Microbeam Resonators Research Articles

Symmetry Breaking and Modal Localization in a System of Parametrically Excited Microbeam Resonators

In this work, we study the nonlinear dynamics of parametrically excited bending vibrations of two weakly coupled beam microresonators under electrothermal excitation. A steady-state harmonic temperature distribution in the volume of the resonators in the frequency domain was obtained. A system of equations for mechanically coupled beam resonators is derived, considering the deposited particle on one of them. Using asymptotic methods of nonlinear dynamics, equations in slow variables were obtained, which were studied by methods of the theory of bifurcations. It is shown that in a perfectly symmetrical system in a certain frequency range, the effect of symmetry breaking is observed – the emergence of a mode with different amplitudes of oscillations of two beam resonators, which can be the basis for a new principle of high-precision measurements of weak disturbances of various physical natures, in particular – measurements of ultra-low masses of deposited particles.

Nonlocal strain gradient-based nonlinear vibration analysis of nonlinear FG three-phase HJCNT/MWCNT/epoxy composite microbeam resonators using a modified micromechanical model

This paper aims to study nonlinear dynamic behavior of functionally graded (FG) three-phase composite microbeam resonators made of an epoxy matrix and two reinforcements namely multi-walled carbon nanotubes (MWCNTs) and hetero-junction carbon nanotubes (HJCNTs). The effective mechanical properties of the composite microbeam are obtained using the modified Halpin-Tsai micromechanical model. The microbeam surrounding medium is simulated using a two-parameter elastic foundation. The von-Karman’s geometric nonlinearity relations are incorporated and the equations of motion are derived based on the nonlocal strain gradient Euler–Bernoulli beam model. A new closed-form analytical solution is obtained using the homotopy perturbation method. The effects of vibration amplitude, nanofiber volume fraction, nanofiber distribution pattern, small-scale parameters and the foundation parameters on the nonlinear frequency and deflection of the FG three-phase composite microbeams are studied in detail. The findings of the paper are valuable for researchers in the field of microbeam resonators.

Period-doubling cascade route to chaos in an initially curved microbeam resonator exposed to fringing-field electrostatic actuation

This paper explores the chaotic dynamics of a piezoelectrically laminated initially curved microbeam resonator subjected to fringing-field electrostatic actuation, for the first time. The resonator is fully clamped at both ends and is coated with two piezoelectric layers, encompassing both the top and bottom surfaces. The nonlinear motion equation which is obtained by considering the nonlinear fringing-field electrostatic force, includes geometric nonlinearities due to the mid-plane stretching and initial curvature. The motion equation is discretized using Galerkin method and the reduced order system is numerically integrated over the time for the time response. The variation of the first three natural frequencies with respect to the applied electrostatic voltage is determined and the frequency response curve is obtained using the combination of shooting and continuation methods. The bifurcation points have been examined and their types have been clarified based on the loci of the Floquet exponents on the complex plane. The period-doubled branches of the frequency response curves originating from the period doubling (PD) bifurcation points are stablished. It's demonstrated that the succession PD cascades leads to chaotic behavior. The chaotic behavior is identified qualitatively by constructing the corresponding Poincaré section and analyzing the response's associated frequency components. The bifurcation diagram is obtained for a wide range of excitation frequency and thus the exact range in which chaotic behavior occurs for the system is determined. The chaotic response of the system is regularized and controlled by applying an appropriate piezoelectric voltage which shifts the frequency response curve along the frequency axis.

Size-dependent thermoelastic damping analysis in functionally graded bi-layered microbeam resonators considering the nonlocal dual-phase-lag heat conduction model

Size-dependent thermoelastic damping analysis in functionally graded bi-layered microbeam resonators considering the nonlocal dual-phase-lag heat conduction model

Effects of humidity and temperature on quality factor of micro-beam resonators in atmospheric pressure and gas rarefaction

At atmospheric pressure (p=101325 Pa), the effects of humidity and temperature on moist air become important when discussing the quality factor of micro-cantilever and micro-bridge resonators. The squeeze film damping (SFD) problem, the dominant damping source for micro-beam resonators, is modelled using the modified molecular gas lubrication (MMGL) equation with finite element modelling (FEM) in the eigenvalue problem. The MMGL equation is modified with the effective viscosity of moist air (μeff) to account for the effects of humidity and temperature. Other damping sources, such as thermoelastic damping (TED) and the support loss of micro-beam resonators, are also calculated. The quality factor of micro-beam resonators is then discussed over a wide range of temperatures and relative humidity levels at atmospheric pressure and gas rarefaction. The results show that the quality factor of micro-cantilever and micro-bridge resonators increases as both humidity and temperature rise in atmospheric pressure and gas rarefaction. Furthermore, the quality factor of a micro-bridge resonator with changes in humidity and temperature is significantly higher than that of a micro-cantilever resonator in atmospheric pressure and gas rarefaction.

Stabilization in extensible thermoelastic Timoshenko microbeam based on modified couple stress theory

AbstractIn this article we derive the equations that constitute the nonlinear model of extensible thermoelastic Timoshenko microbeam. The constructed mathematical model is based on the modified couple stress theory which implies prediction of size dependent effects in microbeam resonators, by applying the Hamilton principle to full von Kármán equations. This takes account of the effects of extensibility where the dissipations are entirely contributed by temperature. Based on semigroups theory, we establish existence and uniqueness of weak and strong solutions to the derived problem. By an approach based on the Gearhart‐Herbst‐Prüss‐Huang theorem, we prove that the associated linear semigroup (without extensibility) is not analytic in general. In the absence of additional mechanical dissipations, the system is often not highly stable. Then by adding a damping frictional function to the first equation of the derived model and using the multiplier method, we show that the solutions decay exponentially under a condition on the physical coefficients.

Mass Sensing by Symmetry Breaking and Mode Localization in a System of Parametrically Excited Microbeam Resonators

In this work, we study the nonlinear dynamics of a mode-localized mass detector. A system of equations is obtained for two weakly coupled beam resonators with an alternating electric current flowing through them and taking into account the point mass on one of the resonators. The one-dimensional problem of thermal conductivity is solved, and a steady-state harmonic temperature distribution in the volume of the resonators is obtained. Using the method of multiple scales, a system of equations in slow variables is obtained, on the basis of which instability zones of parametric resonance, amplitude-frequency characteristics, as well as zones of attraction of various branches, are found. It is shown that in a completely symmetrical system (without a deposited particle), the effect of branching of the antiphase branch of the frequency response is observed, which leads to the existence of an oscillation regime with different amplitudes in a certain frequency range. In the presence of a deposited particle, this effect is enhanced, and the branching point and the ratio of the amplitudes of oscillations of the resonators depend on the mass of the deposited particle.

Analysis of thermoelastic dissipation in microbeam resonators covered with multiple partial coatings

Thermoelastic dissipation (TED) is a mechanism for intrinsic energy dissipation that governs the upper limit on microresonators’ quality factor (Q-factor). The demands for high performance and diverse functionality have led to the development of MEMS resonators with increasingly complex structural geometries and materials. This work proposes an analytical framework to evaluate TED behavior in microbeam resonators coated with multiple partial coatings. By determining the one-way thermoelastic coupled equation in the thickness direction, the temperature function is obtained for each region, and TED model is derived by capturing the energy loss of each region within the framework of thermal energy method. The effectiveness of present framework is verified by comparing it with experimental data and finite element method (FEM) results. The developed TED model has an explicit closed-form expression that converges rapidly, and retaining only leading terms yields a simple TED model. Rules for applying the simple TED model to ensure accuracy are discussed. TED behaviors considering finite thermal contact conductance are comprehensively investigated. A methodology for a mixture of multiple partial coatings to suppress TED is proposed. This work is helpful in suppressing the TED and achieving high Q-factors for the microbeam resonators.

Nonlocal dual-phase-lag thermoelastic damping analysis in functionally graded sandwich microbeam resonators utilizing the modified coupled stress theory

Functionally graded (FG) sandwich structures stand as one of the most representative composite structures owing to the thermal resistance and the energy absorption property in non-unfiorm thermal environment. Additionally, accurately estimating thermoelastic damping (TED) is of great importance for the design of high-performance micro/nano-resonators. Nevertheless, the classical TED models fail on the micro/nano-scale structures due to without considering the influences of the spatial size-dependent effects related to heat transfer and elastic deformation. The nonlocal heat conduction model and modified coupled stress theory are responsible for the size-dependent effects. To address this issue, present study aims to conduct the size-dependent TED model of FG sandwich microbeam resonators for TED analysis by incorporating the nonlocal dual-phase-lag (NDPL) heat conduction model and the modified coupled stress theory (MCST). It is assumed that the FG sandwich microbeam resonators consist of a ceramic core and FG surfaces. The energy equation and the transverse motion equation are formulated, and then, the analytical solution is solved by complex frequency method. Exact and closed-form expressions for TED can be obtained through the complex frequency method. The results are validated, and the parameter effects of the nonlocal thermal parameter, the material length-scale parameter, the power-law index and the vibration modes on the TED are analyzed. The results show that the energy dissipation can be reduced by the size-dependent effects resulting in improving the quality factor of microstructures. It is expected that these results may provide a theoretical basis for predicating TED in the design of FG sandwich micro/nano-resonators with high quality factor in extreme heat transfer environment.

On Thermo-Optically Excited Parametric Oscillations of Microbeam Resonators. II

On Thermo-Optically Excited Parametric Oscillations of Microbeam Resonators. II

Damping of 3D-printed polymer microbeam resonators

The emerging high-resolution 3D printing technique called two-photon polymerization (2PP) enables to print devices bottom-up rapidly, contrary to the top–down lithography-based fabrication methods. In this work, various polymer microbeams are 3D printed and their resonant characteristics are analyzed to understand the origin of damping. The 2PP printed polymer resonators have shown less damping than other polymer devices reported earlier, with tensile-stressed clamped–clamped beams reaching a record quality factor of 1819. The resonant energy loss was dominant by bulk friction damping. These results pave the path towards using 3D printed microresonators as mass sensors with improved design and fabrication flexibility.

Thermoelastic damping analysis of functionally graded sandwich microbeam resonators incorporating nonlocal and surface effects

Functionally graded (FG) sandwich microstructures/nanostructures stand as one of the most promising composite structures due to the capability to resist thermal shock and high noise in extreme thermal environments. Meanwhile, accurately prediction quality-factor based on thermoelastic damping (TED) is of great significance for the design of high-performance microresonators /nanoresonators. However, the classical TED models fail to explain the size-dependent thermo-mechanical behavior. The nonlocal elasticity and surface elasticity are responsible for the size-dependent phenomenon. In this article, a modified TED model is proposed to estimate the impact of the size-dependent effect on the TED of FG sandwich microbeam resonators in unsteady heat transfer processes by combining the nonlocal elasticity model, surface elasticity model, and dual-phase-lag (DPL) heat conduction model. It is assumed that the FG sandwich microbeam resonators consist of a ceramic core and FG surfaces. The energy equation and the transverse motion equation are derived. The analytical expression of TED is obtained by the complex frequency method. The influences of the factors including the nonlocal parameter, the surface effect, the power-law index, and the vibration modes on the TED are analyzed. These results provide a reasonable theoretical analysis to predicate TED in the design of FG sandwich microresonators/nanoresonators with high performance.

Nonlinear dynamics of a piezoelectrically laminated initially curved microbeam resonator exposed to fringing-field electrostatic actuation

In this paper, the nonlinear dynamics of a piezoelectrically sandwiched initially curved microbeam subjected to fringing-field electrostatic actuation is investigated. The governing motion equation is derived by minimizing the Hamiltonian over the time and discretized to a reduced-order model using the Galerkin technique. The modelling accounts for nonlinearities due to the fringing-field electrostatic force, initial curvature and mid-plane stretching. The electrostatic force is numerically computed using finite element simulation. The nonlinear dynamics of the microbeam in the vicinity of primary resonance is investigated, and the bifurcation types are determined by investigating the location of the Floquet exponents and their configuration with respect to the unit circle on the complex plane. The branches on the frequency–response curves, which originate from the period-doubling bifurcation points, are introduced, and the transition from period-1 to period-2 response is demonstrated by slight sweep of the excitation frequency over the time. The effect of DC and AC electrostatic excitation and the piezoelectric excitation on the response of the system are examined, and their effect on the bifurcation types is determined. The force response curves assuming the AC voltage as the bifurcation parameter are also introduced; it is illustrated that in contrast to in-plane electrostatic excitation, in fringing field-based resonators the resonator is not limited by pull-in instability, which is substantially confining the amplitude of the motion in in-plane resonators.

Dual-phase-lag thermoelastic damping analysis of a functionally graded sandwich micro-beam resonators incorporating double nonlocal effects

Functionally graded sandwich micro/nano-structures have attracted great attention due to the capability to resist high noise and thermal stress in a non-isothermal environment. Additionally, the design of high quality-factor micro/nano-resonators requires accurate estimation of their thermoelastic damping. However, the classical thermoelastic damping models fail at the micro/nano-scale due to the influences of the size-dependent effects related to heat transfer and elastic deformation. This work aims to investigate the influences of the size-dependent effects on the thermoelastic damping of functionally graded sandwich micro-beam resonators by combining the nonlocal dual-phase-lag heat conduction model and the nonlocal elasticity model. It is assumed that the functionally graded sandwich micro-beam resonators consist of a ceramic core and functionally graded surfaces. The energy equation and the transverse motion equation are derived. The analytical expression of thermoelastic damping is obtained by complex frequency method. Numerical results are analyzed for the effects of the thermal nonlocal parameter, the elastic nonlocal parameter, the power-law index, and the vibration modes on the thermoelastic damping of functionally graded sandwich micro-beam resonators. The results show that the thermoelastic damping of functionally graded sandwich micro-beam resonators can be adjusted by the suitably modified parameters, which strongly depends on the double nonlocal effects and the power-law index.

On Opto-Thermally Excited Parametric Oscillations of Microbeam Resonators. I

On Opto-Thermally Excited Parametric Oscillations of Microbeam Resonators. I

An interconnect-free micro-electromechanical 7-bit arithmetic device for multi-operand programmable computing

Computational power density and interconnection between transistors have grown to be the dominant challenges for the continued scaling of complementary metal–oxide–semiconductor (CMOS) technology due to limited integration density and computing power. Herein, we designed a novel, hardware-efficient, interconnect-free microelectromechanical 7:3 compressor using three microbeam resonators. Each resonator is configured with seven equal-weighted inputs and multiple driven frequencies, thus defining the transformation rules for transmitting resonance frequency to binary outputs, performing summation operations, and displaying outputs in compact binary format. The device achieves low power consumption and excellent switching reliability even after 3 × 103 repeated cycles. These performance improvements, including enhanced computational power capacity and hardware efficiency, are paramount for moderately downscaling devices. Finally, our proposed paradigm shift for circuit design provides an attractive alternative to traditional electronic digital computing and paves the way for multioperand programmable computing based on electromechanical systems.

The role of axial pre-tension in reducing energy dissipation of micro/nano-mechanical resonators

Pre-tension is usually applied to raise the frequency of resonators. Theoretical studies revealed that tensile stress could inhibit thermoelastic damping of the resonators and thus increase the quality factor when thermoelastic damping is dominant. Up to now, there is little theoretical investigation for its impacts on other energy dissipation mechanisms, especially support loss. In this paper, a concise formula for the support loss of doubly clamped beam resonators with pre-tension is derived. Then theoretical analyses are carried out, revealing that the pre-tension decreases thermoelastic damping but increases support loss of micro-beam resonators. Making a trade-off between two competing influences of the pre-tension on support loss and thermoelastic damping by tuning pre-tension, an optimal quality factor can be achieved when these two dissipation mechanisms are dominant. Theoretical results of effects of pre-tension on thermoelastic damping and support loss are verified by finite element simulation, and the theoretical prediction of the quality factor associated with these two energy dissipation mechanisms also has the similar trend to the experimental data of reported researches. As a result, the quality factor can be improved by tuning the axial pre-tension in the case that thermoelastic damping greatly surpasses support loss of the resonator. Application of axial pre-tension is further demonstrated to be an effective approach to improving the quality factor of most polymer-based beam resonators. Hopefully, this work will provide theoretical basis and practical guidelines to the design and development of high-performance micro/nano resonators.

Dynamic analysis of a novel wide-tunable microbeam resonator with a sliding free-of-charge electrode

Dynamic analysis of a novel wide-tunable microbeam resonator with a sliding free-of-charge electrode

On opto-thermally excited parametric oscillations of microbeam resonators. II

This article is the second part of the work devoted to the investigation of the nonlinear dynamics of parametrically excited flexural vibrations of a clamped-clamped microbeam - the basic sensitive element of a promising class of microsensors of various physical quantities - under laser thermooptical action in the form of periodically generated pulses acting on a certain part of the surface of the beam element. The fundamental technical feasibility of laser generation of parametric oscillations of high-Q microresonators without the implementation of scenarios for the loss of elastic stability of the sensitive element or its unacceptable heating is shown. The nature of the zone of the main parametric resonance is analyzed analytically. The resonant characteristics of the system are constructed in a geometrically non-linear formulation corresponding to the Bernoulli - Euler beam model.

On opto-thermally excited parametric oscillations of microbeam resonators. I

The present article is the first part of the work devoted to investigation of the nonlinear dynamics of parametrically excited flexural vibrations of a clamped-clamped microbeam - the basic sensitive element of a promising class of microsensors of various physical quantities - under laser thermooptical action in the form of periodically generated pulses acting on a certain part of the surface of the beam element. An analytical solution of the heat transfer problem is found for the steady harmonic distribution of temperature in the volume of the resonator. The static and dynamic components of temperature-induced axial force and bending moment are determined. Using the Galerkin method, the discretization of nonlinear coupled partial differential equations describing the longitudinal-flexural oscillations of the resonator is performed. Using the asymptotic method of multiple time-scales, an approximate analytical solution is obtained for the nonlinear dynamics problem under the conditions of primary parametric resonance.