^{1}, Zubair Usman

^{1}and Muhammad Sabieh Anwar

^{1,a)}

### Abstract

The article describes a systematic experimental study of a string vibrating nonlinearly. The string is tracked in real time using strategically located cameras; the video tracking enables a remote observation of the oscillator without perturbing its inherent nonlinearities. We show that our technique can help probe the parametrically excited oscillations and study phenomena such as elliptical and circular trajectories near resonance, resonance fold-over, jump, hysteresis, and subharmonic resonance. The experiment has been successfully employed in the advanced physics laboratory.

I. INTRODUCTION

II. MATHEMATICAL PRELIMINARIES

A. Background

B. Setting up the system of coupled Duffing oscillators

C. Approximate solution to the problem of coupled oscillators and predicting the spatio-temporal portraits

III. THE EXPERIMENT

A. Experimental setup

B. Spatial portraits

C. Foldover, jump, and hysteresis

D. Subharmonic resonance

IV. CONCLUSIONS

### Key Topics

- Structural beam vibrations
- 5.0
- Cameras
- 4.0
- Equations of motion
- 4.0
- Oscillators
- 4.0
- Coupled oscillators
- 3.0

##### G09B

## Figures

Experimental setup for (a) phase angle measurement, using only one obliquely positioned camera, and (b) studying nonlinear resonance, using one webcam facing vertically downward and another oriented horizontally.

Experimental setup for (a) phase angle measurement, using only one obliquely positioned camera, and (b) studying nonlinear resonance, using one webcam facing vertically downward and another oriented horizontally.

Planar vibrations collapse at and a gradual increase in *a* sweeps the polarization from elliptic to circular. The dashed line represents the condition *a* = *b*.

Planar vibrations collapse at and a gradual increase in *a* sweeps the polarization from elliptic to circular. The dashed line represents the condition *a* = *b*.

Orthogonal displacements for elliptical trajectories for driving frequencies (a) 29.6 Hz, (b) 30.0 Hz, (c) 30.3 Hz, (d) 30.4 Hz, (e) 30.7 Hz, (f) 30.8 Hz, (g) 31.0 Hz, (h) 31.1 Hz, and (i) 31.2 Hz. The theoretically predicted resonance frequency is 31.1 Hz.

Orthogonal displacements for elliptical trajectories for driving frequencies (a) 29.6 Hz, (b) 30.0 Hz, (c) 30.3 Hz, (d) 30.4 Hz, (e) 30.7 Hz, (f) 30.8 Hz, (g) 31.0 Hz, (h) 31.1 Hz, and (i) 31.2 Hz. The theoretically predicted resonance frequency is 31.1 Hz.

(Color online) (a) Illustration of the fold-over effect. Resonance curves for signal amplitudes *f* of 3, 5, and 7 V acquired as frequency is swept in the upward direction. The horizontal axis is the driving frequency in Hz and the vertical axis shows the peak-to-peak amplitude. In our case, Hz and resonance is achieved at 30.5 Hz, 30.68 Hz, and 30.71 Hz. (b) Schematic illustration of the fold-over, emphasizing the origin of jump and hysteresis. The unstable region is indicated by the thicker line. (c) Resonance curve for *x* displacement with increasing and decreasing drive frequencies. (d) Resonance curve for *y* displacement with increasing and decreasing drive frequencies.

(Color online) (a) Illustration of the fold-over effect. Resonance curves for signal amplitudes *f* of 3, 5, and 7 V acquired as frequency is swept in the upward direction. The horizontal axis is the driving frequency in Hz and the vertical axis shows the peak-to-peak amplitude. In our case, Hz and resonance is achieved at 30.5 Hz, 30.68 Hz, and 30.71 Hz. (b) Schematic illustration of the fold-over, emphasizing the origin of jump and hysteresis. The unstable region is indicated by the thicker line. (c) Resonance curve for *x* displacement with increasing and decreasing drive frequencies. (d) Resonance curve for *y* displacement with increasing and decreasing drive frequencies.

Experimental trajectory of the string in the condition of subharmonic resonance with *A*, *B*, *C*, *D* values: (a) 3.68, 3.91, 3.50, 3.62; (b) 3.36, 4.21, 3.45, 3.62; and (c) 1.43, 4.60, 1.15, 3.73. The corresponding simulation results are shown in (d), (e), and (f).

Experimental trajectory of the string in the condition of subharmonic resonance with *A*, *B*, *C*, *D* values: (a) 3.68, 3.91, 3.50, 3.62; (b) 3.36, 4.21, 3.45, 3.62; and (c) 1.43, 4.60, 1.15, 3.73. The corresponding simulation results are shown in (d), (e), and (f).

(a) Frequency response curve for the undamped Duffing oscillator near resonance. The curve uses the parameters *g* = 1 and . (b) A plot of the solution for a damped Duffing oscillator with *a* = 1, , , and .

(a) Frequency response curve for the undamped Duffing oscillator near resonance. The curve uses the parameters *g* = 1 and . (b) A plot of the solution for a damped Duffing oscillator with *a* = 1, , , and .

(a) Directly captured webcam image. (b) Geometrical analysis for finding the bisector of region *M* and calculating the phase angle. (c) Bisector determination on the captured image. The left vertical line is divided into half and the middle pixel is identified with coordinates (58, 147). Likewise, another middle point is found on the right vertical line, which has the center point (271, 128). Joining both of these points gives the bisector. (d) Morphological (skeleton) image of the elliptical trajectory. These trajectories were captured at a driving frequency of 31.1 Hz.

(a) Directly captured webcam image. (b) Geometrical analysis for finding the bisector of region *M* and calculating the phase angle. (c) Bisector determination on the captured image. The left vertical line is divided into half and the middle pixel is identified with coordinates (58, 147). Likewise, another middle point is found on the right vertical line, which has the center point (271, 128). Joining both of these points gives the bisector. (d) Morphological (skeleton) image of the elliptical trajectory. These trajectories were captured at a driving frequency of 31.1 Hz.

## Tables

matlab image processing commands used for this work.

matlab image processing commands used for this work.

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