Synchronization of Active Mechanical Oscillators by an Inertial Load

Motivated by the operation of myogenic (self-oscillatory) insect flight muscle, we study a model consisting of a large number of identical oscillatory contractile elements joined in a chain, whose end is attached to a damped mass-spring oscillator. When the inertial load is small, the serial coupling favors an antisynchronous state in which the extension of one oscillator is compensated by the contraction of another, in order to preserve the total length. However, a sufficiently massive load can synchronize the oscillators and can even induce oscillation in situations where isolated elements would be stable. The system has a complex phase diagram displaying quiescent, synchronous and antisynchronous phases, as well as an unusual asynchronous phase in which the total length of the chain oscillates at a different frequency from the individual active elements.

Figure 1

Model system: A chain of serially coupled active Hopf oscillators is attached to a damped inertial oscillator.

Figure 2

(a) Phase diagram (Q, quiescent; S, synchronous; Anti, antisynchronous; A, asynchronous) as a function of the control parameter $ϵ$ of the active elements and the frequency $\Omega$ of the inertial oscillator. Other parameters have the values $\zeta /NM\omega =0.3125$ and $\gamma /\omega =0.2$. The diagram was established by numerical solution of Eqs. (1, 3). The phase boundary of the synchronous state is given by Eq. (4) for $ϵ<0$ and can be determined by perturbation theory for $ϵ>0$. (b) Magnification of the boxed region in (a).

Figure 3

The asynchronous state. (a) Phase ${\varphi }_i=\arg (z_i)$ of each of the $N=128$ active elements as a function of time. The phase velocity ${\dot{\varphi }}_i$ of a particular element is slow for a number of periods of oscillation, then speeds up over a few periods, then slows again. The average frequency of the active elements is therefore faster than the speed of propagation of the high-density area in the phase space, which is the frequency of the inertial load. (b) Total extension $x={\Sigma }_i\mathrm{R}\mathrm{e}{z}_i$ (in units $N\sqrt{\omega /B}$) as a function of time, illustrating the lower frequency of the inertial oscillator. Parameters: $\zeta /NM\omega =0.3125$, $\gamma /\omega =0.2$, $ϵ/\omega =0.6$, and $\Omega /\omega =0.83$.

Figure 4

Frequency and amplitude of oscillations as a function of $\Omega$ for a fixed, positive value of the control parameter ($ϵ/\omega =0.6$). These parameters correspond to moving up the right vertical axis of Fig. 2a. The system transits from the synchronous (S), to the asynchronous (A), to the antisynchronous (Anti) phase. Black curves show the frequencies of the inertial load (continuous) and of the active elements (dashed). Gray curves show the rms amplitude of the mass (continuous) in units $N\sqrt{\omega /B}$ and that of each active element (dashed) in units $\sqrt{\omega /B}$. Other parameters: $\zeta /NM\omega =0.3125$, $\gamma /\omega =0.2$.