The Torsional Harmonic Oscillator by

TeachSpin

Amanda Hunter and Jeremy Rys

May 18, 2012

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

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

Theory

The Torsional Harmonic Oscillator device by TeachSpin can simulate damped,driven simple harmonic motion. Starting with Newton's Second Law we can

work our way to the equation for simple harmonic motion in Cartesiancoordinates. The equations of motion for linear oscillations are identical to the

ones for angular oscillations:

∑~Fnet = m~a = mx (1)

The most common introduction to simple harmonic motion is through Hooke'sLaw, or the force exerted by a spring F = −kx when displaced a linear distancex. This force is analogous to the linear restoring torque τ = −κθ in the torsion�ber system with angular displacement θ. The equations of motion are the sameand their sum is

1

−kx = mx =⇒ x+k

mx = 0

When we set km = ω2

0 we get the textbook di�erential equation for simpleharmonic motion

x+ ω20x = 0 (2)

Damping forces F = −bv = −bx resist or slow the velocity of the system.This damping force can be added to our di�erential equation above. When weset b

m = 2βd we get the di�erential equation for damped harmonic motion

x+ 2βdx+ ω20x = 0 (3)

In addition to these forces we can include an external driving force to drive thesystem at a certain frequency. This gives us the equation for driven damped

harmonic motion

x+ 2βdx+ ω20x =

F

msin(ωdt) (4)

The Torsional Oscillator device uses one-dimensional rotational motion of arotor about a vertical axis described by a singular angular coordinate θ(t).

This system is best described using the equivalent of Newton's Second Law forrotational motion

~τnet = I~α = Iθ (5)

where I is the rotor's rotational inertia, which will be experimentallymeasured, and ~τnet is the net torque applied to the system

Three main sources of torque in system:

1. Torsion: A linear restoring torque due to twisting in the torsion �ber,

τ1 = −κsθ (6)

where the torsion �ber constant κ is derived experimentally by measuringthe angular displacement of the rotor for given torque values.

2

2. Damping torque: This torque resists the angular velocity and removesenergy from the system. Retractable magnets placed above and below thecopper rotor disc allow for an even linear deceleration.

τ2 = −bθ (7)

3. The system allows for a driving torque, τ = µB, by an external electriccurrent,i(t), due to a magnetic interaction, B. Current, i, produces amagnetic �eld, B = ki, through a set of Helmholtz coils. That magnetic�eld, ki, acts on a magnetic dipole, µ, mounted on the rotor disc, givinga driving torque:

τ3 = µB = µki (8)

Putting these all together we have:

~τnet = Iθ = −bθ − κsθ + µki (9)

This can be rewritten to look like the di�erential equation for damped, driven,simple harmonic motion:

θ + 2βθ + ω20θ =

µki

I(10)

Where β = b2I and ω0 =

√κs

I .According to our theory the motion exhibited by the rotor system of our

torsional harmonic oscillator will be a solution to this di�erential equation. Themost basic solutions look like plane travelling waves. Here is the solution for asimple oscillator with no damping or driving forces

θ(t) =

Cei(ωt+θ)

cos(ωt) + sin(ωt)cos(ωt+ δ)

(11)

The general solution to equation (10) is a linear combination of the homo-geneous and particular solutions

x (t) = xh (t) + xp (t) (12)

The homogeneous solution for the damped harmonic oscillator equation θ+2βθ + ω2

0θ = 0 with resonant frequency ω0 is

xh (t) = e−βt[A1e

(√β2−ω2

0t)

+A2e

(−√β2−ω2

0t)]

(13)

3

The particular solution for a given driving force F = Acos(ωt) with drivingfrequency ω is

xp (t) =A√

(ω20 − ω2)

2+ 4β2ω2

cos (ωt− δ)

where

δ = arctan

(2βω

ω20 − ω2

)The constants in this equation represent the limits placed on the motion of

this system. The most important constants however are the damping term b andthe linear restoring torque constant κ. The ratio of these two values determinesthe shape of the solution graphs which are separated into 3 distinct solutioncases known as over damped, critically damped, and under damped. The �rststep of our experiment is to come up with values for each of these constants. 1

Results:

Torsion Constant

The net torque Στ = Iθ on the rotor-�ber system is proportional to a linearrestoring force τ = −κθ supplied by the torsion in the �ber.

We began by measuring the linear restoring force of the torsion �ber in orderto measure torsional sti�ness constant κ.

To balance the lateral forces on the torsional �ber, two suspended masses wereused m = 2M on opposite sides to apply even, balanced forces on the systemat right angles to the radius r of the cylindrical piece shown in the picture

below where the string was attached.

Figure 1: This is the set up to measure the torsion constant on the �ber

1Van Baak, David. "A Brief Introduction to Resonance in the Damped, Driven Simple

Harmonic Oscillator." July 2009.

4

τ = F × r = Fr sin(90) = mgr = −κθ (14)

Torque vs. angular displacement were graphed to solve for the torsionconstant from the slope of the graph

Figure 2: Torsion Constant

From the slope of this graph we measured the static torsion constant to be

κs = 0.05965± 0.00015 (15)

Rotational Inertia

In engineering problems, such as the Tacoma Narrows Bridge disaster, thegeometry of mass distribution is where most of the physics takes place. Justlike the frequency of a pendulum is dependent upon it's length, the resonantfrequencies 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 ofvarying mass, density, and spatial orientation. Fortunately the large copperdisc at the top of the rotor supplies the majority of the rotational inertia forthe system, so it makes a good �rst approximation for the magnitude of therotor's moment of inertia. More precise measurements were conducted byadding a series of brass quadrants of known rotational inertia ∆I and

measuring the change in oscillation frequency for the system.

5

Figure 3: Brass quadrants and set up to measure the rotational inertia

Attaching the brass quadrants in pairs allows us to balance the change inthe rotational inertia, ∆I which can be calculated using the equation

∆I =M(R2

1 +R22

)2

= 0.000291 (16)

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

I = n∆I + I0 (17)

We also know that the period depends on the restoring torque and the ro-tational inertia of the system

T = 2π

√I

κs(18)

We can solve for, I, the rotational inertia of the system using equation (20)and replace this into equation (19) giving us(

T

2π

)2

= n

(∆I

κd

)+

(I0κd

)(19)

We can measure the period of the oscillation and plot in excel to solve for I

6

Figure 4: Plot to measure the rotational inertia with results

From the slope of this graph (m) we can calculate the dynamic torsion con-stant

κd =∆I

m= 0.0616 (20)

Where ∆I is the change in rotational inertia we calculated using equation(18).

From the intercept of this graph (b) we can solve for I0

I0 = b · κd = 0.00215 (21)

In the graph above are the calculate values for I for each quadrant, n, added.

Damping in the system

Damping forces counteract the motion of an oscillating system causing theamplitude of the oscillations to decrease over a period of time.

Variable damping is introduced to the system through a series of magnetsattached to a retractable assembly resembling brake calipers. The magnets are

spaced above and below the copper rotor disc so that they will causeresistance in the motion of the disc through eddy currents induced inside ofthe copper by the still positioned magnets. As the copper disc rotates, the

magnets induce a resistance force which is linearly proportional to the velocityof the rotor as it passes through the magnets. The picture on the right inFigure 3 shows these dampers above and below the copper rotor disk.

7

τ = −bθ (22)

This damping term can be added to our di�erential equation giving us:

Iθ = −bθ − κθ =⇒ θ +b

Iθ +

κ

Iθ = 0 (23)

Which can be written in the standard form

θ + 2βdθ + ω20θ = 0 (24)

Where the damping parameter is

2βd =b

I

and the angular frequency is

ω20 =

κ

I

Damped Cases:

There are three di�erent cases when we have damped oscillations. The threedamped solutions are based on a ratio of the damping parameter to the angularfrequency squared.

1. Underdamped

Underdamped oscillations occur when the damping parameter is less than theangular frequency squared:

2βd < ω20 (25)

This system oscillates about the equilibrium until the amplitude graduallydecreases to zero as shown in the graph below

8

Figure 5: Underdamped oscillation

2. Overdamped

Overdamped oscillations occur when the damping parameter is larger than theangular frequency squared.

2βd > ω20 (26)

The larger the damping parameter the slower the system returns to equilib-rium. This system experiences an exponential decay to the equilibrium withoutoscillating.

Figure 6: Overdamped oscillation

9

3. Critically Damped

Critically damped oscillations occur when the damping parameter is equal tothe angular frequency squared.

2βd = ω20 (27)

This system also does not oscillate about the equilibrium and returns to zeroquickly as possible.

Figure 7: Critically Damped oscillation

Damped Driven Oscillations

I did not measure the driving force in order to solve for the constants discussedin the theory. I did measure the Q-Factor and transient solution for dampeddriven harmonic oscillations and will be discussing these below.

Measuring the Q-Factor

The Q-factor, or quality factor, of a system is

Q =ω0ε

P(28)

a ratio of the energy stored, ε, to the power lost, P .

To measure this response around the resonant frequency we experimented witha range of frequencies for the driving oscillations and measured the amplitudeof the response to each frequency. Extra �ne measurements were performedaround the peak resonance to enhance data resolution around the resonant

frequency.

10

Figure 8: High Q and Low Q data

Measurements were taken for two separate damping parameters which a�ectedthe width or range as well as amplitude of frequency responses. We found thatthe resonant frequency for this system is approximately 860 mHz. That wasthe closest approximation we could get without possibly snapping the torsion

�ber in the device.

High Q

A system with a high Q factor has Q 12 . This system is said to be

underdamped, which means it will oscillate for a while until damping causesthe oscillations to die out. If the Q factor is increased, damping is decreased

causing the system to possibly oscillate out of control.

Low Q

A system with a low Q factor has Q � 12 . This system is said to be

overdamped, which means it will not oscillate but will experience anexponential decay back to its equilibrium. Normally we do not have to worryabout systems with low Q factors oscillating out of control as we do with a

high Q factor.

The Q-Factor is especially important since it determines whether a givensystem will oscillate violently out of control, much like the Tacoma NarrowsBridge disaster. As long as the energy pumped into the oscillator each cycle isgreater than the energy dissipated from the oscillator through damping, thenthe amplitude of the oscillations will continue to increase until the energy

stored becomes too large for the system to contain and something eventuallybreaks.

11

Transient and Steady State Solutions

The initial behavior of a damped driven oscillator can be quite complex. Whena driving force �rst begins to operate on an oscillatory system there is a periodof instability called the �transient� which is caused by the initial position andvelocity of the system. Over time this transient dies out, and the steady statesolution generated by the driving force is all that remains.

For a given driving force Fcsin (ωt), our di�erential equation becomes

Iθ = −bθ − κθ + Fcsin (ωt) =⇒ θ +b

Iθ +

κ

Iθ =

FcIsin (ωt) (29)

Which can be rewritten in the standard form

θ + 2βdθ + ω20θ =

FcIsin (ωt) (30)

The transient solution is determined by the initial position and velocity ofthe harmonic oscillator and is independent of the driving force. The steadystate solution, on the other hand, is independent of the starting conditions andis determined by the driving force as well as the motion constants 2βd and ω2

0

for a given system. The resulting waveform is a superposition (sum) of thetransient and steady-state solutions.

Figure 9: Transient Solution

The driving force was terminated at the end of the graph to show the naturaloscillation frequency and exponential decay envelope.

12

Conclusions

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

Sources:

Van Baak, David. "A Brief Introduction to Resonance in the Damped,Driven Simple Harmonic Oscillator." July 2009.

Van Baak, David. Torsional Oscillator Instructor's Manual. Vol. 1 and 2.Bu�alo, NY: Teach Spin, 2008. Print. TO1-A. <http://www.TeachSpin.com>

13