# Linear and non-linear modelling and control in Matlab / Simulink

Linear and non-linear modelling and control in Matlab / Simulink
A basic vehicle simulation model for the evaluation of ride comfort is characterised by four degrees of freedom: vertical displacement and pitch angle of the sprung mass, and vertical displacement of the unsprung masses. Figure 1 is the schematic of the system.

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The report (maximum length: 6 A4 pages excluding Appendices) which is required for this assignment should cover the following points:

• Equations describing system dynamics in linear conditions;
• Conversion of the equations into a state space formulation, where the inputs are assigned in terms of vertical displacements of the contact points between tyre and road;
• Implementation of the state space formulation by using Simulink;
• Simulation of a step input from the road (amplitude 2 cm, at vehicle velocities of 50, 90 and
130 km/h);
• Simulation of the system in the same manoeuvres, by supposing non-linear springs and dampers (typical characteristics to be found from the relevant literature);
• Critical analysis of the properties of the linear and non-linear models when sinusoidal inputs are applied;
• Simulation of the system behaviour when a sky-hook controller (formulation to be found from the literature) is introduced in order to vary the damping coefficient of the shock absorbers.
Figure 1 – Schematic of a basic ride comfort model for a passenger car

Main vehicle parameters (to be integrated with plausible data from the literature)
ms = 1,500 kg mUS,F = 100 kg mUS,R = 80 kg mLOAD,FP = 80 kg mLOAD,RP = 65 kg Jys = 900 kgm2
as = 1.2 m
bs = 1.42 m
kSPRING,F = 44,000 N/m kSPRING,R = 40,000 N/m cSHOCK,F/R = 2,800 Ns/m P = 300,000 N/m
cP = 200 Ns/m
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