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Math 4360 (Spring 2023/2024)

Project 1: Mechanical vibrations

Due on 2024-March-04

Consider a linear oscillator with linear friction:

(a) Show that the energy  is a decreasing function of time.

(b) Let  and show that dv cv kx .

(c) Show that if c2 < 4mk (i.e., in the underdamped regime), then v = λx is not a solution in the phase plane. (The phase plane is spanned by the position x and the velocity y, which are both real.)

(d) If m = 1, c = 1, k = 1, then roughly sketch the solution in the phase plane. (Use known information about the time-dependent solution to improve your sketch.)

(e) Show that if c2 < 4mk (i.e., in the overdamped regime), then v = λx is a solution in the phase plane for two different values of λ. Show that both values of λ are negative.

(f) If m = 1, c = 3, k = 1, then roughly sketch the solution in the phase plane. (Use known information about the time-dependent solution to improve your sketch.)

(g) Explain the qualitative differences between parts (d) and (f).

(h) Carry out numerical computation to plot the numerical solutions in the phase plane for (d) and (f) respectively. (Note that for either (d) or (f), different initial conditions give different solutions that show the same salient feature.)

PS: Your Project Report shall be submitted to Canvas in one single PDF file, in which the original codes are included as an appendix. Make sure that your name and student ID are shown on the first page.