from the application of type-synthesis techniques. Therefore, with modifications to allow for inclusions of compliance, type synthesis is seen as a useful tool in the design of compliant mechanisms. Previous efforts have developed methods that result in a large number of possible design solutions to a given problem. This paper deals primarily with the development of a simplified compliant-mechanism type-synthesis methodology that limits the number of design solutions considered. The techniques are derived by modifying existing compliant mechanism type-synthesis techniques to yield a simpler model with greater pragmatic value.

Compliant mechanisms are extensively employed in many fields due to their advantages over their rigid-body counterparts. They transmit force, motion, and energy through the elastic deformations of their flexible parts, that often consist of beams with uniform or variable cross-section, with straight or initially-curved axis. Although extensive efforts have been made in the analysis of the deflections, fewer studies dealt with the kinematic aspects associated to the deflection problem. Recently, some investigations focused on the behavior of the pole of the displacements, which characterizes the rigid displacements or on the inflection circle for developing pseudo-rigid-body models. In this paper, the instantaneous geometric invariants are applied to the description of the motion generated by straight flexures. The developed analytical formulation gives kinematic insights on the geometric characteristic of motion up to the fourth order. The invariants lead to the determination of fundamental geometric entities, that are the inflection circle, the cubic of stationary curvature and its derivative, the Ball’s point, and the Burmester’s points. In particular, Ball’s and Burmester’s points identify the special points on the plane that approximate straight paths to the third order, and circular paths to the fourth order, respectively. A possible implementation of the geometric invariants to the design of compliant mechanisms is briefly discussed.

Contact-Aided Compliant Cellular Mechanisms (C³M) are compliant cellular structures with integrated contact mechanisms. The focus of the paper is on the design, fabrication, and testing of C³M with curved walls for high strain applications. It is shown that global strains were increased by replacing straight walls with curved walls in the traditional honeycomb structure, while the addition of contact mechanisms increased cell performance via stress relief in some cases. Furthermore, curved walls are beneficial for fabrication at the meso-scale. The basic curved honeycomb cell geometry is defined by a set of variables. These variables were optimized using Matlab and finite element analysis to find the best non-contact and contact-aided curved cell geometries as well as the cell geometry that provides the greatest stress relief. Currently, the most effective contact-aided curved honeycomb cell can withstand global strains approximately 160% greater than the most effective contact-aided, non-curved cell. Four different designs were fabricated via the Lost Mold-Rapid Infiltration Forming (LM-RIF) process. An array of the contact-aided optimized curved cell was then mechanically tested using a custom designed test rig, and the results were found to have a higher modulus of elasticity and lower global strain than the predictions. Despite these discrepancies, a high-strength highstrain cellular structure was developed, for potential use in morphing aircraft applications.

Morphing is a technology with high potential to reduce emissions in aviation, since it enables wings to adapt their shape to operate at a higher efficiency over the full range of flight conditions. This paper is presenting a concept to adapt camber by drooping the nose. The scope is the setup and bench top testing of a full scale wing tip leading edge wind tunnel model with a morphing droop nose. The complete model features a span of 1.3 m and a strong taper from the root to the tip. For completeness, the design approach is covered as well. The design comprises a GFRP skin to be drooped by two compliant mechanisms, which are driven by linear motors. The compliant morphing devices are “designed-through-optimization”, with the optimization algorithms including Simplex optimization for composite compliant skin design, continuum-based and load path representation topology optimization methods for compliant internal substructure design. The compliant mechanism is manufactured by nickel-titanium alloy to allow high strains in the order of several percent, which is shown to be critical in the design of such compliant mechanisms. In order to validate the models, strains within the mechanisms are measured while drooping the nose in the bench top test. This is done after installing the mechanisms into the leading edge skin. It can be shown, that the simulation for the inboard mechanism is close to the experimental results. The comparison of strain levels in the skin and in the mechanism during droop reveals that the stiffness distribution between these two components is quite different. As a result this ratio can be taken into account in future design processes in order to distribute strains more evenly. Moreover the 3D shapes of the morphed and clean skin are measured and their comparison with the target shapes is presented as well. Finally, the bench top tests are a proof of concept for the overall concept and design which resultes in a “go” for the following low speed subsonic wind tunnel tests.

While much has been contributed to techniques for enumerating and identifying rigid-body mechanisms in the past decades, proportionally little has been accomplished in this regard in compliant mechanisms design. This paper deals primarily with identification and discussion of important kinematic properties of compliant mechanisms. To facilitate these appropriate terminology is developed at the very fundamental level. The conventional degrees-of-freedom concept for a rigid-body chain is briefly reviewed. It is then used to help define a compliance number (or degrees-of-compliance) concept for characterizing compliant mechanisms. Finally, a systematic and convenient approach is presented, enabling the type synthesis of this class of mechanisms.

The analysis and design of compliant mechanisms, undergoing large (geometrically nonlinear) deflections, have been assisted by the Newton-Raphson method to find the loads which satisfy a prescribed set of force and displacement boundary conditions. This paper introduces a graphical, user-driven Newton-Raphson technique that allows easy access to good initial design variable estimates, and subsequently accurate and expeditious solutions. These design variables may include loads as well as material and geometric properties of the beam segments composing the mechanism. A line search step-restriction technique is included to enhance the stability of the method. The method uses six-degree-of-freedom planar beam elements in a chain calculation that cumulatively evaluates the large deflections corresponding to a given load set.