Compliance Matrices Research Articles

Stiffness Modeling of Parallel Mechanisms at Limb and Joint/Link Levels

Drawing on screw theory and the virtual joint method, this paper presents a general and hierarchical approach for semianalytical stiffness modeling of parallel mechanisms. The stiffness model is built by two essential steps: 1) formulating the map between the stiffness matrices of platform and limbs using the duality of wrench and twist of the platform; and 2) formulating the map between stiffness matrices of a limb and a number of elastic elements in that limb using the duality of the wrench attributed to the limb and the twist of the endlink of that limb. By merging these two threads, the Cartesian stiffness matrix can be explicitly expressed in terms of the compliance matrices of joints and links. The proposed approach bridges the gap between two currently available approaches and is thereby very useful for evaluating stiffness over the entire workspace and investigating the influences of joint/link compliances on those of the platform in a quick and precise manner. A stiffness analysis for a 3- P RS parallel mechanism is presented as an example to illustrate the effectiveness of the proposed approach.

Synthesis of Point Planar Elastic Behaviors Using Three-Joint Serial Mechanisms of Specified Construction

This paper presents methods for the realization of 2 × 2 translational compliance matrices using serial mechanisms having three joints, each either revolute or prismatic and each with selectable compliance. The geometry of the mechanism and the location of the compliance frame relative to the mechanism base are each arbitrary but specified. Necessary and sufficient conditions for the realization of a given compliance with a given mechanism are obtained. We show that, for an appropriately constructed serial mechanism having at least one revolute joint, any single 2 × 2 compliance matrix can be realized by properly choosing the joint compliances and the mechanism configuration. For each type of three-joint combination, requirements on the redundant mechanism geometry are identified for the realization of every point planar elastic behavior at a given location, just by changing the mechanism configuration and the joint compliances.

Standardized Compliance Matrices for General Anisotropic Materials and a Simple Measure of Anisotropy Degree Based on Shear-Extension Coupling Coefficient

The compliance matrix for a general anisotropic material is usually expressed in an arbitrarily chosen coordinate system, which brings some confusion or inconvenience in identifying independent elastic material constants and comparing elastic properties between different materials. In this paper, a unique stiffest orientation-based standardized compliance matrix is established, and 18 independent elastic material constants are clearly shown. During the searching process for the stiffest orientation, it is interesting to find from our theoretical analysis and an example that a material with isotropic tensile stiffness does not definitely possess isotropic elasticity. Therefore, the ratio between the maximum and minimum tensile stiffnesses, although widely used, is not a correct measure of anisotropy degree. Alternatively, a simple and correct measure of anisotropy degree based on the maximum shear-extension coupling coefficient in all orientations is proposed. However, for a two-dimensional constitutive relation, both the stiffness ratio and the shear-extension coupling coefficient can be adopted as proper measures of anisotropy degree.

Modeling the Human Tibiofemoral Joint Using Ex Vivo Determined Compliance Matrices.

Several approaches have been used to devise a model of the human tibiofemoral joint for embedment in lower limb musculoskeletal models. However, no study has considered the use of cadaveric 6 × 6 compliance (or stiffness) matrices to model the tibiofemoral joint under normal or pathological conditions. The aim of this paper is to present a method to determine the compliance matrix of an ex vivo tibiofemoral joint for any given equilibrium pose. Experiments were carried out on a single ex vivo knee, first intact and, then, with the anterior cruciate ligament (ACL) transected. Controlled linear and angular displacements were imposed in single degree-of-freedom (DoF) tests to the specimen, and the resulting forces and moments were measured using an instrumented robotic arm. This was done starting from seven equilibrium poses characterized by the following flexion angles: 0 deg, 15 deg, 30 deg, 45 deg, 60 deg, 75 deg, and 90 deg. A compliance matrix for each of the selected equilibrium poses and for both the intact and ACL-deficient specimen was calculated. The matrix, embedding the experimental load-displacement relationship of the examined DoFs, was calculated using a linear least squares inversion based on a QR decomposition, assuming symmetric and positive-defined matrices. Single compliance matrix terms were in agreement with the literature. Results showed an overall increase of the compliance matrix terms due to the ACL transection (2.6 ratio for rotational terms at full extension) confirming its role in the joint stabilization. Validation experiments were carried out by performing a Lachman test (the tibia is pulled forward) under load control on both the intact and ACL-deficient knee and assessing the difference (error) between measured linear and angular displacements and those estimated using the appropriate compliance matrix. This error increased nonlinearly with respect to the values of the load. In particular, when an incremental posterior-anterior force up to 6 N was applied to the tibia of the intact specimen, the errors on the estimated linear and angular displacements were up to 0.6 mm and 1.5 deg, while for a force up to 18 N, the errors were 1.5 mm and 10.5 deg, respectively. In conclusion, the method used in this study may be a viable alternative to characterize the tibiofemoral load-dependent behavior in several applications.

Realization of point planar elastic behaviors using revolute joint serial mechanisms having specified link lengths

This paper presents methods for the realization of 2 × 2 translational compliance matrices using serial mechanisms having only revolute joints, each with selectable compliance. The link lengths of the mechanism and the location of the compliant frame relative to the mechanism base are arbitrary but specified. The realizability of a given compliant behavior is investigated, and necessary and sufficient conditions for the realization of a given compliance with a given mechanism are obtained. These realization conditions are interpreted in terms of geometric relationships among the joints. We show that, for an appropriately sized 3R serial mechanism, any single 2 × 2 compliance matrix can be realized by properly choosing the joint compliances and the mechanism configuration. Requirements on mechanism geometry to realize every particle planar elastic behavior at a given location just by changing the mechanism configuration are also identified.

Program Tools for Residual Stress Evaluation by Ring-Core Method

Residual stress determination requirements of production and research users grow rapidly. Commercially available programs enable relatively quick residual stress evaluation with certain level of accuracy and with limited user access to used compliance matrices and calculations. However precise analyses require sensitive approach to compliance matrices determined for individual case e.g. for specific specimen dimensions. Therefore program tools for complex residual stress evaluation by Ring-Core method were developed using finite element analysis and Visual Basic scripts.

In-plane mechanical properties of carbon nanotube films fabricated by floating catalyst chemical vapor decomposition

Large-scale carbon nanotube (CNT) films fabricated by floating catalyst chemical vapor deposition (FCCVD) are promising reinforcement materials for high performance composites. However, little research has been reported on the four independent in-plane engineering constants of FCCVD CNT films, which are essential for understanding and prediction of mechanical behavior of FCCVD CNT film-based structures. The tensile strength in CNT-oriented direction is 127 MPa and the tensile modulus in oriented and transverse directions are 3.0 and 0.3 GPa, respectively. These mechanical properties are mainly attributed to the as-grown CNT-to-CNT contacts in the films. The Poisson’s ratio in the oriented direction at 5 % strain is 0.75. A negative Poisson’s ratio of −0.99 is observed at 0.1 % strain in CNT-oriented direction. The in-plane shear modulus is 0.57 GPa, which is derived from the coordinate transformation between the on-axis and 45° off-axis compliance matrices. The in-situ scanning electron microscopy is adopted to observe the microstructure at different tensile strains. During the tensile testing, the reorientation of CNT bundles in CNT film is evaluated by numerical image processing and Raman spectroscopy.

A Compliance-Based Parameterization Approach for Type Synthesis of Flexure Mechanisms

This paper presents an approach based on parameterized compliance for type synthesis of flexure mechanisms with serial, parallel, or hybrid topologies. The parameterized compliance matrices have been derived for commonly used flexure elements, which are significantly influenced by flexure parameters including material and geometric properties. Different parameters of flexure elements generate different degree of freedom (DOF) characteristic of types. Enlightened by the compliance analysis of flexure elements, a parameterization approach with detailed processes and steps is introduced in this paper to help analyze and synthesize flexure mechanisms with the case study as serial chains, parallel chains, and combination hybrid chains. For a hybrid flexure, the results of finite element (FE) modeling simulations are compared to analytical compliance elements characteristic. Under linear deformations, the maximum compliance errors of analytical models are less than 6% compared with the FE models. The final goal of this work is to provide a parameterized approach for type synthesis of flexure mechanisms, which is used to configure and change the parameters of flexure mechanisms to achieve the desired DOF requirements of types initially.

Fracture characterization using converted waves

We present a method for inversion of fracture compliance matrix components from wide-azimuth noisy synthetic PS reflection data and quantitatively show that reflection amplitude variations with offset and azimuth for converted PS-waves are more informative than P-waves for fracture characterization. We consider monoclinic symmetry for fractured reservoir (parameters chosen from Woodford Shale), which can be formed by two or more sets of vertical fractures embedded in a vertically transverse isotropic background. Components of effective fracture compliance matrices for a medium with monoclinic symmetry are related to the characteristics of the fractured medium. Monte Carlo simulation results show that inversion of PS reflection data is more robust than that of PP reflection data to uncertainties in our a priori knowledge (vertically transverse isotropic parameters of unfractured rock) than PP reflection data. We also show that, while inversion of PP reflections is sensitive to contrasts in elastic properties of upper and lower media, inversion of PS reflections is robust with respect to such contrasts.

Modeling and design of planar parallel-connection flexible hinges for in- and out-of-plane mechanism applications

The paper studies the planar parallel-connection, small-deformation flexible hinge chains formed of serially-coupled individual segments with variable cross-sections. It introduces the concept of virtual flexible hinge that is quasi-statically equivalent to the actual parallel-hinge configuration. General compliance and stiffness matrices are formulated for the virtual hinge under in-plane and out-of-plane loads by combining the transformed compliances of the individual hinge segments. Two classes are specifically analyzed: one comprises geometrically parallel, straight-axis hinge designs and the other includes concentric, circular-axis hinge configurations. From each class, particular designs with identical and transversely symmetric hinges of right circularly corner-filleted geometry are further investigated. Specifically, the behavior of parallelogram mechanisms with straight-axis hinges and of stage devices with circular-axis hinges is analyzed. Their elastic responses are validated by finite element analysis and their stiffnesses are subsequently studied in terms of offset geometric parameters.

Three-Dimensional Deformation of a Multilayer Plate with Elastic Links Between its Layers

We proposed a method for the determination of the stress–strain state of a multilayer plate with elastic links between the layers in the case of its three-dimensional deformation. The solution of the problem is based on the method of compliance functions with application of the two-dimensional Fourier integral transformation. We construct recursive relations for finding the compliance matrices. For a two-layer plate subjected to the action of normal concentrated loads, we analyze the influence of the coefficients of elastic links on the distribution of stresses and displacements in the layers.

Bipod flexure for 1-m primary mirror system.

We present an analytical formulation of the bipod flexure for mounting the 1-m primary mirror in a space telescope. Compliance and stiffness matrices of the bipod flexure are derived to estimate theoretical performance and to make initial design guidelines. We use finite element analysis to optimize the bipod design satisfying the application requirements. Experimental verification is achieved by vibration test with a dummy mirror system.

Compliance analysis of a 3-SPR parallel mechanism with consideration of gravity

By taking gravity and joint/link compliances into account, this paper presents a semi-analytical approach for compliance analysis of a 3-SPR parallel mechanism which forms the main body of a 5-DOF hybrid manipulator especially designed for high-speed machining and forced assembling in the aircraft industry. The approach is implemented in three steps: (1) kinetostatic analysis that considers both the externally applied wrench imposed upon the platform and the gravity of all moving components; (2) deflection analysis that takes both joint and link compliances into account; and (3) formulation of the component compliance matrices using a semi-analytical approach. The advantage of this approach is that the deflections of the platform caused by both the payload and gravity within the given task workspace can be evaluated in an effective manner. The computational results show that the deflection arising from gravity of the moving components may have significant influence on the pose accuracy of the end-effector.

A simple compliance modeling method for flexure hinges

Various types of flexure hinges have been introduced and implemented in a variety of fields due to their superior performances. The Castigliano’s second theorem, the Euler-Bernoulli beam theory based direct integration method and the unit-load method have been employed to analytically describe the elastic behavior of flexure hinges. However, all these methods require prior-knowledge of the beam theory and need to execute laborious integration operations for each term of the compliance matrix, thus highly decreasing the modeling efficiency and blocking practical applications of the modeling methods. In this paper, a novel finite beam based matrix modeling (FBMM) method is proposed to numerically obtain compliance matrices of flexure hinges with various shapes. The main concept of the method is to treat flexure hinges as serial connections of finite micro-beams, and the shearing and torsion effects of the hinges are especially considered to enhance the modeling accuracy. By means of matrix calculations, complete compliance matrices of flexure hinges can be derived effectively in one calculation process. A large number of numerical calculations are conducted for various types of flexure hinges with different shapes, and the results are compared with the ones obtained by conventional modeling methods. It demonstrates that the proposed modeling method is not only efficient but also accurate, and it is a more universal and more robust tool for describing elastic behavior of flexure hinges.

In-Plane Compliances of Planar Flexure Hinges With Serially Connected Straight- and Circular-Axis Segments

The paper introduces a new category of planar flexure hinges that are formed by serially connecting variable cross-sectional segments of straight longitudinal axes with segments of circular longitudinal axes. The small-displacement compliance analytical model is derived for a general hinge configuration using a matrix approach that sums the transformed local-frame compliance matrices of individual component segments. The particular class of antisymmetric flexure hinges is studied using the general model and the corresponding global-frame compliance matrix is calculated as a linear combination of compliances defining the half-hinge configuration. A serpentine (folded) flexure hinge is introduced to illustrate the generic antisymmetric design and model. Finite element simulation is used to validate the analytic compliances of this particular configuration and the compliance sensitivity to geometric parameters variation is further analyzed. The translation stiffnesses of a planar-motion stage with two identical serpentine hinges are calculated based on hinge compliances. The optimum hinge design is subsequently identified, which realizes minimum-resistance motion along the stage axial motion direction.

Mobility criteria of compliant mechanisms based on decomposition of compliance matrices

Mobility analysis is an important task in the conceptual design stage of kinematic mechanisms. Not like in rigid body mechanisms, identifying the mobility of compliant mechanisms is particularly challenging as their motion is determined by both kinematic pairs and deformable compliant joints. In this paper, we investigate the use of the eigentwist and eigenwrench decomposition of compliance matrices to identify the mobility of spatial compliant mechanisms. We first prove that the eigencompliances are invariant to the coordinate transformation. We then introduce the characteristic length to scale the eigencompliances and compliance matrices to compare translational compliances with rotational ones. We propose two mobility criteria for the compliance matrix of any given compliant mechanism. We have also proposed two guidelines for choosing the characteristic length for serial open chains and closed loop mechanisms respectively. The robustness of the chosen characteristic length is discussed. A general procedure for determining the mobility for any compliant mechanism is presented. Finally, one case study is provided to verify our method.

On the choice of the reference frame for beam section stiffness properties

This work discusses the choice of a reference frame for beam section stiffness properties. Established concepts as the center of elasticity, the center of stiffness and the center of compliance are discussed and contextualized. An interpretation of univocally defined generalized strain transformations is given in terms of minimization of appropriate norms of the stiffness and compliance matrices of the beam section that univocally define special reference points. Transformations of generalized strain perturbations that preserve the angular strain are sought. They are subsequently constrained to represent a change of reference point, and further restricted to lie in the plane of the section. Each transformation is univocally defined and given a clear mathematical and geometrical interpretation. It is recognized that transformations that decouple forces and linear strains from moments and angular strains cannot be described as a mere change of reference point.

Multi-objective optimization of a 3-DOF translational parallel kinematic machine

In this paper, stiffness modelling and analysis of a typical 3-DOF parallel kinematic machine (PKM) that provides translational motion along X, Y and Z axes is presented. The mechanism consists of three limbs each having an arm and a forearm with prismatic-revoluterevolute-revolute joints (PRRR). The joint arrangement is in such a way that the moving or tool platform maintains same orientation in the entire workspace. Through inverse kinematics, the joint angles for a given position of tool platform necessary for the stiffness modelling and analysis are obtained. The stiffness modelling is based on the compliance matrices of arm and forearm of each limb. Typical non-dimensional performance indices, namely, workspace volume index (WVI), global translational stiffness index (GTSI), and global rotational stiffness index (GRSI), are introduced and used to study the influence of dimensions. Attempts are also made to find the optimal dimensions of the translational PKM using multi-objective optimization based on the genetic algorithms (MOGA) in MATLAB. The methodology presented and the results obtained are useful for predicting the performance capability of the PKM under study.

Anisotropy of the elastic properties of crystalline cellulose Iβ from first principles density functional theory with Van der Waals interactions

In spite of the significant potential of cellulose nanocrystals as functional nanoparticles for numerous applications, a fundamental understanding of the mechanical properties of defect-free, crystalline cellulose is still lacking. In this paper, the elasticity matrix for cellulose Iβ with hydrogen bonding network A was calculated using ab initio density functional theory with a semi-empirical correction for van der Waals interactions. The computed Young’s modulus is found to be 206 GPa along [001] (c-axis), 98 GPa along [010] (b-axis), and 19 GPa along [100] (a-axis). Full compliance matrices are reported for 1.0, 1.5 and 2.0 % applied strains Color contour surfaces that show variations of the Young’s modulus and average Poisson’s ratio with crystallographic direction revealed the extreme anisotropies of these important mechanical properties. The sensitivity of the elastic parameters to misalignments in the crystal were examined with 2D polar plots within selected planes containing specific bonding characteristics; these are used to explain the substantial variability in the reported experimental Young’s moduli values. Results for the lattice directions [001], [010] and [100] are within the range of reported experimental and other numerical values.

Metasolid with anisotropic mass density

We have developed a homogenization theory for solid phononic crystals which is valid for arbitrary Bravais lattice and any form of inclusions in the unit cell. The theory provides explicit expressions for the tensors of the effective nonlocal elastic response, namely the effective dynamic mass density and compliance matrices. It is shown how the derived expressions can also be applied to phononic crystals with liquid components. We demonstrate that rectangular two-dimensional lattices of water-filled holes in an elastic host matrix exhibit solid-like behavior with strongly anisotropic mass density in the low-frequency limit.