Part 1: Design and Initial Solutions

Vehicle: 2020 honda civic

Mass (m): 1695 kg

Spring Rate (k): 117680 N/m

dampening coefficient(c): 8000

Choose and state a vehicle from the assigned class above. Next, to construct your mathematical model, you may assume that the mass of the vehicle is distributed evenly throughout the car (for simplicity). Here, m (in kg) will represent the mass of the vehicle, the dampener attached to the suspension will have a damping coefficient of c (in N/(m/s)), and the spring constant, otherwise known as the spring rate, of the coil in the suspension application will be k (in N/m). Without an external force acting on the suspension, the second order differential equation governing this system is given below.

(m/4)y" + cy’ + ky = 0

B) Write this second order differential equation as a first order matrix-vector system of differential equations. Show your work in detail. Now, suppose the front left portion of your vehicle drifts off a curb of height 0.15 meters without an initial velocity. Write an appropriate initial condition using proper notation describing this event and pair it with your first order system.

C) Create a phase portrait of your first order system using the Phase Portrait script in MATLAB. Provide your plot and code in your final submission. Classify the type and stability of the critical point at the origin, then describe why your phase portrait matches your solution determined using the IVP Solver.

Part 2: External Forces, Resonance

A) Return to your second order differential equation with your values for m, c, and k entered into it, and note the initial conditions stated earlier. Solve this differential equation using any appropriate method discussed in class.

B) Based on the solution to your IVP above, identify a function that if acting on the system as an external force (like your vehicle driving over a bumpy road) would cause mechanical resonance to occur in the system. Justify why this choice would cause resonance to occur.

C) Set up a slightly modified second order differential equation of the form below. m/4 y^''+ cy^'+ky=F(t) Here, the function on the right-hand side of the equation should be the external force identified at the end of the prior page. This system should experience mechanical resonance if the choice made above is correct. Write this second order differential equation as an inhomogeneous first order matrix-vector system of differential equations. Show your work in detail. Next, use the IVP Solver script in MATLAB to plot the solution to this differential equation. Provide your plot and code in your final submission. Describe the changes in the solutions for your system with and without the presence of the external force.

Part 3: Relationship with Circuits

A) Virtually every portion of the results above can be duplicated in the setting for a closed circuit with an external voltage. Recall the second order differential equation for a closed RLC circuit, given here.

Lq^''+Rq^'+ 1/C q=e(t)

q(0)=q_0

q^' (0)= i(0)=i_0

Determine the values of L, R, C, the function, e(t), and the initial conditions q0 and i0 that would create an exact duplicate of the mass-spring work above with respect to circuits. Ensure that proper and correct units are included in your results throughout.

B) If a circuit undergoes resonance, describe what this means in context. Lastly, further explain the differences and similarities between resonance occurring in a mass-spring system versus resonance occurring in a closed circuit, and take particular care in describing the phenomena scientifically.