The body, suspension and engine mounting system of a vehicle may be idealised as a three degree of freedom system consisting of a bar of mass 1400 kg and the centre of mass at G, 1 m from the front and 2 m from the rear ends as shown in the Figure Q1. The suspension system consists of springs and dampers, the front suspension consists of a spring of stiffness of k_{F }_{=} 25 kN/m and damper of damping coefficient c and the rear with stiffness of k_{R }_{=} 20 kN/m and damping coefficient of c. The moment of inertia of the vehicle about point G is1200 kgm^{2}. The engine and its mount is idealised as mass-spring-damper attached to the vehicle at G to isolate the engine vibration transmitted to the vehicle, the engine has a mass of 400 kg, the isolator has stiffness k_{E}= 10 kN/m and damping coefficient of c.
(a) Draw the free body diagrams of the system and show that the equation of motion for the undamped system is given by:
(b) Use eigen functions in MATLAB to calculate all the mass-normalised modal vectors and associated undamped natural frequencies. Hence sketch the modeshapes illustrating the dynamic behaviour of the system. Use hand calculations wherever possible to verify the results obtained from MATLAB.
(c) If the damping due to the suspension and engine mount isolator are to be included in the model, modify the equation of motion given in part (a) to include the damping matrix, use the damping coefficient c.
(d) The engine produces a sinusoidal excitation force of magnitude 150 N at frequency of 30 Hz, further excitation forces from the road are in the form of unit impulses applied to the front and rear ends of the vehicle, using Modal Analysis calculate the response of the system at the front suspension. using MATLAB plot the response in time domain. Assume a damping ratio of 0.1 for all modes.
(e) Calculate one Frequency Response Function in Inertance form and plot it in the frequency range of interest using MATLAB.
Include all your MATLAB outputs and label them clearly. State all the necessary assumptions clearly. You may Export the MATLAB plots into word documents.