Energy Dissipation And Complex Dynamic Behavior Of Micromechanical Resonator

Micro-electro-mechanical systems (MEMS) technology is a new research field that isdeveloping rapidly in the21Stcentulry, and producing an enormous influence on several kindsof research fields. But the rapid development of MEMS technology has broughtunprecedented opportunities and challenges to the subjects of mechanical dynamics andvibration. Micromechanics resonator offers many novel applications in precisionmeasurements and provides the unprecedented opportunity for gaining insight into physicalphenomena, has become one of the focuses and hot spots of current research in this field.Micro-structures are widely used as the key components of various micromechanics sensingand actuation systems. Their relatively simple geometries make them very advantageous bothfrom a design and microfabrication point of view. In the wide range of applications rangingfrom the mean residual stress measurement, the Young’ modulus determination, microscopy,environment monitoring and real-time clock systems, mass flow sensors to bio-medical orDNA analysis, the sensing mechanism depends upon the sensitivity or response of the MEMSstructures to some applied excitation.At present, the work frequency of the micromechanical resonator can reach MHz evenGHz, the quality factor is in the range of102-105. Thus, the resonator not only has ultrahighsensitivity and resolution, but also exhibits "unstable" parameter vibration, self-excitedvibration, thermal vibration and the phenomenon of frequency drift. Working principle of themicromechanical resonator involves complex energy conversion process, and its nonlinearbehaviors are very obvious. Regarding MEMS resonators, the scaling effect will bring themicro scientific question (micro mechanics, micro optics, microelectronics, microaerodynamics, slight fever mechanics, micro chemistry and so on), the multi-energy bandcoupling will cause the multi-disciplinary overlapping questions (machine, electricity,magnetism, light, sound, thermal, chemistry and so on). In addition, in order to improve the sensitivity, signal-to-noise ratio, resolution and measurement accuracy of micromechanicalresonator, the surface/interface function, micromachining technology, the scale effect, energydissipation mechanism and the inherent nonlinear driving force have become the core anddifficulties of its research, also are the key scientific problems that urgently need to be studied.But at present, the micromechanical resonator’s energy dissipation mechanism and control,nonlinear vibration characteristics and its law of evolution and transfer are lack of in-depthunderstanding. Thus, as the rapid development of ultrahigh frequency, multi-function andultra-precision micromechanical resonator, its frequency stability, the stability of movement,the mechanism and control method of energy dissipation is an urgent subject, has importantacademic value and application potential.This paper will explore the key scientific issues related to dynamics design, analyticaland experimental methods of micromechanical resonators at trans-scale, including boundaryeffect, micromachining process error, nonlinear dynamics theories, energy loss mechanismsand its control, methods to reduce the energy loss and optimum design of the resonantstructures, explanations of the scaling effect and surface/interfacial effect.In real system applications, the support type that resembles best the behavior is selected.However, microfabrication methods and limitations can lead to boundary support conditionsfor suspended MEMS beams not rigidly clamped. Real system behavior may deviate from theidealized support conditions. The qualitative dynamical behavior of the microresonators issignificantly affected by the non-ideal boundary conditions. First, this paper studies thesupport-induced losses in generic mechanical resonators due to the tunneling of mesoscopicphonons between the CNT and its supports. After formulating the problem, the resultingdifferential equations are solved analytically using the method of multiple scales, and a closedform solution is obtained. The results reveal that the Young’ modulus, density and geometricparameters of the CNT not only influence the resonant frequency shift and the systemstiffness, but also affect the system vibration amplitude. By employing the principle of energyequivalence, rigorous theoretical solutions of the tangential and rotational equivalent stiffnessare derived based on the Boussinesq’s and Cerruti’s displacement equations. The proposedtheoretical results remained a good situation consistency with the experimental data from thepublished literatures. The non-dimensional differential partial equation of the motion, as wellas coupled boundary conditions, is solved analytically using the method of multiple timescales. The closed-form solution provides a direct insight into the relationship between the boundary conditions and vibration characteristics of the dynamic system, in which resonancefrequencies increase with the nonlinear mechanical spring effect but decrease with the effectof flexible supports. It is demonstrated that the proposed model with the flexible supportsboundary conditions has significant effect on the rigorous quantitative dynamical analysis ofthe MEMS beams. Moreover, the proposed analytical solutions are in good agreement withthose obtained from finite element analyses.The comb fingers of high aspect ratio structures fabricated by micromachiningtechnology are usually not parallel. Effects of the inclination of the fingers and edge effect onthe capacitance, driving electrostatic force and electrostatic spring constant are studied. Thecomplex nonlinear air damping in the three-dimensional (3-D) resonators is also determinedaccurately. The governing equations are presented to describe the complex dynamic problemwith taking both linear and nonlinear mechanical spring stiffness constants into account. Thedynamic responses of the micro-resonator driven by electrostatic combs are investigated usingmulti scale method. Stability analysis is presented using the maximum Lyapunov index map,and effects of vacuum pressure on the frequency tuning and stability are also discussed. Thecomparisons show that the numerical results agree well with the experimental data reported inliterature, and it verify the validity of presented dynamic model. The results also demonstratethat the inclination of the fingers causes the resonance frequency increasing and theelectrostatic spring hardening under applied dc voltage. Therefore, it can provide an effectiveapproach to balance the traditional resonance frequency decreasing and stiffness softeningfrom driving electrostatic force. The inclination of the fingers can be helpful for strengtheningthe stability of the MEMS resonators, and avoiding the occurrence of pull-in.As an inherent energy dissipation mechanism, the thermoelastic damping imposes anupper limit on the quality factors of micromechanical resonators. The dynamic properties andbehaviors of certain materials are always dependent on its internal structures, which havebeen experimentally shown to be size dependent. Based on the modified couple stress theory,the size-dependent thermoelastic damping in microbeam and microplate resonators arerespectively investigated. The governing equations of motion are derived by using Hamiltonprinciple. The analytical expression of thermoelastic damping is obtained by solving the heatdiffusion equation of the thermal flow across the microstructure. Agreement of thethermoelastic frequencies is achieved between the present results and reported values. Theresults show that the size effect has significant impact on the thermoelastic damping when the thickness of microstructure has a similar value to the material length scale parameter. It alsodemonstrates that the thermoelastic damping can be suppressed and the quality factor can beimproved as the material length scale parameter increases. The quality factor is improved byseveral orders of magnitude as the representative temperature drop from500K to80K.However, the size-dependent quality factor at400K can go beyond the quality factor at293Kwhen the plate thickness has a similar value to the material length scale parameter. Similarly,the size-dependent quality factor of PolySi0.35Ge0.65-microplate can go beyond the qualityfactor of SiC-microplate and even the value of polysilicon-microplate. In addition, the criticalmodals that the maximum damping occurs for silicon-microbeam resonantors are greater thanthose of Polysilicon ones.First, this paper discusses the origin of the optical gradient force in detail and summarizesthe rapid development in this field. The nano-waveguides resonator driven by the tunableoptical gradient force has numerous unique properties and can easily enter into the nonlinearoscillation regime where the resonance frequency will shift. In this work, a continuum elasticmodel of the opto-resonator is presented and solved analytically using the method of multiplescales. The effects of the optical gradient force on the resonance frequency and dynamicbehavior are investigated. The results theoretically figure out why and when the nonlinearbehavior of spring softening and spring hardening can occur. The proposed solutions areverified with the reported experimental results. Additionally, it is indicated that thediversification of the width, the length and the initial gap of opto-resonators have significanceinfluence not only on the optical gradient force, but also on the system stiffness, the maximumvibration amplitude, and certainly on the resonance frequency shift. The presented model canbe useful for the dynamical design and optimization of nano-waveguides resonators derivedby the optical gradient force.In addition, the complex distribution of injected optical power makes the temperaturedistribution in the optical waveguide more complex. The nature of thermoelastic damping ofthe opto-resonators would be significantly transformed. In this work, with considering thebolometric effect of injected optical power, the governing equation of opto-waveguideresonator is derived based on the theory of Euler-Bernoulli. The theory model ofthermoelastic damping is derived through solving the heat diffusion equation. The resultsshow that the injected optical power, representative temperatures, waveguide material andgeometries significantly affect the thermoelastic damping in the opto-resonator. The peak damping increases with the injected optical power. Further increase in the optical power hasalmost no effect on the increase of the intrinsic energy dissipation. As2uW optical power isinjected, the ranking of the peak damping value forSi0.35Ge0.65is increased by about five times,furthermore, the value is increased by almost fifteen times for PolySilicon. In addition, it isfound that these properties are size-dependent. When the optical power is larger than itscharacteristic value, the effect of thermoelastic damping weakens as the waveguide thicknessincreases, but strengthens as the waveguide length increases. We believe that the theory modeland results would have immediate applications in designing the high-performance opticalresonators.