Question

Derive/show the second order differential equation that describes the system and resonance.

Answer 1

Let us consider to the example of a mass on a spring. We now examine the case of forced oscillations, which we did not yet handle. That is, we consider the equation

mx?+cx?+kx=F(t)

for some nonzero F(t). The setup is again: m is mass, c is friction, k is the spring constant, and F(t) is an external force acting on the mass.

What we are interested in is periodic forcing, such as noncentered rotating parts, or perhaps loud sounds, or other sources of periodic force. Once we learn about Fourier series in chapter 4, we will see that we cover all periodic functions by simply considering F(t)=F0cos(wt) (or sine instead of cosineFirst let us consider undamped c=0 motion for simplicity. We have the equation

mx?+kx=F0cos(wt)

This equation has the complementary solution (solution to the associated homogeneous equation)

xc=C1cos(w0t)+C2sin(w0t)

where w0=?km is the natural frequency (angular), which is the frequency at which the system