Illustration 16.4: Forced and Damped Motion
Please wait for the animation to completely load.
A 1-kg mass on a spring is shown (position is given in meters and time is given in seconds) initially at its equilibrium position. Various parameters related to the spring and the initial conditions of the motion are also given. Restart. Once a variable or the velocity box is changed, you must reinitialize your choice by clicking the "set values, then drag the ball" button. Once you have clicked this button, drag the ball to the initial position you desire (by default it starts at its equilibrium position) and then press "play." When you get a good-looking graph, right-click on it to clone the graph and resize it for a better view.
We have thus far considered ideal motion of a mass on a spring: a perfect spring obeying Hooke's law and no additional varying force or damping. This Illustration will discuss what happens to a mass on a spring subject to a varying additional force and/or a damping force. Specifically, the damping force is -b v and the driving force is F0 cos(ωt).
First, what is the natural frequency of oscillation of the mass? Look at the animation with no additional forces or damping. Drag the ball to 3 m and let go. Pause the animation and measure the period (about 4.45 seconds from peak to peak). The frequency is one over this, or 0.225 Hz. The angular frequency is 2πf or 1.41 rad/sec. Since the angular frequency squared (here 2) is equal to the ratio of k/m, we know that k = 2 N/m.
What happens to the motion of the mass when a driving force is turned on? Try it and find out. Vary the angular frequency of the driving force. What happens when the angular frequency of the oscillation is close to or far from that of the driving force? How sensitive is the motion to this parameter? When the natural and driving frequencies are the same, it is called resonance.
There are three types of damped motion you should also investigate:
- Under Damped: the damping is so small there are many oscillations before motion is stopped.
- Over Damped: the damping is rather large; the motion takes a long time to get back to equilibrium.
- Critically Damped: a special case in which the time to get back to equilibrium is minimized.