COMPUTER ASSIGNMENT 1

Due February 4

A mass is attached to a bungy--cord spring and a viscous damper as shown below. Gravity acts downward in the vertical direction. There is no friction between the mass and the slope.

Click here for figure

The force of spring the bungy-cord spring on the mass is given by the relation:

F(x) = - Ko x exp(alpha |x|)

where x is the displacement of the mass 'along the slope'. The force of the damper acting on the mass is minus C times the velocity. The fixed parameters in the problem are the mass M = 10kg and the slope of the incline theta = 45degrees and the value of Ko = 1000N/m.The values of C and alpha will be varied for different parts of the problem.

Use the initial conditions that x(t=0)=0 and dx/dt(t=0) =0.

a.) Derive the equations of motion for the mass.

b.) Use MATLAB ode45 to plot the displacement (x) as a function of time for alpha = 0.1,1,10 and for C = 0 and 5 N-s/m over a 3 second interval (i.e., six curves).

c.) When C = 0 you can analytically compute the distance the mass travels until it first stops. Compute this distance analytically (for various alpha) and compare those analytic results with those obtained from the matlab simulation. Describe the effect of increasing C and alpha on the stopping distance. (Note. For C not equal to 0, there is no analytic solution.)

d.) The parameter alpha represents the strength of the nonlinearity of the bungy spring (if alpha = 0 the spring is a linear spring). For the linear system, the frequency of oscillation (f) is given by 2 Pi f = square root(K_0/ M) = 10 corresponding to a period of 0.628 s. Determine the period of oscillation for the computed time histories and compare to 0.628 s -- discuss the effect of alpha on the nonlinear period of oscillation.