Engineering Mathematics

Select appropriate mathematical or modelling techniques and apply them to the solution of an engineering problem;

Work and learn independently and communicate results effectively.

The suspension system can be modelled using the following differential equation, where m is the mass (kg), c is the damping coefficient of the damper (N s/m) and k is the spring stiffness (N/m).

(a) The mass carried by a suspension system is 200kg, the damping coefficient of the viscous damper is 100Ns/m and the spring stiffness is 12N/m. Solve the differential equation for this system to obtain a solution for the displacement, , given the initial conditions at . [10 marks]

(b) Plot a graph of displacement, , against time, , to show the time response for the system if it is subjected to a unit step input. [5 marks]

(c) Calculate the Fourier Transform of your solution for , and plot the Amplitude and Phase Spectra to show the frequency response of the system. [15 marks]

(d) The car is travelling at 5mph when it hits a series of bumps in the road. This subjects the suspension system to an externally applied force which can be described by the forcing function,

Calculate the first four non-zero terms of the Fourier series for this function and thus identify the fundamental frequency that the suspension system will be subjected to. [20 marks]

Plot a graph of amplitiude against frequency for your Fourier Series and, referring to your Amplitude and Phase Spectra from (c), comment on how the suspension system will react to the externally applied force. [15 marks]

(e) The car is now travelling at 100mph and hits the same series of bumps in the road. The forcing function is now

Calculate the first four non-zero terms of the Fourier series and thus identify the new fundamental frequency that the suspension system will be subjected to. [20 marks]

Plot a graph of amplitiude against frequency for your Fourier Series and, referring to your Amplitude and Phase Spectra from (c), comment on how the suspension system will react to the externally applied force. [15 marks]

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