Dynamic Stress Analysis of a Bus Systems

This paper presents the effective method for dynamic stress analysis of structural components of bus systems or general mechanical systems. The proposed method is the hybrid superposition method that combined finite element static and eigenvalue analysis with flexible multibody dynamic analysis. In the stress recovery, dynamic stresses are calculated through sum of pseudostatic stresses and modal acceleration stresses, which are obtained by applying the principle of linear superposition to the modal acceleration method. The proposed method is more effective than conventional methods, that is, the mode displacement method or the mode acceleration method. Numerical example of bus systems estimates the efficiency and accuracy of the proposed method. * Author : Tel: +82-0339-369-6179, e-mail: [email protected] # Coauthor INTRODUCTION In structural component design of bus systems, the appropriate criterion for fatigue failure should be based on consideration of failure modes of the specific component being designed. If the structural component is to withstand millions of cycles of load application, a criterion for fatigue failure must be used. The fatigue damage caused by repeated dynamic loads depends on the number of cycles and the frequency of the occurrence of significant stresses. Therefore, for accurate prediction of fatigue failure in structural components, accurate prediction of dynamic stress time histories is required. Experimental testing method may be the most exact for evaluation of dynamic stresses or strains in components of bus systems. The experimental method, however, requires at least one prototype bus system. The structural components in the prototype are then subjected to laboratory durability tests or real vehicle durability tests on proving grounds or fields in which the loading life cycle is applied at an accelerated rate. Owing to growth in the power of digital computers, several analytical approaches for dynamic stress calculation in structural components have been developed to speed design cycles. One method is Quasi-static Method combined with Rigid Multibody Dynamic Simulation. In this method, a bus system simulated by modeling its components as rigid bodies. From the rigid multibody dynamic simulation of bus systems, loads in joints and D’Alembert acceleration loads on structural components of concern are generated. These loads are applied the finite element models of structural components, so that quasi-static stress time histories that used for dynamic stress time histories may be computed. However, it does not consider the flexibility of structural components and the effects of such flexibility on dynamically induced loads, such as joint reaction loads and inertia loads. Therefore, this method may be useful for very stiff systems but inaccurate for flexible systems To consider flexibility effects in dynamic stress calculation, another method can be used ; i.e., Mode Superposition Method combined with Flexible Multibody Dynamic Simulation. In this method, instead of full nodal coordinates, a few component modes are used to represent the deformation field in each structural component for flexible multibody dynamic analysis. Dynamic stress time histories in the structural component are then calculated by linear superposition of modal stresses, multiplied by corresponding modal coordinate time histories that are generated from flexible multibody dynamic analysis of the bus systems. When the modal stress superposition method are used for stress calculation, accuracy of stress is dependent on the components modes that are used in flexible multibody dynamic analysis. This means that inappropriate modes are selected, the resulting stresses will be inaccurate. Therefore, this method must consider the selection of component modes in order to improve accuracy. To improve the accuracy of dynamic stress calculation, the other method can be used ; i.e., Mode Acceleration Method combined with Flexible Multibody Dynamic Simulation. In this method, dynamic stress time histor ies in the component are calculated by summing pseudostatic stresses and modal acceleration stresses at each integration time step. Pseudostatic stresses are obtained from static analysis of structural components using dynamic loads at each integration time step and modal acceleration stresses are calculated by linear superposition. This method is more accurate than the previous method because of static correction of deleted vibration normal modes, but less efficient than the previous one because of increased computational cost; i.e. computational time and hardware space. Therefore, new methodology must be considered in order to improve efficiency. The above three conventional methods are combined to form the hybrid superposition method that improves the accuracy and efficiency of dynamic stress prediction. A brief theory and A conceptual procedure of the hybrid superposition method are presented. Numerical example of bus structures shows the effectiveness of the proposed method. HYBRID SUPERPOSITION METHOD The hybrid superposition method is defined as a computational dynamic stress analysis that employs hybrid sum of the pseudostatic stress and the modal acceleration stress, which are obtained by the principle of linear superposition in order to improve the efficiency of the modal acceleration method. The structural component in bus systems can be modeled in equilibrium with body loads including inertia loads, Coriolis loads, centrifugal loads, gravity forces by ignoring corresponding loads resulted from elastic deformation and surface loads including joint reaction loads, externally applied loads, etc. The so called dynamic equilibrium equation for a structural component is then obtained in the form body rigid surface f f u K u C u M + = + + & & (1) where M , C and K are mass matrix, damping matrix, stiffness matrix. u , u& and u& are displacement, velocity and acceleration vector as nodal coordinates. body rigid f and surface f are body loads and surface loads vector, respectively. If body loads and surface loads that act on the component at time t are known, stresses in the component at time t can be calculated by the static analysis ; i.e., the dynamic problem is reduced to an equivalent static problem, in the conventional manner, by the application of D’Alembert’s principle. Therefore, the static equilibrium equation is converted from the dynamic equilibrium equation of Eq. 1 as u M u C f f u K & &− − + = body rigid surface (2) Consider a component that is in static equilibrium under D’Alembert body loads and surface loads at time t. Since the component is in statically equilibrium, stress in the component at time t can be calculated by executing conventional finite element static analysis with D’Alembert body loads and surface loads. If boundary conditions of the component have free conditions, Static analyses with inertia relief scheme are executed. Stress i σ at node i with viscous damping assumption at time t can