you will simulate real mechanical systems with a simplified mass$-$spring damper system, from which the mathematical model will be derived. The response of the system will be determined using analytical equations. Finally, the mechanical system will be optimized by altering the stiffnesses and the damping coefficients until the desired response is achieved.

Design tasks:

$(1)$ Select an actual mechanical system for modeling. The system should have only one degree of freedom $(1$ DOF$).$ And the project is about $(($ Designing a Suspension System for a School Bus $))$

$(2)$ Model the mechanical system as a mass$-$spring$-$damper system. Note that you are expected to use realistic masses, spring stiffnesses, and damping coefficients of the actual system.

$(3)$ Derive the mathematical model of the mass$-$spring$-$damper system.

$(4)$ Determine the response of the dynamic system using analytical equations. Also, use the appropriate initial conditions.

$(5)$ Optimize the stiffnesses and damping coefficients and determine the lowest damping coefficients that can be used to dissipate $9396\%$ of the maximum recorded displacements after $10$ cycles.

$(6)$ Plot the displacement$/$time$,$ velocity$/$time$,$ and acceleration$/$time graphs. Also, determine the natural frequency, damped natural frequency, damping ratio, maximum displacement, maximum velocity, and the maximum acceleration of the mechanical system.