Showing papers on "Direct stiffness method published in 2008"

Variable stiffness composite panels : Effects of stiffness variation on the in-plane and buckling response

Abstract: Descriptions of fiber orientation variation for flat rectangular composite laminates that possess variable stiffness properties are introduced. The simplest definition employs a unidirectional variation based on a linear function for the fiber orientation angle of the individual layers. Analyses of variable stiffness panels for in-plane and buckling responses are developed and demonstrated for two distinct cases of stiffness variations. The first case assumes a stiffness variation in the direction of the loading, and numerical results indicate small improvements in buckling load for some panel configurations due to favorable distribution of the transverse stresses over the panel planform. The second case varies the stiffness perpendicular to the loading, and provides a much higher degree of improvement due to the re-distribution of the applied loads. It is also demonstrated that the variable stiffness concept provides a flexibility to the designer for trade-offs between overall panel stiffness and buckling load, in that there exist many configurations with equal buckling loads yet different global stiffness values, or vice versa.

Stiffness identification and damage localization via differential evolution algorithms

Abstract: The goal of structural health monitoring is to identify which discrepancies between the actual behaviour of a structure and its reference undamaged state are indicative of damage. For this purpose, an objective function, which minimizes the difference between the measured and theoretical modal characteristics of the structure, is formulated. By selecting the stiffness parameters as optimization variables, a differential evolution algorithm is applied to create successive generations that better reflect the measured response, until a certain tolerance is met. At each step of the algorithm, the current modal parameters are re-calculated from the new generation of stiffness matrices to estimate the value of the objective function. This procedure represents a favourable path to solve the so-called ‘inverse problem’. Furthermore, the comparison of the identified stiffness matrix with the initial one allows for damage detection and localization. A numerical example, where a generic structure is discretized into finite elements, is provided. Copyright © 2008 John Wiley & Sons, Ltd.

An investigation on mobility and stiffness of a 3-DOF translational parallel manipulator via screw theory

Abstract: This paper analyzes the mobility and stiffness of a three-prismatic-revolute-cylindrical (3-PRC) translational parallel manipulator (TPM). Firstly, the original 3-PRC TPM is converted into a non-overconstrained manipulator since there exist some practical problems for the overconstrained mechanism. By resorting to the screw theory, it is demonstrated that the conversion brings no influences to the mobility and kinematics of the manipulator. Secondly, the stiffness matrix is derived intuitively via an alternative approach based upon screw theory with the consideration of actuations and constraints, and the compliances subject to both actuators and legs are taken into account to establish the stiffness model. Furthermore, the stiffness performance of the manipulator is evaluated by utilizing the extremum stiffness values over the usable workspace, and the influences of design parameters on stiffness properties are presented, which will be helpful for the architecture design of the TPM.

Cartesian compliance model for industrial robots using virtual joints

Abstract: Industrial robots represent a promising, cost-saving and flexible alternative for machining applications. Due to the kinematics of a vertical articulated robot the system behavior is quite different compared to a conventional machine tool. The robot’s stiffness is not only much smaller but also position dependent in a non-linear way. This article describes the modeling of the robot structure and the identification of its parameters with focus on the analysis of the system’s stiffness. Therefore a method for the calculation of the Cartesian stiffness based on the polar stiffness and the use of the Jacobian matrix is introduced. Furthermore, so called virtual joints are used. With this method it is possible to model each joint of the robot with three degrees of freedom. Beside the gear stiffness the method allows the consideration of the tilting rigidity of the bearing and the link deformations to improve the model accuracy. Based on the results of the parameter identification and the calculation of the Cartesian stiffness the experimental model validation is done.

Stiffness Matrix of Compliant Parallel Mehanisms

Abstract: Starting from the definition of a stiffness matrix, the authors present the Cartesian stiffness matrix of parallel compliant mechanisms. The proposed formulation is more general than any other stiffness matrix found in the literature since it can take into account the stiffness of the passive joints and remains valid for large displacements. Then, the conservative property, the validity,and the positive definiteness of this matrix are discussed.

A 3-D hybrid finite-difference—finite-element viscoelastic modelling of seismic wave motion

Abstract: SUMMARY We have developed a new hybrid numerical method for 3-D viscoelastic modelling of seismic wave propagation and earthquake motion in heterogeneous media. The method is based on a combination of the fourth-order velocity–stress staggered-grid finite-difference (FD) scheme, that covers a major part of a computational domain, with the second-order finite-element (FE) method which can be applied to one or several relatively small subdomains. The FD and FE parts causally communicate at each time level in the FD–FE transition zone consisting of the FE Dirichlet boundary, FD–FE averaging zone and FD Dirichlet zone. The implemented FE formulation makes use of the concept of the global restoring-force vector which significantly reduces memory requirements compared to the standard formulation based on the global stiffness matrix. The realistic attenuation in the whole medium is incorporated using the rheology of the generalized Maxwell body in a definition equivalent to the generalized Zener body. The FE subdomains can comprise extended kinematic or dynamic models of the earthquake source or the free-surface topography. The kinematic source can be simulated using the body-force term in the equation of motion. The traction-at-split-node method is implemented in the FE method for simulation of the spontaneous rupture propagation. The hybrid method can be applied to a variety of problems related to the numerical modelling of earthquake ground motion in structurally complex media and source dynamics.

A parameterized numerical model for the evaluation of gear mesh stiffness variation of a helical gear pair

Abstract: One of the most common challenges in gear drive design is to determine the best combination of gear geometry parameters. These parameters should be capable of being varied effectively and related to gear mesh stiffness variation in advanced excitation and vibration analysis. Accurate prediction of gear mesh stiffness and transmission error requires an efficient numerical method. The parameterized numerical model was developed for the evaluation of excitation induced by mesh stiffness variation for helical gear design purposes. The model uses linear finite-element (FE) method to calculate tooth deflections, including tooth foundation flexibility. The model combines Hertzian contact analysis with structural analysis to avoid large FE meshes. Thus, mesh stiffness variation was obtained in the time and frequency domains, which gives flexibility if comparison is made with measured spectrums. Calculations showed that a fairly low number of elements suffice for the estimation of mesh stiffness variation. A reaso...

Stiffness research on a high-precision, large-workspace parallel mechanism with compliant joints

Abstract: Parallel-structure mechanisms, especially the non-backlash compliant parallel mechanisms, excel serial-structure ones in many indexes. This paper explores a novel six-strut compliant parallel mechanism based on the development of wide-range flexure hinges, and in this system the repeatability and resolution of sub-micron scale can be achieved over cubic centimeter motion range. The system stiffness, as a very important performance for compliant parallel mechanisms, directly influences the workspace, load-carrying capacity and driving-load capacity, etc. The system stiffness depends on the parallel mechanism's geometric dimensions and spatial layout, which is discussed in detail in this paper. The stiffness equation of individual flexure hinge is established firstly, and then the stiffness of the whole mechanism is modeled via assembling stiffness matrices and formulating constraint equations. Finally, the system stiffness influence plots are presented and discussed. The stiffness research on the six-strut compliant parallel mechanism provides further theoretical principles for designing and developing this kind of precision parallel devices.

Using the PiP Model for Fast Calculation of Vibration from a Railway Tunnel in a Multi-layered Half-Space

Abstract: This paper presents a new method for calculating vibration from underground railways buried in a multi-layered half-space. The method assumes that the tunnel’s near-field displacements are controlled by the dynamics of the tunnel and the layer that contains the tunnel, and not by layers further away. Therefore the displacements at the tunnel-soil interface can be calculated using a model of a tunnel embedded in a full space. The Pipe-in-Pipe (PiP) model is used for this purpose, where the tunnel wall and its surrounding ground are modelled as two concentric pipes using elastic continuum theory. The PiP model is computationally efficient on account of uniformity along and around the tunnel. The far-field displacement is calculated by using another computationally efficient model that calculates Green’s functions for a multi-layered half-space using the direct stiffness method. The model is based on the exact solution of Navier’s equations for a horizontally layered half-space in the frequency-wavenumber domain.

On prestress stiffness analysis of bolt-plate contact assemblies

Abstract: Bolt connections are among the most important connections used in structures. The stiffnesses of the bolt and of the connected members are the primary qualities that control the lifetime of the connection. The stiffness of the bolt can be estimated rather easily, in contrast to the member stiffness, but with finite element (FE) and contact analysis, it is possible to find the stiffness of the member. In the case of many connections and for practical applications, it is not suitable to make a full FE analysis. The purpose of the present paper is to find simplified expressions for the stiffness of the member, including the case when the width of the member is limited. The calculation of the stiffness is based on the FE, including the solution to the contact problem, and we express the stiffness as a function of the elastic energy in the structure, whereby the definition of the displacements related to the stiffness is circumvented. The contact analysis is performed using a method where iterations are not necessary, and the results are compared to alternative available results. New practical formulas for the stiffnesses are suggested.

A systematic approach to stiffness analysis of parallel mechanisms

Abstract: A systematic approach to compute total stiffness of parallel manipulators is proposed. Link stiffness and joint stiffness can be calculated simultaneously. Stiffness of passive joint (joint without actuator) is temporally assumed as nonzero variable in the procedure and infinite stiffness can be treated. A simple parallel manipulator is used as an example to illustrate the process and its correctness. Then validity of the proposed approach is confirmed by comparing to FEA and conventional joint stiffness method. The total link compliance (inverses of total stiffness) calculated by the developed approach and its counterpart obtained by FEA model are very close. The total joint stiffness computed by the developed approach and its correspondence derived from conventional joint stiffness method are totally the same. Stiffness performance of parallel force redundancy mechanisms is evaluated. These results indicate that this scheme is feasible and effective.

Exact integration of the stiffness matrix of an 8‐node plane elastic finite element by symbolic computation

Abstract: Computer algebra systems (CAS) are powerful tools for obtaining analytical expressions for many engineering applications in both academic and industrial environments. CAS have been used in this paper to generate exact expressions for the stiffness matrix of an 8-node plane elastic finite element. The Maple software system was used to identify six basic formulas from which all the terms of the stiffness matrix could be obtained. The formulas are functions of the Cartesian coordinates of the corner nodes of the element, and elastic parameters Young’s modulus and Poisson’s ratio. Many algebraic manipulations were performed on the formulas to optimize their efficiency. The redaction in CPU time using the exact expressions as opposed to the classical Gauss–Legendre numerical integration approach was over 50%. In an additional study of accuracy, it was shown that the numerical approach could lead to quite significant errors as compared with the exact approach, especially as element distortion was increased. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 24: 249–261, 2008

General parametric stiffness excitation – anti-resonance frequency and symmetry

Abstract: Stability investigations on vibration cancelling employing the concept of actuators with general stiffness elements are presented Systems with an arbitrary number of degrees of freedom with linear spring- and damping elements are considered, that are subject to self-excitation as well as parametric excitation by stiffness variations with arbitrary phase relations General conditions for full vibration suppression are derived analytically by applying a singular perturbation technique These conditions naturally lead to the terms of parametric resonance and anti-resonance and enable a stability classification with respect to the parametric excitation matrices and their symmetry properties The results are compared to former investigations of systems with a single or synchronous stiffness variation in time and geometrical interpretations are given These basic results obtained can be used for design of a control strategy for actuators with periodically actuated stiffness elements and arbitrary phase relations

Stiffness analysis of 3-d.o.f. overconstrained translational parallel manipulators

Abstract: The paper presents a new stiffness modelling method for overconstrained parallel manipulators, which is applied to 3-d.o.f. translational mechanisms. It is based on a multidimensional lumped-parameter model that replaces the link flexibility by localized 6-d.o.f. virtual springs. In contrast to other works, the method includes a FEA-based link stiffness evaluation and employs a new solution strategy of the kinetostatic equations, which allows computing the stiffness matrix for the overconstrained architectures and for the singular manipulator postures. The advantages of the developed technique are confirmed by application examples, which deal with comparative stiffness analysis of two translational parallel manipulators.

Hybrid Simulation with Improved Operator-Splitting Integration Using Experimental Tangent Stiffness Matrix Estimation

Abstract: Improved numerical integration procedures are essential for the extension of hybrid numerical and experimental simulation to large and complex structural systems While implicit integration algorithms are widely used in pure numerical simulations for their superior stability and accuracy, their direct application to hybrid simulation has been partially limited by difficulties in estimating the tangent stiffness matrix of multi-degree-of-freedom experimental substructures Current applications of hybrid simulation using integrators with improved stability have mostly resorted to methods that are noniterative or utilize the initial stiffness matrix for iterative corrections To improve the accuracy of integration procedures for hybrid simulation, a new method for online estimation of experimental tangent stiffness is proposed The stiffness estimation procedure is tailored for fast online applications by transforming the measurements into a coordinate system, which reduces the number of unknown stiffness coefficients that need to be updated during the simulation The updated experimental stiffness matrix is used in a modified operator-splitting integration scheme to improve the accuracy of hybrid simulations with highly nonlinear experimental substructures The application and effectiveness of the proposed approach is demonstrated through hybrid simulations with multi-degree-of-freedom experimental substructures

3D FEA simulation of segmented reinforcement variable stiffness composites

Abstract: Reconfigurable and morphing structures may provide significant improvement in overall platform performance through optimization over broad operating conditions. The realization of this concept requires structures, which can accommodate the large deformations necessary with modest weight and strength penalties. Other studies suggest morphing structures need new materials to realize the benefits that morphing may provide. To help meet this need, we have developed novel composite materials based on specially designed segmented reinforcement and shape memory polymer matrices that provide unique combinations of deformation and stiffness properties. To tailor and optimize the design and fabrication of these materials for particular structural applications, one must understand the envelope of morphing material properties as a function of microstructural architecture and constituent properties. Here we extend our previous simulations of these materials by using 3D models to predict stiffness and deformation properties in variable stiffness segmented composite materials. To understand the effect of various geometry tradeoffs and constituent properties on the elastic stiffness in both the high and low stiffness states, we have performed a trade study using a commercial FEA analysis package. The modulus tensor is constructed and deformation properties are computed from representative volume elements (RVE) in which all (6) basic loading conditions are applied. Our test matrix consisted of four composite RVE geometries modeled using combinations of 5 SMP and 3 reinforcement elastic moduli. Effective composite stiffness and deformation results confirm earlier evidence of the essential performance tradeoffs of reduced stiffness for increasing reversible strain accommodation with especially heavy dependencies on matrix modulus and microstructural architecture. Furthermore, our results show these laminar materials are generally orthotropic and indicate that previous calculations of matrix gap and interlaminar strains based on kinematic approximations are accurate to within 10-20% for many material systems. We compare these models with experimental results for a narrow geometry and material set to show the accuracy of the models as compared to physical materials. Our simulations indicate that improved shape memory polymer materials could enable a composite material that can accommodate ~30% strain with a cold state stiffness of ~30GPa. This would improve the current state of the art 5-10x and significantly reduce the weight and stiffness costs of using a morphing component.

The statistical two-order and two-scale method for predicting the mechanics parameters of core-shell particle-filled polymer composites

Abstract: The statistical two-order and two-scale method is developed for predicting the mechanics parameters, such as stiffness and strength of core-shell particle-filled polymer composites. The representation and simulation on meso-configuration of random particle-filled polymers are stated. And the major statistical two-order and two-scale analysis formulation is briefly given. The two-order and two-scale expressions for the strains and stresses of conventionally strength experimental components, including the tensional or compressive column, the twist bar and the bending beam, are developed by means of their classical solutions with orthogonal-anisotropic coefficients. Then a new effective mesh generation algorithm is presented. The mechanics parameters of core-shell particle-filled polymer composites, including the expected stiffness parameters, minimum stiffness parameters, and the expected elasticity limit strength and the minimum elasticity limit strength, are defined by means of the stiffness coefficients and elasticity strength criterions for core, shell and matrix. Finally, the numerical results for predicting both stiffness and elasticity limit strength parameters are compared with the experimental data.

Eight‐node membrane element with drilling degrees of freedom for analysis of in‐plane stiffness of thick floor plates

Abstract: The development of eight-node arbitrary quadrilateral membrane elements with drilling degrees of freedom is presented using the compatible displacement interpolation within the element. The element is considered to develop specifically for analysing the in-plane stiffness of thick floor plates in building systems, particularly the transfer plates in tall buildings as well as pile caps. With a new set of shape functions and following the displacement-based element procedure, the element stiffness and force vector are derived and nodal displacements are obtained after solving the simultaneous equations; the element stresses are then determined. A wide range of patch tests is conducted to evaluate the consistency and stability of the proposed element. The test results show very good agreement with the exact solutions of the beam theory. Numerical investigations are carried out, showing that the analyses using the proposed elements provide better results than those from the existing methods. Copyright © 2008 John Wiley & Sons, Ltd.

Theoretical aspects of the internal element connectivity parameterization approach for topology optimization

Abstract: The internal element connectivity parameterization (I-ECP) method is an alternative approach to overcome numerical instabilities associated with low-stiffness element states in non-linear problems. In I-ECP, elements are connected by zero-length links while their link stiffness values are varied. Therefore, it is important to interpolate link stiffness properly to obtain stably converging results. The main objective of this work is two-fold (1) the investigation of the relationship between the link stiffness and the stiffness of a domain-discretizing patch by using a discrete model and a homogenized model and (2) the suggestion of link stiffness interpolation functions. The effects of link stiffness penalization on solution convergence are then tested with several numerical examples. The developed homogenized I-ECP model can also be used to physically interpret an intermediate design variable state.

Exact dynamic stiffness for axial-torsional buckling of structural frames

Abstract: A dynamic axial-torsion buckling theory is proposed for analysis in structural mechanics. Second order effects of the axial force and torque are considered. The consistent natural boundary moments and forces are given to ensure the symmetry of the dynamic stiffness matrix in fulfilling the requirement of the reciprocal theorem. The exact dynamic stiffness matrix is obtained for the first time using power series expansion. Generally distributed axial force can be analyzed without difficulty. It is pointed out that non-uniform sections may not be effectively analyzed due to the convergent problem. The interaction diagrams due to vibration frequency, axial force and torque are studied in details.