Supplements for Chapter 10
Damped and Driven Oscillation
An excellent model of a spring-mass resonator by Walter Fendt with nice controls found here.
Remember to wait untill the system settles down, i.e. wait until the transient behavior after a change in parameters has died out, to see the relative phase of drive and system. If you are anxious for this to happen, make the damping larger.
An important question (we use multiple choice) which everyone should answer correctly in order to move forward is given below:
A sinusoidal drive with a frequency f acts for a long time on a damped oscillator system; the oscillator itself has a natural vibration frequency g. We suppose the drive is still acting, and triansients have decayed and the system has settled down to a steady behavior.The oscillator will be found to be oscillating at
a) the drive frequency f
b) its own natural frequency g
c) both f and g will be present (2 frequencies)
d) a frequency somewhere between f and g (1 frequency).
Find the answer with the app above (the red ball is the drive and it always moves sinusoidally at the drive frequency), but remember to wait till the system settles down.
By watching the oscillation (blue) and the drive (red) (Perhaps use Slow motion) you can see that velocity of the blue mass, and the force on it (the spring when short is exrting downward force, and when the spring is long it is exerting upwardforce). are in phase of you set f to a resonant frequency (with initial settings this is f=3.19; set the damping higher to keep the mass frpm going too wild..
A movie is viewable below showing a damped oscillator (spring, mass, and "dashpot") driven by sinusoidally oscillating force, that slowly increases its frequency starting from below resonance and continuing through and above resonance. The force is indicated by a red arrow acting on the blue ball that can be a little hard to see. The power provided by the force is shown in the little window at the left; increased power is shown in the downward direction. In the large window are plotted the force due to the drive (red) and the displacement of the ball (blue). The effects of passing through resonance are easy to see. Click on the image below to see the animation.
Next, and even better, an interactive resonance machine, courtesy of the incredible Wolfram Demonstrations Project (free CDF Player needed to run it). A wheel (black circle) rotates with a frequency you can set, causing a rigid rod to oscillate the position of a wall (red); attached to this wall is a spring with a variable spring constant, connected to a variable mass. The mass (green) slides on a table with a friction constant that is also variable. A key point on resonance, (forcing frquency about 1) is that the force exerted at the drive point by the drive (the rotating black disk and the rod) is always aligned with the velocity of the drive point. This is a little subtle here, since you have to notice with care that when the rod is pushing in to the right, the spring is compressed, so the force the rod exerts is to the right, and its velocity is too. At the other end of the stroke, the spring is at maximum extension if you look carefully, and the force and velocity are again alinged as the rod moves to the left.
Next, a different applet run on the Wolfram CDF Player shows changes in the amplitude of oscillation as a function of frequency (upper left window) and phase (upper right, with a sign difference from our definition of phase) as a function of the Q of the oscillator, that you adjust with your mouse.
Resonance Lineshapes of a Driven Damped Harmonic Oscillator from the Wolfram Demonstrations Project by Antoine Weis
Passing through resonance
This illustration of a driven damped oscillator, with the drive frequency passing from below resonance to above, shows nicely the phase of the drive and the oscillator evolving through the resonance. Here the wall (green) is oscillating back and forth, with a frequency that is increasing with time, acting as the drive.
This video shows a case of cooperation between one of the natural frequencies of a spring and bead system (a little different than a stretched string but it shares many traits) and the natural frequency of a driven and damped rotor, that at the top is attached to the first bead of the spring and bead system. At the bottom, and in green on the graph, the unfettered (unattached) rate of change of the angle of the drive for the driven and damped rotor is shown; the resultant rate of change of the angle of the drive motion for the drive + spring and bead system is shown in red. It soon settles down into periodic motion that we can see is not periodic at the free drive frequency, as seen by comparing the red and green data.
The drive consists of a constant force on the little arm sticking out of the wheel, opposed by the friction due to the "dashpot", shown, that is supposed to act like a piston moving through oil. The arm is heavy and there is gravity, too.