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 . 

1.      Consider all the actions on the system and derive the differential equation that represents the system. Describe the type of it and explain the role of each one of the terms (driving, restoring, restoring and energy absorbing forces etc)

2.      Explain the simplifications we have applied and what a more accurate simulation would be like.

3.      Express your differential equation in terms of natural frequency (ωn) and damping ratio (ζ) and explain which physical parameters (i.e. mass, spring stiffness, damper viscosity) of the problem affect these specific two dynamic properties and in which way.

4.      Perform a forward Laplace transform and establish the transfer function for the system.

5.      Assuming that the external excitation y(t) is an impulse, perform an inverse Laplace transform to derive the response of the system u(t) in the time domain. What is the type of the expected response? If possible, use symbolic expressions up to this point (i.e. the natural frequency, wn the damped natural frequency, wD and the damping ratio z).

6.      Insert the function u(t) you derived in (5) in an Excel spreadsheet and calculate/plot some impulse time responses, using the data for your group shown in Table 1. How is the transient response affected by changing values of stiffness k and damping ratio z?

7.      Express the transfer function you derived in (4) into the Fourier frequency domain and produce the expression for its magnitude. Insert this expression in Excel and calculate/plot the results for the data given in Table 1. How is the magnitude affected by changing values of stiffness k and damping ratio z?  

8.      If the road surface is sine wave function y(x) = y0 sin(2πx/L), where x is the horizontal distance and L the wavelength, what is now the expression for the system response in the Laplace domain, U(s)?  Note that you need to express the sine function in the time domain first, using the constant driving velocity of the car. What is the effect of the driving velocity of the car on the cyclic frequency of the excitation?