On Mechanical Resonance Theory applying to free, damped and forced

oscillations

Benjamin SheldanDepartment of Physics, Simon Fraser University, Burnaby, BC,

V5A 1S6, Canada

(Dated: November 19th, 2014)

Abstract

In order to test whether mechanical resonance theory (the theory stating that a mechanical system has a

tendency to respond at a greater amplitude when its frequency of oscillation nears that of the system’s natural

frequency of vibration) is an acceptable model during free, damped, and forced oscillatory motion, we preformed

the following experiment. We attached an accelerometer to the end of a hacksaw blade and allowed it to oscillate

freely, then under the effects of damping, using a magnet and copper brick and, finally, under the effects of a

vibration driver (both with and without effective damping). We observed acceleration amplitudes reach peak

values when the frequency of oscillation neared the natural frequency of the system. Additionally, we fit curves

based on the harmonic oscillator equation to our graphs. The resulting agreement between theoretical curve fits

and experimental data as well as the peak amplitudes occurring near the natural frequency conclusively show that

mechanical resonance theory is an acceptable model for the hacksaw, accelerometer, and magnet system.

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Introduction Mechanical resonance theory predicts that a system will respond with greater

amplitudes when it oscillates at a frequency close to that of its natural frequency of vibration. However, in all systems there is a finite amount of stress they can undergo before they break down. In this experiment we tested when mechanical resonance theory was an acceptable model for describing the behaviour of our vibrating hacksaw blade system and at which frequencies and damping situations it broke down. We chose to measure the acceleration of our hacksaw blade to observe how the amplitude of our system was reacting and the phase angle between our hacksaw blade and wave driver to observe when the two were in and out of phase.

TheoryWhen a free harmonically oscillating system, x ' '+2 γ x '+ω0

2 x=01, where x is the position of an object and ω is it’s natural angular frequency, is solved, the result is an exponentially decaying sinusoidal function, x (t )=x0 e

−γt cos (ω1+φ)1, where x0 is an object’s initial position, γ is the damping coefficient, ω1 is the object’s angular frequency, φ is the phase angle and t is the independent variable time, that states the amplitude of position of a mechanically oscillating system will gradually decrease until it reaches zero as time increases; that is, if we take the limit as t approaches infinity the result will be zero. If we differentiate the function twice to find acceleration, a ( t )=Y 0+X0 e

−γt cos (2πft+φ)1, where ω is now written as 2πf and f is the frequency of oscillation, the same result is stated. If a system has an initial value other than

zero that it will oscillate about, for example if the reference point of a system is 5 ms2

, then a

constant term Y 0 is added to the equation so that this offset from zero is reflected. Similarly, if

a forced harmonically oscillating system,x ' '+2 γ x '+ω02 x=

F0m

eiωt1, where F0 is an initial force

applied to an object and m is it’s mass, is solved in terms of an input frequency,

x0 (ω)=

F0m

ω02−ω2+2iγω

1, where i is the imaginary number √−1, the position amplitude is clearly

shown to reach a maximum when the input frequency is equal to that of the natural frequency of the system. Again, differentiating the function and taking the real value states the same

argument for the amplitude of acceleration, |a0(ω)|=ω2

F0m

√(ω¿¿02−ω2)2+4 γ2ω2¿

1. Lastly, the

phase angle φ between the amplitude of the driving force applied to the system and the

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corresponding amplitude of the system itself, tan (φ )= 2 γωω2−ω0

21, decreases as the driving

frequency increases. Calculating the quality factor of a system can be done in two ways,

Q=ω02 γ

=π f 0γ

=f 0∆ f

1, where f 0 is the natural frequency. Simplifying the first expression shows

that the quality factor can be calculated when the natural frequency of a system and the damping coefficient are known. Secondly, if a plot of amplitude (position, velocity, or acceleration) versus frequency is given, the quality factor can be determined by the natural frequency of the system (where amplitude is highest) and the difference between two

frequencies at ‘half-power’, that is 1√2 of the maximum amplitude.

ProcedureTo determine whether the amplitude of a mechanical system is varying, a setup (Fig. 1)

is needed to measure either the position, velocity, or acceleration of the system and the frequency that the system is being driven with. The driving frequency is also required to calculate the phase angle, φ, between amplitudes of the wave driver and the system being driven. The first observation, the amplitude, could be qualitatively seen by simply observing the system. However, we would like to record values for the amplitude so we can then go on to calculate the quality factor for each damping situation.

First, for our setup, an accelerometer and magnet are attached to the end of a hacksaw blade and this composition is fastened to a heavy brick. Next, a mechanical wave driver is positioned against the base of the hacksaw to deliver a force at arbitrarily chosen frequencies both near to resonance, within 0.1 Hz, and far from resonance, up to 1 Hz greater or less than, to the system. Also, a copper brick is fastened to a stand that can be placed at varying distances from the magnet on the hacksaw. To obtain values for the amplitude of voltage, which we

would later convert to ms2

, and the frequency driving the system, our data acquisition device or

DAQ is wired in parallel to both the accelerometer and sinusoidal wave driver. Lastly, we wrote a program in National Instruments LabVIEW to collect the data from our DAQ in order to graph our results.

In order to be able to convert the data being taken in voltage to an understood unit of

acceleration, [ms2

], we had to calibrate our accelerometer and DAQ. First, we took a

measurement of the system when the acceleration was zero, that is, when the hacksaw was at an angle of 0 radians with respect to the table. Next, we took another measurement at a known

acceleration, 9.8 ms2

, when our hacksaw was at angle of π2 with respect to the table. The

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difference in voltage from 0 to 9.8 ms2

was 0.289V. Using this calibration we were able to

construct a conversion factor that allowed us to calculate acceleration from our acquired data of voltage.

9.8m /s2

0.289V=

[Acceleration ]AcquiredVoltage

Our LabVIEW program consisted of two key components. First, we programmed the DAQ to acquire signals of voltage from two differential channels, the accelerometer and the wave driver, which we would later convert to acceleration in units of meters per second squared. Next we added a control to alter the sampling rate at which we acquire data. The DAQ has a max sampling rate of 400 kHz, and therefore, two differential signals can be read at a sampling rate of 200 kHz.

After our setup was built and we had a program running to collect our data we begun testing taking data by allowing the system to oscillate freely with no driving force and no damping present. Using this free oscillation, we were able to measure the natural oscillation frequency of the hacksaw system. For our second test, we now ran the mechanical wave driver at certain frequencies programmed by our digital function generator, again with no damping present (Fig 2.). For the following tests, we ran our mechanical wave driver at a series of different frequency values, both close to the value of the natural frequency, and several at values far from the natural frequency, under the effects of increasingly greater damping (Fig. 3, 4, 5). The raw data collected by our DAQ and LabVIEW program were then analyzed using IGOR pro software from Wavemetrics.

Apparatus

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Fig. 2 Fig. 3 Fig. 4 Fig. 5

From left to right, top to bottom respectively: The DAQ device acquiring data in the form of voltage from the accelerometer and mechanical wave driver. The copper brick interacting with a magnet on the hacksaw blade used eddy currents to damp our system and was placed in 4 distinct positions as shown in fig.2 through 5. On the computer we had a LabVIEW program running to analyze the intake of data. The accelerometer for measuring amplitude was attached to the end of the hacksaw blade. Our mechanical wave driver placed at the base of the hacksaw blade applied a force at chosen frequencies.

Figures 2 through 5 show the position used for maximum, moderate, minimum, and no damping respectively.

ResultsOur first task in the analysis of our experiment testing the mechanical resonance theory

was to check whether our initial theoretical equations for non-driven, damped oscillations were reflected in observations. And, indeed, we observed that as we increased the damping, a system will decay to equilibrium much quicker. We also noticed that when the system is under maximum damping its amplitude is approximately ninety percent that of the minimum damping condition.

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Fig. 1

Fig. 6 shows acceleration of our hacksaw blade system versus time for a minimum damping condition (fig. 4). X2=27324

Fig. 7 shows acceleration of our hacksaw blade system versus time for a moderate damping condition (fig. 3). X2=8484

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Fig. 8 shows acceleration of our hacksaw blade system versus time for a maximum damping condition (fig. 2). X2=5352

Table 1. Chi squared, number of samples taken and number of free fit parameters used to assess the quality of our fit.

Condition X2 N Samples Taken

Free Fit Parameters

Minimum Damping

27324 30000 5

Moderate Damping

8484 10000 5

Maximum Damping

5352 10000 5

a ( t )=Y 0+X0 e−γt cos (2πft+φ)

1.However, we did not want to simply qualitatively observe the decay of our system. In

order to test the theory developed for free, damped oscillators, we applied curve fits using equation 1, where Y 0 is the mean acceleration our system oscillates about, X 0 is the initial acceleration value from the mean, γ is the damping coefficient dependant on where the copper brick is placed, f is the frequency of oscillation, and φ is the phase angle, to each plot of acceleration versus time. To test whether the curve fits accurately described the decaying oscillation of our system we observed the values obtained for chi squared. In practice chi squared should behave as follows; X2=N−F3 where N is the number of experimental points taken and F is the number of free fit parameters. When N becomes very, very large compared to F, we can simply say that chi squared should equal the number of samples taken and, as seen on table 1, our X2 roughly follows this trend. Chi squared, however, is also described by

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Table 2. Damping coefficients and corresponding uncertainties for free, damped oscillations.

X2=∑ (Theory−Calculated )2

σ2 3, where σ is the standard deviation, which states that chi

squared should equal the number of points taken because each point, on average, should lie one standard deviation from the theoretical fit. Minimum and moderate damping situations

reflect this pattern but X2 for maximum damping is only about 12N which would mean that

each point only lies about half of one standard deviation from the curve fit. This could simply be explained in that the uncertainties we used to calculate the standard deviation were overstated.

Using our equation we derived for the acceleration amplitude of an oscillating system, we fit curves to three damping conditions. The following increase in values determined for the damping coefficient, γ, as well as the decrease in time the system takes to decay to equilibrium agree with our theoretical predictions of how this system should behave. The uncertainty for each damping coefficient is one standard deviation as given by our graphical software.

Additionally, in order for us to be able to measure the phase angle, we would need to compare data of voltage versus time through both the accelerometer and wave driver. We would then use cursors and our second conversion factor to change the measured phase angle from seconds to radians.

2πPeriod [seconds ]

=φ [radians ]φ[ seconds]

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Condition Damping Coefficient (1/s) Uncertainty (+1/s)Minimum Damping 0.6723 + 0.0121Moderate Damping 1.0471 + 0.0258Maximum Damping 10.603 + 0.071

Fig. 9 shows voltage amplitude versus time obtained from our accelerator (upper wave) and wave driver (lower wave).

The core of our experiment was testing whether the mechanical resonance theory was a good model for our oscillating system and whether the equations derived accurately described the behaviour of our system. To test this, we picked four different damping conditions (Fig. 2 through 5) to observe how the amplitude of a system changed when we drove it at a frequency near that of resonance. Based on the derived equations we expected to see a peak value when we drove the system at resonance and the lower the damping coefficient, the sharper the acceleration would peak at resonance.

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Table 3. Chi squared, number of samples taken and number of free fit parameters used to assess the quality of our fit.

|a0(ω)|=ω2

F0m

√(ω¿¿02−ω2)2+4 γ2ω2 ¿

2.Using equation 2, where we defined F0 to be the initial force applied by the wave driver

to our hacksaw-blade, magnet, accelerometer system, m to be our oscillating mass, γ to be the damping coefficient, ω0 to be the natural frequency of vibration for our system, and ω to be the driving frequency, we applied fits to our data of acceleration versus frequency. As with our free, damped oscillation data, we again observed chi squared, our number of samples taken and number of free fit parameters to evaluate whether we had achieved a reasonable fit.

Condition X2 N Samples Taken

Free Fit Parameters

No Damping 73 12 4Minimum Damping

7.1 10 4

Moderate Damping

6.6 10 4

Maximum Damping

6.2 10 4

During our first trial for acquiring data we discovered that we had placed the wave driver to far from the base of the hacksaw and when the driving frequency neared resonance, the hacksaw would leave contact with the wave driver and the resulting data was unusable as our model did not account for the additional vibrations from the force of the hacksaw hitting the wave driver when the two objects made contact again. From this evidence we can deduce that our simple mechanical resonance model breaks down when additional oscillations are introduced to the system. After, we adjusted the position of the wave driver and retook our previous data. Although we corrected the situation as best we could, the apparatus was still not ideal, and the same scenario could likely have happened again. This is a possible explanation of why, in the minimum damping scenario we noticed two of our data points did not align with our theoretical curve fit. We noticed that we did not have this problem in any of the other damping scenarios because the magnetic force kept the hacksaws amplitude of acceleration from reaching the same peak values. Since both points lie outside 3 standard deviations we could have ignored them in plotting this data, however, we did not specifically record any bouncing between the wave driver and hacksaw in this trial, and as a result we included these two points in our analysis and let the uncertainties (Table 4), which were larger in this curve fit

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Fig 10. Acceleration amplitudes plotted against driving frequency, along with their corresponding fit. Blue data corresponds

to a no damping condition, X No Damp2 =73, red data to minimum damping, XMinDamp

2 =7.1, green data to moderate

damping, XMod Damp2 =6.6, and purple data to maximum damping, XMax Damp

2 =6.2.

Table 4. Damping coefficients, natural frequencies, and corresponding uncertainties for forced, damped oscillations.

than in any other, reflect that our model is not ideal in explaining this situation. As we also noticed in the oscillations without a driving force, our model gets more accurate in fitting the experimental data as we increase the damping present.

Condition Damping Coefficient (1/s)

Uncertainty (+1/s)

Natural Frequency (Hz)

Uncertainty (+Hz)

No Damping 0.314 + 0.236 12.299 + 0.053Minimum Damping

0.482 + 0.132 12.349 + 0.011

Moderate Damping

1.704 + 0.172 12.355 + 0.024

Maximum Damping

2.7076 + 0.0568 12.343 + 0.0078

The next process in order to test whether these were accurate fits, is to graph the residuals along with the acceleration amplitude plot. For a residual to be a good fit, the plot of residual points must reflect only ‘noise’. If there had been an underlying pattern to the residual points, such as a sine wave, then our fit would clearly not have taken into account some

physical phenomenon.

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Fig 11. Acceleration amplitude and corresponding residuals for the maximum damping condition plotted versus frequency.

X Max Damp2 =6.2.

As stated previously, a fit is only accurate if it takes all effects into account and the residuals only represents noise. Our residuals for acceleration versus frequency do not have underlying patterns to them, and our values for chi squared that follows, X2=N−F3, are in agreement that our fits, based on mechanical resonance theory are a good model for these

oscillations.

Another key point in examining whether or not mechanical resonance theory held during our experimental oscillations was to observe the phase angle between the frequency of the wave driver and the resulting frequency of our hacksaw blade system. As we stated in theory, the phase angle should increase as our driving frequency drops below the natural frequency of oscillation. We should notice a steep curve as our driving frequency nears the natural frequency as the denominator of our theoretical equation approaches zero. Finally, our phase angle should decrease as our driving frequency becomes larger than our system’s natural oscillation frequency.

tan (φ )= 2 γωω2−ω0

2

3.Condition X2 N Samples

TakenFree Fit Parameters

No Damping 14.3 12 2

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Fig 12. Angle phi in radians plotted against driving frequency, along with corresponding fits. Blue data corresponds to a no

damping condition, X No Damp2 =14.3, red data to minimum damping, XMinDamp

2 =8.7, green data to moderate

damping, XMod Damp2 =7.4, and purple data to maximum damping, X Max Damp

2 =7.8.

Table 5. Chi squared, number of samples taken and number of free fit parameters used to assess the quality of our fit.

Table 6. Damping coefficients, natural frequencies, and corresponding uncertainties for forced, damped oscillations.

Minimum Damping

8.7 10 2

Moderate Damping

7.4 10 2

Maximum Damping

7.8 10 2

Using equation 3, where γ is the damping coefficient, ω is the driving frequency, and ω0 is the natural frequency of vibration of our system, we applied curve fits to each plot of phi versus frequency. As with our previous data sets, chi squared was used to assess the quality of our fits in order to determine whether our model was accurately describing the physical situation of our hacksaw blade system.

Again, our first trial of observing the oscillations of our hacksaw system in a scenario with no damping resulted in several points that were outside of two standard deviations from our curve fit. One plausible explanation for this result is that reading cursors became very inaccurate at frequencies much less than the natural frequency. When the frequency of our wave driver was reduced to well below the natural frequency of our system, the amplitude also became very small. Our cursors in our data analysis software had to be placed, or dragged along the curve and because of human error it was hard to accurately place them. A possible solution for obtaining accurate phase angles would be for a program to allow us to manually enter a number for an amplitude value, and for the program to then tell us the corresponding value for that point along the x-axis in time. That being said, for the remaining damping conditions, minimum to maximum, our chi squared followed X2=N−F3. Also, as with previous fits, we observed the general trend that as we increased the damping on our system, our fits became more accurate.

Condition Damping Coefficient (1/s)

Uncertainty (+1/s)

Natural Frequency (Hz)

Uncertainty (+Hz)

No Damping 0.313 + 0.153 12.309 + 0.012Minimum Damping

0.536 + 0.194 12.272 + 0.016

Moderate Damping

1.727 + 0.241 12.271 + 0.023

Maximum Damping

2.876 + 0.057 12.177 + 0.041

After we had obtained values for the differing damping coefficients and the natural frequency from both acceleration amplitude and phase angle fits we compared to make sure they agreed. Indeed, in all 8 of our fit cases, our values agree within their corresponding

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uncertainty range. One interesting note is that in our acceleration fits the natural frequency appears to increase very slightly rather than just fluctuate within a given range as the damping situation is increased and, contrastingly, our system’s natural frequency appears to represent a very slight decreasing trend in the phase angle fits. A possible explanation for a trend appearing in the natural frequency is that as our wave driver applied an oscillating force, our hacksaw blade could have become slightly loosened and if the length that the accelerometer and magnet are oscillating at changes, the natural frequency of their vibrations will certainly change.

Our residual values for our phi fits were slightly larger than those of our acceleration fits. We also noticed a clear pattern that before the resonance frequency, our residual values increase, as well as after and when near the resonance value the residuals suddenly decrease. Our value for chi squared still obeyed the formula based on number of samples taken and free fit parameters but the clear pattern indicates that there is an underlying phenomenon occurring that our fit does not take into account.

Discussion When we begin looking at the quality factor, the mechanical resonance theory states we

should see a decrease in this quantity as damping increases and the two terms are inversely

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Fig 13. Phase angle and corresponding residuals for the maximum damping condition plotted versus frequency.

XMax Damp2 =7.8.

proportional according to equation: Q=ω02 γ

=π f 0γ

=f 0∆ f

. We both calculated the quality factor

using the damping coefficient and the range of driving frequencies at ‘half power’. Tables 7 and 8 show the results based on the damping coefficient method, and width at half power method respectively. For the damping coefficient and natural frequency, we used an average based on the fit coefficients from our acceleration amplitude and phase angle curves. We estimated the uncertainty on the damping coefficient and natural frequency to be the standard deviation, and the overall uncertainty was then calculated using quadrature sums based on the standard deviation.

Condition

Quality Factor Uncertainty

No Damping 123.31 + 0.0255Minimum Damping 75.99 + 0.0636Moderate Damping 22.55 + 0.0213Maximum Damping 13.80 + 0.0159

Condition

Quality Factor Uncertainty

No Damping 128.54 + 2.62Minimum Damping 81.56 + 2.79Moderate Damping 21.78 + 0.39Maximum Damping 13.76 + 0.02

The uncertainties in figure 17 were calculated based on the half-range of Q factors obtained, as we were using cursors to graphically analyze our data. This demonstrated that, again, cursors were an inaccurate way to acquire data. As our numerical range on the axis for amplitude of acceleration grew, it became harder to accurately read the cursors.

As we observed qualitatively in our graphs of acceleration amplitude versus driving frequency, the quality factor tells us numerically that each subsequent increase in damping results in a more rounded curve with lower peak values. The quality factor in an important value to be able to calculate and measure. For example, a large quality factor reflects a sharp curve in amplitude of acceleration. Differentiating the acceleration tell us the impulse a system is receiving. Impulse is an important concept for propulsion systems. A small quality factor reflects a soft curve and therefore a small impulse. Shock absorbing systems are based upon reducing the force felt during an impulse. Both of the previous examples are entirely different studies on their own, and our point was to emphasize the importance of being aware of and measuring the quality factor.

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Table 7. Quality factor and corresponding uncertainty calculated using damping coefficient and natural frequency method.

Table 8. Quality factor and corresponding uncertainty calculated using half-power method.

As mentioned before, one way to improve the accuracy of the data take during this experiment would be to improve the ‘cursor’ method for measurements. IGOR’s current edition has two options when using cursors: fixed, meaning you can drag the cursor along the curve and it will display the corresponding x and y coordinate values, and free-standing, which allow you to place the curve anywhere on the graph. The first method, fixed cursors, is inaccurate because the cursors only stop at certain values as you drag it along, the corresponding coordinate values are not continuous. The second method, using free standing is hard to place at the exact value you wish to observe. A solution to this would be to have programmable cursors. That is, if you wish to measure a root of a curve, being able to input an exact value of 0.00 for the y component and consequently be told the x value, this would reduce the margin for uncertainty in measuring with cursors.

As we discussed during the results, our first fit for acceleration amplitude versus frequency has two points that are extremely far from the curve resulting a large value for chi squared. The others, however, are much closer to the expected values when we consider the equation we are using to fit them. When we graphed a plot of ten points, with a fit curve using five parameters, we would expect our chi squared value to be similar to our degrees of freedom: 5. Also, we noticed that like our plots for free oscillations, when we increase the damping on our system our curve fits get increasingly accurate. By not allowing the system to oscillate as greatly when at resonance, we are reducing the chances that our curve fit will not be compensating for additional physical interactions, as in additional vibrations if the wave driver was bouncing on and off the hacksaw.

Conclusion We began this experiment to test the validity of mechanical resonance theory for small

oscillations. Our apparatus consisted of an accelerometer to measure the amplitude of our system, a hacksaw to provide a medium to oscillate, a wave driver to provide a force to induce oscillatory motion, a copper brick to introduce damping, and a DAQ to measure voltage from our accelerometer. After the assembly of our system, we wrote a simple program in N.I. LabVIEW software to allow us to display the signals we were receiving of voltage from both our wave driver and accelerometer. We first allowed our hacksaw to oscillate freely in order to test our theoretical equations for decaying motion and also to measure the natural frequency of our system. Our second step was to record the amplitude of acceleration at selected frequencies both above and below resonance. Finally, we measured the phase angle between our wave driver and our oscillating hacksaw.

We discovered that, in accordance with our theory, when nearing resonance our system’s acceleration amplitude increased greatly. As our equation predicted, increasing the damping caused our quality factor to decrease. We intuitively knew this must be the case as damping decreases the amplitude of acceleration and, hence, must also effect the sharpness of

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our curve for amplitude versus frequency. When studying our resulting data for the phase angle our observations again agreed with our predictions. Our phase angle data only fit well when we used a formula that only displayed plotted real angles of phi. This was exactly what our theory predicted as the denominator of our equation for phi went to zero as the driving frequency approached that of the system’s natural frequency.

When the oscillations become too large other forces from additional vibrations come into effect and our simple harmonics oscillatory equations break down. Our data shows that frequencies very near to resonance can still be studied as long as a system is under the effect of damping because this reduces those unwanted circumstances and prevents the oscillations near resonance from becoming too violent. As a result, we can conclusively show mechanical resonance theory holds for small oscillations.

Acknowledgements I would like to thank Michael Embuido for being my lab partner for this project.

1. SFU Physics 231, Lab Script 5; http://www.sfu.ca/phys/231/143/2. SFU Physics 231, Reference Manual; http://www.sfu.ca/phys/231/143/Ref_Manual14-3.pdf 3. Introduction to Error Analysis, Second Edition by John R. Taylor

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