214-2010: Statistical Modeling of the Mechanical Behavior of Spring, Mass, and Damper Assembly Using SAS®

Dampers in mechanical systems provide safety and comfort against dynamical and impact forces transmitted by external profiles. Since dampers are made of diverse elements, have a large range of performance variables. Therefore, analytical solution for predicting damper behavior is difficult. In the absence of analytical models, one of the popular and reliable methods is statistical modeling. In this study, a modeling approach for a continuously variable damper was successfully used with variable discrete time model through Euler's approximation. Effect of changing spring constants, forces, damper constants and initial displacement in a springmass-damper assembly for their effects upon damper characteristics were closely investigated via simulations using PROC MODEL. Initial values were specified in a VAR statement. The mass, time step, spring constants, and damper constants were declared and initialized by a CONTROL statement. Damping force equation was written in SAS® language and solved through a SOLVE statement. Multiple runs of simulation were compared by merging data sets and overlaying plots at discrete time intervals. Error monitoring and optimization were performed in PROC IML. INTRODUCTION PROBLEM FORMULATION Damper is an important component of any mechanical system. In this particular analysis a simple model of four springs in series, mass (three cars connected in series) damper (suspension) system in series is used which is shown in Figure 1. A torque is input into the system at the motor which exerts a force on the first car which changes the position of the cars. The change in position of the car is measured by an encoder. There are shutdown safety switches to stop unstable operation. In this particular analysis, the force F(t) applied to the first car is the only input to the system and the displacement of the car (X(t)) is the output of concern. The goal of this project is to design a controller which can control the displacement of all the cars at each instant of time (t) for a specified force input. The value of displacement at the specified force input at each instant of time is estimated. In terms of a cars‟ suspension, when the car hits a bump a force will act on the tire. The tire will flex a certain amount to reduce the effect of the force on the passengers. The design goal is a suspension that will react to the force and reach equilibrium in the tire quickly. To get the system to respond quickly to the force, a series of springs are used with appropriate values of springs. This is equivalent to changing stiffness of the coil spring in the suspension system. The force input F(t) can take various forms and can be modeled readily by standard mathematical functions: i) an unforced system which is modeled by using F(t) =0, ii) a constant applied force using F(t) = constant (F), iii) a constantly changing force (ramp input) with F(t) = Bt + C (B & C are constants), iv) a quadratically changing force, F(t) = At 2 + Bt + C (A, B & C are constants), v) an oscillating force (sinusoidal input), F(t) = A sin ωt + Bcos ωt ( here A,B & ω are constants where ω is the angular frequency of the applied oscillations), vi) an exponentially changing input, F(t) = Ae Bt (A & B are constants). Extensions of the model to unforced, constantforce, ramp, sinusoidal and exponential force inputs are compared in this paper.