Dynamic Finite Element Formulation Research Articles

Active control of FGM shells subjected to a temperature gradient via piezoelectric sensor/actuator patches

AbstractA generic static and dynamic finite element formulation is derived for the modelling and control of piezoelectric shell laminates under coupled displacement, temperature and electric potential fields. The base shell is of functionally graded material (FGM) type, which consists of combined ceramic–metal materials with different mixing ratios of the ceramic and metal constituents. A multi‐input–multi‐output (MIMO) system is applied to provide active feedback control of the laminated shell using self‐monitoring sensors and self‐controlling actuators through a close loop. Numerical studies clearly show the influence of the positional configurations of sensor/actuator pairs on the effectiveness of static and dynamic control for the shell laminates. The effects of the constituent volume fractions on the static and dynamic responses of the shell laminate are also elucidated. Copyright © 2002 John Wiley & Sons, Ltd.

Non-linear static–dynamic finite element formulation for composite shells

Three non-linear finite element formulations for a composite shell are discussed. They are the simplified large rotation (SLR), the large displacement large rotation (LDLR), and the Jaumann analysis of general shells (JAGS). The SLR and the LDLR theories are based on total Lagrangian approach, and the JAGS is based on a co-rotational approach. Both the SLR and LDLR theories represent the in-plane strains exactly the same as Green's strain–displacement relations, whereas, only linear displacement terms are used to represent the transverse shear strain. However, a higher order kinematic through the thickness assumption is used in the SLR theory, which leads to parabolic transverse shear stress distribution compared to a first order kinematic through the thickness relationship used in the LDLR theory that leads to linear transverse shear stress distribution. Furthermore, the LDLR theory uses an Euler-like angle in the kinematics to account for the large displacement and rotation. The JAGS theory decomposes the deformation into stretches and rigid body rotations, where an orthogonal coordinate system translates and rotates with the deformed infinitesimal volume element. The Jaumann stresses and strains are used. Layer-wise stretching and shear warping through the thickness functions are used to model the three-dimensional behavior of the shell, where displacement and stress continuities are enforced along the ply interfaces. The kinematic behavior is related to the original undeformed coordinate system using the global displacements and their derivatives. Numerical analyses of composite shells are performed to compare the three theories. The commercial code ABAQUS is also used in this investigation as a comparison.

Free vibrational analysis of axially loaded bending-torsion coupled beams: a dynamic finite element

In this article, a dynamic finite element formulation for the free vibration analysis of axially loaded bending-torsion coupled beams is presented. Based on the Euler–Bernoulli and St. Venant beam theories, the exact solutions of the differential equations governing the uncoupled vibrations of an axially loaded uniform beam are found. Then, employing these solutions as basis functions, the analytical expressions for uncoupled bending and torsional dynamic shape functions are derived. Exploiting the principle of virtual work, together with the variable approximations based on the resulting shape functions, leads to a single frequency dependent element matrix which has both mass and stiffness properties. The application of the theory is demonstrated by an illustrative example of a bending-torsion coupled beam with cantilever end conditions, for which the influence of axial force on the natural frequencies is studied. The correctness of the theory is confirmed by the published results and numerical checks.

A Dynamic Finite Element (DFE) method for free vibrations of bending-torsion coupled beams

In this paper, a Dynamic Finite Element (DFE) formulation for the free vibration analysis of bending-torsion coupled beams is presented. First, the exact solutions of the differential equations governing the uncoupled vibrations of a uniform beam are found. The employment of these solutions as basis functions leads to the appropriate frequency dependent shape functions which can then be utilized to find the nodal approximations of variables. By exploiting the Principle of Virtual Work (PVW), the elementary Dynamic Stiffness Matrix (DSM) is then obtained which has both mass and stiffness properties. The implementation of the derived DFE matrices in a program is discussed with a particular reference to the Wittrick–Williams algorithm. The application of the theory is demonstrated by illustrative examples wherein a substantial amount of coupling between bending and torsion is highlighted. The correctness of the theory is confirmed, to a high degree of accuracy, by published results and numerical checks.

A NEW DYNAMIC FINITE ELEMENT (DFE) FORMULATION FOR LATERAL FREE VIBRATIONS OF EULER–BERNOULLI SPINNING BEAMS USING TRIGONOMETRIC SHAPE FUNCTIONS

This paper presents a new Dynamic Finite Element (DFE) formulation for the vibrational analysis of spinning beams. A non-dimensional formulation is adopted, and the frequency dependent trigonometric shape functions are used to find a simple frequency dependent element stiffness matrix which has both mass and stiffness properties. An appropriate bisection method, based on a Sturm sequence root counting technique, is used and the flexural natural frequencies of cantilevered beams, for a variety of configurations, are studied. The results are compared to those found by the Dynamic Stiffness Matrix and the classical Finite Elements Method, using “Hermite” beam elements. Much better convergency rates are found using the proposed DFE method.

Slow-dynamic finite element simulation of manufacturing processes

Explicit time integration and dynamic finite element formulations are increasingly being used to analyze nonlinear static problems in solid and structural mechanics. This is particularly true in the simulation of sheet metal manufacturing processes. Employment of slow-dynamic, quasi-static techniques in static problems can introduce undesirable dynamic effects that originate from the inertia forces of the governing equations. In this paper, techniques and guidelines are presented, analyzed and demonstrated, which enable the minimization of the undesirable dynamic effects. The effect of the duration and functional form of the time histories of the loads and boundary conditions is quantified by the analysis of a linear spring mass oscillator. The resulting guidelines and techniques are successfully demonstrated in the nonlinear finite element simulation of a sheet metal deep drawing operation. The accuracy of the quasi-static, slow-dynamic finite element analyses is evaluated by comparison to results of laboratory experiments and purely static analyses. Various measures that quantify the dynamic effects, including kinetic energy, also are discussed.

Dynamic plane-strain finite element simulation of industrial sheet-metal forming processes

A plane-strain implicit dynamic finite element formulation is applied for the analysis of sheet-forming processes. The bending model developed here uses an updated Lagrangian formulation based on incremental nonlinear shell theory which neglects the shear deformation but takes very large displacements and rotations into account. Hill's normally anisotropic yield criterion and associated flow rule are employed. The material is assumed to follow a power law of hardening with strain-rate sensitivity once the initial elastic limit strain is reached. The modified Coulomb law is used to model the interfacial friction. The frictional contact is treated by imposing the constraints directly into the tangent stiffness matrix. The Newton-Raphson algorithm is employed by considering the change in normal due to incremental displacements for contact nodes. Hermite cubic elements are used for the in-plane and out-of-plane displacements, resulting in four degrees of freedom at each node. The results from the developed formulation are found to be in good agreement with other numerical solutions and measured data. Applications are made to industry-scale problems using complex tool geometries with multiple curvatures.

Stress wave effects in a finite element analysis of an impulsively loaded articular joint.

A dynamic contact finite element formulation was used to study transient stresses in the impulsively loaded rabbit knee, an established experimental model of mechanically induced osteoarthrosis. The computations were used to test the hypothesis that stress wave propagation and reflection, from juxtarticular interfaces of material property discontinuity, could be responsible for markedly increased levels of transient local cartilage stress. The finite element results demonstrated intuitively credible stress wave propagation and interfacial reflection phenomena. However, the magnitude of these waves was not nearly large enough to appreciably alter the quasi-static stress distributions otherwise prevailing. Thus, local stress wave reflection from interfaces of modulus discontinuity (for example the cartilage/subchondral plate) probably does not contribute appreciably to the heightened tissue sensitivity to impulsive loading experimentally observed in this animal model.

Nonlinear Dynamic Slope Stability Analysis

A procedure is described for performing nonlinear dynamic slope stability analysis. The method combines updated Lagrangian kinematic relations with a cap‐type, strain‐hardening soil plasticity model in an explicit nonlinear dynamic finite element formulation. A detailed analysis of the progressive failure of an embankment is performed and comparisons with conventional methods for seismic slope stability analysis are presented. The results presented indicated the applicability and accuracy of the proposed method.