Nonlinear dynamics of a rack-pinion-rack device powered by the Casimir force

Using the lateral Casimir force—a manifestation of the quantum fluctuations of the electromagnetic field between objects with corrugated surfaces—as the main force transduction mechanism, a nanomechanical device with rich dynamical behaviors is proposed. The device is made of two parallel racks that are moving in the same direction and a pinion in the middle that couples with both racks via the noncontact lateral Casimir force. The built-in frustration in the device causes it to be very sensitive and react dramatically to minute changes in the geometrical parameters and initial conditions of the system. The noncontact nature of the proposed device could help with the ubiquitous wear problem in nanoscale mechanical systems.

Figure 1

(Color online) The schematics of the rack-pinion-rack device. The racks and the pinion have sinusoidal corrugations of wavelength $\lambda$ and amplitudes $a$ and $b$, respectively. The choice of parallel rack velocities $V_R$ (rather than opposite) frustrates the system, which is only possible because of the noncontact design. The pinion velocity $V_P$ is taken as positive if it is in the direction shown, as a convention.

Figure 2

(Color online) Phase diagram of the dynamical system. The following notation is used in the labels of the different phases. 0 means that the average pinion velocity is zero, while 2s, 3s, and 5s denote the number of possible different states (outcomes) in each region depending on the initial conditions. The subscript “f” denotes no oscillations around the terminal value of (mean) velocity, while the subscript “i” denotes intermittency in the pinion velocity time series, both for the case when the average velocity is zero. The sequence of phases are ($0_{\text{f}}$, $3{\text{s}}_{\text{f}}$, 3s, 2s, 0, $0_{\text{i}}$) for $\epsilon =0.1$, ($0_{\text{f}}$, 3s, 5s, 2s, $0_{\text{i}}$, 0, $0_{\text{i}}$) for $\epsilon =0.2$, ($0_{\text{f}}$, 0, $0_{\text{i}}$, $3{\text{s}}_{\text{i}}$, 2s) for $\epsilon =0.5$, and ($0_{\text{f}}$, 0, $0_{\text{i}}$, 2s) for $\epsilon =1.0$.

Figure 3

(Color online) Various types of signals observed for $v\left(t\right)$ for different values of $f$ and $\epsilon$. (a) corresponds to $f=0.1$ and $\epsilon =0.1$, which belongs to the domain denoted as $0_{\text{f}}$ in Fig. 2, where the pinion velocity vanishes in the long-time limit. (b) and (c) correspond to $f=1$ and $\epsilon =0.1$ and represent the cases where the pinion is locked onto one of the racks and the velocity oscillates about $+V_R$ or $-V_R$. (d) corresponds to $f=2$ and $\epsilon =0.1$, which belongs to the domain denoted as 0 in Fig. 2, and shows a case where the pinion velocity oscillates about zero. (e) corresponds to $f=2$ and $\epsilon =0.2$, which belongs to the domain denoted as $0_{\text{i}}$ in Fig. 2, and shows a case where the pinion velocity oscillates in an intermittent manner about zero. (f) and (g) correspond to $f=0.6$ and $\epsilon =0.2$, which belongs to the domain denoted as 5s in Fig. 2, and represent the cases where the pinion average velocity could be $-0.33V_R$ or $+0.33V_R$, respectively.

Figure 4

(Color online) In many of the regions in the phase diagram of Fig. 2, different initial conditions lead to different average velocities. $x_0$ is between $-\lambda$ and $\lambda$ (20 points) and $v_0$ is between $-3V_R$ and $3V_R$ (20 points). Top: $f=1.5$ and $\epsilon =0.1$ (2s), white means $V_P=V_R$, and black means $V_P=-V_R$. Middle: $f=0.3$ and $\epsilon =0.1$ $\left(3{\text{s}}_{\text{f}}\right)$, red means $V_P=0$, white means $V_P=V_R$, and black means $V_P=-V_R$. Bottom: $f=0.6$ and $\epsilon =0.2$ (5s), white means $V_P=0$, black means $V_P=V_R$, gray means $V_P=-V_R$, and red means $V_P=0.33V_R$ if $v_0>0$ and $V_P=-0.33V_R$ if $v_0<0$.

Figure 5

(Color online) (a) The intermittent velocity signal for $f=2$ and $\epsilon =0.2$ (within the $0_{\text{i}}$ phase) corresponding to $x_0=3\lambda /\left(2\pi \right)$ and $v_0=3V_R$. (b) shows a portion of the time series, indicating large correlation times. In (c), the correlation time $t_c$ is calculated for the section of the time series between $0.75\times 20000$ and $20000$ time steps. In (d), the histogram of correlation times $N\left(t_c\right)$ is plotted vs the correlation time in a log-log scale. The line that goes through the data has a slope of $-2.7$ and is a guide to the eyes.