Assignment 6

Name:
Student Number:

Q1

Consider two harmonic oscillators, one damped linearly, the other damped quadratically.  They are governed by the following equations,
x''+γ x' + ((Subscript[ω, 0])^2) x =0  (Linear case) and
x''+β x'|x'| + ((Subscript[ω, 0])^2) x =0  (Quadratic case).
(Note the absolute value, which you can get using Abs[x’].)  Both oscillators have (Subscript[ω, 0])= 2 rad/s , mass of 1 kg, and have the same initial conditions x’(0)=0, x(0)=1.00 m.

a)

For  γ = 0.400 (s^(-1))and β = 0.400 (m^(-1))(s^(-1)), plot the solutions for the two oscillators (Subscript[x, linear])(t) and (Subscript[x, quad])(t) along with A(t)=exp(-γt/2), the predicted decay for the linearly damped case, for a time long enough to see the decay.  Comment on the decay of (Subscript[x, quad])(t) relative to (Subscript[x, linear])(t).  Does it decay exponentially?

b)

Now let each oscillator be driven by a force F=(Subscript[F, 0])cos(ωt), (Subscript[F, 0])=1 kg m/(s^2).  Using the exact expression for the steady-state amplitude (from class) as a function of ω for the linear case, find the driving frequency (Subscript[ω, R]) that maximizes the amplitute, i.e., the resonant frequency.  Plot (Subscript[x, linear])(t) and (Subscript[x, quad])(t) for the case ω=(Subscript[ω, R]).  Which oscillator has the larger amplitude?  Does this make sense?

γ=0.4;  β=0.4; ω0=2.0;m=1;F0=1;

c)

Still for  γ = 0.400 (s^(-1)), β = 0.400 (m^(-1))(s^(-1))and F=(Subscript[F, 0])cos(ωt), estimate <E(ω)> by writing a Do loop that calculates <E(ω)(Subscript[>, APPROX])=(1/10)(∫_20^30)((1/2)((m(x'(t)))^2)+(1/2)k (((x(t)))^2))dt  for both the linear and quadratic cases for several values of ω from ω=0.1 to 6 rad/s.  Plot  <(Subscript[E, linear])(ω)(Subscript[>, APPROX]) and <(Subscript[E, quad])(ω)(Subscript[>, APPROX]) along with the exact expression for <E(ω)> for the linear case.  Which oscillator is more sensitive to the driving frequency?

Q2

Repeat the numerical treatment for obaining <E(ω)(Subscript[>, APPROX]) in Q1(c) for the case of two coupled oscillators (2 masses, three springs - as in class and in the lab exercise) for linear damping.  As in the lab exercise, only one mass should feel the driving force directly.  Take the masses, spring constants (and (Subscript[ω, 0])), damping coefficient and forcing term as the same as for the linearly damped oscillator in Q1.  Compare <E(ω)(Subscript[>, APPROX]) with the exact expression for a single oscillator.  

Comment on anything interesting you find.

You may need to simulate for longer to obtain the steady state.  (Try out a test case before writing a loop.)

You will need to consider the kinetic energy of both masses, as well as the compression of all three springs in calculating the potential energy.