Non-Linear Spring Oscillations: A Representation of Duffng's Equation

7/30/2019 Non-Linear Spring Oscillations: A Representation of Duffng's Equation

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Non-Linear Spring Oscillations:2

A Representation of Duffings Equation3

Esther Campbell and Meena Sharma4

University of the Fraser Valley, Department of Physics, Abbotsford, BC, V2S 7M85

(Dated: April 23, 2010)6

Abstract

Duffings equation is a representation of chaotic motion. The equation itself can be represented in various

ways and used in a variety of experiments with nonlinearities. In this equation, the cubic form of Duffings

equation was used to find a mathematical relationship between the force and position of an air track glider

attached to two nonlinear springs. Oscillations of three different sizes were analyzed to verify the relationship.

The data were inconclusive; it was shown that the force could be represented as a cubic function of displacement,

but the error calculated in the coefficients was too high to make any predictions about the actual equation;

the data were also represented linearly to compare.

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I. INTRODUCTION7

For beginner or aspiring physicists, Hookes law is sufficient for representing the relationship between8

force and displacement of a spring from its unstretched length, and is written as9

F = k x, (1)

where F is the force, k is the spring constant (in units of kg /s2), and x is the displacement. However,10

when exploring more complicated situations, Hookes law can be a very inaccurate representation.111

Duffings equation was derived as a possible alternate representation in cases where oscillations were12

too chaotic to represent linearly. There are many uses of this equation within the spectrum of nonlinear13

oscillations, and the equations do not look the same in every case. For example, Gottwald et al.2 used14

the cubic nonlinearity when mirroring Duffings motion equation for a ball in a double well. There are15

many other experiments which also use the cubic form of Duffings equation; most of the applications16

of Duffings equation employ the cubic nonlinearity (which will be the case in this experiment) but a17

few also use the quintic. The quintic form of Duffings equation has been studied much less extensively18

because of the level of difficulty in solving it.3 There are also varying levels of difficulty in the solution19

depending on if the damped or forced versions of the equation are used. The following equation is an20

example of a more complicated form of the Duffing equation:21

d2 u

dt2+ u + u3 + u5 = sin t. (2)

In this case, , , and are constants. This is a forced, undamped version of Duffings equation.22

The right hand side of the equation gives the forcing term. The damped equation would also have a23

first order derivative of u with respect to t.24

In this experiment, two non-linear hard springs were forced to oscillate. This was for the purpose25

of creating a situation where Duffings equation could be used to estimate the information gathered26

from the chaotic behaviour. The experiment was designed to test the ability of Duffings equation to27

represent the relationship between the force and displacement of an object attached to two nonlinear28

hard springs.29

This article describes the theory behind the experiment by elaborating on which equations were30

used. It also details how the apparatus was set up and what materials were used. The procedure31

describes how the data were analyzed by noting the smoothing algorithms used. The Results section32

states the findings of the experiment and then the discussion and conclusion interpret those results, as33

well as suggest improvements for further experimentation in this area of study.34

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II. THEORY35

As was mentioned before, the most basic and fundamental equation used in this experiment was36

Duffings equation. The form of Duffings equation that was used in this experiment looks like37

d2 u

dt2= a u3 + b u2 + c u + d, (3)

where a, b, c, and d are all constants, and a is in units of 1/(m2s2), b is in units of 1/(ms2), c is in units38

of 1/s2 and d is in units of m/s2.39

The apparatus used was an air-track glider. The glider was set in the center of the track, attached40

to a hard spring on either side. The springs were circular metal strips stretched into an ellipse and41

attached to one end of the air track and one end of the glider as shown in FIG 1.142

FIG. 1: Basic setup of the apparatus showing the air track, glider(m) and two springs.

Also, a high resolution video camera (Casio High Speed, EX-FH20) was placed on a tri-pod and43

used to record the data.44

The basic assumption was that since the springs used were non-linear, the data could be fit to a45

cubic function in order to model Duffings equation.46

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III. PROCEDURE47

The air track was leveled before any data were taken. The springs were raised to ensure they were48

not touching any part of the airtrack or glider to which they were not already attached, to ensure that49

no extra friction force was involved.50

After the apparatus, camera and tri-pod were set up, the glider was pulled to one side of the track51

until one spring was almost circular and the other pulled into an ellipse. The camera was turned on and52

the glider was released to oscillate freely until it came to rest while the camera recorded the oscillations.53

Once the recording was taken, the video was opened in a video analysis program for the data to be54

analyzed4. Three separate oscillations were analyzed, one large, one medium, and one small, all relative55

to the size of the first and largest oscillation. For each of these three oscillations, the position of the56

glider was recorded frame by frame. The program then calculated the instantaneous velocities at each57

position (starting with v1) by dividing the difference in position by the change in time between each58

frame. For example, if each position was labeled x1, x2, x3... etc. the program would use the formula59

v1 =x2 x1

t, (4)

where t is the change in time between frames (note that the frame rate is 210 frames per second). The60

instantaneous acceleration (starting at a1) was calculated similarly using velocities instead of positions.61

So in the same way, if the velocities were labeled v1, v2, v3... etc. the formula would look almost62

identical:63

a1 =v2 v1

t. (5)

Once these were calculated, the data were graphed, showing force versus position. The force was64

found simply by multiplying the acceleration by 0.299 kg, which was found to be the mass of the glider.65

Since the data were simply fit to a cubic function, no differential equations needed to be solved to66

find the coefficients in Duffings equation. Since the graphs showed points which were quite spread67

out, smoothing algorithms were used on the curves to make it easier to fit it to a cubic function; the68

smoothing algorithms made it easier to fit to Duffings equation. Two smoothing algorithms were used.69

The first was an average of three accelerations and the second was an average of five. For averaging70

three accelerations (where the accelerations are labeled a1, a2, a3... etc.) the formula71

a1 + a2 + a33

(6)

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was used. This equation is very similar to the average of five accelerations equation which looks like72

a1 + a2 + a3 + a4 + a55

. (7)

Averaging over a large number of points (3 or 5) meant discarding a few data points at the beginning73

of the oscillation and a few at the end, however this was not seen as a problem because of the large74

number of data points collected to begin with.75

The smoothed data were fit to a cubic function using the least-squares fitting technique. A sample76

graph showing the raw data before any smoothing occured is shown in FIG 2.77

FIG. 2: Sample graph of force vs. position for a medium oscillation of no smoothing showing the best fit line.

This graph can be compared to the graph in FIG 3 which shows the same data set for a smoothing78

algorithm of five data points.79

Three graphs were made with the best fit curve for each oscillation size (recall that data were taken80

for small, medium and large oscillations), one best fit for the raw data (no smoothing), one for the81

smoothing with an average over three data points and one for the smoothing with an average over five;82

nine graphs were made in total. Once these graphs were made the coefficients (with error) could be83

found for the best fit curve.84

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FIG. 3: Sample graph of force vs. position for a medium oscillation of smoothing over five data points showing

the best fit line.

IV. RESULTS85

The results of the experiment were inconclusive; the data were fit to a cubic curve in every case,86

however the estimated percent error for each of the coefficients was so large that the estimates were87

extremely unreliable. Therefore we were unable to show that Duffings equation was the best model for88

these data. TABLE 1 is a sample of the coefficients generated for the best fit curve with the smallest89

percent error of all the data collected. In this table, a, b, c, and d are the same coefficients shown in90

equation (3), the version of Duffings equation that was used in this experiment.91

Since the data were so spread out, the data were also fitted to a linear best fit line (for the same92

smoothed data as in FIG 3) for comparison, which is shown in FIG 4. As shown in this figure, the data93

can be represented just as well by Hookes law as it can by Duffings equation.94

It should also be noted that the smoothing algorithms were used to reduce the error in the fit;95

averaging over 3 or 5 points reduced the scatter in the curve, making the cubic function easier to fit to96

it. However for the large oscillation data set averaging over five data points actually made the fit worse.97

Smoothing over three data points did decrease the error in the fit in all three cases but the smoothing98

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Coefficient Estimate with Error Error (%)

a -689.03 356.80 51.78

b -2983.18 1678.00 56.25

c 2339.45 1372.00 58.64

d 73.75 29.26 39.68

TABLE I: Coefficient estimates of the cubic Duffing equation (3) with error for the large oscillation using a

smoothing average of three data points.

FIG. 4: Sample graph of force vs. position for a medium oscillation of smoothing over five data points showing

the linear best fit line.

did not sufficiently reduce the error to produce an equation that could definitively be represented by99

Duffings equation.100

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V. DISCUSSION101

The data collected from this experiment were not sufficient to show that the data could be rep-102

resented by Duffings equation; the percentage error in the coefficients generated through computer103

analysis were so large that no useful information could be gathered.104

It is most probable that the large error found in the data was due to observational error. In105

video analysis, it was difficult to accurately pinpoint the position of the glider because the pixels were106

repeatedly changing from frame to frame and the exact position of the corner of the glider that was107

being referenced was difficult to see.108

If this experiment were to be repeated, it would be useful to take more data. Data were taken from109

only one oscillation at each amplitude in this experiment. In the future, taking multiple oscillations110

from each amplitude would be beneficial.111

Another suggestion for producing a more useful experiment would be to use a higher resolution112

camera (although this may be quite expensive), however the same effect would occur simply by zooming113

in to the oscillations so that they take up the entire field of the camera, thus reducing the fractional114

error and the absolute position error.115

VI. CONCLUSION116

Although the springs used in this experiment were supposedly non-linear, the data from this ex-117

periment were not sufficient to verify a mathematical relationship between the force and position of118

the glider on the airtrack. However, the experiment did show that the experimental instantaneous119

acceleration data could be represented by a cubic function (through Duffings equation) as predicted,120

even though it could also be represented linearly. The nature of this cubic function (i.e. the value of121

the coefficients in the function) could not be deduced in this experiment but possibly could be through122

repetition and reduction of error through methods mentioned in the discussion section. If the error123

could be sufficiently reduced it may also be possible to show that Duffings equation is a better predictor124

of chaotic motion than Hookes law, which this experiment did not show.125

1 Richard H. Enns and George C. McGuire, Nonlinear Physics with Maple for Scientists and Engineers (2009).126

2 T. Pirbodaghi, S. H. Hoseini, M. T. Ahmadian, and G. H. Farrahi, Duffing equations with cubic and quintic127

nonlinearities, Computers & Mathematics with Applications 57(3), 500506 (2009).128

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3 J. A. Gottwald, L. N. Virgin, and E. H. Dowell, Experimental Mimicry of Duffings Equation, J. Sound129

and Vibration 158(3), 447467 (1992).130

4 Tracker. http://www.cabrillo.edu/ dbrown/tracker/ Retrieved on February 8, 2010.131

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Non-Linear Spring Oscillations: A Representation of Duffng's Equation

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