The quarter car model (QCM), shown below, is a model for the response of the rig

Question

The quarter car model (QCM), shown below, is a model for the response of the right front suspension of a car to road forcing. In this project, you will consider the simpler case of a passive QCM, where no active controller is used to enhance the damping efficiency of the suspension. The governing equations of the QCM are equations (1) and (2) below, where xu, xs, and q are the displacements of the sprung (car chassis), unsprung mass, and road (from a given reference level), respecively. The values of mass, stiffness, and damping coefficients are in the table below.

You will have to compute the coefficiats k1, k2, k3, and c1, c2 through a least sqares curve-fit outlined below.

Fsp:

Fd:

Perform a polynomial least squares fit on each dataset to comput the values of coefficients k1, k2, k3 and c1, c2. On two separate plots, show the Fsp and Fd data and the corresponding least-squares fits.

b) Now, convert equations (1) and (2) into a system of four 1st order ODE's. Write out clearly this systme of ODE's.

c) You will now numerically solve the above system of ODEs using a 4th order Runge Kutta method: to this end, you will adapt the code given below. Note: you can use built-in Matlab commands or Simulink as alternative ways to build this solver. Critial in this direction is that you use 4th order accurate Runge-Kutta solver that uses a fixed timestep. Your choise for this is ode4, which we cannot help you with when implementing it to solve a systme of ODEs.

Do not use ode45! Even if you get your ouput in uniformly spaced times, this will involve a lower-accuracy interpolation that will degrade the accuracy of the 4th order Runge-Kutta method.

d) One way towards building a "sanity test" for your numerical solver (both Runge-Kutta and Euler) is to build an analytical solution to test against. You can only do this via Laplace Transforms for the linearized problem, where you would retain only the terms for the spring and damping forces in equations (3) and (4). The linearized problem is only valid for small displacements and velocities but can serve as a very useful comparison to compare your numerical model to.

Simulation:

e) You will run your model for 2 different car velocities values of 10 and 40 km/hr. Your simulation will end at time t=4s. Select your simulation timestep, h, such that you have at least 50 timesteps per the characteristic time scale of your problem, i.e. the time it takes the car to go over the bump.

Perform a timestep independence test: run first with h=T/50. Then repeat a simulation with h=T/100. Compare your result for xu and xs for each of the two timesteps. If these are visually indistinguishable (no need for any quantitavtive tests), your RK4 based simulations h=T/50 will be those you will analyze. Otherwise continue reducing your timestep by 1/2 until you obtain timestep independence.

The initial values for all 4 of your unknown variable should be set to zero.

f) For your 2 different values of car velocity V, plot xu, xs, dxu, and dxs as a function of time. Use one plot with four Matlab subplots, one for each variable. How do the amplitude and structure (period of oscillation, time of decay to very small values) of xs and dxs change with V?

g) Write a simplified forward Euler code that solves the system of ODEs you constructed in part b. Outline explicitly the calculations needed to advance your vector of unknown variable from time ti to ti+1.

Now, focus on the case of V=40km/hr. Run your forward Euler code with the time step you used in part e. Using four Matlab subplots, one for each variable, plot xu, xs, dxu, and dxs as a fucntion of time contrsting the forward Euler method to the RK4 technique. At the time t=2T, what is the absolute difference between teh two methods for the four unknown variables?

Through trial and error, determine how much smaller you have to make h, for the above absolute difference tot drop below 10^-3 for all four unknown variables. How many total time steps did you have to run your forward Euler code for in this case? Determine whether, for this value of time step h, your forward Euler code is computationally less costly than your RK4 code. Can you explain your answer?

Hint: Think how many Right Hand Side evaluations of each method involves over the course of a full simulation.

Data Analysis: (focus on RK4 results)

h) For your 2 different values of car velocity V, plot the sprung mass acceleration, d2x, as a function of time, t. Compute d2xs using an O(h2) centered finite difference scheme.

Now construct a diagram of the value of maximum recorded value of sprung mass acceleration as a function of car velocity V across all 2 runs. Taking into account the table below which shows the four passenger comfort ranges as a function of acceleration, indicate in which comfort range each of the 2 values of V rsides within (see table below).

i) Repeat the exercise above using a O(h)-accurate finite difference scheme. Is your assessment of passenger comfort range for each value of car velocity impacted by the accuracy of your numerical derivative computation?

j) For each of the 2 car velocities, what is the first zer-crossing time for the sprung mass displacement, xs, timeseries?

To this end, sample 6 data points to the left and right of th evisually identified zero crossin gpoint, construct the 5th degree interpolating polynomial, P5(t), in Lagrange form for these points. Outline your steps in building this polynomial.

P5(t) will be the function whose zero crossing you will have to compute using the nonlinear root finding technique of your choice. Compute this crossing time with an approximate absolute error tolerance of 10-3. Justify why you chose the particular technique.

k) For each of the 2 car velocities, compute numerically how much energy is lost due to damping by the shock absorber over the time interval T the car takes to go over the bump. This energy lost due to damping is given by the integral:

Perform your numerical integration using both composite trapezoidal and 1/3 Simpson's rules ensuring that you have an odd number of points. What is the relative difference between the two different estimates of Ed? Does the use of the composite 1/3 Simpson's rule produce radically different results?

If you needed more accuracy could you use Gauss Legendre quadrature to compute the integral in equation (7)?

-5.1467E3 -3.0863E3 -1.6938E3 -816.6333 -302.7525 -0.1097 243.7155 580.7339 1.1631E3 2.1431E3 3.6729E3 -0.1000 -0.0800 -0.0600 -0.0400 -0.0200 0 0.0200 0.0400 0.0600 0.0800 0.1000 Sprung Mass Suspension Suspension Spring Damper Unsprung Mass Tire Spring Damper Road Profile Fig. 1: Two degree of freedom passive quarter car model.

Explanation / Answer

Please try to understand this is a very long formate of question . Will you please provide the question in a short format for simplicity and to get a good and satisfactory answer of your question.thankyu

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