Parviz E. Nikravesh

A contact force model with hysteresis damping for impact analysis of multibody systems

A continuous contact force model for the impact analysis of a two-particle collision is presented. The model uses the general trend of the Hertz contact law. A hysteresis damping function is incorporated in the model which represents the dissipated energy in impact. The parameters in the model are determined, and the validity of the model is established. The model is then generalized to the impact analysis between two bodies of a multibody system. A continuous analysis is performed using the equations of motion of either the multibody system or an equivalent two-particle model of the colliding bodies. For the latter, the concept of effective mass is presented in order to compensate for the effects of joint forces in the system. For illustration, the impact situation between a slider-crank mechanism and another sliding block is considered.

Continuous contact force models for impact analysis in multibody systems

One method for predicting the impact response of a multibody system is based on the assumption that the impacting bodies undergo local deformations and the contact forces are continuous. In a continuous analysis, the integration of the system equations of motion is carried out during the period of contact; therefore, a model for evaluating the contact forces is required. In this paper, two such contact force models are presented, both Hertzian in nature and based upon the direct-central impact of two solid particles.At low impact velocities, the energy dissipation during impact can be represented by material damping. A model is constructed based on the general trend of the Hertz contact law in conjuction with a hysteresis damping function. The unknown parameters are determined in terms of a given coefficient of restitution and the impact velocity. When local plasticity effects are the dominant factor accounting for the dissipation of energy at high impact velocities, a Hertzian contact force model with permanent indentation is constructed. Utilizing energy and momentum considerations, the unknown parameters in the model are again evaluated. The two particle models are generalized to an impact analysis between two bodies of a multibody system.

Application of Euler Parameters to the Dynamic Analysis of Three-Dimensional Constrained Mechanical Systems

This paper presents a computer-based method for formulation and efficient solution of nonlinear, constrained differential equations of motion for spatial dynamic analysis of mechanical systems. Nonlinear holonomic constraint equations and differential equations of motion are written in terms of a maximal set of Cartesian generalized coordinates, three translational and four rotational coordinates for each rigid body in the system, where the rotational coordinates are the Euler parameters. Euler parameters, in contrast to Euler angles or any other set of three rotational generalized coordinates, have no critical singular cases. The maximal set of generalized coordinates facilitates the general formulation of constraints and forcing functions. A Gaussian elimination algorithm with full pivoting decomposes the constraint Jacobian matrix, identifies dependent variables, and constructs an influence coefficient matrix relating variations in dependent and indpendent variables. This information is employed to numerically construct a reduced system of differential equations of motion whose solution yields the total system dynamic response. A numerical integration algorithm with positive-error control, employing a predictor-corrector algorithm with variable order and step size, integrates for only the independent variables, yet effectively determines dependent variables.

Systematic construction of the equations of motion for multibody systems containing closed kinematic loops

This papers presents a systematic method for deriving the minimum number of equations of motion for multibody system containing closed kinematic loops. A set of joint or natural coordinates is used to describe the configuration of the system. The constraint equations associated with the closed kinematic loops are found systematically in terms of the joint coordinates. These constraints and their corresponding elements are constructed from known block matrices representing different kinematic joints. The Jacobian matrix associated with these constraints is further used to find a velocity transformation matrix. The equations of motions are initially written in terms of the dependent joint coordinates using the Lagrange multiplier technique. Then the velocity transformation matrix is used to derive a minimum number of equations of motion in terms of a set of independent joint coordinates. An illustrative example and numerical results are presented, and the advantages and disadvantages of the method are discussed.

Systems identification for material properties of the intervertebral joint

Abstract An identification of the elastic solid properties of the intervertebral disc was achieved by the finite element method utilizing experimental data from the axial loading of a lumbar intervertebral joint. By assuming orthotropic material properties for the vertebra and disc, a 3-D finite element model resembling the test specimens was constructed and exercised. The model deformations were then compared to the experimental measurements and an initial correlation found between them. Finally, using an efficient optimization scheme, the overall material constants of the disc were optimally determined by minimizing the error between the experimental data and the predicted deformations from the finite element analysis. The material constants thus identified show the orthotropic elastic moduli of the lumbar intervertebral disc to be independent of segment level but decrease directly as a function of the severity of degeneration.

Computerized three-dimensional finite element reconstruction of the left ventricle from cross-sectional echocardiograms

A computerized method for the generation of a three-dimensional finite element mesh of left ventricular geometry is presented. The technique employs two dimensional echocardiographic images of the left ventricle. The echocardiographic transducer is attached to an articulated, computer-assisted, position registration arm with six degrees-of-freedom. These six degrees-of-freedom record the location and orientation of the transducer, when images are obtained, referenced to an external point. Hence, the images are digitized and aligned relative to one another, then several interpolation and curve fitting steps are used to reconstruct a three-dimensional finite element model of the left ventricle. The finite element model can be used for volume determination, stress analysis, material property identification, and other applications.

Elasto-plastic deformations in multibody dynamics

The problem of formulating and numerically solving the equations of motion for a multibody system undergoing large motion and clasto-plastic deformations is considered here. Based on the principles of continuum mechanics and the finite element method, the equations of motion for a flexible body are derived. It is shown that the use of a lumped mass formulation and the description of the nodal accelerations relative to a nonmoving reference frame lead to a simple form of these equations. In order to reduce the number of coordinates that describe a deformable body, a Guyan condensation technique is used. The equations of motion of the complete multibody system are then formulated in terms of joint coordinates between the rigid bodies. The kinematic constraints that involve flexible bodies are introduced in the equations of motion through the use of Lagrange multipliers.In this paper the following general rules will apply:(a)Matrices and higher order tensors are in boldface upper-case characters.(b)Column and algebraic vectors are in boldface lower-case characters.(c)Scalars are in lightface characters.(d)Summation convention is applied when tensors are written on component form.(e)Left superscript denotes the configuration in which an event occurs.(f)Left subscript denotes the configuration to which an event is refered to.

Some Methods for Dynamic Analysis of Constrained Mechanical Systems: A Survey

Three algorithms are presented for dynamic analysis of constrained mechanical systems. The first algorithm integrates the differential equations of motion without any consideration for constraint violation. The other two algorithms consider the violation of the kinematic constraints and correct the violation in two different ways. A brief comparison between these algorithms is also provided.

Plastic hinge approach to vehicle crash simulation

Abstract This paper presents a computer-based method for formulation and efficient solution of nonlinear, constrained differential equations of motion for spatial dynamic analysis of mechanical systems and its application to vehicle crash simulations. The program can be used to analyze plastic deformations of structures by employing a plastic hinge concept. A structure is divided into several components connected by plastic hinges. A plastic hinge is modeled by a joint-spring combination to represent the structural characteristics. The spring characteristics, which are obtained from experimental results, need to be modified to account for the elastic effects and to compensate for the element length difference between experiment and simulation. Dynamic correction factors are incorporated into the spring characteristics to account for the strain rate effects in the simulation. The plastic hinge technique is applied to a torque box crash event. It is found that the program, with the plastic hinge concept, provides relatively accurate results for the crash simulation, subject to availability of the elasto-plastic response characteristics and the dynamic correction factors for structural components.

Systematic reduction of multibody equations of motion to a minimal set

Abstract This paper presents a two-step process to convert the equations of motion for closed-loop systems from a large set of absolute coordinates to a minimal set of relative joint coordinates. Initially, absolute coordinates are used to define the position of each body, the kinematic joints, and the forces acting on the bodies. Before numerical integration of the equations of motion, the equations are converted to a minimal set to gain computational efficiency. A simple example is provided to illustrate the conversion steps. It is also shown that the equations of motion can be expressed in terms of the time derivative of the system momenta, instead of the accelerations, to reduce numerical integration error and gain computational stability.