Computer aided analysis - assignment solution of non-linear

Question

Computer Aided Analysis - Assignment (Solution of non-linear equations)

Consider a car suspension system which consists of a shock-absorber and spring at each wheel. The shock-absorber provides a damping effect that is proportional to the vertical speed (ie up and down motion) of the wheel, and the spring's resistance is proportional to the vertical displacement of the wheel.

Therefore, the equation of motion of the wheel in the vertical direction can be expressed as,

m d²y/dt² = -c dy/dt - ky or d²y/dt² + c/m dy/dt + k/m y = 0

where m is the mass of the wheel, k is the spring stiffness and c is the damping coefficient.

This is a second order differential equation, and if for example, the car hits a hole at t = 0, such that it is displaced from its equilibrium position with y = y₀, and dy/dt = 0, it will have a solution of the form,

Y(t) = e^-nt(y₀cos(pt) + y₀(n/p) sin(pt))

where p = √((k/m) - (c²/4m²)) and n = c/2m provided k/m > c²/4m².

Submission - Produce your findings in a report, complete with graph/s, results and MATLAB scripts to investigate the following:

1. Calculate the first 4 four times the wheel passes through the equilibrium position when the suspension system has the following specifications:

mass acting on each wheel = 3.6 x 10³ kg

c = 2 x 10³ Ns/m

k = 2 x 10⁴ N/m

initial displacement = 0.3 m

2. In addition, the response of the system is thought to be too slow, and it is decided that,

• the first oscillation should take less than 2 seconds, but
• the maximum displacement after three oscillations should only be (approximately) 20% of the initial displacement.

Determine a suitable combination of spring stiffness and damping coefficient that will achieve this performance.

Make sure all sections of the MATLAB script are annotated to explain the role of each line within the script.