Showing papers on "Tangent stiffness matrix published in 2015"

Consistent tangent operator for an exact Kirchhoff rod model

Abstract: In the paper, it is considered an exact spatial Kirchhoff rod structural model. The configuration space for this model that has dimension 4 is obtained considering an ad hoc split of the rotation operator that implicitly enforces the constraints on the directors. The tangent stiffness operator, essential for the nonlinear numerical simulations, has been studied. It has been obtained as second covariant gradient of the internal energy functional for the considered structural model that preserves symmetry for any configuration, either equilibrated or not. The result has been reached evaluating the Levi-Civita connection for the tangent space of the configuration manifold. The results obtained extend to the case of Kirchoff -Love rods those presented by Simo (Comput Methods Appl Mech Eng 49:55–70, 1985) for Timoshenko rods. Given the different structure of the tangent spaces in this case, it has been necessary to introduce a specific metric that accounts for the rotation of the intrinsic triad due to the change of the position of the centroid axis of the rod.

Non-linear dynamic analysis of tensegrity structures using a co-rotational method

Abstract: A new formulation is presented for the non-linear dynamic analysis of space truss structures. The formulation is based on the dynamics of 3D co-rotational rods. In the co-rotation method, the rigid body modes are assumed to be separated from the total deformations at the local element level. In this paper a new co-rotational formulation is proposed based on the direct derivation of the inertia force vector and the tangent dynamic matrix. A closed-form equation is derived for the calculation of the inertia force, the tangent dynamic matrix, the mass matrix and the gyroscopic matrix. The new formulation is used to perform dynamic analysis of example tensegrity structures. The developed formulation is applicable to tensegrity structures with non-linear effects due to internal mechanisms or geometric non-linearities, and is applied to two numerical examples. The efficiency of the proposed approach is compared to the conventional Lagrangian method, and savings in computation of about 55%, 54% and 37% were achieved.

Numerical calculation of thermo-mechanical problems at large strains based on complex step derivative approximation of tangent stiffness matrices

Abstract: In this paper a robust approximation scheme for the numerical calculation of tangent stiffness matrices is presented in the context of nonlinear thermo-mechanical finite element problems and its performance is analyzed. The scheme extends the approach proposed in Kim et al. (Comput Methods Appl Mech Eng 200:403---413, 2011) and Tanaka et al. (Comput Methods Appl Mech Eng 269:454---470, 2014 and bases on applying the complex-step-derivative approximation to the linearizations of the weak forms of the balance of linear momentum and the balance of energy. By incorporating consistent perturbations along the imaginary axis to the displacement as well as thermal degrees of freedom, we demonstrate that numerical tangent stiffness matrices can be obtained with accuracy up to computer precision leading to quadratically converging schemes. The main advantage of this approach is that contrary to the classical forward difference scheme no round-off errors due to floating-point arithmetics exist within the calculation of the tangent stiffness. This enables arbitrarily small perturbation values and therefore leads to robust schemes even when choosing small values. An efficient algorithmic treatment is presented which enables a straightforward implementation of the method in any standard finite-element program. By means of thermo-elastic and thermo-elastoplastic boundary value problems at finite strains the performance of the proposed approach is analyzed.

On the effect of the bulk tangent matrix in partitioned solution schemes for nearly incompressible fluids

Abstract: The purpose of this paper is to study the effect of the bulk modulus on the iterative solution of free surface quasi-incompressible fluids using a mixed partitioned scheme. A practical rule to set up the value of a pseudo-bulk modulus a priori in the tangent bulk stiffness matrix for improving the conditioning of the linear system of algebraic equations is also given. The efficiency of the proposed strategy is tested in several problems analyzing the advantage of the modified bulk tangent matrix with regard to the stability of the pressure field, the convergence rate and the computational speed of the analyses. The technique has been tested on a finite calculus/particle finite element method Lagrangian formulation, but it can be easily extended to other quasi-incompressible stabilized finite element formulations. Copyright (C) 2014 John Wiley & Sons, Ltd.

Frictional beam-to-beam multiple-point contact finite element

Abstract: The present paper is the extension of author's earlier research devoted to more accurate numerical modelling of beam-to-beam contact in the cases when beam axes form acute angles in the contact zone. In such situation with beam deformations taken into account, the contact cannot be considered as point-wise but it extends to a certain area. To cover such a case in a more realistic way, two additional pairs of contact points are introduced to accompany the original single pair of contact points from the point-wise formulation. The Coulomb friction model is introduced and advantage is taken from the analogy to plasticity. The penalty method is used to enforce the contact and friction constraints. The appropriate kinematic variables for tangential contact and their finite element approximation are derived. Basing on the weak form for frictional contact and its linearisation, the tangent stiffness matrix and the residual vector are derived. The enhanced element is tested using author's computer programs and comparisons with the point-wise contact elements are made.

On the algorithmic and implementational aspects of a Discontinuous Galerkin method at finite strains

Abstract: In this work, algorithmic modifications are proposed and analyzed for a recently developed stabilized finite strain Discontinuous Galerkin (DG) method. The distinguishing feature of the original method, referred to as VMDG, is a consistently derived expression for the numerical flux and stability tensor that account for evolving material and geometric nonlinearity in the vicinity of the interface. Herein, the proposed modifications involve simplifications to the residual force vector and tangent stiffness matrix of the VMDG method that lead to formulations similar to other existing DG methods but retain the enhanced definition for the stability parameters. The primary objective is to reduce the costs associated with implementing the method as well as executing simulations while retaining accuracy and flexibility, thereby making the formulation amenable to boarder material classes such as inelasticity. Each simplification has associated implications on the mathematical and algorithmic properties of the method, such as L 2 convergence rate, solution accuracy, continuity enforcement, and stability of the nonlinear equation solver. These implications are carefully quantified and assessed through a comprehensive numerical performance study. The range of two and three dimensional problems under consideration involves both isotropic and anisotropic materials. Both triangular and quadrilateral element types are employed along with h and p refinement. The ability of the proposed methods to produce stable and accurate results for such a broad class of problems is highlighted.

Local continuum shape sensitivity with spatial gradient reconstruction for nonlinear analysis

Abstract: Gradient-based optimization for large-scale, multidisciplinary design problems requires accurate and efficient sensitivity analysis to compute design derivatives. Presented here is a nonintrusive analytic sensitivity method, that is relatively easy to implement. Furthermore, it can be as accurate as conventional analytic sensitivity methods, which are intrusive and tend to be difficult, if not infeasible, to implement. The nonintrusive local continuum shape sensitivity method with spatial gradient reconstruction (SGR) is formulated for nonlinear systems. This is an extension of the formulation previously published for linear systems. SGR, a numerical technique used to approximate spatial derivatives, can be leveraged to implement the sensitivity method in a nonintrusive manner. The method is used to compute design derivatives for a variety of applications, including nonlinear static beam bending, nonlinear transient gust response of a 2-D beam structure, and nonlinear static bending of rectangular plates. To demonstrate that the method is nonintrusive, all analyses are conducted using black box solvers. One limiting requirement of the method is that it requires the converged Jacobian or tangent stiffness matrix as output from the analysis tool. For each example the design derivatives of the structural displacement response are verified with finite difference calculations.

Effects of partial slip on the peristaltic transport of a hyperbolic tangent fluid model in an asymmetric channel

Abstract: In the present analysis, we have modeled the governing equations of two dimensional hyperbolic tangent fluid model under the assumptions of long wavelength and low Reynolds number. The flow is investigated in a wave frame of reference moving with the velocity of the wave. The governing equations of hyperbolic tangent fluid have been solved using regular perturbation method. The expression for pressure rise has been calculated using numerical integrations. The behavior of different physical parameters have been discussed graphically.

A Standard Basis Free Algorithm for Computing the Tangent Cones of a Space Curve

Abstract: We outline a method for computing the tangent cone of a space curve at any of its points. We rely on the theory of regular chains and Puiseux series expansions. Our approach is novel in that it explicitly constructs the tangent cone at arbitrary and possibly irrational points without using a standard basis.

The tangent stiffness matrix in rigid multibody vehicle dynamics

Abstract: In the development of the equations of motion of a rigid multibody system, particularly vehicles, it is quite common to linearize the equations after they are derived, or even to ignore the non-linear terms from the outset. When doing so, the tangent stiffness matrix, i.e., the stiffness term that results from preload of the system rather than physical flexibility, is often ignored. The motion analysis of preloaded mechanical systems, e.g., the ride quality analysis of vehicle suspensions, may be significantly altered by this omission. Explicit expressions for the tangent stiffness matrix for a few of the common constraint types, including the revolute joint and the rolling wheel, are derived in this article. These expressions are coded into software and included in an open-source linear equation of motion generator for rigid multibody systems. A sample automotive suspension system is analysed, comparing the results with and without the tangent stiffness matrix effects; additionally, a benchmark solution ...

Elasto-Plastic Stability Analysis of the Frame Structures Using the Tangent Modulus Approach

Abstract: This paper presents the procedure for stability analysis of frames in elastic-plastic domain using the concept of the tangent modulus. When the buckling of structure occurs in plastic domain, it is necessary to replace the constant modulus of elasticity E with the tangent modulus Et. Tangent modulus is stress dependent function and takes into account the changes of the member stiffness in the inelastic range. Formulation of the corresponding stiffness matrices is based upon the solution of the equation of bending of the beam according to the second order theory. Numerical analysis was performed using the code ALIN, developed in the C++ programming language

Analysis of Dynamic Stiffness Characteristics of Double-row Angular Contact Ball Bearings

Abstract: A simulation and analysis model is established to describe the dynamic stiffness of double-row angular contact ball bearings based on the dynamics analysis. Fine integral method and predict-correct Adams-Bashforth-Moulton multi-step method are used to solve the dynamic stiffness model. The influences of bearing structural parameters and operating conditions on the dynamic stiffness are investigated.The research provides theoretical basis to analyze the dynamic stiffness and optimize the geometry parameters of double-row angular contact ball bearings. The results show that the groove radius of curvature has a small effect on the bearing dynamic stiffness. With the increase in inner and outer groove radii of curvature,the radial stiffness increases slightly,while the axial stiffness and angular stiffness decrease. Thedynamic stiffness can be improved by increasing the number of balls. A big axial preload contributes to higher dynamic stiffness,while excess preload makes the bearing life decrease,so the axial preload should be selected properly. With the increase in rotating speed,the dynamic stiffness decreases first and then increases. The bearing dynamic stiffness can be replaced by the contact stiffness approximately when the bearing works at a low speed,while at a high speed,the bearing dynamic stiffness should be calculated by combining the contact stiffness with the film stiffness. External loads have an obvious effect on the dynamic stiffness,which gains greatly with the increase in radial and axial loads. The dynamic stiffness changes complexly with the increase in tilting moment,and decreases first and then increases trend on the whole.

The geometry of tangent conjugate connections

Abstract: The notion of conjugate connection is introduced in the almost tangent geometry and its properties are studied from a global point of view. Two variants for this type of connections are also considered in order to find the linear connections making parallel a given almost tangent structure. 2000 AMS Classification: 53C15; 53C05; 53C10; 53C07.

Stiffness identification of four-point-elastic-support rigid plate

Abstract: As the stiffness of the elastic support varies with the physical-chemical erosion and mechanical friction, model catastrophe of a single degree-of-freedom (DOF) isolation system may occur. A 3-DOF four-point-elastic-support rigid plate (FERP) structure is presented to describe the catastrophic isolation system. Based on the newly-established structure, theoretical derivation for stiffness matrix calculation by free response (SMCbyFR) and the method of stiffness identification by stiffness matrix disassembly (SIbySMD) are proposed. By integrating the SMCbyFR and the SIbySMD and defining the stiffness assurance criterion (SAC), the procedures for stiffness identification of a FERP structure (SIFERP) are summarized. Then, a numerical example is adopted for the SIFERP validation, in which the simulated tested free response data are generated by the numerical methods, and operation for filtering noise is conducted to imitate the practical application. Results in the numerical example demonstrate the feasibility and accuracy of the developed SIFERP for stiffness identification.

Semi-Analytical Approach for Stiffness Estimation of 3-DOF PKM

Abstract: Due to their advantageous of high stiffness, high speed, large load carrying capacity and complicated surface processing ability, PKMs (Parallel Kinematic Manipulators) have been applied to machine tools. This paper mainly addresses the issue of stiffness formulation of a three-prismatic- revolute-spherical PKM (3-PRS PKM). The manipulators consist of three kinematic limbs of identical topology structure, and each limb is composed of an actuated prismatic-revolute-spherical. In order to build up the stiffness model, kinematics, Jacobian and finite element analysis are also performed as the basis. Main works in this paper can be outlined as follows. By use of approaches of vector, inverse position analysis of 3-PRS PKM is conducted. When the independent position and orientation parameters of the end-effectors are provided, the translational distances of active prismatic joints can be determined. Then with the aid of the wrench and reciprocal screw theory, the overall Jacobian of this manipulator is formulated quickly, and which is a six by six dimensional matrix and can reflect all information about actuation and constraint singularity. After for- mulating the position analysis and Jacobian matrix, the next step is stiffness analysis. Analytical stiffness model, a function of Jacobian matrix and components stiffness matrix, is obtained first using the principle of virtual work. Stiffness model is also a six by six dimensional matrix and can provide the information of actuation and constraint stiffness simultaneously. For the complex geometry shape of some components, it is impossible to know their stiffness distributions with the varying configuration. Therefore, ANSYS technology has to be applied to compute the stiffness coefficients of these components at different configurations. Then, the computed data are used to obtain the stiffness distribution by use of the numerical fitting method. Up to now, the semi-ana- lytical stiffness model of the manipulator is completely formulated and can be applied to estimate the system stiffness of 3-PRS PKM. The model enables the stiffness of a 3-PRS PKM to be quickly estimated. Provided with the geometry parameters and load situation on tool tip, the stiffness of 3-PRS PKM system is estimated based on the stiffness matrix about tool tip which is obtained by transforming the point from the center of circle composed of three S joints to the tool tip. Then, the stiffness of system along x, y and z directions can be solved. In order to testify the correctness, the corresponding stiffness is also obtained by use of FEA software. The stress distribution and fre- quency of system are also gained by solving the FEA model.

The construction and approximation of feedforward neural network with hyperbolic tangent function

Abstract: In this paper, we discuss some analytic properties of hyperbolic tangent function and estimate some approximation errors of neural network operators with the hyperbolic tangent activation function Firstly, an equation of partitions of unity for the hyperbolic tangent function is given Then, two kinds of quasi-interpolation type neural network operators are constructed to approximate univariate and bivariate functions, respectively Also, the errors of the approximation are estimated by means of the modulus of continuity of function Moreover, for approximated functions with high order derivatives, the approximation errors of the constructed operators are estimated

The decoupling of the Cartesian stiffness matrix in the design of microaccelerometers

Abstract: The 6×6 Cartesian stiffness matrix obtained through finite element analysis for compliant mechanical structures may lead to spurious coupling that stems from discretization error. The coupling may lead, in turn, to inaccurate results of the translational and rotational displacement analysis of the structure, for which reason a reliable decoupling technique becomes essential. In this paper, the authors resort to a decoupling technique of the Cartesian stiffness matrix, reported elsewhere, which is applied to the stiffness matrix of a class of accelerometers. In doing this, the generalized eigenvalue problem is first recalled as a powerful tool that is pertinent to the design task at hand (Ding and Selig in Int. J. Mech. Sci. 46(5): 703–727, 2004). The decoupled submatrices are then investigated by means of eigenvalue analysis. As a consequence, the translational and rotational stiffness matrices can be analyzed independently. Meanwhile, the decoupled stiffness matrices reveal compliance along the sensitive axes and high off-axis stiffness, thereby satisfying the ultimate design objectives for microaccelerometers with isotropic, monolithic structure.

Semi-analytical tools for the analysis of laminated composite cylindrical and conical imperfect shells under various loading and boundary conditions

Abstract: The non-linear buckling of unstiffened laminated composite cones and cylinders will be investigated and new semi-analytical models capable to predict the static and the instability response of these shells under various loads and boundary conditions will be proposed. An introduction is given to the reader in order to present and discuss some of the main deterministic approaches currently used for the design of imperfection sensitive structures. From this introduction it will become clear the need for non-linear tools that can consider both geometric and load imperfections, which are recognized to be among the main factors affecting the load carrying capacity of the shells under discussion. The complete non-linear strain equations are derived using two Equivalent Single-Layer Theories: the Classical Laminated Plate Theory and the First-order Shear Deformation Theory. The non-linear terms will be identified corresponding to Donnell’s, Sanders’ and Timoshenko and Gere’ assumptions, but the discussion will focus on Donnell’s and Sanders’ equations. The resulting strain-displacement equations will then be applied to the stationary conditions of the total potential energy in order to obtain the non-linear static equations and the eigenvalue problem that can be used to predict the instability behavior, the latter using the neutral equilibrium criterion. The Ritz method is selected to solve the non-linear set of equations and a new set of appropriate approximation functions for the displacement field is proposed, in order to simulate axial compression, torsion, pressure, load asymmetry, any arbitrary surface or concentrated loads, and any load case combining these loads. Elastic constraints are used to produce a wide range of boundary conditions, covering the four types of boundary conditions commonly used in the literature. Two methods to solve the non-linear static equations are discussed: Newton-Raphson with line search and Riks; both presented in the full form, where the tangent stiffness matrix is calculated at every iteration, and in the modified form, where the tangent stiffness matrix is updated at the beginning of each load increment (or arc-length increment) and kept constant along the iterations. An analytical integration scheme is proposed for the linear stiffness matrices and a numerical integration scheme proposed for the non-linear stiffness matrices. The analytical integration schemes assume constant laminate properties over the whole conical/cylindrical surface. For conical shells a novel approximation is proposed in order to efficiently perform the analytical integration of the linear stiffness matrices. Detailed convergence analyses are presented and the proposed models are verified with finite element results, models available from the literature and test results from the literature. All the developed tools and algorithms are presented in detail and made available to the reader online. Keywords: Ritz method, Linear, Non-Linear, Static, Buckling, Composite, Conical, Cylindrical, Pressure, Torsion, Axial Compression, Donnell, Sanders, Geometric Imperfection, Load Imperfection

Integrated Stiffness Analysis of Redundant Parallel Manipulator Based on Finite Element Method

Abstract: An integrated stiffness model is established for a Planar Parallel Manipulator (PPM) with actuation redundancy based on Finite Element Method (FEM), and the static stiffness, dynamitic stiffness and moving stiffness of the PPM are analyzed according to the integrated stiffness model. Firstly, a dynamic model of flexible plane beam element is created as a basic unit for branches. Secondly, each branch is assembled in generalized coordinates, and the integrated stiffness model of the PPM is established. Then calculation and simulation for the static stiffness, dynamitic stiffness and moving stiffness are carried out. The results show that the static stiffness and dynamitic stiffness are related with the position and posture of the PPM. The moving stiffness shows that the elastic deformations cause the oscillation of the PPM. In this paper, three stiffness models are unified in the integrated stiffness model, which improves the efficiency of the stiffness calculation and mechanism design.

A phenomenological intra-laminar plasticity model for FRP composite materials

Abstract: The nonlinearity of fibre-reinforced polymer (FRP) composites have significant effects on the analysis of composite structures. This article proposes a phenomenological intralaminar plasticity model to represent the nonlinearity of FRP composite materials. Based on the model presented by Ladeveze et al., the plastic potential and hardening functions are improved to give a more rational description of phenomenological nonlinearity behavior. A four-parameter hardening model is built to capture important features of the hardening curve and consequently gives the good matching of the experiments. Within the frame of plasticity theory, the detailed constitutive model, the numerical algorithm and the derivation of the tangent stiffness matrix are presented in this study to improve model robustness. This phenomenological model achieved excellent agreement between the experimental and simulation results in element scale respectively for glass fibre-reinforced polymer (GFRP) and carbon fibre-reinforced polymer (CFRP). Moreover, the model is capable of simulating the nonlinear phenomenon of laminates, and good agreement is achieved in nearly all cases.

A modal strategy devoted to the hidden state variables method with large interfaces

Abstract: In many mechanical engineering applications, the interactions of a structure through its boundary is modelled by a dynamic boundary stiffness matrix. Nevertheless, it is well known that the solution of such computational model is very sensitive to the modelling uncertainties on the dynamic boundary stiffness matrix. In a recent work, the "hidden state variables method" is used to identify mass, stiffness and damping matrices associated with a given deterministic dynamic boundary stiffness matrix which can be constructed by using experimental measurements. Such an identification allows the construction of the probabilistic model of a random boundary stiffness matrix by substituting those identified mass, stiffness and damping matrices by random matrices. Nevertheless, the numerical cost of the "hidden state variables method" increases drastically with the dimension (number of degrees of freedom) of the interface. We then propose an enhanced approach which consists in a truncated spectral representation of the displacements on the boundary and with a partition of the frequency band of analysis. A collection of mass, stiffness and damping matrices is then identified for each sub-frequency band of analysis. A probabilistic model is constructed in substituting each of those matrices by random matrices. A numerical application is proposed.

Structural Stiffness of Tire Calculated From Strain Energy

Abstract: The vertical stiffness of a tire is the ratio of the vertical force to the deflection; it can be expressed as the summation of the structural stiffness and air stiffness. However, the calculation of the structural stiffness is a challenging topic. This paper presents a new methodology for extracting the structural stiffness from the strain energy of a regular tire. In order to verify our proposed method, the vertical force-deflection results from the finite element method is compared with those from the strain energy method at zero air pressure. Also the results for an inflated tire are compared to calculate the structural stiffness. Finally, we calculated the contribution ratio of the tire components and used an alternative way of extracting the structural stiffness based on changing the Young’s modulus.Copyright © 2015 by ASME

Newton Iteration Method for Analysis of Suspension Cable

Abstract: A Newton iteration method is proposed based on the analytic method of segmental catenary for the suspension cable. The method is developed from the Newton iteration method for the nonlinear equations according to the analysis of different common methods. The tangent stiffness matrix of the iteration method is presented and the characteristics of the stiffness matrix are discussed. Then the iterative calculation process of suspension cable's form-finding for the specific engineering problem is listed. The method offers significant improvements in accuracy, efficiency and calculation amount of the solution, which can meet the needs of practical engineering. The results of the Newton iteration method are compared with the numerical results in other literatures and the contrast verifies the reliability of this method. The method is suitable for the analysis of suspension bridges, cargo ropeways and other various types of cable structures.

A Tetrahedral Decomposition Method for Computing Tangent Curves of 3D Vector Fields

Abstract: This paper presents the development of certain highly efficient and accurate method for computing tangent curves for three-dimensional vector fields. Unlike conventional methods, such as Runge-Kutta method, for computing tangent curves which produce only approximations, the method developed herein produces exact values on the tangent curves based upon piecewise linear variation over a tetrahedral domain in 3D. This new method assumes that the vector field is piecewise linearly defined over a tetrahedron in 3D domain. It is also required to decompose the hexahedral cell into five or six tetrahedral cells for three-dimensional vector fields. The critical points can be easily found by solving a simple linear system for each tetrahedron. This method is to find exit points by producing a sequence of points on the curve with the computation of each subsequent point based on the previous. Because points on the tangent curves are calculated by the explicit solution for each tetrahedron, this new method provides correct topology in visualizing 3D vector fields.