Coupled Oscillators

Andy Chmilenko, Nick Kuzmin

Instructor: Jeff Gardiner Section 1

(Dated: 1:30 pm Monday May 26, 2014)

I. ABSTRACT II. INTRODUCTION

III. THEORETICAL BACKGROUND IV. EXPERIMENTAL DESIGN AND PROCEDURE

V. ANALYSIS

Recorded Masses of Pendulums:

m_{1} = 2931 g m_{2} = 2946 g

Sample Calculations for \overline{m}

\overline{m} = ^{\frac{∑_{i}^{n} d_{i}}{n}} \overline{m} = \frac{m_{1}+m_{2}}{2} \overline{m} = \frac{2931g+2946g}{2} \overline{m} = 2939g

Mounting Location Length from Pivot Average Length
(±0.1cm)  (±0.1cm)
1  13.4, 13.5  13.4
2  22.2, 22.1  22.2
3  30.5, 30.2  30.4
4  41.6, 41.6  41.6
(to mass)  55.9, 56.1  56.0

TABLE I: Measurements of the mounting lengths and the distance to the center of mass from the pivot point of each pendulum.

Sample Calculations for Average Length using row 1 of Table I

\overline{d} = ^{\frac{∑_{i}^{n} d_{i}}{n}}

\overline{d} = \frac{13.4cm+13.5cm}{2}
\overline{d} = 13.4cm

Sample Calculations for Δ Average Length using row 1 of Table I

Δ\overline{d} = \sqrt{Δd^{2} × 2}

Δ\overline{d} = \sqrt{0.1cm^{2} × 2}
Δ\overline{d} = 0.14cm
Δ\overline{d} ≈ 0.1cm

A. Determination of Spring Constants

Spring 1  Spring 2  Spring 3  Spring 4
Mass Displacement Mass Displacement Mass Displacement Mass Displacement
(g) (± 0.05 cm) (g) (± 0.05 cm) (g) (± 0.05 cm) (g) (± 0.05 cm)
50  27.6  50  34.3  50  31.6  50  34.9
70  30.8  70  35.0  100  31.9  90  35.7
90  34.1  90  35.6  150  32.2  130  36.4
110  37.3  110  36.2  200  32.5  180  37.3
130  40.6  130  36.8  280  33.0  1 220  38.1
–  –  150  37.5  550  34.6  280  39.1

TABLE II: Masses applied and associated displacements to Springs 1-4

FIG. 1: Spring Displacement versus Force Applied using masses, for Springs 1-4.

B. Normal Modes and Coupling Frequency

Spring ℓ  ν_{1}  Δν_{1}  ω_{1}  Δω_{1}  ν_{2}  Δν_{2}  ω_{2}  Δω_{2}  \overline{ω}  Δ\overline{ω}  % error
No.  (Hz)  (Hz)  ( \frac{rad}{sec}) ( \frac{rad}{sec})  (Hz)  (Hz)  ( \frac{rad}{sec}) ( \frac{rad}{sec})  ( \frac{rad}{sec})  ( \frac{rad}{sec})
1  1 0.662870 0.000571 4.164935 0.003588 0.661430 0.000618 4.155887 0.003883 4.16041115 0.00528671 0.00597854
2 0.659420 0.000650 4.143258 0.004084 0.655640 0.000776 4.119508 0.004876 4.13138284 0.00636023 0.01291409
3 0.656870 0.000687 4.127236 0.004317 0.657080 0.000680 4.128555 0.004273 4.12789567 0.0060735 0.01374726
4 0.663890 0.000603 4.171344 0.003789 0.665940 0.000627 4.184224 0.003940 4.17778416 0.00546579 0.00182771
2  1 0.654530 0.000821 4.112533 0.005158 0.657440 0.000716 4.130817 0.004499 4.12167531 0.00684463 0.01523345
2 0.661180 0.000635 4.154316 0.003990 0.661730 0.000596 4.157772 0.003745 4.15604434 0.00547193 0.00702187
3 0.660970 0.000641 4.152997 0.004028 0.659750 0.000717 4.145332 0.004505 4.14916425 0.00604288 0.00866569
4 0.663770 0.000706 4.170590 0.004436 0.664580 0.000502 4.175679 0.003154 4.1731346 0.00544299 0.0029386
3  1 0.673690 0.000987 4.232919 0.006202 0.672950 0.000876 4.228270 0.005504 4.23059433 0.00829177 -0.0107899
2 0.675000 0.000860 4.241150 0.005404 0.676080 0.000975 4.247936 0.006126 4.244543 0.00816868 -0.01412257
3 0.668340 0.000694 4.199304 0.004361 0.668970 0.000685 4.203262 0.004304 4.20128327 0.00612687 -0.00378679
4 0.666700 0.000744 4.189000 0.004675 0.666770 0.000792 4.189439 0.004976 4.18921956 0.0068276 -0.00090448
4  1 0.665650 0.000912 4.182402 0.005730 0.660370 0.000757 4.149227 0.004756 4.16581469 0.00744708 0.0046875
2 0.661250 0.000684 4.154756 0.004298 0.660430 0.000716 4.149604 0.004499 4.15218018 0.00622166 0.00794511
3 0.670570 0.000582 4.213316 0.003657 0.669330 0.000732 4.205524 0.004599 4.20942 0.00587586 -0.00573084
4 0.660180 0.000725 4.148033 0.004555 0.660000 0.000744 4.146902 0.004675 4.14746779 0.00652714 0.00907102

TABLE III: Frequency values from Fourier Analysis of amplitude data for Pendulum 1 2 (ω_{2}) for Even modes.

Spring No. Mounting Location  ν_{1}  ν_{2}
(Hz)  (Hz)
1 1 0.67354 ±0.00099 0.67356 ±0.000927
2  0.68261 ±0.00166 0.68407 ±0.0017
3  0.6935 ±0.00116 0.69443 ±0.00112
4  0.69955 ±0.000239 0.69887 ±0.000215
2 1 0.69405 ±0.000999 0.69493 ±0.00091
2  0.71182 ±0.00213 0.70124 ±0.000814
3  0.76329 ±0.00388 0.77023 ±0.00376
4  0.79165 ±0.00123 0.79109 ±0.00116
3 1 0.77166 ±0.000668 0.77467 ±0.00118
2  0.88002 ±0.00159 0.87788 ±0.00156
3  1.02200 ±0.00143 1.0231 ±0.00127
4  1.31860 ±0.00277 1.3219 ±0.00225
4 1 0.70376 ±0.000541 0.69953 ±0.000408
2  0.76033 ±0.00117 0.76406 ±0.00209
3  0.80902 ±0.00113 0.80696 ±0.000954
4  0.92830 ±0.00123 0.92827 ±0.00131

TABLE IV: Frequency values from Fourier Analysis of amplitude data for Pendulum 1 (ω_{1}) and Pendulum 2 (ω_{2}) for Odd modes.

C. Beat Frequency

VI. CONCLUSION

VII. REFERENCES

Jeff Gardiner. Coupled Oscillators. Waterloo, Ontario: University of Waterloo; c2014. 4 p.