Photographs of (a) the experimental setup and (b) the hot side of the stack. A schematic diagram of the complete experimental setup is shown in (c).
Arnold tongues obtained in the experiments as a function of the driving frequency f and of the root-mean-square amplitude of the loudspeaker voltage for: (a) d = 5 mm, ds = 8 cm; (b) d = 5 mm, ds = 19 cm; and (c) d = 1 mm, ds = 8 cm. The regions are labelled as follows: PS corresponds to perfect synchronization; QP corresponds to quasi-periodicity (loss of synchronization); IPL corresponds to imperfect phase locking, for which the signal looks quasi-periodic but the phase difference stays bounded; and BD corresponds to “beating death,” for which the self-sustained oscillations are almost reduced to silence.
Transition to synchronization in the case of weak forcing ( mV): (a) acoustic pressure p(t) and frequency spectra p(f) measured for the two operating points labeled (I) and (II) in Fig. 2(a) ; (b) time evolution of the instantaneous phase difference , and time evolution of the real part of the instantaneous amplitude as a function of its imaginary part in the frame rotating at angular frequency ; (c) normalized amplitude modulation , and normalized time-average phase difference , as functions of the frequency detuning.
Transition to synchronization in the case of strong forcing ( V): (a) acoustic pressure p(t) and frequency spectra p(f) measured for the three operating points (III), (IV), and (V) in Fig. 2(b) ; (b) time evolution of the instantaneous phase difference and representation in phase space of the operating points (III) to (V); (c) normalized amplitude modulation and normalized time-average phase difference as functions of the frequency detuning.
Illustration of beating death in the case f = 110 Hz. (a) Measured difference between the sound pressure level due to forcing, and the sound pressure level due to self-sustained oscillations, as a function of the driving voltage . (b) Frequency spectra corresponding to the operating points labeled (VI) and (VII) in Fig. 2(c) . Note that the peaks labeled correspond to measurement “artifacts” related to electromagnetic interference; these peaks are located at the electrical network frequency (50 Hz) and its harmonics, except for a peak at Hz.
Effect of a feedback loop. (a) Sketch of the experimental setup. (b) Steady-state acoustic pressure as a function of the assigned phase shift between the loudspeaker and the microphone signals, for different values of the voltage gain , where and refer to the voltages at the input and the output of the audio power amplifier, respectively.
Synchronization in a relaxation regime. The stack position is fixed at ds = 25 cm while the coupling distance d = 4 cm; the heater power supply Q 0 is fixed at 24.5 W so that a spontaneous and periodic onset/damping of self-sustained oscillations takes place. From the upper graph to the lower graph, the effect of external forcing (frequency f = 176.9 Hz) on the dynamics of wave amplitude evolution is investigated with an increasing loudspeaker voltage . Except for the first (no forcing) and the last graph, the loudspeaker is switched on at time t = 100 s and switched off at time t = 150 s.
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