If an undamped mass on a spring is influenced by an external force
F(t) = F0 cos wtthen the equation of motion is
m x´´ + k x = F0 cos wtwhere m is the mass and k is the spring constant. This type of system is descibed in Section 3.6 of Edwards and Penney (1996).
The natural frequency of the spring is
w0 = sqrt (k/m).
One way of producing the external force on the mass is to drive the other end of the spring so that its displacement is
y(t) = F(t) / k.
The first animation shows the behavior of the system with m = 1, w0 = 3, w = 5 and F0 = 80. The large orange circle represents the mass; the small blue circle is the driven end of the spring. For the animations, the motion of the spring was calculated using a Runge-Kutta method.
When the driving frequency w is close to the natural frequency w0 the system beats. The next animation shows this behaviour with m = 1, w0 = 8, w = 9 and F0 = 80.
The fast oscillation has frequency w0 + w. The slowly varying amplitude has frequency w0 - w.
When the driving frequency w is the same as the natural frequency w0 the system resonates. The final animation shows this behaviour with m = 4, w0 = 8, w = 8 and F0 = 80.
