Description

Engineering Mathematics 2

Introduction

The challenge is mainly comprised of the analysis of equations of motion of mechanical systems. This process involves modelling & analysing the systems in the time and frequency domains. In particular:

1.      Analysis of a second order dynamic system to gain an in-depth understanding of system dynamics and to develop your mathematical skills.

2.      Analysis and modification/design of a more complicated 2nd order system with a view to system performance and improvement.

3.      You just use the data shown for your challenge group shown below

The systems that you will be analysing and modifying are mass-spring systems (containing a viscous dashpot) and so the modelling of such systems will involve developing Newton’s laws of motion. The analysis of the systems will require conversion of system models into different model types (transfer functions, state-space models etc).  The engineering system within the challenge are basic, simplified models of a vehicle suspension system. A vehicle suspension system essentially acts as a low-pass filter; its purpose is to damp high frequency oscillations caused by a road surface to produce a more comfortable ride, and more importantly, to maintain vehicle stability. Two key parameters in maintaining vehicle comfort and stability are the spring stiffness constant k and the viscosity of the dashpot damper c; if these two parameters are not correct, the suspension system may respond too slowly to an uneven road surface leading to a loss of control of the vehicle. Analysis of natural frequency (ωn) and damping ratio (ζ) will be key in system design.

Task 1: Development of system model & analysis of system characteristics

Consider the following basic model of a spring-dashpot shock absorber. For task 1, you will develop the deferential equation of motion and you will analyse the properties of the system without interference from a road surface. Once you have analysed the system dynamics and how spring stiffness and viscosity effect it, you will move on to see how the system behaves over a simulated road surface in task 2.

Note that u(t) represents the displacement of the car body, which includes also the road unevenness y(t) – so the compression/extension of the suspension is the relative displacement u(t) – y(t). Same principle applies to velocities; the suspension responds to the relative velocity .