Video-based spatial portraits of a nonlinear vibrating string
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.
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.
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) 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.
matlab image processing commands used for this work.
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