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Free vibration and sympathetic vibration are demonstrated by two sets of interactive gadgets. The theme of these gadgets are that motions of one, somehow, can transmit and moves the other. How can that be? The visual clarity of the effect always give children an "a-ha" moment of genuine joy and wonder.
(The motion of the aluminum beam, marked yellow, excites the other aluminum beam)
The first gadget is an eight pendulum set with slightly different cable lengths, resulting in a gradual change in vibration period. If excited at once, the small differences manifest themselves as a visible wave form.
The second set of gadgets are inverted cantilever pendulums with (1) different beam material and (2) different beam lengths, with a concentrated mass on the top. Each beam material and length comes in sets of two. As the first cantilever pendulum starts to vibrate, it excites its counterpart some distance away. The first cantilever vibrates at its natural frequency, and motion at that frequency excited the other idle beam (of similar natural frequency). This phenomenon is known as sympathetic vibration.
The characteristic frequencies of a cantilever is dependent on beam length, Young's modulus, and cross-sectional geometry. By keeping the cross-sectional geometry constant, the vibration is only characterized by beam material and length.
One system consist of beams of three different lengths. The other system consist of beams of three different materials (aluminum, titanium, and stainless steel). The system can be modeled as a concentrated mass on a massless spring, which is about 80-85% accurate in capturing the natural frequency. For better results, beam material mass, modeling of the foundation as a deformable stiff torsional spring, are all needed to better describe the system.
Striking a tuning fork (with the identical tuning fork nearby) also demonstrates this effect. The second tuning fork will start to vibrate and produced sound of the identical pitch.
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