Mohamed A. Omar

Multibody System Modeling of Leaf Springs

Leaf springs are essential elements in the suspension systems of vehicles including sport utility vehicles, trucks, and railroad vehicles. Accurate modeling of the leaf springs is necessary in evaluating ride comfort, braking performance, vibration characteristics, and stability. In order to accurately model the deformations and vibrations of the leaf springs, nonlinear finite-element procedures, which account for the dynamic coupling between different modes of displacement, are employed. Two finite-element methods that take into account the effect of the distributed inertia and elasticity are discussed in this investigation to model the dynamics of leaf springs. The first is based on a floating frame of reference formulation, while the second is an absolute nodal coordinate formulation. The floating frame of reference formulation allows for using a reduced-order model by employing component mode synthesis techniques, while the absolute nodal coordinate formulation enables more detailed finite-element models for the large deformation of very flexible leaf springs. Methods for modeling the contact and friction between the leaves of the spring are discussed. A comparison is also presented between the results obtained using the proposed method and simplified approaches presented in the literature. While there are many issues that can be important in leaf spring modeling, the analysis presented in this paper is focused on a few key issues that include the computer implementation, the effect of the dynamic load on the spring stiffness, the selection of the vibration modes in the reduced-order model, and the effect of the structural damping on the response of the leaf spring.

Static Analysis of Large-Scale Multibody System Using Joint Coordinates and Spatial Algebra Operator

Initial transient oscillations inhibited in the dynamic simulations responses of multibody systems can lead to inaccurate results, unrealistic load prediction, or simulation failure. These transients could result from incompatible initial conditions, initial constraints violation, and inadequate kinematic assembly. Performing static equilibrium analysis before the dynamic simulation can eliminate these transients and lead to stable simulation. Most exiting multibody formulations determine the static equilibrium position by minimizing the system potential energy. This paper presents a new general purpose approach for solving the static equilibrium in large-scale articulated multibody. The proposed approach introduces an energy drainage mechanism based on Baumgarte constraint stabilization approach to determine the static equilibrium position. The spatial algebra operator is used to express the kinematic and dynamic equations of the closed-loop multibody system. The proposed multibody system formulation utilizes the joint coordinates and modal elastic coordinates as the system generalized coordinates. The recursive nonlinear equations of motion are formulated using the Cartesian coordinates and the joint coordinates to form an augmented set of differential algebraic equations. Then system connectivity matrix is derived from the system topological relations and used to project the Cartesian quantities into the joint subspace leading to minimum set of differential equations.

Modeling Flexible Bodies in Multibody Systems in Joint-Coordinates Formulation Using Spatial Algebra

This paper presents an efficient approach of using spatial algebra operator to formulate the kinematic and dynamic equations for developing capabilities to model flexible bodies within general purpose multibody dynamics solver. The proposed approach utilizes the joint coordinates and modal elastic coordinates as the system generalized coordinates. The recursive nonlinear equations of motion are initially formulated using the Cartesian body coordinates (CBC) and the joint coordinates (JC) to form an augmented set of differential algebraic equations. The system connectivity matrix is derived from system topological relations and is used to project the Cartesian quantities into the joint subspace leading to minimum set of differential equations. The modal transformation matrix is used to describe the finite element kinematics in terms of a small set of generalized modal coordinates. Although the resulting stiffness matrix is constant, the mass matrix depends on the generalized elastic modal coordinates and needs to be updated at each time step. To reduce the computational efforts, a set of precomputed inertia shape invariants (ISI) can be identified and used to update the flexible body mass matrix. In this proposed joint-coordinates formulation, the transformation operations required for the flexible body inertia matrix are different from those in case of CBC formulation. The necessary ISI and the algorithm to reconstruct the modal mass matrix will be presented in this paper.

Design and Implementation of a Capstone Course to Satisfy the Industry Needs of Virtual Product Development and ABET Engineering Criteria

Over the past two decades, computer aided engineering (CAE) processes and procedures became an integral part of the product development cycle. Virtual product development (VPD) refers to procedures that integrate the CAE tools in a unified approach that spans all the product development phases. Current industrial trends utilize VPD tools and procedures to reduce the product development time without jeopardizing the product quality. These trends led to an increasing demand for engineers with computer skills, multidisciplinary engineering knowledge, and acquaintance with VPD tools. ABET program outcomes emphasize providing courses with an accumulated background of curricular components to solve realistic open-ended engineering problems. Capstone design project (CDP) course has been regarded as important learning activity that could be designed to provide senior engineering student an opportunity to solve such problems. A major objective of the CDP course is to simulate industrial setting and allow students to experience real-life engineering practice. This paper presents an implementation of the VPD procedures in a mechanical engineering CDP course. This integration simulates the industrial environment through multidisciplinary teams working together in subsystems to produce one product using standard commercial VPD tools. This course implementation is demonstrated using a case study of teams working to design and build a solar car.

A nonlinear finite element model for leaf springs

This paper presents a nonlinear finite element model for the leaf spring that can be used in multibody applications and vehicle dynamic simulations. The floating frame of reference formulation is used in this investigation to model leaf spring nonlinear dynamics. This formulation accounts for the coupling between different modes of deformation as well as the nonlinear coupling between the rigid body motion and the elastic deformation. By employing component mode synthesis techniques, a reduced order model is obtained for the leaf spring while maintaining a good degree of accuracy. The inertia shape integrals can be calculated once in advance using a preprocessor and then stored to be used to automatically generate the nonlinear equations of motion of the leaf spring. The use of a preprocessor to evaluate the inertia shape integrals before the dynamic simulation leads to considerable saving in CPU time and allows the utilization of existing finite element computer codes to obtain the data required for the flexible body simulation. This reduced order model is implemented in a general multibody algorithm in order to examine the effectiveness and robustness of the proposed techniques. As an application, the wind-up deformation of the front suspension system of a typical sport utility vehicle under severe braking condition is investigated.© 2003 ASME