This section reports the experimental measurement of extensive and sustained homoclinic chaotic response obtained for the first time from fabricated cc-beams resonators in the range of the MHz. The procedure of chaotic behavior seeking requires the application of specific biasing conditions for each resonator under test. In this case, the procedure with the specific resonator C4(17)-k15 is reported.
For a bias voltage beyond the lower bias boundary value, once the bistable region has been reached, the analytic equation (6.2) provides no real frequency; however numerical
sim-ulations of the complete system for a frequency sweep provide a new resonance frequency which grows again with the applied bias voltage until the upper bias boundary value (Vpiw), from which the collapse of the beam caused by the pull-in effect would occur. The growth of the resonance frequency in the bistable region has been also experimentally observed (as-suming the pull-in danger when the frequency sweep makes the system undergo thought its resonance frequency at so high bias voltages), and compared with the numerically obtained one in figure 6-18)a). The experimental frequencies start to grow with the bias voltage for a DC value smaller than the analytically expected value from equation (6.2)(Vpi0); the assumption that the linear stiffness (either in mechanical and electrical domain) dominates in the frequency over the other nonlinear stiffness terms becomes non acceptable when the linear stiffness becomes suffciently small. The result is that the system steps into the bistable region and the corresponding homoclinic structure arises for lower DC values than theoretically expected. This effect can also be seen in the experimental coupling bifurcation diagram, figure 6-18b).
From the conducted experiment for bistable behavior of the system, referred in previous section and effectuated with the specific resonator under test (figure 6-18), it is known that for a bias voltage greater than approxVDC = 121.5V the system exhibit a two-well potential distribution and, with this condition, it is susceptible to provide chaotic response.
Figure 6-18: a) Experimental and numerical resonance frequency as a function of the bias voltage, and b) experimental bistability and numerical pitchfork bifurcation. Both plots refer data obtained under a temperature of 120◦C.
The fact that the homoclinic structure appears for a lower bias voltage than the ana-lytically predicted one prevents the correct application of the formulated procedure of the Melnikov method. A further correction of the Melnikov method taking into account this situation would be needed. The experimental output is a voltage signal. In the experimen-tal procedure the Setup#1 is used in order to minimize the parasitic capacitive current. To find the generation of chaotic behavior, the bias voltage is set in a value which implies the two-potential well distribution, and then the nonlinear and chaotic behavior are tracked by sweeping the excitation amplitude and frequency. When the proper excitation frequency is found, the presence of nonlinear effects impose their influence and by increasing the excita-tion amplitude the system eventually exhibits extensive chaotic behavior (figures 6-21, and 6-23). The nonlinear behavior corresponding to the period doubling bifurcation has been also measured and reported in figures 6-19 and 6-20.
Experimental chaos is found to be more evasive than in numerical simulations, but finally it was obtained. Another control variable that can also be modified during the chaotic behavior search process is VD1. All the experimental measurements of period doubling bifurcation and chaotic response have been performed under an operating temperature of 120◦C. Besides the signal time series, the obtained chaotic behavior is also represented with its Poincare map (figures 6-21b), and 6-23b)), showing a typical shape of the chaotic series, which may be compared with the numerically obtained one, in figure 5-12. All the experimentally obtained signals are filtered in order to remove the high frequency noise.
After this filtering, the maximal Lyapunov exponent of the time series depicted in figures 6-21a), and 6-23a) have been calculated by means of the algorithm introduced by Wolf et.al in [77] and successively improved until the implementation of the latest version for the Matlab environment in 2016 (see section 2.5.3). The positive and finite value of the maximal Lyapunov exponent given by the Wolf algorithm, and shown in figures 6-22 and 6-24 corroborates the presence of extensive chaotic essence in the measured time series.
Finally, figure 6-25 shows the lab experimental setup, in a moment where a chaotic response was experimentally measured.
Figure 6-19: Experimental time series of the system showing nonlinear behavior and period doubling bifurcation. The parameter values are a) (up) VMEMS = 127 V, VD1 = 2.9 V, fexc = 1.387 MHz and VAC = 15.05 dBm (bottom) VMEMS = 127(V), VD1 = 4 (V), fexc= 1.38 MHz andVAC = 18.3 dBm and b) (up) VMEMS= 127 V, VD1= 4 V,fexc = 1.42 MHz and VAC = 15.05 dBm (bottom) VMEMS = 127(V), VD1 = 2.8 (V), fexc = 1.38 MHz and VAC = 15.05 dBm.
Figure 6-20: Experimental time series of the system showing nonlinear behavior and period doubling bifurcation. The parameter values are a) (up) VMEMS = 126 V, VD1 = 4 V, fexc = 1.415 MHz and VAC = 15.05 dBm (bottom) VMEMS = 128(V), VD1 = 4 (V), fexc = 1.43 MHz and VAC = 15.05 dBm and b) VMEMS = 127 V, VD1 = 4 V, fexc = 1.386 MHz and VAC = 15.05 dBm.
Figure 6-21: Experimental chaotic time series a) and Poincare map b) forVMEMS= 128 V, VD1 = 3.24 V, an excitation amplitude of 15.05 dBm and a driving frequency of 1.39 MHz
Figure 6-22: Maximal lyapunov exponent, numerically obtained with Wolf algorithm from the chaotic time series depiced in figure 6-21.
Figure 6-23: Experimental chaotic time series a) and Poincare map b) forVMEMS= 128 V, VD1 = 3.72 V, an excitation amplitude of 15.05 dBm and a driving frequency of 1.39 MHz.
Figure 6-24: Maximal lyapunov exponent, numerically obtained with Wolf algorithm from the chaotic time series depiced in figure 6-23 .
Figure 6-25: Photograph of the lab facilities, and of the experimental detection of chaotic behavior in MEMS resonator.
6.5 Discussion and conclusions
The first conclusions of this chapter is the verification of the hypothesis formulated in previous chapters, especially in chapter 5. The nonlinear model based on the finite difference method from a near real deflection profile and taking into account the second order nonlinear effects has been experimentally proved to be accurate enough. To reach this conclusion, the experimental variation of resonance frequency with the bias voltage and temperature has been observed to have the shape predicted by the model equations, and the experimental values of the second order constants have been seen to keep respectively similar enough
values between them. The experimental measurements have been performed on devices fabricated in two different generations (Run 2015 and 2017 sets). The slight differences between the values of the constants of the second-order non-linear effects can be explained from the tolerances of the nominal fabrication process and chemical attack for the release of the structure. However, in the specific case of the residual stress, as stated in chapter 5, its variation has a great influence on the results and produces a large divergence on the bias voltage values needed to obtain two-well potential distribution.
On the other hand, the second order parameter values of the tungsten resonators have been experimentally obtained. The measured average value of the tungsten residual stress disables the achievement of the bistability at reasonable values of temperature and/or bias voltage.
The bistable behavior reported in this chapter has been experimentally measured for the first time in straight and non axially forced cc-beam resonators. The measured bistable behaviors have been experimentally obtained for the bias voltage predicted by the nonlinear model (as function of the respective values of the second order effects constants) in several fabricated devices. The homoclinic structure has been found to arise for a bias voltage slightly lower than the predicted one, because in practice the resonance frequency does not reach the zero value with the growing bias voltage, but starts to grow again from a small but greater than zero value. Furthermore, from the situation of bistability, generated in a fabricated device, extensive chaotic motion, based on the homoclinic structure proper of the achieved two-well potential distribution, has been experimentally measured for the first time in a cc-beam resonator providing, in addition, important improvements in terms of frequency (MHz range) and bandwidth with respect to the previous experimental works reported in literature. The chaotic nature of the measured signals have been verified by means of a numerically obtained positive and finite value of the maximal Lyapunov exponent from a trustworthy algorithm.