Figure 1

Schematic of the experimental apparatus. The inverted pendulum is a four-layer piezoelectric beam made by lead zirconate titanate (PSI-5A4E) 60 mm of free length, clamped at one end. The piezoelectric beam has a width of 5 mm and a thickness of 0.86 mm. The pendulum mass is a steel cylinder 140.0 mm long and with diameter of 4.0 mm, with three magnets attached (each magnetic dipole moment is $0.051\mathrm{A}{\mathrm{m}}^2$). The inverted pendulum resonance frequency (in the linear regime) is 6.67 Hz. The displacement $x$ is measured via an optical readout. The voltage signal from the piezo is measured through a load resistor $R_L$ placed in parallel. The actual magnitude of the standard deviation of the vibrational force applied in the three cases is $\sigma =3\times {10}^{-4}\mathrm{N}$, $6\times {10}^{-4}\mathrm{N}$, $12\times {10}^{-4}\mathrm{N}$. The load resistance is $R=R_L=100\mathrm{M}\Omega$ and the piezo capacitance $C=112nF$. The effective mass is $m=0.0155\mathrm{kg}$. The damping constant is $\gamma =0.016\mathrm{Hz}$.Reuse & Permissions

Figure 2

Piezoelectric oscillator mean electric power (upper panel) and position $x_{\mathrm{rms}}$ (lower panel) as a function of $\Delta$ for three different values of the noise standard deviation $\sigma$. The symbols correspond to experimental values measured from the apparatus in Fig. 1. The continuous curves have been obtained from the numerical solution of the stochastic differential equation (1). Both in the experiment and in the numerical solution, the stochastic force has the same statistical properties with correlation time $\tau =0.1\mathrm{s}$. Every data point is obtained from averaging the rms values of ten time series sampled at a frequency of 1 kHz for 200 s. The rms is computed after zero averaging the time series. The expected relative error in the numerical solution is within 10%.Reuse & Permissions

Figure 3

Inverted pendulum potential function $U(x)$. Five different plots of the potential function (2) corresponding to five different values of the parameter $\Delta$, representing the distance between the external magnet and the tip magnet (see Fig. 1) are shown (vertical scale in $J$). On decreasing $\Delta$ the potential changes from monostable to bistable.Reuse & Permissions

Figure 4

Root mean squared (rms) values of the Duffing oscillator position $x_{\mathrm{rms}}$ as a function of $a$ and the noise variance ${\sigma }^2$. The values plotted here have been obtained from the numerical solution of (4). rms is computed after zero averaging the $x(t)$. Inset: contour plot of $x_{\mathrm{rms}}$. The continuous line represents the theoretical prediction for the maximum: i.e. $x_{\mathrm{rms}}$ has a maximum when $a=\sqrt{b{\sigma }^2\tau }$. The expected relative error in the numerical solution is within 10%.Reuse & Permissions