Sources:Van Baak, David. Torsional Oscillator Intructor's Manual. Vol. 1 and 2. Buffalo, NY: TeachSpin, 2008. Print.

TO1-A. <http://www.TeachSpin.com>Image Sources:

Photographer Unknown. Howard Clifford Running off the Tacoma Narrows Bridge during Collapse, Tacoma, Washington. Digital image. University of Washington Tacoma Narrows Bridge Collection. 7 Nov. 1940. Web. 17 Apr. 2012. < http://content.lib.washington.edu/u?/farquharson,27 >.

Bashford and Thompson. Tacoma Narrows Bridge midsection collapsing into the waters of the Tacoma Narrows. Digital image. University of Washington Tacoma Narrows Bridge Collection. 7 Nov. 1940. Web. 17 Apr. 2012. <http://content.lib.washington.edu/u?/farquharson,19>

Tesla’s Earthquake MachineAmanda HunterJeremy RysMentor: Dr. Edward Deveney

Abstract What became known as "Tesla's Earthquake Machine" was actually just a simple harmonic oscillator which could be tuned to a resonant frequency for a given object, particularly a large object such as a bridge, a building, or even a tectonic plate. The small harmonic vibrations from the machine create a standing wave which can build up over time and eventually produce some intense shaking. Our experiment was designed to investigate the science behind this phenomenon, the science of damped and driven harmonic oscillators.

Introduction

Conclusions

Damped Oscillations

Transient and Steady State Solutions

The Q FactorThe Q-Factor is a measure of the sharpness of the peak resonant response of a given oscillatory system.

The Q-Factor is essentially a ratio of the energy stored ε to the power P lost per cycle.

To measure this response around the resonant frequency we experimented with a range of frequencies for the driving oscillations and measured the amplitude of the response to each frequency. Extra fine measurements were performed around the peak resonance to enhance data resolution around the resonant frequency.

The initial behavior of a damped driven oscillator can be quite complex. When a driving force first begins to operate on an oscillatory system there is a period of instability called the “transient” which is caused by the initial position and velocity of the system. Over time this transient dies out, and the steady state solution generated by the driving force is all that remains.

The transient solution is determined by the initial position and velocity of the harmonic oscillator and is independent of the driving force. The steady state solution, on the other hand, is independent of the starting conditions and is determined by the driving force as well as the motion constants and for a given system. The resulting waveform is a superposition (sum) of the transient and steady-state solutions.

For a given driving force , the differential equation becomes

The Torsional Oscillator Machine by TeachSpin is a precision instrument for experimenting with resonances and the basic physics behind how things vibrate. These concepts allow engineers to build better shock absorbers for cars, or bridges that don't shake violently when a big truck goes over them.

The net torque on the rotor-fiber system is proportional to a linear restoring force supplied by the torsion in the fiber.

We began by measuring the linear restoring force of the torsion fiber in order to measure the torsional stiffness constant .

The net torque on the system is proportional to a linear restoring forcewhich can be rearranged in the form of a differential equation,

This takes the form of the Helmholtz equation for a simple harmonic oscillator

The solution to this differential equation takes the form of a sine wave

The angular frequency of this sine wave is:

The collapse of the Tacoma Narrows bridge was the result of resonant growth of large-amplitude oscillations, which shook the bridge back and forth at it’s resonant frequency until it collapsed.

The (THO) Torsional Harmonic Oscillator Device by TeachSpin

Damping forces counteract the motion of an oscillating system causing the amplitude of the oscillations to decrease over a period of time. Variable damping can be introduced to the THO through a series of magnets attached to a retractable assembly resembling brake calipers. The magnets are spaced above and below the copper rotor disc so that they do not touch but will cause resistance in the motion of the disc through eddy currents induced inside of the copper by the still positioned magnets. As the copper disc rotates, the magnets induce a resistance force which is linearly proportional to the velocity of the rotor as it passes through the magnets.

Which can be written in the standard form

Measurement of the Torsion Fiber Constant

This damping term can be added to our differential equation giving us:

Where and

The 3 damped solutions are based on a ratio of the damping parameter to the angular frequency squared.

The 3 Solutions for Damped Oscillations

The equations which model the physical behavior of oscillating systems have an extremely large range of applications in science and engineering from electrical systems (RLC Circuits) to civil and structural engineering. Through an understanding of resonance and damping we are helping to create safer buildings, bridges, and cars.

Measurement of the Rotational Inertia

In engineering problems, such as the Tacoma Narrows Bridge disaster, the geometry of mass distribution is where most of the physics takes place. Just like the frequency of a pendulum is dependent upon it’s length, the resonant frequencies of other oscillating systems too are highly restricted by their geometry.

However the geometry of real life objects, such as bridges, buildings, or the THO rotor shaft are often quite complex. The rotor shaft in our experiment was comprised of a variety of components of varying mass, density, and spatial orientation. Fortunately the large copper disc at the top of the rotor supplies the majority of the rotational inertia for the system, so it makes a good first approximation for the magnitude of the rotor’s moment of inertia.

More precise measurements were conducted by adding a series of brass quadrants of known rotational inertia ∆I and measuring the change in oscillation frequency for the system.

To balance the lateral forces on the torsion fiber, two suspended masses were used on opposite sides to apply even, balanced forces on the system at exact right angles to the radius r of the cylindrical piece shown in the picture above where the string was attached.

Mass vs. angular displacement for this arrangement were graphed accordingly:

Measurements were taken for two separate damping parameters which affected the width or range as well as amplitude of frequency responses.

The total inertia of the system is the inertia of the rotor shaft assembly plus the inertia of the n added weights.

Using the brass quadrants we can compute ∆I that each quadrant contributes

The period depends on the restoring torque & the rotational inertia of the system

Attaching the brass quadrants in pairs allows us to balance the change in the rotational inertia.

The driving force was terminated at the end of the graph to show

the natural oscillation frequency and exponential decay envelope