Castigliano's Theorem Research Articles

Design methodology and performance analysis of application-oriented flexure hinges

Flexure hinges are widely employed in miniature and ultra-precision equipments with the development of precision engineering. In this paper, Castigliano's theorem and Labotto quadrature formula are utilized to derive the compliance matrix. An exclusive longitudinal section of corner-filleted flexure hinges is proposed and the curve of the cross-section is expressed by basic design parameters (r0, d, and l) and machining error parameter (η). The influences of parameters are analyzed by the defined evaluation indexes, i.e., rotation capacity, relative flexibility, and relative rotation error, which are used to assess the quality of flexure hinges. The analysis results indicate small d and optimized values of r0, l, and η will improve flexure hinges' performances. The proposed method is verified by finite element method and experiments within a 5% margin of error.

Nanomechanical studies and materials characterization of metal/polymer bilayer MEMS cantilevers

Abstract Bilayer microelectromechanical components such as microcantilevers, microbridges or micromembranes are usually used in microtransduction for actuation and sensing. One layer achieves the structural and elastic recovery function and the other layer acts as the active part by deforming under actuations. This paper describes the studies of mechanical characteristics of flexible bilayer microcantilevers fabricated from the polymer SU8 with a reflective nano-metallic layer on the top. The mechanical characteristics investigated are stiffness, modulus of elasticity, resonant frequency, bending strain and stress. The analytical relationships for bending stiffness of bilayer microcantilevers are determined by using Castigliano's second theorem. The first bending resonant frequency is computed based on the lumped-parameter model. Experimental tests of mechanical characteristics and materials characterization are developed using atomic force microscopy and nanoidentation. Finite element analysis is used to determine the maximum stress in the sample layers and their mechanical responses.

Two-Port Network Models for Compliant Rhomboidal Strain Amplifiers

Piezoelectric stack actuators have the advantages of zero backlash and no acoustic noise, but their stroke is too small to actuate robotic links directly. Because the force available is often more than the required, the stroke of the piezoelectric stack can be amplified by a compliant mechanism at the expense of force. It is not always clear what the geometry of this compliant mechanism should be. Compliant mechanisms have parallels in biology in that they describe two-way interactions between the actuator and the environment. In this paper, we employ the concept of a two-port network model from circuit theory to describe this two-way interaction and present a method to obtain each element of the two-port model as an analytical function of physical geometric parameters for a wide class of geometries. This method makes use of Castigliano's theorem and Euler-Bernoulli linearly elastic beam theory. To our knowledge, this is the first two-port representation of a compliant mechanism that is based on analytical expressions of geometric parameters. This analytical model agrees well with finite-element method calculations. We also examine a representative case experimentally and achieve accuracies better than 18%.

A model for a constrained, finitely deforming, elastic solid with rotation gradient dependent strain energy, and its specialization to von Kármán plates and beams

The aim of this paper is to develop the governing equations for a fully constrained finitely deforming hyperelastic Cosserat continuum where the directors are constrained to rotate with the body rotation. This is the generalization of small deformation couple stress theories and would be useful for developing mathematical models for an elastic material with embedded stiff short fibers or inclusions (e.g., materials with carbon nanotubes or nematic elastomers, cellular materials with oriented hard phases, open cell foams, and other similar materials), that account for certain longer range interactions. The theory is developed as a limiting case of a regular Cosserat elastic material where the directors are allowed to rotate freely by considering the case of a high “rotational mismatch energy”. The theory is developed using the formalism of Lagrangian mechanics, with the static case being based on Castigliano's first theorem. By considering the stretch U and the rotation R as additional independent variables and using the polar decomposition theorem as an additional constraint equation, we obtain the governing and as well as the boundary conditions for finite deformations. The resulting equations are further specialized for plane strain and axisymmetric finite deformations, deformations of beams and plates with small strain and moderate rotation, and for small deformation theories. We also show that the boundary conditions for this theory involve “surface tension” like terms due to the higher gradients in the strain energy function. For beams and plates, the rotational gradient dependent strain energy does not require additional variables (unlike Cosserat theories) and additional differential equations; nor do they raise the order of the differential equations, thus allowing us to include a material length scale dependent response at no extra “computational cost” even for finite deformation beam/plate theories

Effects of an edge crack on the free vibration and lateral buckling of a cantilever laminated composite slender beam

The objective of this paper is to investigate the effects of laminate scheme, crack ratio (a/h) and crack position (L1/L) on the free vibration and lateral buckling of a cantilever slender rectangular beam such as an airplane wing or a wind turbine blade with a single edge crack. The local flexibility approach is adopted to model the crack, based on linear fracture mechanics and the Castigliano theorem. The governing matrix equations for free vibration and buckling load are derived from the standard beam element and cracked beam element combined with the local flexibility concept. The results are obtained numerically using the finite element method, based on the energy approach and are given for crack ratios up to 0.5 at crack positions up to 0.9 ratio. Polynomial approximation for displacements gives results with reasonable accuracy.

Simulation and Fabrication of a Bistable Electromagnetic Micro-Actuator

This article proposes a bistable electromagnetic micro-actuator consisting of a planar suspended platform and an integrated micro-coil, which was fabricated by micro-electromechanical systems technology. Its operational principle is based on the equilibrium between the magnetic force and elastic force. The elastic force of the suspended platform is analyzed by Castigliano's theorem, and the magnetic force is analyzed by the magnetic circuit method. Both the mechanical and magnetic characteristics were simulated by the finite-element method and verified by testing. Test results showed that the bistable electromagnetic micro-actuator could be realized at a voltage pulse of 5 V with 380 μm of large displacement, and the response time is about 4.96 ms.

Stress energy minimization as a tool in the material layout design of shallow shells

The present research deals with the compliance minimization problem of an elastic thin shallow shell subjected to simultaneous in-plane and bending loads. In this context, our goal is to lay out a given amount of material in the volume of a shell assuming that the distribution in the direction transversal to its middle surface S is homogeneous. The discussion hence reduces to the question of finding the optimal material arrangement on S. Similar problems were solved in the framework of two dimensional elasticity or Kirchhoff plate theory and the present research attempts to generalize these results. Following the pattern emerging from the above mentioned considerations, our research starts from the minimum compliance problem of a structure made of two elastic materials whose volumetric fractions are fixed. The existence of a solution to thus posed optimization task is guaranteed if the fine-scale microstructural composites are admitted in the analysis. Their constitutive tensors can be obtained by certain averaging ensuing from the theory of homogenization for periodic media. Additionally, by the Castigliano Theorem, the compliance minimization problem is equivalent to the one for structural stress energy. In turn, the lower estimation of the energy is achieved in two steps: (i) its modification by a certain energy-like functional, and (ii) utilizing the quasiconvexity property of thus obtained expression. As a result, formulae describing the effective stress energy of one-material shallow shell and the material distribution function are explicitly derived.

An investigation on stiffness of a 3-PSP spatial parallel mechanism with flexible moving platform using invariant form

In this paper, the stiffness of a 3-PSP spatial parallel manipulator is investigated. Unlike traditional stiffness analysis, the moving platform is assumed to be flexible. Two analytical methods are used in finding the robot stiffness. In the first method, robot is modeled as lumped system and principle of virtual work is used. In the second method, the robot is modeled as a distributed system and strain energy of robot main components as well as Castigliano's theorem are used. Force analysis is also presented and reaction forces at the joints as well as internal forces/moments are obtained. For each of the main robot components, a matrix called Wrench Compliant Module Jacobian, WCMJ, is introduced. These matrices will allow mapping the applied external wrench on the moving platform to corresponding reaction forces for the corresponding compliant module. All analysis is presented using invariant form. To evaluate accuracy of the two methods, finite element analysis is used. Finally, using the distributed method, maximum and minimum eigenvalues of the stiffness matrix are obtained and values of kinematic stiffness index are presented.

3自由度精密定位平台的运动特性和优化设计

To achieve the precise motion of a 3-RRR compliant parallel precision positioning stage,a closed-form exact motion model was established and the optimized design of structure parameters was investigated.The Castigliano's second theorem was applied to establishment of the closed-form compliance model for the precision positioning stage.According to the structural characteristics of compliant parallel mechanisms,the system was divided into three symmetrical motion sub-chains.Combining the compliance equations of flexure hinge with the force transmission relations of mechanisms,the stiffness model of each sub-chain was obtained,and the stiffness of the entire system was calculated by summing the stiffness of three sub-chains in the same coordinate system.The proposed stiffness model took the hinge flexibility as the independent variables in the closed form.According to the flexibility matrix,the Jacobian matrix to reflect the relationship between input displacement and output one could be derived.By comparing the kineamatic model between theoretical analysis and FEA,the results show that the errors are within 1.0%~9.5%,which illurastrates that the proposed kinematic model is correct and precise.According to the closed-form Jacobian matrix,its sensitivity to structureal parameters was analyzed,then the design variables with greater impact on the kinematic properties were chosen.By taking the maxmium workspace as a target and the hinge strength,maxmium input forces,geometric dimensions and input coupling as the constrains,an optimal model was proposed.The results show that the optimized structural parameters can obtain more output displacements,and the proposed model can meet the design requirement.

Modeling of wave propagation in one-dimension structures using B-spline wavelet on interval finite element

This paper presents an application of B-spline wavelet on interval (BSWI) finite element for one-dimension elastic wave propagation problems. Using central difference method in time domain, numerical results involving wave propagations in rod and beam are obtained and compared with results calculated by classical finite element method. Numerical examples are used to validate the accuracy and efficiency of BSWI element. Based on the Castigliano theorem, the cracked rod and cracked beam BSWI element are constructed in this paper. Then the wave propagation in cracked one-dimension structures is simulated.

Biomechanical reevaluation of orthodontic asymmetric headgear

To investigate the distribution of distal and lateral forces produced by orthodontic asymmetric headgear (AHG) using mathematical models to assess periodontal ligament (PDL) influence and to attempt to resolve apparent inconsistencies in the literature. Mechanical models for AHG were constructed to calculate AHG force magnitudes and direction using the theory of elasticity. The PDL was simulated by elastic springs attached to the inner-bow terminals of the AHG. The total storage energy (E(t)) of the AHG and the supporting springs was integrated to evaluate the distal and lateral forces produced by minimizing E(t) (Castigliano's theorem). All analytical solutions were derived symbolically. The spring-supported headgear model (SSHG) predicted the magnitude and distribution of distal forces consistent with our data and the published data of others. The SSHG model revealed that the lateral forces delivered to the inner-bow terminals were not equal, and the spring constant (stiffness of the PDL) affected the magnitude and direction of the resultant lateral forces. Changing the stiffness of the PDL produced a greater biomechanical effect than did altering the face-bow design. The PDL spring model appeared to help resolve inconsistencies in the literature between laboratory in vitro experiments and clinical in vivo studies. Force magnitude and direction of AHG were predicted precisely using the present model and may be applied to improve the design of AHG to minimize unwanted lateral tooth movement.

Stiffness analysis of a compliant precision positioning stage

SUMMARYIn this paper a 3-degree of freedom compliant parallel positioning stage utilizing flexure hinges is explored for nanoimprint lithography. The performance of the stage is extensively analyzed using a pseudo-rigid body model and finite element method. The position and velocity models are established. Accordingly, the stiffness at driving point of the active arm is obtained on the basis of Castigliano's first theorem. The system stiffness of the compliant stage is explored using the compliant matrix methodology and matrix transformation, and the influence of the geometry parameters on stiffness factors in three directions has been graphically evaluated as well. Finite element analysis has been conducted to verify that the established models faithfully predict device performance.

On the Evaluation of the Energy Release Rate in Edge Cracked Beams

In recent years, significant patents have been devoted to developing non-destructive techniques for damage identification in structures. A crack in a structural member induces a local flexibility which affects the dynamic and static behavior of the whole structure. In this paper, an approach that is able to derive the flexibility parameters in beams with an edge crack based on the strain energy density theory is presented. The method takes into account that the crack in a beam does not propagate in a collinear manner, but due to the presence of the mixed mode, it changes its direction. This allows a description of the phenomenon of the non collinear propagation of the crack similar to the experimental evidence. The crack is assumed to be vertical before the loading, and then it runs in the direction of the minimum of the strain energy density. Then, the crack is supposed to follow the direction of the principal tensile stress in the section. Comparisons are made to the evaluation of the energy released during the whole crack process in the case of incorrect collinear propagation of the crack in mixed mode. Preliminary results are presented for different positions and initial length of the crack. Keywords: Castigliano theorem, edge crack, energy released, flexibility parameter, mixed mode, principal stress, Strain energy density, three-point bending, collinear propagation of the edge-crack, non-collinear propagation of the edge-crack

Optimization of Multilayer Probe Card Using Strain Energy-Based Analytical Model and Multiobjective Programming Algorithm

Wafer testing in the semiconductor industry is generally performed using a multilayer probe card. In this paper, Castigliano's second theorem is used to derive analytical formulae for the contact force and scrub mark length generated during the probing test. The formulations are then integrated with a multiobjective programming algorithm to optimize the probe needle parameters in such a way as to ensure a uniform contact force and a minimum scrub mark length. The validity of the analytical model is confirmed by comparing the solutions obtained for the contact force and scrub mark length with the equivalent results obtained from finite element simulations. The effectiveness of the analytical model and optimization procedure is demonstrated by optimizing the needle parameters of a commercial four-layer probe card. It is shown that the optimized probe card not only produces a more uniform contact force than the original probe card but also yields a shorter and more uniform scrub mark length.

Structure synthesis and statics analysis of a foldable stair

This paper proposes a foldable stair that can be deployed for use and folded for conservation and investigates the geometry and statics of the structure. It not only has the moving flexibility of a mechanism but also has the high stiffness and high strength of statically indeterminate structures. The primary linkage is a deployable scissor-like element which forms the staircase after expanding. This paper starts with the analysis of the geometry of planar four- and six-bar mechanisms and processes to synthesize the foldable stair and then investigates its statics. The paper further investigates redundant inner forces of every link based on the second Castigliano theorem, develops a foldable stair model, assembles and manufactures. Working experiments verify the theoretical results and the outstanding advantages over the general fixed stairs.

Estimation of internal defect size by means of radial deformations in pipes subjected to internal pressure

Metals like carbon steel are used extensively in pipeline industries due to simplicity and economy. However, flaws such as cracks, pitting, local wall thinning can be generated by corrosion, erosion, and environmental exposure to various substances. In this paper, an analysis of asymmetric radial deformation in pipe with local wall thinning under internal pressure is proposed. The deformation is asymmetric with respect to the axis and consists of a radial displacement of all points in the wall of the pipe. Here, only the vertical and horizontal displacements in pipe are taken into account and they are calculated by means of strain energy method based on the classical Castigliano's theorem. In order to validate the results, a commercially available finite element code is used. The agreement between the theoretical predictions and numerical results is very close. It is found that the internal defect sizes can be estimated knowing the horizontal and vertical displacements.

Accuracy and stiffness analysis of a 3-RRP spherical parallel manipulator

SUMMARYIn this paper, accuracy and stiffness analysis of a 3-RRP spherical parallel manipulator (SPM) (Enferadi and Tootoonchi, A novel spherical parallel manipulator: Forward position problem, singularity analysis and isotropy design, Robotica, vol. 27, 2009, pp. 663–676) with symmetrical geometry is investigated. At first, the 3-RRP SPM is introduced and its inverse kinematics analysis is performed. Isotropic design, because of its design superiority, is selected and workspace of the manipulator is obtained. The kinematics conditioning index (KCI) is evaluated on the workspace. Global conditioning index (GCI) of the manipulator is calculated and compared with another SPM. Unlike traditional stiffness analysis, the moving platform is assumed to be flexible. A continuous method is used for obtaining mathematical model of the manipulator stiffness matrix. This method is based on strain energy and Castigliano's theorem. The mathematical model is verified by finite element model. Finally, using mathematical model, kinematics stiffness index (KSI), and global stiffness index (GSI) are evaluated.

Precision of Rotation Analysis of Two-axis Rectangular Cross-section Corner-filleted Flexure Hinges

A two–axis rectangular cross–section corner–filleted flexure hinge as an alternative to the conventional flexure hinges for 3D multi–degrees of freedom compliant mechanisms is presented. The 3D mechanical model of two–axis flexure hinge is established,and the precision of rotation of the flexure hinge is developed on the basis of Castigliano's theorem. The closed–form formulas for the flexure hinge are obtained by the method of substitution. Aiming at errors arising from rotation center deviation of the flexure hinge,the results show that compared with analytical model and the finite element simulation,the error margins are within 7%. It is prove that the analytical model is accurate. Finally,the influence of the structural parameters on the precision of rotation of the flexure hinge is analyzed,and it is concluded that the precision is proportional to the Young's modulus or the shear modulus of the material. When the minimum thickness δ and the minimum width b of the flexure hinge are defined,whether the precision of rotation of the flexure hinge reduces or improves depending on the increment or decrement of the radius R or the beam length l. But the influence of the beam length l is more obvious. When the radius R and the beam length l of the flexure hinge are defined,whether the precision of rotation of the flexure hinge is improved or reduced depends on the increment or decrement of the minimum thickness δ or the minimum width b. Some theoretical principles for engineering design of flexure hinges are provided from the above conclusions.

Closed-form compliance equations of filleted V-shaped flexure hinges for compliant mechanism design

This paper presents the closed-form compliance equations for the filleted V-shaped flexure hinges. The in-plane and out-of-plane compliances of the flexure hinges are developed based on the Castigliano's second theorem. The accuracy of motion, denoted by the midpoint compliance of the flexure hinges, is also derived for optimized geometric design. The influences of the geometric parameters on the characteristics of the flexure hinges are investigated. It is noted that the filleted V-shaped flexure hinges have diverse ranges of compliance corresponding to different filleted radius R and angle θ. These types of hinges can provide both higher and lower stiffnesses than circular flexure hinges. This makes filleted V-shaped flexure hinges very useful for wide potential applications with different requirements. The finite element analysis is used to verify the established closed-form compliance equations for these filleted V-shaped flexure hinges.

Introduction to engineering mechanics: a continuum approach

Introduction A Motivating Example: Remodeling an Underwater Structure Newton's Laws: The First Principles of Mechanics Equilibrium Definition of a Continuum Some Mathematical Basics: Scalars and Vectors Problem-solving Examples Strain and Stress in One Dimension Kinematics: Strain The Method of Sections and Stress Stress-Strain Relationships Limiting Behavior Equilibrium Stress in Axially Loaded Bars Deformation of Axially Loaded Bars Equilibrium of an Axially Loaded Bar Statically Indeterminate Bars Thermal Effects Saint-Venant's Principle and Stress Concentrations Strain Energy in One Dimension Properties of Engineering Materials A Road Map for Strength of Materials Examples Case Study 1: Collapse of the Kansas City Hyatt Regency Walkways Strain and Stress in Higher Dimensions Poisson's Ratio The Strain Tensor The Stress Tensor Generalized Hooke's Law Equilibrium Formulating Two-Dimensional Elasticity Problems Examples Applying Strain and Stress in Multiple Dimensions Torsion Pressure Vessels Transformation of Stress and Strain Failure Prediction Criteria Examples Case Study 2: Pressure Vessels Why Pressure Vessels Are Spheres and Cylinders? Why Do Pressure Vessels Fail? Beams Calculation of Reactions Method of Sections: Axial Force, Shear, Bending Moment Shear and Bending Moment Diagrams Integration Methods for Shear and Bending Moment Normal Stresses in Beams and Geometric Properties of Sections Shear Stresses in Beams Examples Case Study 3: Physiological Levers and Repairs The Forearm Is Connected to the Elbow Joint Fixing an Intertrochanteric Fracture Beam Deflections Governing Equation Boundary Conditions Beam Deflections by Integration and by Superposition Discontinuity Functions Beams with Non-Constant Cross-Section Statically Indeterminate Beams Beams with Elastic Supports Strain Energy for Bent Beams Deflections by Castigliano's Second Theorem Examples Case Study 4: Truss-Braced Airplane Wings Modeling and Analysis What Does Our Model Tell Us? Conclusions Instability: Column Buckling Euler's Formula Effect of Eccentricity Examples Case Study 5: Hartford Civic Arena Connecting Solid and Fluid Mechanics Pressure Viscosity Surface Tension Governing Laws Motion and Deformation of Fluids Examples Case Study 6: Mechanics of Biomaterials Nonlinearity Composite Materials Viscoelasticity Case Study 7: Engineered Composite Materials Concrete Plastics Ceramics Fluid Statics Local Pressure Force due to Pressure Fluids at Rest Forces on Submerged Surfaces Buoyancy Examples Case Study 8: St. Francis Dam Fluid Dynamics: Governing Equations Description of Fluid Motion Equations of Fluid Motion Integral Equations of Motion Differential Equations of Motion Bernoulli Equation Examples Case Study 9: China's Three Gorges Dam Fluid Dynamics: Applications How Do We Classify Fluid Flows? What Is Going on Inside Pipes? Why Can an Airplane Fly? Why Does a Curveball Curve? Case Study 10: Living with Water, and the Role of Technological Culture Solid Dynamics: Governing Equations Continuity, or Mass Conservation Newton's Second Law, or Momentum Conservation Constitutive Laws: Elasticity References Appendix A: Second Moments of Area Appendix B: A Quick Look at the del Operator Appendix C: Property Tables Appendix D: All the Equations Index