An automobile, traveling on a rough road can be modeled for pitch and bounce motions as shown in the figure below

An automobile, traveling on a rough road can be modeled for pitch and bounce motions as shown in the figure below. The free-body diagram of the automobile leads to the following equations 
ma + k,-)x + (412 — kfli)0 = kyf + k,y, and for motion along 0 (pitch) Job + likf)x + (k,11+ k11 )11 = k,12y, — kflipJ 
(1) () 
where nt = mass of the automobile, Jo = mass moment A inertia A the automobile, kf(kr) = stiffness A the front (rear) suspension and tires, /i(/2) = distances of the front (rear) wheels from the center of gravity, and w(y,.) = downward ground displacement of the front (rear) wheels. 
Given: m = 1000 kg, Jo = 800 kg-m2, kJ = 20 kN/m, k5 = 25 ./m, = 1.0 m, 12 = 1.5 m, w(t) = 0.05sin (10 t) m, and yr(t) = 0.05 sin rovs 121 m .11 velocity = v = 15 m/s and d = 10 m. 
Bounce 
Pitch 
AMA 

tb) 
Figure: automobile traveling on a rough road 
Assignment Specifically 1. Give a little background and description of the problem. 2. Derive equations (1) and (2). Explain the mathematical model, including all assumptions. 3. Clearly state all inputs to your model and all outputs. 4. Solve the above equations for bounce and pitch. 5. Present sufficient data (in graphical form?) and discuss your results. Communicate your results in a brief memo. A schematic, table, and/or figure will be useful to present your results. Attach all relevant calculations, programs, or spreadsheets in an appendix Include references, properly formated.