1.

Consider the second-order linear system presented in class:

mx + cx + kx = F cos(@t);

m, c, k > 0.

Normalize the above system to

*+25wx+x=(F/m) cos(@t), w

and find the complete solution for arbitrary initial conditions of x(0)= x₁, x(0)=x, assuming that

0<5<1. The algebra is tedious and so you can use Maple or Mathematica. Do include all the steps in

the submission.

Now, specialize the solution to the undamped case of = 0, and set x to zero. This solution consists of

two frequencies, co and on. We now want to explore the nature of this solution.

i) Let x₁ = 0.5, F₁ /m=1.0, ₁ = 5.0 and @=2.0. Plot the solution components x(t), x(t) as a

function of time; Plot the solution in the phase plane (i.e., (x,x) plane); Now, pick the initial condition

Xo such that no contribution comes from the free response, and again plot the solution x(t) as a

function of time. Comment on all these solution plots.

ii) Repeat the work in part (i) above with @, = 0.5 and @=4.0.