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• Non-axisymmetric oscillations... Comparison of axisymmetric oscillations and non-axisymmetric oscillations observed for frequencies 40Hz/60Hz and 50Hz respectively, for the same actuation voltage 74Vrms. This figure depicts the existence of a local minimum at 50Hz where non-axisymmetric modes can be observed at lower voltages. ... Number of cycles required for mixing of droplets at different actuation frequencies for 115Vrms using non-axisymmetric modes. Below 55Hz, only k=2 mode oscillations exist and the number of cycles required for mixing increase with the frequency. Beyond 55Hz, other higher oscillation modes exist. ... Mixing of droplets of DI water (8μl) and diluted orange food colour droplet (2μl) using non-asymmetric oscillations with 115Vrms and frequencies (A) 35Hz (mode k=2) and (B) 85Hz (mode k=3). [supplementary videos available]. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) ... Oscillation patterns (top view) of a 8μl droplet at different voltages and frequencies (a) mode k=0 at 35Hz, 74Vrms; (b) mode k=2 at 35Hz, 117Vrms; (c) mode k=3 at 100Hz, 117Vrms. [supplementary videos available]. ... Change in base radius of an drop at different frequencies for 35Vrms. Axisymmetric oscillations at 35Vrms voltage are found to have a resonance peak at 25Hz (having an average contact angle θa∼113°).
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• Coupled oscillators... FRCs of the normalised relative displacement W as a function of the normalised frequency, Ω, for γ=2×10−3, ζs=0.046,ζ=0.026 and for different values of the normalised primary resonance of the oscillator: (a) ω0=1.4, (b) ω0=1.1, (c) ω0=0.7, (d) ω0=0.5, (e) ω0=0.3, (f) ω0=0.1. Stable solution (blue solid line), unstable solution (red dashed line). Numerical solution by integrating Eqs. (4a) and (4b) for μ=0.001 (black ‘∘’). ... Supplimentary material to “On the interaction of the responses at the resonance frequencies of a nonlinear two degrees-of-freedom system”.
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• (a) Time series for the first chrono-mode of the POD, a1(t), for the three different forcings with vin=0.4m/s (Re=3.1in×103, N=0.02). (b) Power spectra of the chrono-modes a1(t). Frequency peaks are found at fPOD=0.027Hz (FL0). The values of the frequency peaks are in reasonable agreement with the frequencies found for the free surface fluctuations, fTS. ... (a–c) Profiles of the turbulence kinetic energy kturb,2D. (d–f) Profiles of the kinetic energy associated with the large-scale oscillations kosc,2D. The inlet velocity is vin=0.4m/s (Rein=3.1×103, N=0.02). ... Amplitude A and frequency fTS of the free surface oscillation at a monitoring point at x=0.175m for the three different forcings (Rein=3.1×103, N=0.02). Dominant frequency fPOD from the power spectrum of the first chrono mode of the POD. ... Self-sustained oscillations
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• Analytically obtained frequency-response curves defined by Eq. (35) of the oscillator governed by Eq. (1) for van der Pol damping, α=4/5, ε=0.2 and different values of F. Numerical results are depicted by dots. The lines and zones related to tr2−4det, tr and det are defined by Eqs. (39) and (40). A backbone curve is plotted as a dashed–dotted line. The mark ‘∗’ stands for the characteristics of free limit cycle oscillations. ... Frequency-response curves of the oscillator governed by Eq. (1) for van der Pol damping, α=3, ε=0.566 and two different values of F. Analytical results from  are depicted by a dashed–dotted line, analytical results from this paper by a solid and dotted line and numerical results by dots. The lines and zones related to tr and det are defined by Eqs. (39) and (40). The mark ‘∗’ stands for the characteristics of free limit cycle oscillations. ... Period of oscillations TexND given by Eq. (7). ... Frequency-response curves of the oscillator governed by Eq. (1) for linear viscous damping, F=ε=0.1 and for: (a) α=1/4; (b) α=4. Analytical results (49) are depicted by a solid line (stable) and by a dotted line (unstable); a shaded region (instability zone) is bounded by the curves defined by Eq. (53); a backbone curve (48) is shown as a dashed–dotted line; numerical results are depicted by dots. ... Frequency-response curve
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• (a) Diffusion signal for different waveforms: square with 90° phase, apodised cosine and apodised trapezoid as a function of oscillation frequency for four different sizes of the restricted compartment; (b) corresponding extracted ADC values. The diffusion signal and ADC for apodised trapezoid and square wave are very similar and are plotted on top of each other. ... Oscillating gradient... (a) Average signal difference between square and sine approximations and the full trapezoidal expressions as a function of α for R=2μm and 10μm. (b) Diffusion signal for R=5μm for the three waveforms with gradient strength G=60mT/m and 200mT/m as a function of oscillation frequency. ... (a) Average signal difference between square and sine approximations and the full trapezoidal expression considering: I – same amplitude, II – same area under the curves, III – same area under the squared curves and IV – same b value per oscillation. (b) Difference between square and sine approximations and the full trapezoidal expressions with SR=200T/m/s as a function of n for all data points with R=5μm. ... Restricted diffusion signal as a function of oscillation frequency for (a) several values of Δ, R=5μm and G=0.1T/m; (b) several gradient strengths, R=5μm and Δ=25ms. In (a) and (b) the filled markers indicate waveforms with integer number of oscillations. Restricted diffusion as a function of (c) gradient strength for several frequencies, R=5μm and Δ=45ms; (d) cylinder radius for several frequencies, G=0.1T/m and Δ=45ms. The markers show the MC simulation and the solid lines are the GPD approximations. The vertical bar separates different scales on the x-axis. ... Square wave oscillations
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• Center of mass displacement (green curve) and acceleration (blue curve) relative to tail position (red, shown for reference) in L. macrochirus swimming at 1.5BL/s. Surge acceleration shows a double peak oscillation at twice the frequency of the tail beat which is absent in the sway data. ... Comparison of COM oscillation magnitude for different animals. COM oscillation magnitude (peak-to-peak) in each direction was averaged across speeds for terrestrial species to give a single number generally representative of each species. Non-fish data were taken from the literature, and represent COM heave (vertical) motion only since this is the most commonly reported direction of COM oscillation, except for the sandfish point which represents sway motion. Fish data (green, red, and blue points) are from the present study and excursions are shown for all three different measured directions: heave, surge, and sway. The linear fit for data on terrestrial animals shown is 1.44x−6.07 (R2=0.88), indicating that COM displacement scales positively with mass, and that larger animals display a larger vertical oscillation than would be expected for their size. Fish have significantly lower heave COM oscillations than terrestrial animals. The orange point represents lizard sandfish moving in a granular medium and is derived from Ding et al. (2012). ... Two-way ANOVA table to show F-values for speed and species effects on each of the three dimensions of center of mass (COM) oscillation. ... Fast Fourier transforms of surge COM acceleration and displacement data for locomotion at 1.5BL/s in (A) L. macrochirus, (B) A. rostrata, and (C) N. chitala. The frequency components of the tail beat are shown for comparison in red. At this speed, L. macrochirus uses body and caudal fin undulation and so body undulations are comparable to those of the other species. Grey bars mark the tail beat and double tail beat frequencies. Both L. macrochirus and A. rostrata show COM acceleration frequency peaks at greater power for double the tail beat frequency than for the single beat frequency, and significant power for displacement at double the tail beat frequency. N. chitala, on the other hand, shows minimal power at double the tail beat frequency for acceleration, and negligible power for displacement. ... Speed vs. frequency for three species swimming in four swimming modes. Frequency had a significant positive relationship (pfrequency was substituted for tail beat frequency during labriform propulsion at 0.5BL/s as there was no body undulation at this speed.
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• Bistable bilinear oscillator: (a) bilinear restoring force (solid curve) and (b) numerical solution of (1) for γt=0.118, ζt=0.071, and xs=0.0073 (solid curve). ... NL bistable oscillator. (a) Bistable oscillator composed of a linear oscillator and a NL element. (b) The dimensionless restoring force of a bistable oscillator for different values of α (α=0, solid curve; α=0.1, dotted curve; α=0.3, dashed curve; α=1, dot-dashed curve) in the case of k˜NL=1 and β=1. (c) Typical response of bistable oscillator to harmonic excitation in a highly pre-resonance regime. The displacement (solid curve), potential (dashed curve) and restoring force (dot-dashed curve) are taken from the experimental setup, see Appendix. ... Bistable energy harvester prototype. (a) Close-up on coil and MP, (b) the prototype oscillator, and (c) excitation base comprised of a motorized lead screw base. ... (a) Power ratio of a bistable oscillator vs. linear oscillator as a function of (γ,ζ) for xs=±0.0073. Analytical results represented by 2-dimensional surface; numerical results represented by the contour plot at the bottom of the figure. (b) Power ratio numeric simulation of a wide barrier bistable oscillator. The restoring force used on the numerical simulation is extracted from the bistable prototype; (c) the extracted force. The excitation amplitude used is Y=50[mm]. For description of the prototype, see Appendix. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
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• Oscillation detection in a single electrode with weak alpha. The electrode was selected from the same subject as in Figs. 2 and 4. (A) The 256-electrode array with the selected electrode highlighted in yellow. (B) Background wavelet power spectrum mean and standard deviation (blue), and the linear regression fit to the background (green). (C) Oscillations detected across all frequencies by the oscillatory episode detection method. Red vertical lines indicate when participants were instructed to close their eyes and black vertical lines indicate when participants were instructed to open their eyes. (D) The proportion of time (Pepisode) during the eyes -closed condition (red) and eyes-open condition (black) that oscillations were detected at each frequency. (E) The raw signal from the chosen electrode, with detected oscillations at the peak alpha frequency (9.5Hz) highlighted in red. Vertical lines are the same as above. (F) An expansion of the highlighted section in E, to show the spindle-like appearance of the alpha oscillation. ... Temporal independence of two alpha components. (A) An 8-s epoch from the alpha component shown in Fig. 2, with detected alpha-frequency oscillations highlighted in red. (B) The same time segment as in A, from the alpha component in Fig. 6. Note the alpha oscillation is maximal in B when the oscillation is at a minimum in A, demonstrating why these were extracted as temporally independent components. ... Lateralized alpha component. From the same subject as Figs. 2 and 4–5. (A) The spline-interpolated scalp distribution of an alpha component extracted by ICA. Color scale denotes electrode weight (unitless). (B) Background wavelet power spectrum mean and standard deviation (blue), and the linear regression fit to the background (green). (C) Oscillations detected across all frequencies by the oscillatory episode detection method. Red vertical lines indicate when participants were instructed to close their eyes and black vertical lines indicate when participants were instructed to open their eyes. (D) The proportion of time (Pepisode) during the eyes-closed condition (red) and eyes-open condition (black) that oscillations were detected at each frequency. (E) The time-domain representation of the chosen component, with detected oscillations at the peak alpha frequency (9.5Hz) highlighted in red. Vertical lines are the same as above. (F) An expansion of the highlighted section in E. ... Oscillation detection in an ICA alpha component. (A) The spline-interpolated scalp distribution of an alpha component extracted by ICA. Color scale denotes electrode weight (unitless). (B) Background wavelet power spectrum mean and standard deviation (blue) and the linear regression fit to the background (green). (C) Oscillations detected across all frequencies by the oscillatory episode detection method. Red vertical lines indicate when participants were instructed to close their eyes and black vertical lines indicate when participants were instructed to open their eyes. (D) The proportion of time (Pepisode) during the eyes-closed condition (red) and eyes-open condition (black) that oscillations were detected at each frequency. (E) The time-domain representation of the chosen component, with detected oscillations at the peak alpha frequency (9.5Hz) highlighted in red. Vertical lines are the same as above. (F) An expansion of the highlighted section in E, to show the spindle-like appearance of the alpha oscillation. ... Oscillation... Oscillation detection in a single electrode with strong alpha. The electrode was selected from the same subject as in Fig. 2. (A) The 256-electrode array with the selected electrode highlighted in yellow. (B) Background wavelet power spectrum mean and standard deviation (blue), and the linear regression fit to the background (green). (C) Oscillations detected across all frequencies by the oscillatory episode detection method. Red vertical lines indicate when participants were instructed to close their eyes and black vertical lines indicate when participants were instructed to open their eyes. (D) The proportion of time (Pepisode) during the eyes-closed condition (red) and eyes-open condition (black) that oscillations were detected at each frequency. (E) The raw signal from the chosen electrode, with detected oscillations at the peak alpha frequency (9.5Hz) highlighted in red. Vertical lines are the same as above. (F) An expansion of the highlighted section in E to show the spindle-like appearance of the alpha oscillation.
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• Bi-stable oscillator... Frequency up-conversion... Velocity decomposition. (Right) Fast component shown in a 3D form: projections show the instantaneous frequency (IF) vs. time, the envelope vs. time and IF vs. envelope on the far projection. (Left) Velocity components: instantaneous amplitude vs. instantaneous frequency. ... Simulated response of a bi-stable energy harvester under constant frequency base excitation ω≈0.1ωn () ωn=7Hz, ζ=0.07. Right: nonlinear (bi-stable) potential (––) and force (–––) functions of the system taken from experimental system. ... Fine resolution spectrum of the bi-stable oscillator response to pure sine excitation.
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