Question 4 (25 of 100 Marks) Car Body Attaches Here Car Body . Wheel and Tire Spring and Damper ЧАНННННИМИР Kz b2 ЯННННН

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Question 4 (25 of 100 Marks) Car Body Attaches Here Car Body . Wheel and Tire Spring and Damper ЧАНННННИМИР Kz b2 ЯНННННННИ & E Wheel/Tyre Hub Assembly ki 501 1 b. 1 Xroad Road Surface The figure above shows a simplified dynamic model for the wheel, suspension system and car body for one-quarter of a car (often referred to as a quarter car model). The upper mass m2 represents the mass of one quarter of the car (car body) and the lower mass mi represents the mass of the wheel and tyre. k2 and b2 represent the spring and damper coefficients of the vehicle's suspension system and ki and bı represent the spring and damper coefficient of the tyre, based on the tyre's material properties and tyre pressure. The coordinate Xroad represents the vertical profile of the road, which varies over time depending on the road properties and vehicle speed.
i. ii. For the quarter car model shown, draw two free body diagrams, one for the car body and one for the wheel/tyre (5 Marks) Derive the equations of motion for the vertical motion of each section of the car, using the position coordinates Xi and x2 which are the displacements of each section relative to their equilibrium positions under the weight of the car. These equations of motion will be differential equations for či and ïa as a function of x1, x2, X1, X2, Xroad, ki, bı, k2, b2, mi and m2. Ensure your equations account for the vertical profile of the road represented by the position coordinate Xroad, to which the wheel rests against (interaction modelled using the spring and damper ki and b1.) (10 Marks) Develop a MATLAB or Python script that uses the fourth-order Runge- Kutta numerical integration method to solve for the vertical displacements and velocities of the two car sections under a simulated driving scenario. During the scenario, the road vertical profile follows the function Xroad = 0.05sin(2(21)t) metres for t = 0 to t = 10 seconds (i.e. a bumpy section of road) after which the road flattens out to Xroad = 0 for t> 10 seconds. Use the system parameters k2 = 12 kN/m, b2 = 1500 Ns/m, kų = 29 kN/m, bı = 1400 Ns/m, m2 = 400 kg and mı = 25 kg. Your model should simulate 15 seconds of motion for the system. You will need to produce a single plot containing three lines as iii.
functions of time: the vertical displacement of the car body, the vertical displacement of the wheel tyre and the road vertical displacement/profile. Make sure the lines are properly defined using a legend. Also produce a second plot containing the vertical velocity of the car body and vertical velocity of the wheel/tyre as functions of time (i.e. two curves in the one plot). In your report, make sure you include all of your commented source code and results figures. (10 Marks) Hint: Matlab and Python have standard functions for Runge-Kutta methods and you may use those rather than generating your own code for the Runge-Kutta method them from scratch.