Body-in-White Weight Reduction via Probabilistic Modeling of Manufacturing Variations

A design is robust when it is not sensitive to variations in noise parameters such as manufacturing tolerances, material properties, environmental temperature, humidity, etc. In recent years several robust design concepts have been introduced in an effort to obtain optimum designs and minimize the variation in the product characteristics [1,2]. In this study, a probabilistic design analysis was performed in order to develop a robust design with the mean value of the resulting stress at target, and minimum standard deviation. The methodology for implementing robust design used in this research effort is summarized in a reusable workflow diagram. INTRODUCTION Currently, to account for manufacturing variations, auto body designs are based on the nominal or worst case scenario values, which leads to over-designed components. If the scatter in material properties, thickness and dimensions is accounted for in the finite element analysis stress prediction, it is expected that lighter designs will be produced. This type of stochastic approach can be used to investigate the robustness and sensitivity of a proposed solution and to minimize the risk of failing corporate and consumer tests. In addition, it can potentially reduce the cost by allowing more variation in components that are not critical to performance requirements. In this research effort, probabilistic modeling of manufacturing variations for a structural auto body component (battery tray of an SUV) is performed to determine the sensitivity and the response distribution (stress, stiffness, fatigue life) due to the scatter of the random variables. The scatter of the modulus of elasticity and the thickness and loading are defined in terms of probability distribution functions. Monte Carlo and response surface sampling techniques are implemented in determining the response distribution. Six sigma design criteria are established to size the component and compare this design to the one developed using the traditional nominal value criteria. The Parametric Deterministic FEA Model For this study, the battery tray (FE model shown in figure 1) was selected. The tray is made from a composite material SMC and is supporting the battery (FE model shown in figure 2). Figure 1. FE Model Of The Battery Tray The battery is modeled with an elastic isotropic material of uniform density appropriately adjusted to produce the battery weight. The battery is supported by the tray with a set of springs at appropriate locations. The parametric model contains 2105 shell, 324 solid and 48 spring elements. Figure 2. FE Model of the Battery and Tray Assembly It is assumed that the tray is rigidly fixed at the support locations. The input parameters of the FEA model can be any dimension, material property and loading. For this study three parameters were considered: the wall thickness (t), the modulus of elasticity (E) and the vertical loading (q). We call these model parameters and for any set of values of the three model parameters a solution of the FEA model can produce two model output variables: the maximum Von Mises stress (max e) and the maximum equivalent strain (max e). The Probabilistic FEA Model Uncertainty in the input parameters of the FEA model can be introduced by assuming certain randomness in the input parameters. In this study, it was assumed that the thickness (t), and the modulus of elasticity (E) are characterized by a Gaussian distribution and that the vertical loading (q) is characterized by a lognormal distribution. These assumptions are based on historical data. The distribution parameters (mean values and standard deviations) can be specified to define a set of random values for the model parameters. The mean value of the thickness was considered as a controllable parameter and it was declared as an optimization design variable. The rest of the distribution parameters (mean values of E & q) and the standard deviation of t, E and q were considered uncontrollable or noise parameters. Figures 3, 4 and 5 show the probability distributions and the probability of the input variables, namely, thickness, modulus of elasticity and vertical loading. The ANSYS probabilistic Design System was used to generate the values from the distribution parameters. A set of 100 points from these distributions was used to perform FEA analysis on the tray. It was assumed that the thickness exhibits a Gaussian distribution with a mean value of 3.0 mm and a standard deviation of 0.3 mm. In the optimization model, the mean value of the thickness was an unknown design variable. It was also assumed that the modulus of elasticity exhibits a Gaussian distribution with a mean value of 5723 N/mm and a standard deviation 570 N/mm. 2 3 4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Thickness (mm) P ro b ab ili ty D en si ty Propability D ensity of Thickness 1 2 3 4 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Thickness (mm) P ro ba bi lit y Propability of Th ickness (Input Variable t) Figure 3. Probability Distribution of Thickness t (input variable) 3000 4000 5000 6000 7000 8000 0 0.5 1 1.5 2 2.5 3 3.5 x 10 -4 Modulus of Elastic ity (N/mm ) P ro b ab ili ty D en si ty P ropability Density of Modulus of Elasticity 2000 4000 6000 800