Thermoelastic damping in torsion microresonators with coupling effect between torsion and bending

Predicting thermoelastic damping (TED) is crucial in the design of high Q MEMS resonators. In the past, there have been few works on analytical modeling of thermoelastic damping in torsion microresonators. This could be related to the assumption of pure torsional mode for the supporting beams in the torsion devices. The pure torsional modes of rectangular supporting beams involve no local volume change, and therefore, they do not suffer any thermoelastic loss. However, the coupled motion of torsion and bending usually exists in the torsion microresonator when it is not excited by pure torque. The bending component of the coupled motion causes flexural vibrations of supporting beams which may result in significant thermoelastic damping for the microresonator. This paper presents an analytical model for thermoelastic damping in torsion microresonators with the coupling effect between torsion and bending. The theory derives a dynamic model for torsion microresonators considering the coupling effect, and approximates the thermoelastic damping by assuming the energy loss to occur only in supporting beams of flexural vibrations. The thermoelastic damping obtained by the present model is compared to the measured internal friction of single paddle oscillators. It is found that thermoelastic damping contributes significantly to internal friction for the case of the higher modes at room temperature. The present model is validated by comparing its results with the finite-element method (FEM) solutions. The effects of structural dimensions and other parameters on thermoelastic damping are investigated for the representative case of torsion microresonators.     

Thermoelastic vibrations in micro-/nano-scale beam resonators with voids

In this paper the closed form expressions for the transverse vibrations of a homogenous isotropic, thermoelastic thin beam with voids, based on Euler-Bernoulli theory have been derived. The effects of voids, relaxation times, thermomechanical coupling, surface conditions and beam dimensions on energy dissipation induced by thermoelastic damping in (micro-electromechanical systems) MEMS/(nano-electromechanical systems) NEMS resonators are investigated for beams under clamped and simply supported conditions. Analytical expressions for deflection, temperature change, frequency shifts and thermoelastic damping in the beam have been derived. Some numerical results with the help of MATLAB programming software in case of magnesium like material have also been presented. The computer simulated results in respect of damping factor and frequency shift have been presented graphically.     

Thermoelastic damping in rectangular and circular microplate resonators

Predicting thermoelastic damping (TED) is crucial in the design of high Q micro-electro-mechanical systems (MEMS) resonators. In the past, some analytical models have been developed for TED in microbeam resonators. Rectangular and circular microplates are also common elements in many micro-resonators. Two analytical models have been developed for TED in the contour-mode vibration and the out-of-plane vibration of circular microplates, respectively. However, there is lack of works that model the TED in the out-of-plane vibration of rectangular microplates. This paper presents an analytical model for the TED in the fully clamped and simply supported rectangular microplates. The quality factor is found by calculating the energy dissipated per cycle of vibration over the volume of the microplate. The derivation in this paper shows that the model for the TED in the fully clamped and simply supported rectangular plates is the same as the model for the TED in the fully clamped and simply supported circular plates. For the rectangular microplates with other boundary conditions, based on Rayleigh's method, this paper presents a set of analytical approximate models to estimate the TED in the rectangular microplates vibrating in the fundamental mode. The present model is validated by comparison with previously reported model and the FEM model.     

Thermoelastic damping of the axisymmetric vibration of circular plate resonators

Thermoelastic damping is recognized as a significant loss mechanism at room temperature in micro-scale circular plate resonators. In this paper, the governing equations of coupled thermoelastic problems are established for axisymmetric out-of-plane vibration of circular plate. Then the analytical expression for thermoelastic damping is obtained. The effects of environmental temperature, plate dimensions and boundary conditions on the thermoelastic damping are studied.     

Analysis of thermoelastic dissipation in circular micro-plate resonators using the generalized thermoelasticity theory of dual-phase-lagging model

This study investigates the thermoelastic dissipation of micro-plate resonators by using the generalized thermoelasticity theory of dual-phase-lagging model. Explicit formulae of thermoelastic damping and frequency shift are derived. Influences of the plate thickness and vibration frequency on the thermoelastic damping are examined. Phenomena distinct from those of classical theory are observed in the numerical results of thermoelastic damping in micro-plate resonators. These results may bring new insights into the study of thermoelastic damping at submicrometer or nanometer scale.     

Thermoelastic damping in thin microrings with two-dimensional heat conduction

Accurate determination of thermoelastic damping (TED) is very challenging in the design of micro-resonators. Microrings are widely used in many micro-resonators. In the past, to model the TED effect on the microrings, some analytical models have been developed. However, in the previous works, the heat conduction within the microring is modeled by using the one-dimensional approach. The governing equation for heat conduction is solved only for the one-dimensional heat conduction along the radial thickness of the microring. This paper presents a simple analytical model for TED in microrings. The two-dimensional heat conduction over the thermoelastic temperature gradients along the radial thickness and the circumferential direction are considered in the present model. A two-dimensional heat conduction equation is developed. The solution of the equation is represented by the product of an assumed sine series along the radial thickness and an assumed trigonometric series along the circumferential direction. The analytical results obtained by the present 2-D model show a good agreement with the numerical (FEM) results. The limitations of the previous 1-D model are assessed.     

Geometric effects on thermoelastic damping in MEMS resonators

The effects of geometry on the energy dissipation induced by thermoelastic damping in MEMS resonators are investigated numerically using a finite element formulation. The perturbation analysis is applied to derive a linear eigenvalue equation for the exponentially decaying rate of the mechanical oscillation. The analysis also involves a Fourier method that reduces the dimensionality of the problem and considerably improves the computational efficiency. The method is first validated by comparing the two-dimensional model to the existing analytical solutions for a simply supported beam system, and then it is extended to a three-dimensional axisymmetric geometry to obtain the energy loss as a function of the geometric parameters in a silicon ring resonator. The computational results reveal that there is a peak value for the resonant frequency when the radial width of the ring varies. In addition, the quality factor (Q-factor) decreases with the radial width as a monotonic function.     

Damping in micro-scale generalized thermoelastic circular plate resonators

The out-of-plane vibrations of a generalized thermoelastic circular plate are studied under different environmental temperature, plate dimensions and boundary conditions. The analytical expressions for thermoelastic damping of vibration and phase velocity of circumferential surface wave modes are obtained. It is noticed that the damping of vibrations and phase velocities of circumferential surface wave modes significantly depend on thermal relaxation time in addition to thermoelastic coupling in circular plates under resonance conditions. The surface conditions also impose significant effects on the vibrations of such resonators. The expressions for displacement and temperature fields in the plate resonator are also derived and obtained. Some numerical results have also been presented for illustration purpose in case of silicon material plate.     

Analysis of Frequency Spectrum of Laser-Induced Vibration of Microbeam Resonators

The vibration phenomenon during pulsed laser heating of micro-beams is investigated. The beam is made of silicon and is heated by a laser pulse with a non-Gaussian temporal profile and with an ultrashort pulse duration of 2ps, which incites vibration due to the thermoelastic damping effect. This coupled thermoelastic problem is solved using an analytical-numerical technique based on the Laplace transformation. The damping ratio and resonant frequency shift ratio of beams due to the air damping effect and the thermoelastic damping effect are also examined and discussed.     

Nonlinear longitudinal waves in the presence of the interaction between the strain and temperature fields and the field of nonequilibrium atomic defects

The propagation of nonlinear longitudinal waves in a plate is studied by taking into account the interaction of the longitudinal displacement component with the temperature field and the field of concentration of nonequilibrium atomic point defects. A nonlinear evolution equation is derived for describing the self-consistent thermoelastic longitudinal strain fields. It is shown that the thermoelastic effect on the strain waves manifests itself in the appearance of dissipative terms, which describe the heat transfer and the thermoelastic interaction caused by the strain-induced heat release due to the recombination of nonequilibrium atomic defects. The soliton solutions to the evolution equation are investigated, and the characteristic features of their damping are considered with allowance for the low-frequency and high-frequency losses.     

Mode-dependent mechanical losses in disc resonators

Mechanical spectroscopy gives information on the structure of solids and their relaxation mechanisms through the measurements of the elastic constants and the mechanical loss angle of materials. One common way to estimate these quantities is the resonant method where the frequency and the characteristic decay time of oscillations are measured. Since many solid materials can be easily found in the shape of thin disc we have investigated the mechanical loss of these resonators and we have found experimentally that the loss angle dependence on the mode is not trivial but rather follow a distribution of modes into families. We give a model that is able to justify the existence of these families and to predict the level of losses in silicon, silica and brass discs. The model considers the thermoelastic effect and the excess damping caused by the condition of the disc edge. The results of this research are relevant to the research on thin films that are deposited on thin discs like the optical coatings used on the mirrors for the gravitational wave detectors.     

Electrical and piezoresistive properties of ion beam deposited DLC films

In present study diamond like carbon (DLC) films were deposited by closed drift ion source from the acetylene gas. The electrical and piezoresistive properties of ion beam synthesized DLC films were investigated. Diode-like current–voltage characteristics were observed both for DLC/nSi and DLC/pSi heterostructures. This fact was explained by high density of the irradiation-induced defects at the DLC/Si interface. Ohmic conductivity was observed for DLC/nSi heterostructure and metal/DLC/metal structure at low electric fields. At higher electric fields forward current transport was explained by Schottky emission and Poole–Frenkel emission for the DLC/nSi heterostructures and by Schottky emission and/or space charge limited currents for the DLC/pSi heterostructures. Strong dependence of the diamond like carbon film resistivity on temperature has been observed. Variable range hopping current transport mechanism at low electric field was revealed. Diamond like carbon piezoresistive elements with a gauge factor in 12–19 range were fabricated.     

Piezoelectric resonators with mechanical damping and resistance in current conduction

A novel design method for high Q piezoelectric resonators was presented and proposed using the 3-D equations of linear piezoelectricity with quasi-electrostatic approximation which include losses attributed to mechanical damping in solid and resistance in current conduction. There is currently no finite element sofware for estimating the Q of a resonator without apriori assumptions of the resonator impedance or damping. There is a necessity for better and more realistic modeling of resonators and filters due to miniaturization and the rapid advances in frequency ranges in telecommunication. We presented new three-dimensional finite element models of quartz and barium titanate resonators with mechanical damping and resistance in current conduction. Lee, Liu and Ballato’s 3-D equations of linear piezoelectricity with quasi-electrostatic approximation which include losses attributed to mechanical damping in solid and resistance in current conduction were formulated in a weak form and implemented in COMSOL. The resulting finite element model could predict the Q and other electrical parameters for any piezoelectric resonator without apriori assumptions of damping or resistance. Forced and free vibration analyses were performed and the results for the Q and other electrical parameters were obtained. Comparisons of the Q and other electrical parameters obtained from the free vibration analysis with their corresponding values from the forced vibration analysis were found to be in excellent agreement. Hence, the frequency spectra obtained from the free vibration analysis could be used for designing high Q resonators. Results for quartz thickness shear AT-cut and SC-cut resonators and thickness stretch poled barium titanate resonators were presented. An unexpected benefit of the model was the prediction of resonator Q with energy losses via the mounting supports.     

Thermoelastic damping in micro-scale circular plate resonators

The governing equations of coupled thermoelastic problems are established for out-of-plane vibration of a circular plate. The analytical expression for thermoelastic damping is obtained. Then the thermoelastic damping is studied under different environmental temperature, plate dimensions and boundary conditions.     

单晶金刚石微杯形谐振子的激光刻蚀制造方法

基于激光刻蚀技术,提出了一种新型单晶金刚石微杯形谐振子的三维制造方法。由于高弹性模量、低热弹性阻尼以及较大的声波传递速度等优异的性质,单晶金刚石是一种理想的谐振子加工材料,但是金刚石硬度大且化学性能稳定,难以采用传统的加工方法成形加工。激光刻蚀因精度高,加工结构对称性好以及破损率低等特点,是一种较好的单晶金刚石加工方法。采用紫外激光加工机,研究了不同激光参数对单晶金刚石刻蚀质量的影响,通过合理设计微杯形谐振结构,规划激光刻蚀加工的轨迹,采用优化后激光刻蚀参数,实现了结构对称性较好的微杯形谐振子的加工,有望应用于高性能微杯形谐振陀螺。     

高功率激光器腔镜热变形对输出光束质量的影响

利用有限元分析法结合Fox-Li迭代法,考虑腔内本征模式与腔镜热形变的相互耦合作用,计算模拟了正支共焦非稳腔的本征模式分布,定量分析了高功率激光器腔镜热变形对输出光束质量的影响,重点讨论了腔镜热变形所引起的腔内本征模式相位特性的变化,并从波前功率谱密度、Zernike像差系数及光束质量值等角度对腔镜发生热形变前后的激光器输出光束的光束特性进行比较分析。研究结果表明:高功率激光器腔镜热形变对输出光束的光束质量会产生一定的影响,且随着激光输出功率的增大,镜面热形变引起的输出光束波前相位高频比例及Zernike高阶像差均会有所增大,波前畸变程度也明显变大,光束质量逐渐变差。     

Efficient prediction of the quality factors of micromechanical resonators

A high quality factor (Q-factor) is one of the major requirements of high-performance resonators. An understanding of the dissipation mechanism is crucial for maximizing the quality factor by reducing the energy loss. Thermoelastic damping has been well-known as the important intrinsic dissipation that affects the quality factor of micro-resonators. In this study, a finite element formulation based on the weak form of fully coupled thermoelastic problems is suggested. The coupled thermoelastic equation usually leads to a large-size complex eigenvalue problem, which is very massive and time-consuming to solve. Therefore, we also applied the model order reduction (MOR) scheme to this coupled multiphysical problem in order to achieve computational efficiency. The present approach is validated by comparing the numerical results and analytical solutions.     

Dynamics of microscale pipes containing internal fluid flow: Damping, frequency shift, and stability

This paper initiates the theoretical analysis of microscale resonators containing internal flow, modelled here as microfabricated pipes conveying fluid, and investigates the effects of flow velocity on damping, stability, and frequency shift. The analysis is conducted within the context of classical continuum mechanics, and the effects of structural dissipation (including thermoelastic damping in hollow beams), boundary conditions, geometry, and flow velocity on vibrations are discussed. A scaling analysis suggests that slender elastomeric micropipes are susceptible to instability by divergence (buckling) and flutter at relatively low flow velocities of ∼10 m/s.     

Thermoelastic damping effect on in-extensional vibration of rotating thin ring

Sensitive devices such as resonant sensors and radio frequency micro-electro-mechanical system (RF-MEMS) filters etc., require high Quality factors (Q-factors) defined as the ratio of total system energy to dissipation that occurs due to various damping mechanisms. Also, thermoelastic damping is considered to be one of the most important factors to elicit energy dissipation due to the irreversible heat flow of oscillating structures in the micro scales. In this study, the Q-factor for thermoelastic damping is investigated in rotating thin rings with in-plane vibration. First, in order to obtain the temperature profile of the model, a heat conduction equation for the thermal flow across the radial direction is solved based on the bending approximation so-called in-extensional approximation of the ring. Using the temperature distribution coupled with a displacement, a governing equation of the ring model can then be derived. Eventually, an eigen-value analysis is performed to obtain the natural frequency of rotating thin rings, and the analytical and numerical values of Q-factors can then be determined by the definition. Furthermore, the effects of rotating speed, dimensions of the ring, mode numbers and ambient temperatures on the Q-factor are discussed in detail.     

Vibration analysis of a partially treated multi-layer beam with magnetorheological fluid

Semi-active vibration control based on magnetorheological (MR) materials offers excellent potential for high bandwidth control through rapid variations in the rheological properties of the fluid under varying magnetic field. Such fluids may be conveniently applied to partial or more critical components of a large structure to realize more efficient and compact vibration control mechanism with variable damping. This study investigates the properties and vibration responses of a partially treated multi-layer MR fluid beam. The governing equations of a partially treated multi-layered MR beam are formulated using finite element method and Ritz formulation. The validity of the proposed finite element formulations is demonstrated by comparing the results with those obtained from the Ritz formulation and the experimental measurements. The properties of different configurations of a partially treated MR-fluid beam are evaluated to investigate the influences of the location and length of the MR-fluid for different boundary conditions. The properties in terms of natural frequencies and loss factors corresponding to various modes are evaluated under different magnetic field intensities and compared with those of the fully treated beams. The effect of location of the fluid treatment on deflection mode shapes is also investigated. The forced vibration responses of the various configurations of partially treated MR sandwich beam are also evaluated under harmonic force excitations. The results suggest that the natural frequencies and transverse displacement response of the partially treated MR beams are strongly influenced not only by the intensity of the applied magnetic field, but also by the location and the length of the fluid pocket. The application of partial treatment could also alter the deflection pattern of the beam, particularly the location of the peak deflection.