Compliance Matrices Research Articles

A simple projection method to correlate the principal mechanical direction with the principal microstructural direction of human osteoporotic femoral heads.

The mechanics of the trabecular bone is related to its structure; this work aimed to propose a simple projection method to clarify the correlation between the principal mechanical direction (PMD) and the principal microstructural direction (PMSD) of trabecular bones fromosteoporotic femoral heads. A total of 529 trabecular cubes were cropped from five osteoporotic femoral heads. The micro computed tomography (μCT) sequential images of each cube were first projected onto the three Cartesiancoordinate planes to have three overlapped images, and the trabecular orientation distribution in the three images was analyzed. The PMSD corresponding to the greatest distribution frequency of the trabecular orientation in the three images was defined. Then, the voxel finite element (FE) models of the cubes were reconstructed and simulated to obtain their compliance matrices, and the matrices were subjected to transversal rotation to find their maximum elastic constants. The PMD corresponding to the maximum elastic constant was defined. Subsequently, the correlation of the defined PMSD and PMD was analyzed. The results showed that PMSD and PMD of the trabecular cubes did not show a significant difference at the xy- and yz-planes except that at the zx-plane. Despite this, the mean PMSD-PMD deviations at the three coordinate planes were close to 0°, and the PMSD-PMD fitting to the line PMSD = PMD demonstrated their high correlation. This study might be helpful to identify the loading direction of anisotropic trabecular bones in experiments by examining the PMSD and also to guide bone scaffold design for bone tissue repair.

Analytical Modeling and Validation of New Prismatic Compliant Joints Based on Zero Poisson’s Ratio Lattice Structures

Abstract The design and analysis of prismatic compliant joints have received less attention compared to that given to revolute compliant joints, thus limiting their implementation in compliant mechanisms beyond translational stages. Lattice structures have been used effectively to increase flexibility and stiffness ratios in compliant joints. Considering these, new prismatic compliant joints based on zero Poisson’s ratio lattice structures (ZP-PCJ) are proposed. Lattices with three different cell arrangements are considered: single cells, 2×2, and 3×3 lattices. Additionally, unit cells with three different geometries are studied: triangular, chamfer, and cosine. The compliance matrices of the ZP-PCJs are assembled analytically using Castigliano’s second theorem and compliance series–parallel simplification. The compliance ratios along the three orthogonal axes of the ZP-PCJs are computed varying their geometric parameters. Finite element models are constructed to validate the analytical results. Experimental tests are performed on additively manufactured ZP-PCJs to corroborate the compliance coefficients. Results showed that analytical models can predict the ZP-PCJ’s elastic properties accurately, differences less than 3% and 12% were obtained when compared to computational and experiments, respectively. Based on the compliance ratios obtained, the ZP-PCJs are suitable for two-dimensional applications. Finally, the ZP-PCJs are implemented in a compliant mechanism to evaluate their behavior, analytically and computationally. The ZP-PCJs have advantages such as eliminating axis drift and high flexibility in motion-direction while maintaining stiffness in other directions. The differences observed when comparing the analytically obtained estimations with simulations and experimental data suggest that ZP-PCJ analytical models are reliable for estimating their performance in compliant systems.

A novel bridge-type compliant displacement amplification mechanism under compound loads based on the topology optimisation of flexure hinge and its application in micro-force sensing

The design and modelling of bridge-type compliant displacement amplification mechanisms (CDAMs) are key components in precision engineering. In this study, a bridge-type CDAM under compound loads with an optimum flexure hinge configuration is designed, analysed, and tested. For the case when the flexure hinge configuration is unknown, the internal force distribution for a bridge-type CDAM under compound loads is analysed, and the topology of the flexure hinge is optimised. By applying different volume constraints, the optimised flexure hinge configurations are all V-shaped. Subsequently, a static model of the V-shaped flexure hinge is established. For a bridge-type CDAM with V-shaped flexure hinges, the compliance matrix of the flexure hinge is combined with the relationship among the local compliance matrices in a serial mechanism; consequently, the analytical relationship between the output displacement, output force, and input force is derived. The CDAM is parametrically optimised to further improve the output performance. Simulations and experiments verify the topology optimisation result, static model, and parametric optimisation result. Finally, the CDAM and its static model are applied to the tensile manipulation and micro-force sensing in a microfiber tensile test.

Nonlinear excitation and mesh characteristics model for spiral bevel gears

Accurate and rapid calculation of nonlinear excitation (NE) and loaded mesh characteristics (LMC) of gear pairs is a key to achieving rapid iterative optimization of gear system designs. Therefore, an analytical model (AM) was proposed for the accurate and numerically efficient calculation of NE and LMC for application to spiral bevel gears (SBG). First, a tooth contact analysis of an SBG was completed using Coons surface technology and mesh theory to determine the contact path and unloaded transmission error (UTE). Thus, an AM for the principal and relative normal curvatures of the SBG was derived using the Euler's formula. Based on the traditional calculation method for a Hertz contact ellipse, the relation between the relative curvatures was introduced to derive the formulas for the major and minor axes using the Muller's method, which could improve the programmability and avoid numerical instability of the program. Then, analytical models for the contact ellipse and pressure distribution on the contact tooth surfaces were derived using the Hertz model and geometric theory. Subsequently, based on a single tooth LMC, infinitesimal method, elasticity, and series stiffness model, a single mesh stiffness (SMS) model for SBGs was developed, which incorporates transverse tooth stiffness, transverse gear foundation stiffness, axial tooth stiffness, axial gear foundation stiffness, and Hertz nonlinear contact stiffness. Based on the SMS and UTE, compliance and load matrices were established, and the time-varying mesh stiffness, loaded transmission error, mesh force, and multi-tooth LMC models of the SBG were determined. Finally, the proposed method was validated against nonlinear FE simulations using examples. Compared to FE simulations, the proposed method requires significantly less computational effort and can be further extended to optimization or system analysis problems.

Stiffness Modeling and Deformation Analysis of Parallel Manipulators Based on the Principal Axes Decomposition of Compliance Matrices

Abstract This paper presents a general equivalent approach to solve the stiffness modeling or load-deformation problem of non-redundant parallel mechanisms. Based on the principal axes decomposition of structure compliance matrices, an equivalent six-degrees-of-freedom (6-DOFs) serial mechanism is established to approximate the load-deformation behavior of each flexible link in the mechanism. Hence, each limb of the parallel mechanism can be equivalent to a serial redundant rigid body mechanism with passive elastic joints, and the load-deformation problem can be transformed to the equilibrium configuration calculation of the equivalent mechanism. The main advantage of the proposed method is that the robotic kinematics and statics, rather than the elastic mechanics, can be directly adopted to solve the equilibrium configuration of the parallel mechanism under external load. Besides, a closed form solution of the corresponding deformation can be obtained, which can be solved by the gradient-based searching algorithm. Therefore, the final deformation will no longer be linear to the external load, which makes this method more accurate and more suitable for the deformation prediction and compensation in real industrial working conditions. In order to verify the effectiveness and correctness of this method, a 3PRRU parallel manipulator will be introduced as an example, to compare the load-deformation results with the finite element analysis (FEA) simulation and matrix calculation methods, so the nonlinearity feature can be shown in an intuitive manner.

Compliance and precision modeling of general notch flexure hinges using a discrete-beam transfer matrix

Compliance and precision are two key aspects in designing flexure hinges for use in compliant mechanisms. A generalized method with the finite-discrete idea and beam transfer matrix is developed for quickly predicting the kinetostatic behaviors of single and multiple-axis notch flexure hinges. The transfer matrix of Timoshenko beams explicitly including the shear deformation effect is derived in a concise form of Taylor series expansion. The compliance and rotational precision matrices of general notch flexure hinges are then derived targeting for arbitrary cutout profiles with a step-by-step modeling procedure. It is easy to operate for comparing and synthesizing different flexure hinges within a unified modeling framework. All a designer has to do is to prepare the concerned profile function without being caught into the laborious and even unsolvable derivation of mechanics. Several groups of single and multiple-axis notch flexure hinges are adopted to verify the feasibility of the presented approach by comparing with the numerical and experimental results available in literature. The comparative results indicate the high prediction accuracy and easy operation of the proposed approach for a wide applicability of complex notch flexure hinges.

Design of compliant mechanisms based on compliant building elements. Part II: Practice

In Part II, we present the use and application of the Compliant Building Elements (CBE) method to generate practical designs for different multiple degree-of-freedom (DOF) compliant mechanisms. We first create a compliant mechanism library based on the combination of prismatic beams, which includes commonly used basic elements (BEs), serial elements (SEs), and parallel elements (PEs). Examples are given during the construction of the library. Based on the library and CBE method, we demonstrate the design of a six-axis nanopositioner, a compliant steering mechanism, and a one-dimensional mechanical metamaterial structure. The outcome of each design process yields a flexure topology that meets the design requirements as well as the corresponding parametric stiffness and compliance matrices, from which we can efficiently obtain the kinematic characteristics of the flexure system and optimize the stiffnesses in selected axes.

Исследование по динамике многоэтажных зданий

The design of multi-storey buildings is a natural trend in the development of a modern metropolis. Obtaining exact solutions when studying their own and forced oscillations within the framework of a continuous homogeneous medium model (continuum mechanics) with an infinite number of degrees of freedom is often difficult to implement. Therefore, in the article (as part of the modernization of the finite element method), the model of a multi-storey building is discretized and endowed with a finite number of degrees of freedom placed in the middle of the finite elements at the nodes (the mass of finite elements is also placed there), which elastically interact with the finite elements of the model that do not have mass. It is believed that the elements of a multi-storey building work only for bending, which is fully justified by comparing the frequencies of its bending and longitudinal oscillations. The resolving system of differential equations of oscillations of a multi-storey building, in which expressions for energies (potential, kinetic and Rayleigh) are written in quadratures, is obtained using Lagrange equations of the second kind. In the article, the problems of free oscillations of 3- and 100-storey buildings are solved using Green’s functions, stiffness, mass, compliance matrices, etc. The approximate results obtained in the article, when compared with the little-known approximate results obtained by other methods, as well as exact results (direct and indirect methods of boundary elements), showed a good correspondence.

Root rotations and root displacements in bimaterial layers and thin films

An elasticity technique to derive compliance coefficients describing crack tip root rotations and displacements in end-loaded, bimaterial, isotropic and orthotropic layers has been recently proposed in Ustinov and Massabò (Int. J. Solids Struct., 2022). The coefficients define the boundary conditions which account for the near tip deformations when using beam (plate) theories to describe the detached layers. The coefficients, collected into 6x6 compliance matrices, relate crack tip kinematic variables describing relative displacements and rotations between the detached and intact parts, and six elementary crack tip loadings, which combine to describe general end loadings. The steps necessary to use the coefficients in the solution of statically determined problems, when the forces acting on the layer are known through equilibrium, is presented here with reference to classical fracture mechanics specimens.

Tripod mechanisms with novel spatial Cartesian flexible hinges

The paper studies the tripod mechanisms that comprise novel spatial Cartesian flexible hinges and can be used in three-dimensional sensing/actuation applications. These hinges are formed of serially-connected straight- and circular-axis elastic segments that are located in planes parallel to the planes of a Cartesian reference frame. Two spatial flexible hinge designs are proposed and their compliance matrices are derived based on closed-form compliances of the basic segments. The hinge compliance matrices are utilized to formulate the stiffness matrix of tripod mechanisms that are parallel combinations of three identical hinges and a rigid link. The hinge analytical compliance model is confirmed through finite element code simulation while the tripod analytical stiffness is verified by both finite element analysis and experimental testing of fabricated specimens. Comprehensive simulation is subsequently performed to study the influence of the geometric parameter values on the hinge and tripod analytical compliance/stiffness.

Automation of the compliance matrix «Discipline – Competence» (by example of the educational masters program «Financial Intermediation»)

The object of this research is the automation of the compliance matrix «Disciplines – Competences», which are the links between the list of compulsory and elective disciplines of the educational program according to the curriculum and the set of competencies of the graduate required by the Standard of higher education. The development of the educational program includes a combination of disciplines with «Program Learning Outcomes», which is listed in the Standard. One of the most problematic places is time-consuming of the process of «drawing-up» the links from «General Competencies» (GC) and «Professional Competencies» (PC) of disciplines to «Program Learning Outcomes» (PO). This problem is considered on the basis of the Educational and Professional Program (OPP) «Financial Intermediation» Academic Degree «Master» specialty 072 «Finance, Banking and Insurance» in the field of science 07 «Management and Administration» of the Department of Banking of Kyiv National University of Trade and Economics (KNUTE, Ukraine). The research methods are to use the design of relationships between logical elements («entities») of the data model (Entity-Relationship Model). To develop a compliance matrix «Disciplines – Competences» in the paper the author proposed a software application based on Excel (hereinafter «Application»), which allows to automate the construction of such links. There is a significant reduction in the time-consuming of preparing educational programs by guarantors and support groups. This is due to the fact that the proposed application has a number of features of use, in particular automates the construction of matrices of correspondence «Discipline – Competence». The method of automation of the compliance matrix «Disciplines – Competences» proposed in the research was successfully tested by the author in the development of educational and professional programs of KNUTE, namely «Financial Intermediation», «Management of Banking Business» and «Financial Brokerage». Thus, the application is universal and can be used by guarantors and support groups to build Compliance Matrices of the educational programs of other specializations and specialties.

Numerical study of fractured rock masses: Transverse isotropy vs. implicit joint-continuum models

Fractures prevail mechanical behavior of a rock mass and confer an overall anisotropic response. Engineering analyses in the elastic regime often use transverse isotropy to model fractured rock masses with a single fracture set. An alternative implicit joint-continuum model combines the mechanical response of the intact rock and fractures by adding their compliance matrices. It can accommodate multiple fracture sets and non-linear fracture response. While the transverse isotropic model is inadequate to model fractured rock media because of its inherent assumptions on the continuity for all stress components, the implicit joint-continuum model is verified against the exact solutions of internal stress distributions and displacement field. The analysis of strip foundations using the implicit joint continuum approach shows that the maximum settlement and tilt will take place when the fracture set strikes quasi-collinear with the strip direction (θJ ≈ ±15°) and the fracture dip angle is either βJ ≈ 40° ± 10° or βJ ≈ 140° ± 10°.

Leveraging Geometry to Enable High-Strength Continuum Robots.

Developing high-strength continuum robots can be challenging without compromising on the overall size of the robot, the complexity of design and the range of motion. In this work, we explore how the load capacity of continuum robots can drastically be improved through a combination of backbone design and convergent actuation path routing. We propose a rhombus-patterned backbone structure composed of thin walled-plates that can be easily fabricated via 3D printing and exhibits high shear and torsional stiffness while allowing bending. We then explore the effect of combined parallel and converging actuation path routing and its influence on continuum robot strength. Experimentally determined compliance matrices are generated for straight, translation and bending configurations for analysis and discussion. A robotic actuation platform is constructed to demonstrate the applicability of these design choices.

Stiffness of out-of-plane-motion stages with large-motion folded fractal flexible hinges.

This paper introduces a new planar flexible hinge of fractal configuration to be incorporated in out-of-plane-motion compliant stages that cover a wide stiffness range. The large-displacement fractal hinge consists of a series of scaled-down, concentric, circular-axis flexible segments that are connected in a folded manner by radial rigid links. In-plane and out-of-plane compliance matrices are derived for fractal hinges with variable planar geometry features. The flexible hinges are assembled in a radial architecture to form compliant stages that can be utilized for piston-type, out-of-plane sensing or actuation. The analytical model calculates the stage active, out-of-plane stiffness, as well as its parasitic, in-plane stiffness. The stage analytical stiffness is confirmed by the experimental testing of a prototype and by finite element simulation. Furthermore, analytical-model simulation is performed to evaluate the variations in the active and parasitic stiffnesses with key geometric parameters, which also enables optimization.

Generalized model for conic-V-shaped flexure hinges.

This paper presents a new class of flexure hinges, namely, conic-V-shaped flexure hinges (CFHs), which can be used as a generalized model for flexure hinges with profiles such as parabolic-V-shape, elliptical-V-shape, and hyperbolic-V-shape. Compliance and precision equations for the CFHs were derived as a set of nonlinear equations using Castigliano's second theorem. The parameters of the nonlinear equations inputted to the compliance and precision matrices were based on the generalized equations used for conic curves in polar coordinates. Furthermore, the compliance equations were verified by means of finite element analysis and experiments. The errors in the finite element and experimental results were within 10% and 8% compared to the analytical results, respectively. Finally, the effects of dimensional parameters on the analytical model could be effectively analyzed by numerical simulations and comparisons.

Stiffness modeling of a variable stiffness compliant link

Recently, a variable stiffness robotic link based on the rotating beam concept has been developed for applications in physical human robot interaction. A substantial challenge for design of such links is the modeling of stiffness behavior to permit stiffness control. In this paper we present a general 3D model of the link stiffness using screw theory and compliance matrices as well as a planar model for the lateral and torsional stiffness. Since axial buckling is a major failure mode, we also derive an analytical model for predicting axial buckling behavior. The analytical models are compared to the finite element method and experimental results. One of the challenges involved in design and analysis of variable stiffness links is the parasitic compliance of the mechanical elements that support and drive the active portion of the mechanism. For the design analyzed in this paper, we use the models we derive to identify the major sources of parasitic compliance and suggest optimizations to minimize their effects. These results can be used as guidelines for designing variable stiffness links.

Design and stiffness modeling of a four-degree-of-freedom nanopositioning stage based on six-branched-chain compliant parallel mechanisms.

Multi-degree-of-freedom (multi-DOF) nanopositioning stages (NPSs) have rapidly growing applications in the spatial micro-/nano-machining and manipulation. Compliant parallel mechanisms (CPMs) demonstrate advantages to achieve a large output stiffness and high payload. A four-DOF NPS based on six-branched-chain CPMs is proposed in this paper. First, a mechanism design approach is introduced. One primary vertical DOF is generated using three parallel-kinematic lever amplifiers. A three-revolute-revolute-revolute mechanism acts as the kinematic configuration to produce three secondary planar DOFs. Three types of single-axis and one type of double-axis notch flexure hinges (NFHs) are employed to realize the nanoscale displacement/movement guiding, transferring, and decoupling. Second, a stiffness modeling approach is derived. Combined with exact compliance matrices of 54 NFHs and 95 flexible beams, a four-DOF high-efficiency stiffness model of the six-branched-chain CPM is built. The calculation procedure of the whole input/output stiffnesses and coupling ratios takes 12.06 ms. Simulation and prototype test results validate the calculation accuracy. For example, the maximum calculation deviation of input stiffnesses is verified to be 4.52% and 8.18%, respectively. The two proposed approaches contribute to the statics parameter optimization of spatial multi-DOF NPSs.

Role of infill parameters on the mechanical performance and weight reduction of PEI Ultem processed by FFF

In comparison with conventional manufacturing technologies, Fused Filament Fabrication (FFF) offers countless benefits. It broadens the horizons of the design of structural components with high geometrical complexity, and lighter elements can be obtained by optimizing the infill of the part. The infill density stands as a manufacturing parameter that plays a significant part in weight reduction purposes. This fact provides FFF with an outstanding competitive advantage as compared to the rest of the additive manufacturing technologies. This work aims to investigate the role of infill parameters on the mechanical performance and weight reduction of ULTEMTM 9085 samples processed by FFF, under tensile, flexural, and shear loading conditions in six different orientations with several solid and sparse configurations. Regarding the effect of the part orientation and the infill settings, the experimental results permit to draw conclusions on stiffness, resilience, maximum stress, and type of failure of the printed parts. Three-dimensional compliance matrices for each infill configuration are provided. The analysis of the results correlates the infill configuration with the mechanical performance considering the intra-layer and inter-layer unions. Finally, this research provides experimental evidence to contribute to the definition of novel design-for-manufacturing strategies for obtaining functional structural elements by FFF.

Simple discrete models for dynamic structure-soil-structure interaction analysis

Simple models are presented for dynamic analyses of Structure-Soil-Structure Interaction (SSSI). Structures built on rigid, circular surface foundations resting on a homogeneous, elastic soil half-space are considered. The proposed models consist of discrete mass and spring elements that represent inertial, stiffness, and damping characteristics of an SSSI system. Frequency-dependent springs are developed to simulate Foundation-Soil Interaction (FSI) and Cross-Interaction (CI) between adjacent foundations, both being affected by the number and arrangement of neighbouring foundations. The stated springs are developed using a novel approach that approximates the through-the-soil interaction between adjacent footings using existing solutions to soil-structure interaction (SSI) analyses. Static stiffness matrices are obtained by means of compliance matrices using well known analytical solutions of surface displacement field to a ‘single-disk’ problem and a suggested displacement averaging rule. Dynamic stiffness matrices are created by applying modification factors to their static counterparts. To this end, a practical expression for dynamic modification factors that relate static and dynamic amplitudes of surface displacement field is provided. Equations of motion are formulated and solved in both time and frequency domains, taking into account time-lagging effects on input motion at separate foundations due to their vibration and wave passage. Validation results against more advanced numerical models prove the effectiveness of the proposed models. Finally, the limitations and potential improvements on the models are discussed.

Analytical elastostatic stiffness modeling of parallel manipulators considering the compliance of the link and joint

This paper discusses the analytical elastostatic stiffness modeling of parallel manipulators (PMs) considering the compliance of the link and joint. The proposed modeling is implemented in three steps: (1) the limb constraint wrenches are formulated based on screw theory; (2) the strain energy of the link and the joint is formulated using material mechanics and a mapping matrix, respectively, and the concentrated limb stiffness matrix corresponding to the constraint wrenches is obtained by summing the strain energy of the links and joints in the limb; and (3) the overall stiffness matrix is assembled based on the deformation compatibility equations. The strain energy factor index (SEFI) is adopted to describe the influence of the elastic components on the stiffness performance of the mechanism. Matrix structural analysis (MSA) using Timoshenko beam elements is applied to obtain analytical expressions for the compliance matrices of different joints through a three-step process: (1) formulate the element stiffness equation for each element; (2) extend the element stiffness equation to obtain the element contribution matrix, allowing the extended overall stiffness matrix to be obtained by summing the element contribution matrices; and (3) determine the stiffness matrices of joints by extracting the node stiffness matrix from the extended overall stiffness matrix and then releasing the degrees of freedom of twist. A comparison with MSA using Euler–Bernoulli beam elements demonstrates the superiority of using Timoshenko beam elements. The 2PRU-UPR PM is presented to illustrate the effectiveness of the proposed approach. Finally, the global SEFI and scatter matrix are used to identify the elastic component with the weakest stiffness performance, providing a new approach for effectively improving the stiffness performance of the mechanism.