For this project you will use MATLAB to model a damped harmo

For this project you will use MATLAB to model a damped harmonic oscillator obeying the equation m doubledot x + b dot x + kx = 0 You will write up a report showing your work, your programs, their output, and a discussion of the results. The report is due Monday of finals week. First, scale the equation to be dimensionless. Use frequency priority as discussed in class. Show your work and your scaling in your report. Create a MATLAB function to solve the scaled equation for x(t) using the following parameters: m = 2.0 kg k = 50.0 N/m b = 0.50 kg/s Use the following initial and range parameters: x(0) = 10 m dot x(0) = 0 Chose a time range to show at least 10 periods of the oscillation. Plot your results, x(t) and v(t) in real units (i.e, convert your scaled results into meters, meters/sec, etc.) vs time (in seconds) as line plots using the subplot feature of MATLAB. Ensure that you can see the decay of the oscillation. Create a new function, modifying your first function to include an event that will detect all maxima of x(t), call the resulting array xmax(t). Make a double plot including x(t) as a smooth line, and xmax(t) as individual markers (\'*\' or o\'). Verify that it gives you what is expected. Again, include a program listing and printout of the graph in your report. The amplitude of the oscillation, your xmax(t). should decay exponentially. Show this by plotting the log of xmax vs time. Calculate the slope of the graph either directly by reading the graph, or by printing out the amplitude data, or as part of your function. (Show your work in your report.) Compare your result to the expected slope of -b/2m. Generate a theoretical decay curve and plot it as a smooth line (\'-\') along with your amplitude data as symbols (\'*\' or \'o\'). Include this graph and discussion in your report.

Solution

solution:

hormonic oscillator:

hormonic oscillator is a system that, when displaced from its equilibrium position , experiences a resorting force, f , proportional to the displacement X

F bar=-kXbar
where k is a positive constant
if f is the only force acting on the system, the system is called a simple harmonic oscillator , and it undergoes simple hormonic motion: sinusoidal oscillations about the equilibrium point, with a constant amplitude and a constant frequency (which does not depend on the amplitude)