INVERSE DESIGN METHODS FOR COMPLIANT MECHANISMS by Alejandro Eduardo Albanesi Dissertation submitted to the Postgraduate Department of the ´ ´ FACULTAD DE INGENIERIA Y CIENCIAS HIDRICAS of the UNIVERSIDAD NACIONAL DEL LITORAL in partial fulfillment of the requirements for the degree of Doctor en Ingenier´ıa - Menci´on Mec´anica Computacional 2011 Contents Abstract xi Resumen extendido xiii 1 Introduction 1 1.0.1 The basic nomenclature behind compliant mechanisms . 4 1.0.2 Design of mechanisms: analysis and synthesis . . . . . . 7 1.1 Classic design methods . . . . . . . . . . . . . . . . . . . . . . . 9 1.2 Inverse design methods . . . . . . . . . . . . . . . . . . . . . . . 18 1.3 Content of this thesis . . . . . . . . . . . . . . . . . . . . . . . . 21 2 Design by optimization 23 2.1 Structural optimization . . . . . . . . . . . . . . . . . . . . . . . 23 2.2 Topology optimization . . . . . . . . . . . . . . . . . . . . . . . 26 2.2.1 Samcef Field(cid:13)R CONLIN algorithm . . . . . . . . . . . . 26 2.2.2 Sigmund’s algorithm . . . . . . . . . . . . . . . . . . . . 27 2.2.3 Topology optimization conclusions . . . . . . . . . . . . 29 2.3 Shape optimization . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.3.1 GMMA Algorithm . . . . . . . . . . . . . . . . . . . . . 30 2.3.2 GCM Algorithm . . . . . . . . . . . . . . . . . . . . . . 31 2.3.3 SQP Algorithm . . . . . . . . . . . . . . . . . . . . . . . 31 2.3.4 Shape optimization example . . . . . . . . . . . . . . . . 32 2.4 Size optimization . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.4.1 Problem description . . . . . . . . . . . . . . . . . . . . 40 2.4.2 Discretization of the mechanism: beam formulations . . . 42 i ii CONTENTS 2.4.3 Synthesis of the Input and Follower Segments . . . . . . 49 2.4.4 Optimization algorithms in MATLAB (cid:13)R . . . . . . . . . 51 2.5 Numerical example . . . . . . . . . . . . . . . . . . . . . . . . . 52 2.5.1 Both free ends hinged to the grounds . . . . . . . . . . . 53 2.5.2 One free end hinged and the other clamped to the ground 55 2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3 Inverse FEM of general 3D solids [FCJ08] 61 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.2 Kinematic description . . . . . . . . . . . . . . . . . . . . . . . 63 3.3 Material description . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.3.1 Anisotropy in inverse analysis . . . . . . . . . . . . . . . 64 3.4 Finite element formulation . . . . . . . . . . . . . . . . . . . . . 65 3.4.1 Computation of strains and stresses in finite elements . . 67 3.4.2 Solution of the nonlinear equilibrium equation . . . . . . 68 3.4.3 Computation of the stress derivatives . . . . . . . . . . . 69 3.5 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 3.5.1 Validation test . . . . . . . . . . . . . . . . . . . . . . . 74 3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4 IFEM for Large-Displacement Beams 83 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.2 Beam Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.2.1 Parametrization of rotations . . . . . . . . . . . . . . . . 86 4.2.2 Spatial deformation measures . . . . . . . . . . . . . . . 87 4.3 Governing Equilibrium Equations . . . . . . . . . . . . . . . . . 88 4.3.1 Constitutive equations . . . . . . . . . . . . . . . . . . . 89 4.4 Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . 89 4.4.1 Discretised equilibrium equations . . . . . . . . . . . . . 91 4.4.2 Computation of deformation and stress in the current finite element . . . . . . . . . . . . . . . . . . . . . . . . 92 4.4.3 Linearization of the discrete equilibrium equations . . . . 92 4.4.4 Derivatives of deformation measures. . . . . . . . . . . . 93 CONTENTS iii 4.5 Validation Examples . . . . . . . . . . . . . . . . . . . . . . . . 95 4.5.1 Bending of a flexible cantilever beam . . . . . . . . . . . 95 4.5.2 Cantilever 45-degrees bend . . . . . . . . . . . . . . . . . 98 4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 5 Designs that exactly fit a desired shape 101 5.1 Compliant joint . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 5.2 Compliant S-clutch . . . . . . . . . . . . . . . . . . . . . . . . . 103 5.3 Compliant gripper . . . . . . . . . . . . . . . . . . . . . . . . . 104 5.4 Compliant Biomedical Instruments . . . . . . . . . . . . . . . . 106 5.4.1 Compliant lens folding device . . . . . . . . . . . . . . . 106 5.4.2 Compliant microvalves . . . . . . . . . . . . . . . . . . . 108 5.4.3 Compliant microgrippers . . . . . . . . . . . . . . . . . . 109 5.5 Advantages and disadvantages of IFEM . . . . . . . . . . . . . . 111 5.5.1 Computational cost . . . . . . . . . . . . . . . . . . . . . 112 5.5.2 Stability check and feasibility of a design: detecting crit- ical points . . . . . . . . . . . . . . . . . . . . . . . . . . 112 5.5.3 Intersections and interpenetrations . . . . . . . . . . . . 115 5.5.4 Violation of the design domain . . . . . . . . . . . . . . 117 6 Closure 119 6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 6.2 Further Research . . . . . . . . . . . . . . . . . . . . . . . . . . 121 List of Figures 1.1 Deformed configurations the optimal boundary shape of a can- tilever beam based on linear elastic analysis (top), and on non- linear elastic analysis (bottom). Tip loads are: 10 N (left), 1000 N (middle) and 1500 N (right), Bruns and Tortorelli [BT01]. . . 9 1.2 Mechanisms with concentrated compliance: a rigid crimping mechanism (left) and its compliant counterpart (right) [How01]. Areas with lower stiffness (i.e. smaller cross-section) are clearly visible in the compliant model (right) and serve as compliant hinges that allow the motion of the mechanism. . . . . . . . . . 10 1.3 Design of a compliant transmission: the inner circle of the joint is fixed to the ground and torque is applied to the outer circle to allow its motion (top left) and a detail view of the multiple cross-sectionsthatformthejoint(topright). Anschematicview of the four bar mechanism used to model the joint (bottom left) and its pseudo-rigid body model (bottom right) [PMBV10]. . . . 11 1.4 Design of a bistable actuator for a landing gear mechanism: the landing gear (top left) and its kinematic chain (top right), the graph representation of the proposed solution (bottom left) and the kinematic chain of this solution (bottom right). References for joint types in graphs are: R = revolute, C = clamped. In sketches, flexible links have a letter F, other links are assumed to be rigid. Pucheta and Cardona [PC10]. . . . . . . . . . . . . 13 1.5 Mechanisms with distributed compliance: a rigid crimping mechanism (left) and its compliant counterpart (right) [How01]. 14 v vi LIST OF FIGURES 1.6 Comparison between continuous material density parametriza- tion and ground structure parametrization [LK06]. . . . . . . . . 16 1.7 Design of a compliant pull-clamp using LSM (taking advantage of symmetry): the initial design domain (top left), intermediate iterations (top right) and (bottom left), and the optimal final design (bottom right) [WCWM05]. . . . . . . . . . . . . . . . . 17 1.8 Theoptimizedtopologyofa3Dinvertingmechanismscomputed with BEVO (left), and the resulting complete model (right), [AVMC10]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.9 Determination of the manufacturing shape of a rubber punch for stamping applications: initial deformed configuration (left), andthemanufacturingshapecomputedthroughinverseanalysis (right) [GM98]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.10 Computation of the unloaded shape of a turbine compressor blade: deformed configuration represented as a mesh surface, and manufacturing shape computed through inverse analysis with hyperelasticity represented as a solid surface [FCJ08]. . . . 20 2.1 The design space, feasible and unfeasible designs [RR05]. . . . . 25 2.2 The active constraints of the optimization problem, [RR05]. . . 25 2.3 The clamped-clamped beam model. . . . . . . . . . . . . . . . . 27 2.4 Results of the clamped-clamped beam topology optimization ˙ performed in Samcef Field(cid:13)RThe objective was to maximize the stiffness while minimizing the material mass. The color-scale indicatesthelocationofthematerialmass(redmeansmaximum density, and blue means minimum density). . . . . . . . . . . . 28 2.5 Topology optimization solved by Sigmund’s algorithm for the clamped-clamped beam shown in Figure 2.3. The color-scale in- dicatesthelocationofthematerialmass(darkcolormeansmax- imum density, and light color means minimum density). Note the similarities with the optimization scheme of Figure 2.4. . . . 29 LIST OF FIGURES vii 2.6 The axial-symmetrical model of the disc: the periodic pattern (left), the 4.86 angle minimum pattern (middle), and the result- ing three-dimensional model (right). . . . . . . . . . . . . . . . . 33 2.7 The design variables of the parametric model are the major and minor radius of the hole, RA and RB respectively. . . . . . . . . 33 2.8 Structured mesh of 6600 hexaedral elements. . . . . . . . . . . . 34 2.9 The boundary conditions of the model: symmetry conditions (left), and imposed displacements and temperatures (right). . . 35 2.10 Ventilation hole optimization results: maximum equivalent stress. 36 2.11 A rigid four-bar mechanism. Link N=4 is grounded link. . . . . 37 2.12 A compliant four-bar mechanism, and the classification of its links. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.13 Mechanism task: the guiding of the flexible coupler-link. . . . . 41 2.14 Specified initial and final configuration of the mechanism, ABCD and AbcD respectively. . . . . . . . . . . . . . . . . . . . 41 2.15 Initially curved beam element, [Cri00]. . . . . . . . . . . . . . . 43 2.16 Detail of the beam element with shear deformation, [Cri00]. . . 43 2.17 Beam kinematics, [CG88, GC00] . . . . . . . . . . . . . . . . . . 46 2.18 The design variables of the problem. . . . . . . . . . . . . . . . 50 2.19 Specified task for the compliant coupler-link [SK01]. . . . . . . . 52 2.20 Hinged-hinged4-barmechanism. Crisfield’sbeamfiniteelement model. Displacementsareinthesamescalethanthedimensions of the mechanism. . . . . . . . . . . . . . . . . . . . . . . . . . . 54 2.21 Hinged-hinged 4-bar mechanism. Cardona and G´eradin’s beam finite element model. Displacements are in the same scale than the dimensions of the mechanism. . . . . . . . . . . . . . . . . . 56 2.22 Hinged-clamped 4-bar mechanism. Crisfield’s beam finite el- ement model. Displacements are in the same scale than the dimensions of the mechanism. . . . . . . . . . . . . . . . . . . . 57 2.23 Hinged-clamped 4-bar mechanism. Cardona and G´eradin’s beam finite element model. Displacements are in the same scale than the dimensions of the mechanism. . . . . . . . . . . . . . . 58 viii LIST OF FIGURES 3.1 Distorted configuration B, domain of inverse analysis, and undistorted configuration B sought as solution. . . . . . . . . . 64 0 3.2 Direct problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.3 Inverse problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 3.4 Displacement modulus from the inverse analysis. . . . . . . . . . 77 3.5 Evolution of the residue norm during the inverse analysis. . . . . 77 4.1 Description of beam kinematics. . . . . . . . . . . . . . . . . . . 84 4.2 Finite element model of the inverse beam. . . . . . . . . . . . . 90 4.3 Plane bending of a flexible cantilever beam: undeformed and deformed neutral axes. Note that the scales for x and z are equal. 97 4.4 Bending of a flexible cantilever beam: errors in the approxima- tion of positions and rotations (measured in L -norm) using the 2 proposed inverse finite element model, as a function of the el- ement size. For 20 finite elements the error in displacement is approximately of 10−3 m, which is very small compared to the dimensions of the beam (2 m long). . . . . . . . . . . . . . . . . 98 4.5 Cantilever 45-degree bend: solutions of direct and inverse anal- yses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 5.1 A compliant joint with distributed compliance proposed in [PMBV10] (left), and the solution computed with the inverse FEM for large-displacement beams (right). The inner circle of the joint is fixed to the ground, and torque is applied to the outer circle in order to deform the model. . . . . . . . . . . . . . 102 5.2 A compliant S-clutch. . . . . . . . . . . . . . . . . . . . . . . . . 103 5.3 Compliant S-clutch: deformed (given design requirement), and undeformed (computed) configuration. The model is fixed to the ground at the center. . . . . . . . . . . . . . . . . . . . . . . 104 5.4 Compliant gripper: deformed (given design requirement) and undeformed (computed) configurations. Comparison with a ref- erence solution [LC07]. The actuation force is P = 24N. . . . . 105