Modeling of Resonators by Two DOF Systems | ASEN 5022, Study notes of Aerospace Engineering

Download Modeling of Resonators by Two DOF Systems | ASEN 5022 and more Study notes Aerospace Engineering in PDF only on Docsity! . 16 Modeling of Resonators by Two-DOF Systems 16–1 Chapter 16: MODELING OF RESONATORS BY TWO-DOF SYSTEMS 16–2 §16.1 A GENERIC RESONATOR A second example where simplified discrete modeling has been found valuable is in the assessment of the performance of micro-electro-mechanical system (MEMS) resonators. Of several available resonator models, we will consider a double-beam resonator. When the input beam which acts as a conductor is placed under alternating voltage field, the input beam vibrates with the exciting voltage frequency, possibly with all the frequency ranges of the input frequency. The motion of the input beam then triggers motion of the output beam through the link element as shown in Fig. 16.1. In doing so, the link acts as a filter and triggers the output beam to resonate with its fundamental frequency, thus extracting an almost single-tone frequency. Consequently, the double beam resonator has an inherent ”frequency filtering” capability. Su b st ra te Su b st ra te Input (Receiving) Resonator Output (Transmitting) Resonator L Ls bbs e Thinkness: h Figure 16.1: A Double Beam Resonator §16.2 ONE-DOF MODELING OF A SINGLE BEAM Consider one of the resonator with fixed-fixed ends. As we are interested in modeling the beam with one degree of freedom, we sample the displacement at the beam center and represent the motion of the beam center with an equivalent mass and stiffness. There are two ways of reducing the beam dynamics: order reduction via the finite element method and mass and stiffness lumping from the continuum beam equation. In this section we will employ the continuum beam solution and adapt it to the present task. To this end, we recall that the transverse beam displacement, w(x, t), is expressed as w(x, t) = C1(t) sinβx + C1(t) cosβx + C3(t) sinhβx + C4(t) coshβx (βL) = 4.71 (16.1) The three unknown coefficients (C2,C3,C4) can be expressed in terms of C1 by applying the bounday conditions w(0, t) = ∂w(0, t) ∂x = w(L , t) = ∂w(L , t) ∂x = 0 (16.2) 16–2 16–5 §16.4 COUPLING OF THE LINK BEAM TO THE TWO RESONATING BEAMS §16.4 COUPLING OF THE LINK BEAM TO THE TWO RESONATING BEAMS We have completed the individual modeling of the two resonator beams and the linking beam. For illustrative purposes the in-phase and out-of-phase motions of the system are shown in Figs. 16.3 and 16.4. Observe that for the case of in-phase mode the linking beam acts as a rigid link, whereas of out-of-phase mode the linking beam undergoes its fundamental mode shape. x w L x e e Link beam Input beam Output beam In-Phase Deformed State of Resonator System (m,k) (m,k) Figure 16.3: In-Phase Deformed State of Double Beam Resonator x w L e e Link beam Input beam Output beam Out-of-Phase Deformed State of Resonator System (m,k) (m,k) Figure 16.4: Out-of-Phase Deformed State of Double Beam Resonator An equivalent mass-spring model that accounts for both motions may be represented as shown in Fig. 16.5. The starting point for the development of an equivalent mass-damper-sparing model from the skeletal model is to obtain the kinetic and potential energies of the two systems: 16–5 Chapter 16: MODELING OF RESONATORS BY TWO-DOF SYSTEMS 16–6 x1 C 2 x2 2K1K C 1 2M1 M x1 1M x2 2M x4 e x3 L/2 Model TransformtionSkeletal Physical Model Equivalent Mass-Damper-Spring Model f1 f1 12K C 12 Figure 16.5: Physical and Equivalent Mass-Damper-Spring Model Skeletal Model: Ts = 12 mbẋ21 + 12 mbẋ22 + 12 [ ẋ3 ẋ4 ]T [mlink] [ ẋ3 ẋ4 ] Vs = 12 kbx21 + 12 kbx22 + 12 [ x3 x4 ]T [klink] [ x3 x4 ] (16.14) Mass-Damper-Spring Model: Tmdk = 12 M1 ẋ21 + 12 M2 ẋ22 Vmdk = 12 K1x21 + 12 K2x22 + 12 K12(x1 − x2)2 (16.15) where (mb, kb) are given by (16.7) and (mlink, klink) are given by (16.14), respectively. Comparing the energy expressions of the two models, it is clear that the degrees of freedom (x3, x4) which represent the transverse displacement at the junctions of the linking beam and the two resonator beams must be substituted by the mid-span displacements of the two beams, (x1, x2). This can be accomplished by using the assumed displacement relation for the resonator beam given by(16.5): x3 = a x1 x4 = a x2 a = N (βL , e) N (βL , e) = 8(e) 8(L/2) , βL = 4.73 8(L/2) = (sinβL/2− sinhβL/2)+ (sinβL − sinhβL) (cosβL − coshβL) (cosβL/2− coshβL/2) 8(e) = (sinβe − sinhβe)+ (sinβL − sinhβL) (cosβL − coshβL) (cosβe − coshβe) (16.16) 16–6 16–7 §16.5 A TWO-DOF MODEL AND EVALUATION OF RESONATOR PERFORMANCE Table 16.1 Link offset amount (e) vs. Link factor (a) Link Offset (e) Link Factor (a) a2 0.0625 0.00906327886493 0.00008214302378 0.1250 0.03946457838609 0.00155745294719 0.1875 0.09611178177002 0.00923747459501 0.2500 0.18424759908145 0.03394717776728 0.3125 0.30988889279730 0.09603112587914 0.3750 0.48050748799022 0.23088744601468 0.4375 0.70599928850552 0.49843499537030 0.5000 1.00000000000000 1.00000000000000 The offset amount (e) vs. the link factor (a) for representative ranges are tabulated in Table 16.1 below. Substituting (x3 = ax1, x4 = ax2)obtained in the above equation into the kinetic and potential energy expression of the skeletal model (16.14), we obtain Ts = 12 mbẋ21 + 12 mbẋ22 + 12 [ ẋ1 ẋ2 ]T a2 [mlink] [ ẋ1 ẋ2 ] Vs = 12 kbx21 + 12 kbx22 + 12 [ x1 x2 ]T a2 [klink] [ x1 x2 ] (16.17) The above equation can be simplified by using (mlink, klink) derived in (16.13) as Ts = 12 (mb + a2ms)ẋ21 + 12 (mb + a2ms)ẋ22 Vs = 12 kbx21 + 12 kbx22 + 12 a2 ks (x1 − x2)2 (16.18) Comparing (16.18) with the equivalent mass-damper-spring model (16.15), we find the following model parameters: M1 = M2 = mb + a2ms = 0.3836 (ρAL)+ 0.24638 (a2ρAs Ls) K1 = K2 = kb = 192E I L3 K12 = a2ks = a2(12E Is L3s ) (16.19) 16–7 Chapter 16: MODELING OF RESONATORS BY TWO-DOF SYSTEMS 16–10 1.07 1.08 1.09 1.1 1.11 1.12 1.13 x 10 7 10 −1 10 0 10 1 10 2 10 3 10 4 10 5 10 6 Driving Frequency (Hz) Double Beam Resonator A m pl itu de Ra tio (X 2 (ω )/ x  st ) Figure 16.7: Closer-Up view of H21(ω) Plot In more realistic high-fidelity modeling, the lower-frequency peak is a little lower than that of the higher resonance. This is because the lower peak is associated with the in-phase mode and the higher with the out-of-phase mode, which causes less loss through the substrate. This is shown in an elaborate simulation that has been correlated with experimental data as shown in Fig.16.8. It should be noted that in the high-fidelity simulation, the electrostatic effects, the interaction with the substrate which absorbs the energy as it acts and flexible ground, etc. have been modeled in an elaborate way. The high-fidelity model and its performance assessment is shown in Fig.16.9. In conclusion, a simple two-DOF model can be used to size up the design, peak frequencies, and the position to how to anchor the beams, and the amount of energy loss. It turns out that resonators can also be designed by exploiting the den Hartog invariant points, this time by maximizing its two peaks rather than by minimizing its peak. This is beyond the scope of this course and left as research topics to those interested in the resonator design. 16–10 16–11§16.5 A TWO-DOF MODEL AND EVALUATION OF RESONATOR PERFORMANCE Figure 16.8: High-fidelity model of a resonator and its correlation with experimental data 16–11 Chapter 16: MODELING OF RESONATORS BY TWO-DOF SYSTEMS 16–12 Figure 16.9: A refined model and the performance curves 16–12