## 5 Conclusions  
Noninvasive tuning of the mechanical resonance frequencies of suspended square drum resonators in monolithic vertical arrays was demonstrated using a simple scheme where the membrane chips were directly mounted on a piezoelectric ring actuator. The application of a piezoelectrically controlled force to the bottom chip allowed for modifying the tensile stress of the membrane resonators and thereby change their frequencies. For membranes with not too different bare mechanical frequencies, tuning to degeneracy by the application of a dc-voltage to the PZT is possible, as the bottom and top membrane experience opposite frequency shifts with the bias voltage. Dynamical actuation of both membranes was also demonstrated by the application of an ac-voltage at twice the mechanical resonance frequencies of the fundamental modes of the membranes and observing the parametric amplification of their thermal fluctuations until the parametric oscillation threshold was reached. The experimental observations are well-accounted for by a simple  
Figure 7: Energy gain as a function of pump detuning \(\delta\) for the fundamental mode of membrane B for \(\Delta/(2\pi)=250\) Hz and for two modulation voltage amplitudes corresponding to \(\epsilon=0.949\) (yellow) and \(\epsilon=0.989\) (orange). The solid lines are the theoretical predictions of Equation (15). phase-averaged subthreshold parametric oscillator model. Last, the amplification was shown to be enhanced by the simultaneous application of the dc-voltage, which results in higher amplification gains and a lowering of the parametric oscillation thresholds. Such a tuning of both the linear and nonlinear response of membrane resonators in vertical arrays is promising for exploring collective effects and investigating phonon dynamics in currently investigated optomechanical arrays [40; 41; 42; 43], or for sensing applications [58; 59; 75]. **Supplementary Materials:** The following are available at www.mdpi.com/xxx/s1, Figure S1: Average noise spectra of membrane B's fundamental mode for \(\Delta/(2\pi)=250\) Hz and different normalized modulation amplitude \(\epsilon\) (top label of each subplot). The solid lines are the results of Lorentzian fits to the data. Figure S2: Average noise spectra of membrane B's fundamental mode for \(\Delta/(2\pi)=30\) Hz and different normalized modulation amplitude \(\epsilon\) (top label of each subplot). The solid lines are the results of Lorentzian fits to the data. Figure S3: Average noise spectra of membrane B's fundamental mode for \(\Delta/(2\pi)=-100\) Hz and different normalized modulation amplitude \(\epsilon\) (top label of each subplot). The solid lines are the results of Lorentzian fits to the data. Figure S4: Average noise spectra of membrane A's fundamental mode for \(\Delta/(2\pi)=250\) Hz and different normalized modulation amplitude \(\epsilon\) (top label of each subplot). The solid lines are the results of Lorentzian fits to the data. Figure S5: Average noise spectra of membrane A's fundamental mode for \(\Delta/(2\pi)=30\) Hz and different normalized modulation amplitude \(\epsilon\) (top label of each subplot). The solid lines are the results of Lorentzian fits to the data. Figure S6: Average noise spectra of membrane A's fundamental mode for \(\Delta/(2\pi)=-100\) Hz and different normalized modulation amplitude \(\epsilon\) (top label of each subplot). The solid lines are the results of Lorentzian fits to the data. Figure S7: Average noise spectra for \(\Delta/(2\pi)=0\) Hz and different normalized modulation amplitude \(\epsilon\) (top label of each subplot). The solid lines are the results of Lorentzian fits to the data. **Author Contributions:** conceptualization, A.D.; investigation, S.N. and A.N. ; data curation, S.N. and A.N. and A.D.; writing--original draft preparation, A.D.; writing--review and editing, S.N. and A.N. ; supervision, A.D., funding acquisition, A.D.  
**Funding:** This research was funded by the Velux Foundations. **Conflicts of Interest:** The authors declare no conflict of interest.