Persistent Response in an Ultrastrongly Driven Mechanical Membrane Resonator

We study experimentally and theoretically the phenomenon of “persistent response” in ultrastrongly driven membrane resonators. The term persistent response denotes the development of a vibrating state with nearly constant amplitude over an extreme wide frequency range. We reveal the underlying mechanism by directly imaging the vibrational state using advanced optical interferometry. We argue that this state is related to the nonlinear interaction between higher-order flexural modes and higher-order overtones of the driven mode. Finally, we propose a stability diagram for the different vibrational states that the membrane can adopt.

Figure 1

Persistent response and higher-order nonlinearities. (a) Response function generated by different drive voltages recorded by imaging white light interferometrey showing the mean amplitude response (normalized to the excitation voltage and averaged over the whole membrane area) of the (1,1) mode with eigenfrequency around 321 kHz. (b) Theoretical curves for the amplitude of the (1,1) mode with different higher-order nonlinear forces.

Figure 2

Persistent response and visualization of deflection patterns. (a) Enlargement of the gray area of Fig. 1 without normalizing the $y$ axis. The plateau reveals small steps and kinks superimposed, some of them being marked by colored areas $A$, $B$, and $C$. The red arrows indicate the positions where we captured the deflection patterns for $V_{\mathrm{exc}}=4.6\mathrm{V}$ shown in the insets. The black arrow indicates the frequency at which the ringdown experiments shown in Fig. 4 are performed. (b) Numerical solutions of the nonlinear coupling model (solid lines, coupled modes; dashed lines, driven modes). Left: model interaction between the mode (1,1) and the mode (2,2). Right: model interaction between the mode (1,1) and the mode (1,2) (see text for details).

Figure 3

The subharmonic response functions of the flexural mode (2,2) [(a),(b)] and the mode (1,2) [(c),(d)]. (a) Subharmonic amplitude response of the (2,2) mode. (b) Frequency spectrum of the ringdown for $V_{\mathrm{exc}}=5.0\mathrm{V}$ and ${\omega }_d/2\pi =322.0\mathrm{kHz}$ (instantly, ${\omega }_{11}/2\pi =319\mathrm{kHz}$) showing three integer overtones of ${\omega }_d$. The drive is turned off at $t=0$. (c) Subharmonic response of the flexural mode (1,2). (d) Same as (b) but for $V_{\mathrm{exc}}=7.0\mathrm{V}$ and ${\omega }_d/2\pi =332.5\mathrm{kHz}$, showing three half-integer overtones of ${\omega }_d$. The energy decay during ringdown is shown in Sec. IV of the Supplemental Material [26].

Figure 4

Ringdown measurement in the flexural mode coupling regime. (a) Frequency spectrum of the ringdown for $V_{\mathrm{exc}}=5\mathrm{V}$ and ${\omega }_d=338.6\mathrm{kHz}$ showing six overtones of ${\omega }_d$. The drive is turned off at $t=0$. For details of the mode assignment and energy decay, see Sec. IV in the Supplemental Material [26]. (b) Stability diagram of the vibrational state of the membrane resonator as a function of the detuning from ${\omega }_{11}$ and drive $V_{\mathrm{exc}}$. The solid lines are determined by combining experimental data from MI (colored dots) and IWLI data shown in Fig. S3 in the Supplemental Material [26]. The inset shows an example of a coexistence of flexural mode coupling and spatial overtones [22]: ringdown recorded for $V_{\mathrm{exc}}=1.77\mathrm{V}$ at ${\omega }_d/2\pi =331\mathrm{kHz}$ (marked by a red circle in the main panel) showing a frequency splitting into ${\omega }_{22}$ and $2{\omega }_{11}$.