Frequency tuning due to electrostatic effect, nonlinear mid-plane stretching, thermomechanical effect, etc. are some of the main techniques used to extend the operating frequency range of these devices. On the other hand, electrostatic effects due to the application of a large DC voltage cause softening of one mode and hardening of the other mode. With a sufficiently large DC voltage, modal coupling of the two modes of the beam is achieved.
It has been shown that the coupling strength of the modes can be controlled by effectively changing the differential gaps between the beam and the side electrodes. Here, to obtain the dynamic behavior of the device, we apply multi-scale method to the original modal equations. So controlling these devices in the non-linear region is equally important from the application point of view as well as performance improvement of the system.
Thus, the collective thermal and electrostatic effects may be of greater importance to obtain the dual benefits of increasing the operating range and sensitivity of the devices through multiple modal interactions.
Literature Survey
Using the temperature dependence of beam stress, Ashok et al. [4] are able to use the concept of frequency change with temperature to propose an AuPd temperature sensor. Furthermore, as the dimensions of the devices are reduced to a smaller scale, they exhibit nonlinearities that significantly change the dynamics of the device. The stretching of the neutral axis due to fixed boundary conditions is taken into account in the nonlinear governing equation.
They electrostatically tuned the frequencies of the device to obtain a two-to-one internal resonance condition between the torsional and bending modes of vibration. They showed that nonlinear dissipation plays an important role along with nonlinear elastic effects on the device dynamics. In these cases, multiple coupling between different modes can be obtained, resulting in a modified dynamic behavior of the system and a more complex energy transfer process.
They have analyzed the nonlinear dynamics of the system using the perturbation technique considering the cubic nonlinearity in the restoring forces.
Outline of the thesis
Until the later stage, both linear and non-linear results for the beam array are discussed. We derive the governing equations using elastic beam theory considering large deformations in the structure. Take the deflection of the beam along in-plane and out-plane as y(x, t) and z(x, t), respectively, as shown in Fig.
Second, we consider the contribution of the inhomogeneous electric field due to beam deflection in the z direction [8]. In the above expressions, k1 represents the contribution of the net effective change in area due to the deflection toz. However, it is assumed to be negligible in the present study. k2 represents the force strength due to the inhomogeneous electric field created due to the z deflection, which turns out to be very effective in the present test case. the difference in DC voltage applied between the beam and the electrodes.
The frequency variations in both directions are obtained from the reduced order model of the single beam system using the Rayleigh-Ritz-Galerkin (RRG) procedure.
Frequency analysis based on linear equations
The variations of frequencies in both directions are obtained from the reduced model of the single beam system using the Rayleigh-Ritz-Galerkin (RRG) procedure. ratio, we obtain the final reduced form. Considering only one condition in the analysis, the static and dynamic deflections of the beam along both planes are assumed to be, . 2.21) Here φ(x) is the first normalized linear undamped mode shape of the fixed fixed beam and P1(t), P2(t) are the non-dimensional modal coordinates. A1, A2 are the static deflections of the beam in two orthogonal directions due to DC bias.
These two equations above show the coupling of static deflections A1 and A2 through the extension of the beam in two directions with the magnitudes α1 and α2. If we solve these equations simultaneously, we get static deflections of the beam corresponding to a fixed applied voltage. At a certain value of DC voltage, the electrostatic attraction exceeds the elastic restoring force, resulting in collapse of the structure.
In matrix form, the modal equations are given by, . where c12 and c21 are coupling terms in the two modal directions and λ1, λ2 are the uncoupled natural frequencies of the beam in both directions given by,.
Nonlinear response of dynamic equation
Numerical solution (based on rk5 or ode45)
Solution based on Method of Multiple Scales
By substituting equations into equations (2.39) and (2.40), and then separating different powers up to third order, we get the following three sets of equations. 2.50) These three sets of coupled homogeneous/inhomogeneous equations are solved separately and the final modal displacement solutions are obtained using equation (2.45). Now the two second-order homogeneous coupled equations given by (2.48) are solved to find the values of x11. A1(T1, T2) and A2(T1, T2) represent both the displacement and phase of the oscillation and are later expressed in polar form.
To obtain a solution for (2.49), we substitute (2.51) into it and, taking into account the solvability conditions, eliminate the secular terms that appear on the right-hand side of the equations. Here an and βn are real functions of time, i.e. A Finally, we convert these above non-autonomous equations into autonomous forms by defining two more new variables. To determine the periodic response of the beam, we solve the equation for the equilibrium solution by setting the time derivatives to zero.
Since the control parameterσ1 is varied, we find the roots of these equations a1, θ1, a2 and θ2 using the Newton-Raphson numerical technique.
Stability Analysis
Results and discussion
Linear frequency analysis
The Jacobian matrix of the modulation equations used to determine the coefficients Ri is given in the appendix. It is found that the strength of modal coupling of the two modes is negligible when r1 = 1, but its strength increases as r1 increases up to 1.3. But if one gap is made sufficiently larger than the other, then only one side electrode affects the beam similar to Kozinsky experiment([12]).
The figure.(2.4)(e) presents the experimental evidence of such a coupling together with the analytical results for r1 = 1.24 when the beam and the bottom electrode are supplied with voltage V and the two side electrodes are grounded.
Numerical solution (based on rk5 or ode45)
Solution based on Method of Multiple Scales
Linear frequency response of a single beam using multi-scale method for VDC =76V, VAC =0.03 V and (c),(d) presents a non-linear frequency response near the coupling region using MMS, showing stable and unstable areas are shown at VDC =76V,VAC =0.06V In this section we extend the analysis of a single beam system to a three beam array. For the analysis of the three-beam system, we consider all beams of length L, width B and thickness H and they are attached at both ends.
Beams are separated from each other and side electrodes by side gap afgn and all beams are separated from the bottom electrode by gapdas shown in fig.(3.1). For simplicity, we normalize the distance between each beam and its neighbors by the first beam-electrode separation g0, giving rn = ggn. The motion of each beam is considered to be in two orthogonal directions in the three-beam array as in the case of a single beam.
Due to the symmetry of the system, the forcings on the first and third beams are similar, but different on the middle beam. Now, neglecting the boundary effects and considering the forces due to the inhomogeneous electric field, the governing equations of motion for the system are written as,. The actual stress N can be expressed as N =N0+N0, where N0 is the prestress in the fabricated device corresponding to the reference temperature, and N0 is the additional stress induced in the beam due to the differential thermal contraction of the beam and substrate [4]. .
E and Ex are Young's modulus of elasticity for all beams in two orthogonal directions, respectively. Iz and Iy are the moment of inertia of the cross-section for y and z deflections, respectively. These two quantities are the same for all three beams due to their similar geometric dimensions.
After the forcing expressions for a single beam, the forcing terms Qy and Qz for all three beams can be written as follows. We make Eqn non-dimensional by substituting the following new variables, as indicated by hat in the governing equations.
Frequency analysis for N = 3 beam
Where cj's are the coupling terms and λij's are the undisturbed natural frequencies of the array of rays given below. 3.49). By appropriately varying the spacing between beams and applying DC bias, we obtain multiple modal coupling in the array.
Results and discussions
With the parameters as given in Table 3.1 and Table 3.2, the variation of two modal frequencies for each beam in the array subjected to different DC bias at a fixed temperature distribution is calculated and compared both experimentally and analytically in Figure.( 3.4). . Figure.(3.5) also presents different modal coupling regions that can be controlled by choosing the right gaps between beams. But each individual beam behaves like that of single beam discussed in the previous section.
We consider that the movement of each beam is influenced by the neighboring beams or electrodes in the plane direction and the lower electrode in the other direction. If we consider the motion of each beam in two perpendicular directions in the array, we get a total of 2N governing equations. We nondimensionalize equation (4.1) and (4.2) by substituting the following new variables as denoted by hat in the governing equations.
In the above expression, i and o denote two different states, e.g. in-plane and out-of-plane modes. The thickness of all the beams in the array is assumed to vary between 2-4µm. The produced beams are separated from their adjacent beams by different gap gi,i= 0, 1, .,N, which are taken in the range from 4.0 µm to 6.0 µm.
We tune the frequencies of the beams electrostatically, by applying potential differences between beams and electrodes and exciting them in two mutually perpendicular directions, namely in-plane and out-of-plane. By varying the voltages in the beams, we again obtain wide natural frequencies of a beam. Thus, by controlling the gaps and induced voltages in the beams, one can achieve a desired frequency.
After performing a frequency analysis based on linear dynamic equations, we study the nonlinear dynamic behavior of the single-beam system near and away from the coupling region. We apply the method of multiple scales to the modal equations to find the dynamic response of a single beam. Kenny, Verification of a phase noise model for MEMS oscillators operating in the nonlinear regime, Conference on Solid State Sensors, Actuators and Microsystems, Transducer.
El-Bassiouny, Three-to-one internal resonance in the nonlinear oscillation of ShallowArch, Physica Scripta.
