Parviz E Nikravesh
 Professor, AerospaceMechanical Engineering
 Professor, Roshan Persian and Iranian Studies  GIDP
Contact
 (520) 6214966
 Aerospace & Mechanical Engr., Rm. N633
 Tucson, AZ 85721
 pen@email.arizona.edu
Awards
 Outstanding Senior Faculty
 Department of Aerospace and Mechanical Engineering, College of Engineering, Spring 2016
Interests
No activities entered.
Courses
202021 Courses

Planar Multibd Dyn/Applc
AME 452 (Fall 2020) 
Planar Multibd Dyn/Applc
AME 552 (Fall 2020) 
Research
AME 900 (Fall 2020)
201920 Courses

Planar Multibd Dyn/Applc
AME 452 (Summer I 2020) 
Comp Multibody Dynamics
AME 553 (Spring 2020) 
Dissertation
AME 920 (Spring 2020) 
Dynamics Of Machines
AME 352 (Spring 2020) 
Planar Multibd Dyn/Applc
AME 452 (Spring 2020) 
Planar Multibd Dyn/Applc
AME 452 (Fall 2019) 
Planar Multibd Dyn/Applc
AME 552 (Fall 2019) 
Research
AME 900 (Fall 2019)
201819 Courses

Comp Multibody Dynamics
AME 553 (Spring 2019) 
Dissertation
AME 920 (Spring 2019) 
Dynamics Of Machines
AME 352 (Spring 2019) 
Research
AME 900 (Spring 2019) 
Planar Multibd Dyn/Applc
AME 452 (Fall 2018) 
Planar Multibd Dyn/Applc
AME 552 (Fall 2018) 
Research
AME 900 (Fall 2018)
201718 Courses

Comp Multibody Dynamics
AME 553 (Spring 2018) 
Graduate Seminar
AME 696G (Spring 2018) 
Research
AME 900 (Spring 2018) 
Dynamics Of Machines
AME 352 (Fall 2017) 
Planar Multibd Dyn/Applc
AME 452 (Fall 2017) 
Planar Multibd Dyn/Applc
AME 552 (Fall 2017)
201617 Courses

Comp Multibody Dynamics
AME 553 (Spring 2017) 
Graduate Seminar
AME 696G (Spring 2017) 
Independent Study
AME 599 (Spring 2017) 
Dynamics Of Machines
AME 352 (Fall 2016) 
Independent Study
AME 599 (Fall 2016) 
Planar Multibd Dyn/Applc
AME 452 (Fall 2016) 
Planar Multibd Dyn/Applc
AME 552 (Fall 2016)
201516 Courses

Comp Multibody Dynamics
AME 553 (Spring 2016)
Scholarly Contributions
Journals/Publications
 Nikravesh, P. E. (2016). Flexible Multibody Simulation Using Hybrid Integration Scheme. Multibody System Dynamics, 37(1), 314.
 Flores, P., & Nikravesh, P. E. (2013). Comparison of different methods to control constraints violation in forward multibody dynamics. Proceedings of the ASME Design Engineering Technical Conference, 7 A.More infoAbstract: The dynamic equations of motion for constrained multibody systems are frequently formulated using the Newton Euler's approach, which is augmented with the acceleration constraint equations. This formulation results in the establishment of a mixed set of differential and algebraic equations, which are solved in order to predict the dynamic behavior of general multibody systems. It is known that the standard resolution of the equations of motion is highly prone to constraints violation because the position and velocity constraint equations are not fulfilled. In this work, a general review of the main methods commonly used to control or eliminate the violation of the constraint equations in the context of multibody dynamics formulation is presented and discussed. Furthermore, a general and comprehensive methodology to eliminate the constraints violation at the position and velocity levels is also presented. The basic idea of this approach is to add corrective terms to the position and velocity vectors with the intent to satisfy the corresponding kinematic constraint equations. These corrective terms are evaluated as function of the MoorePenrose generalized inverse of the Jacobian matrix and of the kinematic constraint equations. Copyright © 2013 by ASME.
 Arenbeck, H., Missoum, S., Basudha, A., & Nikravesh, P. (2010). Reliabilitybased optimal design and tolerancing for multibody systems using explicit design space decomposition. Journal of Mechanical Design, Transactions of the ASME, 132(2), 021010102101011.More infoAbstract: This paper introduces a new approach for the optimal geometric design and tolerancing of multibody systems. The approach optimizes both the nominal system dimensions and the associated tolerances by solving a reliabilitybased design optimization (RDBO) problem under the assumption of truncated normal distributions of the geometric properties. The solution is obtained by first constructing the explicit boundaries of the failure regions (limit state function) using a support vector machine, combined with adaptive sampling and uniform design of experiments. The use of explicit boundaries enables the treatment of systems with discontinuous or binary behaviors. The explicit boundaries also allow for an efficient calculation of the probability of failure using importance sampling. The probability of failure is subsequently approximated over the whole design space (the nominal system dimensions and the associated tolerances), thus making the solution of the RBDO problem straightforward. The proposed approach is applied to the optimization of a web cutter mechanism. Copyright © 2010 by ASME.
 Arenbeck, H., Basudhar, A., Missoum, S., & Nikravesh, P. E. (2008). Simulation based optimal tolerancing for multibody systems. 2007 Proceedings of the ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, DETC2007, 4, 483495.More infoAbstract: This paper introduces a new methodology for probabilistic optimal design of multibody systems. Specifically, the effects of dimensional uncertainties on the behavior of a system are considered. The proposed reliabilitybased optimization method addresses difficulties such as high computational effort and nonsmoothness of the system's responses, for example, as a result of contact events. The approach is based on decomposition of the design space into regions, corresponding to either acceptable or nonacceptable system performance. The boundaries of these regions are defined using Support Vector Machines (SVMs), which are explicit in terms of the design parameters. A SVM can be trained based on a limited number of samples, obtained from a design of experiments, and allows a very efficient estimation of probability of failure, even when Monte Carlo Simulation (MCS) is used. A modularly structured tolerance analysis scheme for automatic estimation of system production cost and probability of system failure is presented. In this scheme, detection of failure is based on multibody system simulation, yielding high computational demand. A S VMbased replication of the failure detection process is derived, which ultimately allows for automatic optimization of tolerance assignments. A simple multibody system, whose performance usually shows high tolerance sensitivity, is chosen as an exemplary system for illustration of the proposed approach. The system is optimally designed for minimum manufacturing cost while satisfying a target performance level with a given probability. Copyright © 2007 by ASME.
 Kim, S., Gim, G., & Nikravesh, P. E. (2008). A transient tire model for uneven road surface. 2007 Proceedings of the ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, DETC2007, 5 PART C, 19411946.More infoAbstract: This study presents a transient tire model for tire vibration characteristics due to uneven road surfaces. The model is composed of two parts  a static circular beam and a dynamic rigid ring. A new contact algorithm is developed based on the circular beam model, which can estimate contact pressure distribution by solving a set of linear equations. Tire vibration characteristics are then represented by combining the rigid ring model to the circular beam. Examples of contact pressure distribution and tire transient behaviors due to cleat tests are demonstrated and compared with measured data. Copyright © 2007 by ASME.
 Kim, S., Nikravesh, P. E., & Gim, G. (2008). A twodimensional tire model on uneven roads for vehicle dynamic simulation. Vehicle System Dynamics, 46(10), 913930.More infoAbstract: This study proposes a comprehensive analytical tire model for handling and ride comfort in low frequency ranges. A contact algorithm that is developed in this study provides a twodimensional contact pressure distribution on even and uneven road surfaces with reasonable computational cost. Shear stresses and strains during cornering and braking are estimated by direct measurement of tread deformations. The model is validated against experimental force and moment data for general handling simulations. Cleat tests are also conducted and validated under different forward velocity and vertical load conditions for a tire vibration study.
 Nikravesh, P. E. (2008). Newtonianbased methodologies in multibody dynamics. Proceedings of the Institution of Mechanical Engineers, Part K: Journal of Multibody Dynamics, 222(4), 277288.More infoAbstract: Three commonly used methods of formulating the equations of motion for multibody systems are reviewed in this paper. Two of these methods are Newtonianbased; i.e. the entire formulations can be derived simply based on the knowledge of Newton's laws of motion. These two methodologies result in formulations that contain many algebraic and differential equations suitable for computational procedures. The number of equations in these formulations increases as the number of moving bodies and the number of kinematic joints increase. The third reviewed methodology provides a systematic process to transform the large number of equations from any of the other two formulations to a smaller set, possibly to as many as the number of degrees of freedom in the system. This process preserves the simplicity of the original Newtonianbased formulation and, at the same time, provides significant improvement in the computational efficiency and numerical stability in any dynamic analyses. © IMechE 2008.
 Nikravesh, P. E. (2007). Initial condition correction in multibody dynamics. Multibody System Dynamics, 18(1), 107115.More infoAbstract: A simple procedure is presented to correct initial conditions for the coordinates and velocities prior to performing a kinematic or forward dynamic analysis of multibody systems. Such corrections are crucial since slight amount of constraint violations at the start of any numerical integration of equations of motion can lead to erroneous results. The correction process is based on the wellknown method of minimizing the sumofsquares of adjustments in the coordinates or velocities. The process provides a solution that is closest to the estimated values. It should be a simple task to implement this methodology as a preprocessing step for any kinematic or forward dynamic analysis program regardless of the formulation. © Springer Science+Business Media B.V. 2007.
 Lin, Y., & Nikravesh, P. E. (2006). Deformable body model reduction with meanaxes. Mechanics Based Design of Structures and Machines, 34(4), 469488.More infoAbstract: Dynamic analysis of multibody systems containing deformable bodies is computationally timeconsuming because of the large number of deformation degreesoffreedom. Transforming the equations of motion to modal space and then truncating a significant number of higher frequency modes can greatly reduce the problem size and hence improve the computation time. There is a misconception that meanaxes cannot be used for constrained systems when model reduction is desired. This paper attempts to clarify this issue and shows that it is perfectly fine to use the standard static and normal modes in conjunction with the meanaxis conditions.
 Nikravesh, P. E. (2005). An overview of several formulations for multibody dynamics. Product Engineering: EcoDesign, Technologies and Green Energy, 189226.More infoAbstract: In this paper several formulations for automatic generation of the equations of motion for rigid and rigiddeformable multibody systems are reviewed. The rigidbody formulations are the bodycoordinate formulation based on NewtonEuler equations, a nonconventional pointcoordinate formulation where rigid bodies are represented as a collection of particles, and the joint coordinate formulation that employs relative coordinates between bodies. A transformation process based on velocity relations is described for easy transformation of one formulation into another. Finally, a brief overview of the equations of motion for deformable bodies in a multibody environment is presented. © 2005 Springer Science + Business Media, Inc.
 Nikravesh, P. E., & Lin, Y. (2005). Comparison of reduction methods for deformable multibody dynamics. Proceedings of the ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference  DETC2005, 6 B, 13031310.More infoAbstract: Several wellknown reduction methods for deformable multibody dynamics have been reviewed and compared. All of these methods have been borrowed from structural analysis. A recently proposed reduction method that is derived from the equations of motion, based on a mode condensation process, is also reviewed and compared against the other methods. The general conclusion of this investigation is that the standard condensation methods that are based only on fixed/nonmoving boundary conditions should not be used in multibody dynamics. However, when the standard normal modes are accompanied by other properly selected modes, accurate results can be achieved. Similar results can also be obtained by the mode condensation method. The paper pays specific attention to two methods known as the CraigBampton reduction and the mode condensation. Copyright © 2005 by ASME.
 Nikravesh, P. E., & Lin, Y. (2005). Use of principal axes as the floating reference frame for a moving deformable body. Multibody System Dynamics, 13(2), 211231.More infoAbstract: In this paper a method to adopt the principal axes of a deformable body as its bodyattached frame is presented. The deformable body in a multibody setting is allowed to deform while it undergoes rigidbody motion. The fundamental concepts of imposing the principal axes as a moving reference frame are that the origin of the frame must remain at the instantaneous mass center and that the three products of inertia must remain zero as the body deforms. These conditions require the construction of several auxiliary matrices that are used in the constraint equations at the position, velocity, and acceleration levels. These auxiliary matrices are constructed only once and remain unchanged through the motion of the deformable body. The presented formulation does not depend on the type of finite element and multibody formulations or any associated assumptions. © Springer 2005.
 Yi, T. Y., & Nikravesh, P. E. (2002). A method of identify vibration characteristics of modified structures for flexible vehicle dynamics. Proceedings of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering, 216(1), 5563.More infoAbstract: This paper presents a new method for predicting vibration characteristics of a structure that is considered to undergo a design change. It is assumed that for the original structure the vibration data are available either as a complete or as an incomplete set. On the basis of the available vibration data and the known data on the design change, this paper discusses methodologies for determining the vibration characteristics of the modified structure. By considering practical situations, the emphasis of this paper is on structures for which a finite element model, and hence a stiffness matrix, is not available. If a finite element model of the structure is already available, then predicting the modal characteristics of the modified or redesigned structure may be a trivial task. However, if a finite element model of the structure is not available, then a more economical procedure would be to predict the vibration characteristics of the modified structure from experimentally obtained vibration data. Here the idea is to use the available vibration data and to combine them with the structural data of the proposed design change, in order to predict the vibration characteristics of the modified structure. When the vibration data for a structure are obtained experimentally, probably only a limited number of mode shapes and their corresponding natural frequencies can be computed (an incomplete set of modal data). This paper presents the necessary formulations for both complete and incomplete mode sets. Although the main portion of the paper concentrates on identifying the structural stiffness matrix, a methodology is also presented for mass matrix identification. For validating the developed formulations, two examples with numerical results are presented.
 Yi Sr., T. Y., & Nikravesh, P. E. (2001). Extraction of freefree modes from constrained vibration data for flexible multibody models. Journal of Vibration and Acoustics, Transactions of the ASME, 123(3), 383389.More infoAbstract: This paper presents a method for identifying the freefree modes of a structure by utilizing the vibration data of the same structure with boundary conditions. In modal formulations for flexible body dynamics, modal data are primary known quantities that are obtained either experimentally or analytically. The vibration measurements may be obtained for a flexible body that is constrained differently than its boundary conditions in a multibody system. For a flexible body model in a multibody system, depending upon the formulation used, we may need a set of freefree modal data or a set of constrained modal data. If a finite element model of the flexible body is available, its vibration data can be obtained analytically under any desired boundary conditions. However, if a finite element model is not available, the vibration data may be determined experimentally. Since experimentally measured vibration data are obtained for a flexible body supported by some form of boundary conditions, we may need to determine its freefree vibration data. The aim of this study is to extract, based on experimentally obtained vibration data, the necessary freefree frequencies and the associated modes for flexible bodies to be used in multibody formulations. The available vibration data may be obtained for a structure supported either by springs or by fixed boundary conditions. Furthermore, the available modes may be either a complete set, having as many modes as the number of degrees of freedom of the associated FE model, or an incomplete set.
 Wang, D., Friswell, M. I., Nikravesh, P. E., & Kuo, E. Y. (1999). Damage identification in structural joints using generic joint elements. Proceedings of the International Modal Analysis Conference  IMAC, 1, 792798.More infoAbstract: This paper considers the identification of damage in joints for a structure where no baseline model is available. Linear equations are formulated to relate the joint parameter variations caused by damage to the measured or simulated responses in the frequency domain. Generic joint elements with only translational degrees of freedom are defined to model the structure with damage at the joints. For a particular type of joint, the generic joint element stiffness matrices are formed by using only the translational components of the mode shapes, which can be approximated intuitively or by a detailed FE analysis of the joint. The modeling with generic joint elements also makes the consideration of the rotation coupling more natural. In addition, using generic joint elements can avoid the geometric complexity involved in defining structural joints in the traditional way. With the assumption that the damage appears only in the joints, the other parts of the structure such as beamlike and platelike components do not need to be modeled in the formulation. The methods are illustrated by a plane portal frame with both Lshape and Tshape joints. The results show that the method is effective in locating the damage in these joints.
 Wang, D., Nikravesh, P. E., Kuo, E. Y., Jung, W., & Shumin, L. i. (1999). An optimization method for selecting physical modes in polyreference modal analysis of vehicle systems. SAE Technical Papers.More infoAbstract: Distinguishing physical modes from mathematical modes in the modal analysis of complex systems, such as full vehicle structures, is a difficult and timeconsuming process. The major tools frequently used are stabilization diagrams, mode indicator functions, or modal participation factors. When closely spaced modes are to be identified, the stabilization diagrams and mode indicator functions are no longer effective. Even the reciprocities of mode shapes and modal participation factors cannot be well satisfied to indicate whether a mode is a physical one, when measurement errors are large. To overcome these difficulties, an optimization procedure is developed, whereby physical modes can be sorted out in a given frequency range while the error between measured and synthesized frequency responses is minimized. An optimal subset selection algorithm is used in this procedure. Mode significant values are also introduced to show the relative significance of each mode in contributing to measured frequency responses. This innovative method proves to be effective in aiding the selection of physical modes for full vehicle structures. Copyright © 1999 Society of Automotive Engineers, Inc.
 Park, J., & Nikravesh, P. E. (1997). A multibody approach to modeling tire longitudinal and lateral flexibility. SAE Technical Papers.More infoAbstract: A tire longitudinal and lateral flexibility model has been developed which shows considerable advantages over the other available models or methods. Most of the available methods indirectly considers tire lateral flexibility effect by the inclusion of a timelag function, whereas the developed model is a multibody representation of a wheel/tire subsystem so that the resultant dynamics are similar to those of an actual physical system. Sample simulations with a realistic threedimensional vehicle model show the effectiveness of the developed model. Copyright © 1997 Society of Automotive Engineers, Inc.
 Park, J., & Nikravesh, P. E. (1997). Effect of steeringhousing rubber bushings on the handling responses of a vehicle. SAE Technical Papers.More infoAbstract: An investigation of the effect of rackhousing rubber bushings on the handling characteristics of a vehicle is presented using a sophisticated threedimensional vehicle model based on multibody dynamic analysis method. Previous research on this problem has been limited to a transfer function model for a simplified one or twodimensional steering subsystem. This paper uses a multibody modeling approach to find the effects of the steeringsystem compliance on the complete vehicle system. Sample simulations for circular cornering and pulse steering show that the steeringsystem compliance is the source of the frequency peak in the yaw rate to handwheel angle response function. Copyright © 1997 Society of Automotive Engineers, Inc.
 Park, J., & Nikravesh, P. E. (1996). A lookahead driver model for autonomous cruising on highways. SAE Technical Papers.More infoAbstract: This paper presents a lookahead driver model with potential application to autonomous cruising of vehicles on highways. The driver model has two physically meaningful parameters: the lookahead distance and the steering sensitivity. The two parameters are scheduled with respect to the vehicle speed using a preprocessor that converts openloop steering response data to pseudoclosedloop steering response data. Sample simulations with a realistic threedimensional multibody vehicle model are presented for a lanechange scenario. The results indicate that the proposed driver model performs well for a wide range of vehicle speeds. © 1996 Society of Automotive Engineers, Inc.
 Pereira, M. S., & Nikravesh, P. (1996). Impact dynamics of multibody systems with frictional contact using joint coordinates and canonical equations of motion. Nonlinear Dynamics, 9(12), 5371.More infoAbstract: This paper presents a methodology in computational dynamics for the analysis of mechanical systems that undergo intermittent motion. A canonical form of the equations of motion is derived with a minimal set of coordinates. These equations are used in a procedure for balancing the momenta of the system over the period of impact, calculating the jump in the body momentum, velocity discontinuities and rebounds. The effect of dry friction is discussed and a contact law is proposed. The present formulation is extended to open and closedloop mechanical systems where the jumps in the constraints' momenta are also solved. The application of this methodology is illustrated with the study of impact of openloop and closedloop examples. © 1996 Kluwer Academic Publishers.
 Lankarani, H. M., & Nikravesh, P. E. (1994). Continuous contact force models for impact analysis in multibody systems. Nonlinear Dynamics, 5(2), 193207.More infoAbstract: 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 directcentral 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 conjunction 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.
 Lee, J. ., & Nikravesh, P. E. (1994). Steadystate analysis of multibody systems with reference to vehicle dynamics. Nonlinear Dynamics, 5(2), 181192.More infoAbstract: This paper presents a systematic methodology and formulation for determining the steadystate response of multibody systems. The equations of motion for a general multibody system are described in terms of a set of relative joint accelerations. Then, the differential equations of motion are converted to a set of algebraic equations for the steadystate response. These equations are derived based upon a set of conditions that must exist for the steady state. The application of this formulation in determining the steadystate response of a vehicle moving in a circular path is shown. The multibody model of the vehicle for two or fourwheel steering is presented. The results of the steadystate simulation are compared with those obtained from a transient dynamic analysis.
 Gim, G., & Nikravesh, P. E. (1993). A three dimensional tire model for steadystate simulations of vehicles. SAE Technical Papers.More infoAbstract: This paper presents an analytical approach for a three dimensional tire model used for the steadystate simulations of vehicles. The functional relationship of slip ratios of the SAE definition and new definition suitable for vehicle dynamics analysis is first studied. New formula for friction is used and a contact pressure change is considered during driving/braking. The compliances of carcass, braker, and tread are employed for longitudinal and lateral elastic deformation. Rolling resistance force, plysteer, and conicity are also included in the explicit formulations of longitudinal force, lateral force, and aligning torque due to comprehensive slips. © Copyright 1993 Society of Automotive Engineers, Inc.
 Gim, G., & Nikravesh, P. E. (1993). Joint coordinate method for analysis and design of multibody systems: Part 1. System equations. KSME Journal, 7(1), 1425.More infoAbstract: This paper presents a method, known as the joint coordinate method, for the computer aided design process of multibody systems. The relationship between joint and absolute velocities is described by a linear velocity transformation matrix representing the system kinematics. Joint coordinate method can generate a minimal set of equations of motion in terms of the generalized joint accelerations. This method can also yield reaction and actuator forces acting at the kinematic joints that are necessary for forward and inverse dynamics analyses. Applications of this method to static equilibrium and design sensitivity analyses are also studied. Comparisons between the absolute and joint coordinate formulations are given in terms of their computational efficiency. © 1993 The Korean Society of Mechanical Engineers (KSME).
 Gim, G., & Nikravesh, P. E. (1993). Joint coordinate method for analysis and design of multibody systems: Part 2. System topology. KSME Journal, 7(1), 2634.More infoAbstract: In Part 1 of this paper, the method of joint coordinate formulation for multibody dynamics was reviewed. The application of this method to forward and inverse dynamics, static equilibrium, and design sensitivity analyses was studied. In Part 2 of the paper, systematic procedures for constructing the necessary matrices for the joint coordinate formulation are discussed in detail. These matrices are; the primary and the secondary path matrices describing the topology of the system, the velocity transformation matrix, and the generalized inertia matrix. The procedures for constructing these matrices and other necessary elements for the joint coordinate formulation can easily be implemented in a computer program for analysis and design process. © 1993 The Korean Society of Mechanical Engineers (KSME).
 Nikravesh, P. E., & Gim, G. (1993). Systematic construction of the equations of motion for multibody systems containing closed kinematic loops. Journal of Mechanical Design, Transactions Of the ASME, 115(1), 143149.More infoAbstract: 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.
 Nikravesh, P. E., & Lee, J. (1993). Optimal fourwheel steering strategy using nonlinear analytical vehicle models. SAE Technical Papers.More infoAbstract: This paper presents a systematic methodology and formulation for determining optimal strategies for fourwheel steering of vehicles. The methodology is based on multibody dynamics, design sensitivity, and optimization techniques, and is applicable to a wide variety of mechanical systems. The particular application discussed in this paper considers a vehicle model with fourwheel steering capability, and the presented methodology determines an optimal steering angle ratio strategy for the vehicle. It is shown that such a strategy can improve the ride stability of the vehicle, during a variety of maneuvers, when compared against similar strategies obtained from linearized and simplified vehicle models. © Copyright 1993 Society of Automotive Engineers, Inc.
 Ambrosio, J. A., & Nikravesh, P. E. (1992). Elastoplastic deformations in multibody dynamics. Nonlinear Dynamics, 3(2), 85104.More infoAbstract: The problem of formulating and numerically solving the equations of motion for a multibody system undergoing large motion and clastoplastic 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 uppercase characters. (b) Column and algebraic vectors are in boldface lowercase 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. © 1992 Kluwer Academic Publishers.
 Lankarani, H. M., & Nikravesh, P. E. (1992). Canonical impulsemomentum equations for impact analysis of multibody systems. Journal of Mechanical Design  Transactions of the ASME, 114(1), 180186.More infoAbstract: For mechanical systems that undergo intermittent motion, the usual formulation of the equations of motion is not valid over the periods of discontinuity, and a procedure for balancing the momenta of these systems is often performed. A canonical form of the equations of motion is used here as the differential equations of motion. A set of momentum balanceimpulse equations is derived in terms of a system total momenta by explicitly integrating the canonical equations. The method is stable when the canonical equations are numerically integrated and it is efficient when the derived momentum balanceimpulse equations are solved. The method shows that the constraint violation phenomenon, which is usually caused by the numerical integration error, can be substantially reduced as compared to the numerical integration of the standard Newtonian form of equations of motion. Examples are provided to illustrate the validity of the method.
 Lankarani, H. M., & Nikravesh, P. E. (1992). Hertz contact force model with permanent indentation in impact analysis of solids. American Society of Mechanical Engineers, Design Engineering Division (Publication) DE, 44 pt 2, 391395.More infoAbstract: A continuous analysis method for the directcentral impact of two solid particles is presented. Based on the assumption that local plasticity effects are the sole factor accounting for the dissipation of energy in impact, a Hertzian contact force model with permanent indentation is constructed. Utilizing energy and momentum considerations, the unknown parameters in the model are analytically evaluated in terms of a given coefficient of restitution and velocities before impact. The equations of motion of the two solids may then be integrated forward in time knowing the variation of the contact force during the contact period. For illustration, an impact of two soft metallic particles is studied.
 Nikravesh, P. E., & Ambrosio, J. A. (1992). Automatic construction of equations of motion for rigidflexible multibody systems. American Society of Mechanical Engineers, Applied Mechanics Division, AMD, 141, 125132.More infoAbstract: A method for the systematic formulation of the equations of motion for multibody systems containing rigid and flexible bodies is presented. The method of joint coordinates for deriving the minimum number of equations of motion is utilized for rigid bodies, and the finite element method is employed for flexible bodies. The equations of motion for flexible bodies are simplified in several steps: lumped mass assumption, static condensation, and modal superposition. The combined rigid and flexible body formulation can be used to simulate dynamic response in a variety of applications, such as ride handling, rollover, and crash analyses of vehicles; space structure analyses; and biomechanical problems.
 O'Heron, P. J., Nikravesh, P. E., Arabyan, A., & Kunz, D. L. (1992). Tiltwing rotorcraft dynamic analysis using multibody formulation. American Society of Mechanical Engineers, Design Engineering Division (Publication) DE, 44 pt 2, 397402.More infoAbstract: A model is presented that can be used to simulate the highly nonlinear transient dynamics associated with advanced rotorcraft conversion processes. Multibody equations of motion of the fuselage, the tilting wing, and the rotor assembly are derived using a minimal set of coordinates. An enhanced aerodynamics model is employed to account for unsteadiness and nonlinearity in the nearwake aerodynamics, with a dynamic uniform inflow to compute the farwake aerodynamics, and a flight control system is employed to compute the blade pitch settings that are necessary to achieve a desired flight path. The model is subjected to a demanding flight path simulation to illustrate that it can perform vertical takeoff, hover, tiltwing conversion, and highspeed forward flight maneuvers effectively.
 Gim, G., & Nikravesh, P. E. (1991). Analytical model of pneumatic tyres for vehicle dynamic simulations. Part 2. Comprehensive slips. International Journal of Vehicle Design, 12(1), 1939.More infoAbstract: This is the second paper in a threepart study presenting an analytical approach for determining tyre dynamic properties. In this study analytical formulations are derived for the tyre dynamic properties as functions of the slip ratio, slip angle, camber angle and other tyre dynamic parameters. These formulae can be used for the general vehicle simulations in braking/traction and steering manoeuvres with a varying camber angle on regular or irregular terrains. In this paper the lateral force and selfaligning torque due to both slip and camber angles are first studied. The longitudinal, lateral forces and selfaligning torque during braking/traction and steering without any camber angle or with a varying camber angle are also analysed. Also, both the friction ellipse concept and the friction circle concept are investigated.
 Gim, G., & Nikravesh, P. E. (1991). Analytical model of pneumatic tyres for vehicle dynamic simulations. Part 3. Validation against experimental data. International Journal of Vehicle Design, 12(2), 217228.More infoAbstract: This is the third paper in a threepart study presenting an analytical approach for determining tyre dynamic properties. In this study analytical formulations are derived for the tyre dynamic properties as functions of the slip ratio, slip angle, camber angle and other tyre dynamic parameters. These formulae can be used for the general vehicle simulations in braking/traction and steering manoeuvres with a varying camber angle on regular or irregular terrains. In this part the formulations that were derived in Parts 1 and 2 are validated against experimental results. First the tyre dynamic properties due to pure slips are validated. Then the tyre dynamic properties due to comprehensive slips during braking and traction are considered. In general the tyre dynamic properties are shown to have nonlinear characteristics as functions of the slip ratio, slip angle, camber angle and other driving conditions.
 Gim, G., & Nikravesh, P. E. (1991). Comprehensive three dimensional models for vehicle dynamic simulations. Proceedings of the 6th International Pacific Conference on Automotive Engingeering, 10891097.More infoAbstract: This paper presents a comprehensive approach for the mathematical and computer modeling of subsystems required in vehicle dynamic simulations. Three dimensional models for tire terrain interaction, traction, braking, steering, and suspension systems are presented. The tireterrain interaction model provides the necessary tire forces and moments which are determined by explicit formulation or experimental data based model. For the suspension subsystem, such elements as bushing, leaf spring, and stabilizer bar are reviewed. For the steering, traction, and braking subsystems, simplified models are discussed. The models developed here can easily be implemented in a three dimensional multibody simulation program.
 Nikravesh, P. E., & Ambrosio, J. A. (1991). Systematic construction of equations of motion for rigidflexible multibody systems containing open and closed kinematic loops. International Journal for Numerical Methods in Engineering, 32(8), 17491766.More infoAbstract: A systematic method for formulating the equations of motion for multibody systems containing rigid and flexible bodies is presented. The method of using joint coordinates to derive the minimum number of equations of motion is utilized for rigid bodies and the finite element method is employed for flexible bodies. The equations of motion for flexible bodies are simplified into several steps using (a) a lumped mass assumption. (b) static condensation and (c) modal superposition. The combined rigid and flexible body formulation can be used to simulate dynamic response in a variety of applications such as ride handling, rollover and the crash analysis of vehicles, space structures, and biomechanical problems.
 Ambrosio, J. A., Nikravesh, P. E., & Pereira, M. S. (1990). Crashworthiness analysis of a truck. Mathematical and Computer Modelling, 14(C), 959964.More infoAbstract: This paper presents a computerbased technique for the dynamic analysis of multibody systems undergoing large motion and nonlinear structural deformations. As a specific example, this formulation is applied to the rollover simulation of a truck in order to determine the deformation of its safety rollbar cage. The multibody equations of motion are generated systematically form the data that describe the system. The safety rollbar cage is modeled by the finite element method. The rigid body motion of the structure is included in the finite element equations of motion. In severe rollover or crash situations, elastoplastic material characteristics are considered. Due to the size and the nonlinearity of the equations of motion, the computational efforts in solving these equations may be considerable. For this reason, a simplification technique for reducing the number of degrees of freedom is discussed. © 1990.
 Gim, G., & Nikravesh, P. E. (1990). Analytical model of pneumatic tyres for vehicle dynamic simulations. Part 1. Pure slips. International Journal of Vehicle Design, 11(6), 589618.More infoAbstract: This study presents an analytical approach for determining tyre dynamic properties in a sequence of three papers. Most of the necessary parameters are determined using tyre geometry, orientation and some experimental data. Explicit formulations are derived analytically for the tyre dynamic properties as functions of the slip ratio, slip angle, camber angle and other tyre dynamic parameters. These formulations can be used for the general vehicle simulations in braking/traction and steering manoeuvres with a varying camber angle on regular or irregular terrains. In the first paper a brief review of some previous studies is given. Tyre modelling, contact, slip and friction properties are also studied. Tyre forces and moments due to pure slips are investigated. An algorithmic procedure is provided, in the summary. for computational purposes. In the second paper, tyre forces and moments due to comprehensive slips are studied. Finally, in the third paper these analytical models are validated against some experimental data.
 Lankarani, H. M., & Nikravesh, P. E. (1990). Contact force model with hysteresis damping for impact analysis of multibody systems. Journal of mechanisms, transmissions, and automation in design, 112(3), 369376.More infoAbstract: A continuous contact force model for the impact analysis of a twoparticle 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 twoparticle 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 slidercrank mechanism and another sliding block is considered.
 Nikravesh, P. E. (1990). Systematic reduction of multibody equations of motion to a minimal set. International Journal of NonLinear Mechanics, 25(23), 143151.More infoAbstract: This paper presents a twostep process to convert the equations of motion for closedloop 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. © 1990.
 Nikravesh, P. E., & Ambrosio, J. A. (1990). Rollover simulation and crashworthiness analysis of trucks. Forensic Engineering, 2(12), 257258.More infoAbstract: The abstract refers to a computerbased technique for the dynamic analysis of multibody systems undergoing large motions and structural deformations. As a specific example, this formulation is applied to the rollover simulation of a truck (multibody system) to determine the deformation of its cab or safety rollbar cage (structure). For the multibody system, the equations of motion are generated in a systematic form from the data that describe the system. The data contains such information as the inertia characteristics of each body in the system, the connectivity between the bodies, and the forces acting on the bodies. The cab or the safety rollbar cage is modeled by the finite element method. Due to the considerable size and also the nonlinearity of the equations of motion, the computational efforts in solving these equations may become a major concern. For this reason, several stages of reduction techniques and simplifications of these equations were made in order to gain computational efficiency.
 Nikravesh, P. E., & Gim, G. (1990). Ride and stability analysis of a sports car using multibody dynamic simulation. Mathematical and Computer Modelling, 14(C), 953958.More infoAbstract: This paper summarizes the joint coordinate formulation for automatic generation of the equations of motion for dynamic analysis of multibody systems. A computer program based on this formulation is used to model a sports car and simulate its dynamic response in several driving scenarios. The model contains necessary elements of the suspension system such as the leafsprings, shock absorbers, bushings, tie rods, and roll stabilizer bars. An analytical model of the pneumatic tires is incorporated in the model to determine all of the necessary interacting forces and moments between a tire and the road. Results from some selected simulations are shown. © 1990.
 Nikravesh, P. E., Ambrosio, J. A., & Pereira, M. S. (1990). Rollover simulation and crashworthiness analysis of trucks. Forensic engineering, 2(3), 387401.More infoAbstract: This paper presents a computerbased technique for the dynamic analysis of multibody systems undergoing large motion and structural deformations. As a specific example, this formulation is applied to the rollover simulation of a truck (multibody system) to determine the deformation of its safety rollbar cage (structure). For a multibody system, the equations of motion are generated in a systematic form from the data that describe the system. The data contain such information as the inertia characteristics of each body in the system, the connectivity between and the forces acting on the bodies. For a vehicle these equations describe the motion of the chassis, the suspension systems, the wheels, the steering system, and other components. The suspension springs and shock absorbers may have linear or nonlinear characteristics. Tire deformation and the corresponding interaction with the ground are also incorporated into the equations. The cab or the safety rollbar cage is modeled by the finite element method. The model may contain beam, plate, or other types of finite elements. The material property of the structure is assumed to be linear for small structural deformations. However, in severe rollover or crash situations the deformation is no longer elastic, therefore, elastoplastic material characteristics must be considered. The equations of motion for the multibody system and for the structure are then combined to form a complete set of equations of motion. For a vehicle, a numerical solution of these equations results in a dynamic response in a rollover or other situations.
 Lankarani, H. M., & Nikravesh, P. E. (1989). Contact force model with hysteresis damping for impact analysis of multibody systems. American Society of Mechanical Engineers, Design Engineering Division (Publication) DE, 193(pt 3), 4551.More infoAbstract: A continuous contact force model for the impact analysis of a twoparticle 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 twoparticle 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 slidercrank mechanism and another sliding block is considered.
 Nikravesh, P. E., & Gim, G. (1989). Systematic construction of the equations of motion for multibody systems containing closed kinematic loops. American Society of Mechanical Engineers, Design Engineering Division (Publication) DE, 193(pt 3), 2733.More infoAbstract: This paper 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.
 Lankarani, H. M., & Nikravesh, P. E. (1988). Application of the canonical equations of motion in problems of constrained multibody systems with intermittent motion. American Society of Mechanical Engineers, Design Engineering Division (Publication) DE, 14, 417423.More infoAbstract: For mechanical systems that undergo intermittent motion, the usual formulation of the equations of motion is not valid over the periods of the discontinuity, and a procedure for balancing the momenta of the system is often performed. A canonical form of the equations of motion is used here as the differential equations of motion. A set of momentum balanceimpulse equations are derived in terms of the system total momenta by explicitly integrating the canonical equations. The method shows to be stable while numerically integrating the canonical equations, and efficient while solving the momentum balanceimpulse equations. Examples are provided to illustrate the validity of the method.
 Chang, C. O., & Nikravesh, P. E. (1985). ADAPTIVE CONSTRAINT VIOLATION STABILIZATION METHOD FOR DYNAMIC ANALYSIS OF MECHANICAL SYSTEMS.. American Society of Mechanical Engineers (Paper).More infoAbstract: The transient dynamic analysis of equations of motion for constrained mechanical systems requires the solution of a mixed set of algebraic and differential equations. A constraint violation stabilization method, based on feedback control theory of linear systems, has been suggested by some researchers for solving these equations. However, since the value of damping parameters for this method are uncertain, the method is to some extent unattractive for generalpurpose use. This paper presents an adaptive mechanism for determining the damping parameters. The results of the simulation for two examples illustrate the improvement in reducing the constraint violations when using this method.
 Chang, C. O., & Nikravesh, P. E. (1985). OPTIMAL DESIGN OF MECHANICAL SYSTEMS WITH CONSTRAINT VIOLATION STABILIZATION METHOD.. American Society of Mechanical Engineers (Paper).More infoAbstract: This paper presents a comprehensive optimal design procedure for constrained dynamic systems. The constraint violation stabilization method for dynamic analysis of mechanical systems is briefly reviewed. A direct differentiation method is used to form the equations of design sensitivity analysis based on a constraint violation stabilization method. The sensitivity equations and the equations of motion are integrated simultaneously to obtain the system response, as well as the state sensitivity matrices. All integrations are performed using a multistep predictorcorrector method. The first order design sensitivity matrix is used to calculate the gradient of cost function and the performance constraint during the optimization procedure. An optimization routine is linked to the analysis/sensitivity algorithm. Two examples are given which illustrate the effectiveness of this method for determining the optimal design of a system.
 Nikravesh, P. E. (1985). DYNAMIC ANALYSIS OF LARGESCALE MECHANICAL SYSTEMS AND ANIMATED GRAPHICS.. Journal of Guidance, Control, and Dynamics, 8(1), 104109.More infoAbstract: This paper presents a computerbased method for formulation and solution of coupled differential and algebraic equations describing largescale mechanical systems, including feedback control, aerodynamic forces, and other multidisciplinary effects that interact with the mechanical system. A Euler parameter representation of the configuration of the mechanical system is employed to obtain singularity free solution for generalized coordinates and a much simplified algebraic formulation of the governing system of equations, compared with the more classical Euler angle generalized coordinate formulation. A generalized coordinate partitioning algorithm is employed to identify independent generalized coordinates automatically and reduce the dimension of the numerical integration problem. Animated graphics output is employed to assist in visualization of dynamic performance of several systems  a parachute descending in air, an aircraft landing on a damaged runway, and a truck with flexible chassis.
 Nikravesh, P. E., & Srinivasan, M. (1985). GENERALIZED COORDINATE PARTITIONING IN STATIC EQUILIBRIUM ANALYSIS OF LARGESCALE MECHANICAL SYSTEMS.. International Journal for Numerical Methods in Engineering, 21(3), 451464.More infoAbstract: This paper presents a computerbased method for automatic formulation and efficient numerical solution of static equilibrium equations for nonlinear constrained mechanical systems with conservative forces. Nonlinear holonomic constraint equations and a potential energy function are written in terms of a maximal set of Cartesian generalized coordinates. A stable static equilibrium configuration is found by minimizing the potential energy of the system, subject to the kinematic constraint equations, i. e. constrained optimization. A Gaussian elimination algorithm with full pivoting decomposes the constraint Jacobian matrix, identifies dependent and independent coordinates and constructs an influence coefficient matrix that relates variations in dependent and independent coordinates. This information is employed to convert the constrained optimization problem to an unconstrained optimization problem. A simple example is presented to illustrate the method. An algorithm that may be used in analysis of largescale systems is presented.
 Nikravesh, P. E., Wehage, R. A., & Kwon, O. K. (1985). EULER PARAMETERS IN COMPUTATIONAL KINEMATICS AND DYNAMICS. PART 1 AND PART 2.. Array, 358369 jmtddk.More infoAbstract: Part I of this paper presents useful and interesting identities between Euler parameters and their time derivatives. Using these identities, kinematic constraints and equations of motion for constrained mechanical systems are derived. These equations can be developed into a computer program to systematically generate all of the necessary equations to model mechanical systems. Part II represents a methodology for formulating kinematic constraint equations and equations of motion for constrained mechanical systems. An algorithm for solving the constrained equations of motion using a constraint stabilization technique is reviewed. Significant reduction in computation time can be achieved with this formulation and the accompanying algorithm as compared with the method presented in Part I.
 Nikravesh, P. E. (1984). APPLICATION OF ANIMATED GRAPHICS IN LARGE SCALE MECHANICAL SYSTEM DYNAMICS.. NATO ASI Series, Series F: Computer and Systems Sciences, 9, 369377.More infoAbstract: A computerized method for response postprocessing in dynamic analysis of mechanical systems is presented. The transient response from planar and spatial dynamic analysis programs are processed. Output in print, plot, singleframe and animated graphics can be obtained. Several methods for generating animated graphic displays are reviewed.
 Nikravesh, P. E. (1984). SOME METHODS FOR DYNAMIC ANALYSIS OF CONSTRAINED MECHANICAL SYSTEMS: A SURVEY.. NATO ASI Series, Series F: Computer and Systems Sciences, 9, 351368.More infoAbstract: 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.
 Nikravesh, P. E. (1984). SPATIAL KINEMATIC AND DYNAMIC ANALYSIS WITH EULER PARAMETERS.. NATO ASI Series, Series F: Computer and Systems Sciences, 9, 261281.More infoAbstract: This paper is devoted to developing the mathematical tools involved in describing angular orientation and equations of motion of rigid bodies, which are of considerable interest in themselves. Euler parameters are employed to derive interesting and useful identities. Physical interpretation of Euler parameters and their corresponding transformation matrices are discussed. Lagrange's equations of motion in terms of Euler parameters for a rigid body are presented.
 Nikravesh, P. E., & Chung, I. S. (1984). STRUCTURAL COLLAPSE AND VEHICULAR CRASH SIMULATION USING A PLASTIC HINGE TECHNIQUE.. Journal of structural mechanics, 12(3), 371400.More infoAbstract: This paper presents a computerbased method for formulation and efficient solution of nonlinear, constrained differential equations of motion for spatial dynamic analysis of mechanical systems and its application to automobile crash simulation. 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 Euler parameters. A numerical integration algorithm with positiveerror control, employing a predictorcorrector algorithm with variable order and step size, integrates for only the independent variables, yet effectively determines dependent variables. The program is used to analyze plastic deformations of structures by employing a plastic hinge concept.
 Nikravesh, P. E., Kwon, O. K., & Wehage, R. A. (1984). EULER PARAMETERS IN COMPUTATIONAL KINEMATICS AND DYNAMICS. PART 2.. American Society of Mechanical Engineers (Paper).More infoAbstract: A methodology for formulating kinematic constraint equations and equations of motion for constrained mechanical systems is presented. Constraint equations and transformation matrices are expressed in terms of Euler parameters. The kinematic velocity and acceleration equations, and the equations of motion are expressed in terms of physical angular velocity of the bodies. An algorithm for solving the constrained equations of motion using a constraint stabilization technique is reviewed. Significant reduction in computation time can be achieved with this formulation and the accompanying algorithm as compared with the method presented in Part 1.
 Nikravesh, P. E., Skorton, D. J., Chandran, K. B., Attarwala, Y. M., Pandian, N., & Kerber, R. E. (1984). Computerized threedimensional finite element reconstruction of the left ventricle from crosssectional echocardiograms. Ultrasonic Imaging, 6(1), 4859.More infoPMID: 6540911;Abstract: A computerized method for the generation of a threedimensional 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, computerassisted, position registration arm with six degreesoffreedom. These six degreesoffreedom 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 threedimensional 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. © 1984.
 Nikravesh, P. E., Wehage, R. A., & Kwon, O. K. (1984). EULER PARAMETERS IN COMPUTATIONAL KINEMATICS AND DYNAMICS. PART 1.. American Society of Mechanical Engineers (Paper).More infoAbstract: Using identities between Euler parameters and their time derivatives, kinematic constraints and equations of motion for constrained mechanical systems are derived. These equations can be developed into a computer program to systematically generate all of the necessary equations to model mechanical systems. The compact form of these equations makes it possible to develop a generalpurpose computer program for dynamic analysis of mechanical systems suitable for operation on small computers with limited memory space.
 Nikravesh, P. E., & Haug, E. J. (1983). GENERALIZED COORDINATE PARTITIONING FOR ANALYSIS OF MECHANICAL SYSTEM WITH NONHOLONOMIC CONSTRAINTS.. Journal of Mechanisms, Transmissions, and Automation in Design, 105(3), 379384.More infoAbstract: A computerbased method is presented for formulation and efficient solution of nonlinear, constrained differential equations of motion for spatial dynamic analysis of mechanical systems with holonomic and nonholonomic constraints. Holonomic and nonholonomic constraint equations and differential equations of motion are written in terms of a maximal set of Cartesian generalized coordinates, three translational and four rotational coordiantes for each rigid body in the system, where the rotational coordinates are Euler parameters. 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 and identifies independent coordinates and velocities.
 Nikravesh, P. E., Chung, I. S., & Benedict, R. L. (1983). Plastic hinge approach to vehicle crash simulation. Computers and Structures, 16(14), 395400.More infoAbstract: This paper presents a computerbased 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 jointspring 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 elastoplastic response characteristics and the dynamic correction factors for structural components. © 1983.
 NIKRAVESH, P. E., & CHUNG, I. S. (1982). APPLICATION OF EULER PARAMETERS TO THE DYNAMIC ANALYSIS OF THREEDIMENSIONAL CONSTRAINED MECHANICAL SYSTEMS.. J MECH DES TRANS ASME, V 104(N 4), 785791.More infoAbstract: THIS PAPER PRESENTS A COMPUTERBASED METHOD FOR FORMULATIONAND 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 BODYIN THE SYSTEM, WHERE THE ROTATIONAL COORDINATES ARE THE EULER PARAMETERS. A NUMERICAL INTEGRATION ALGORITHM WITH POSITIVEERROR CONTROL, EMPLOYING A PREDICTORCORRECTOR ALGORITHM WITH VARIABLE ORDER AND STEP SIZE, INTEGRATES FOR ONLY THE INDEPENDENT VARIABLES, YET EFFECTIVELY DETERMINES DEPENDENT VARIABLES.
 Nikravesh, P. E., & Haug, E. J. (1982). GENERALIZED COORDINATE PARTITIONING FOR ANALYSIS OF MECHANICAL SYSTEMS WITH NONHOLONOMIC CONSTRAINTS.. American Society of Mechanical Engineers (Paper).
 Nikravesh, P. E., & Chung, I. S. (1981). APPLICATION OF EULER PARAMETERS TO THE DYNAMIC ANALYSIS OF THREEDIMENSIONAL CONSTRAINED MECHANICAL SYSTEMS.. American Society of Mechanical Engineers (Paper).More infoAbstract: This paper presents a computerbased 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. A numerical integration algorithm with positiveerror control, employing a predictorcorrector algorithm with variable order and step size, integrates for only the independent variables, yet effectively determines dependent variables.
 Skorton, D. J., Chandran, K. B., Nikravesh, P. E., Pandian, N. G., & Kerber, R. E. (1981). THREEDIMENSIONAL FINITE ELEMENT RECONSTRUCTIONS FROM TWODIMENSIONAL ECHOCARDIOGRAMS FOR ESTIMATION OF MYOCARDIAL ELASTIC PROPERTIES.. Computers in Cardiology, 383386.
 Lin, H. S., Liu, Y. K., Ray, G., & Nikravesh, P. (1978). Systems identification for material properties of the intervertebral joint. Journal of Biomechanics, 11(12), 114.More infoPMID: 659451;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 3D 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.
 Ward, C. C., Nikravesh, P. E., & Thompson, R. B. (1978). Biodynamic finite element models used in brain injury research. Aviation Space and Environmental Medicine, 49(1 II), 136142.More infoPMID: 623576;Abstract: Three dimensional finite element models for the monkey, baboon, and human brains have been developed and are described. Isoparametric brick elements and membrane elements represent the soft tissue and partitioning internal folds of dura, respectively. By specifying the finite element mesh on the skull inner surface, the irregular shape of the brain is generated. Each model is subjected to the same skull acceleration to investigate response relationships between species. Important dynamic response differences are revealed by comparing the computed intracranial pressures. Experimentally derived head injury data are correlated with model dynamic responses. Using the baboon and monkey models, brain injury tests are simulated and model response measures are compared to produced injury. Using the human model, computed stresses are compared to intracranial pressures measured in cadaver impact tests.
Proceedings Publications
 Gil, G., & Nikravesh, P. E. (2014, June 30July 3). Flexible Multibody Simulation Using Hybrid Integration Scheme. In 3rd Joint International Conference on Multibody Systems Dynamics/7th Asian Conference on Multibody Dynamics.
 Kazemi, O., Ribaric, A. P., Nikravesh, P. E., & Kim, S. (2014, June 30July 3). Nonrolling Mesh for a Rolling Finiteelement Tire Model. In 3rd Joint International Conference on Multibody Systems Dynamics/7th Asian Conference on Multibody Dynamics.
Presentations
 Nikravesh, P. E. (2016, May). Challenges in Using a Finite Element Tire Model for Multibody Dynamic Analysis of Vehicles. 4th IMSD Conference. Montreal, Canada: McGill University.More infoKeynote Speaker for Conference