Fixed-fixed Beam Research Articles

RF MEMS tunable filter using micro fixed–fixed beam

AbstractIn this article, a tunable bandpass filter using micro fixed‐fixed beam structure is designed, simulated, and fabricated. The filter is designed by micromachined capacitive bridges, with an original passive topology. The fabricated filter occupies a chip area of 12 × 3 mm2 and achieved an insertion loss of less than 2 dB and return loss of less than −15 dB throughout the operation band, thereby tuning the filter centre frequency by 11% between 8.76 and 9.8 GHz. © 2010 Wiley Periodicals, Inc. Microwave Opt Technol Lett 52: 592–597, 2010; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.24981

Modeling and Characterization of ALD Grown ZnO Nanotubes and their Application to Sub-Micron Devices

Characteristic properties and dimensions of ZnO nanowires grown by ALD and template technology have been used to model a nanowire into a fixed-fixed beam structure using conventional microelectromechanical systems (MEMS) design techniques. The frequency response of the structure as a result of piezoelectric actuation has been studied. Percentage error between the simulation and theoretical model has been found to be 0.97%.

Widely Tunable MEMS-Based Fabry–Perot Filter

This paper describes the use of strain stiffening in fixed-fixed beam actuators to extend the tuning range of microelectromechanical-systems-based Fabry-Perot filters. The measured wavelength tuning range of 1.615-2.425 mum is the largest reported for such a filter. Curvature in the movable mirror was corrected using a low-power oxygen plasma to controllably alter the stress gradient in the mirror. After curvature correction, the linewidth of a filter was 52 nm, close to the theoretical minimum for our mirror design. As a proof of concept, a filter was bonded to a broadband infrared detector, realizing a wavelength-tunable infrared detector. All measured data have been compared to theoretical models of the optics and mechanics of the filters, with excellent agreement between theory and measurement demonstrated in all cases. Finally, the Young's modulus and stress of the actuator materials were extracted directly from the measured voltage-displacement curves, demonstrating a novel technique for material property measurement.

Extraction of residual stress and dimensions from electrical measurements on surface micromachined test structures

The deposited thin films in surface micromachining have a lot of residual stress, and it is essential to measure this for both process development and monitoring. We estimate residual stress by electrical measurements on a series of fixed-fixed polysilicon beams designed to deflect laterally due to stress. To minimize errors in estimation during parameter extraction, the device dimensions also have to be measured accurately. Surface micromachining of an oxide-anchored polysilicon cantilever beam can result in beam undercut, reduction in beam thickness, and increase in the gap between the beam and the substrate. The undercut in the beam is estimated from the resonance frequency of the cantilever beam and also by using polysilicon resistors. Final device thickness is obtained by measuring the resistance in fixed-fixed beams.

On the Convergence of Random Differential Quadrature (RDQ) Method and Its Application in Solving Nonlinear Differential Equations in Mechanics

Differential Quadrature (DQ) is one of the efficient derivative approximation techniques but it requires a regular domain with all the points distributed only along straight lines. This severely restricts the DQ while solving the irregular domain problems discretized by the random field nodes. This limitation of the DQ method is overcome in a proposed novel strong-form meshless method, called the random differential quadrature (RDQ) method. The RDQ method extends the applicability of the DQ technique over the irregular or regular domains discretized using the random field nodes by approximating a function value with the fixed reproducing kernel particle method (fixed RUM), and discretizing a governing differential equation by the locally applied DQ method. A superconvergence condition is developed for the RDQ method, which gives more than O(h(p+1)) function value convergence for the uniform as well as random field nodes scattered in the domain. The RDQ method convergence analysis is carried out, and the superconvergence condition is verified by solving several 1D, 2D and elasticity problems. The applicability of the RDQ method to solve the nonlinear governing differential equations is successfully demonstrated by solving the fixed-fixed and cantilever beams for deflection due to the nonlinear electrostatic loading. It is concluded that the RDQ method effectively handles the irregular or regular domains discretized by the uniform or random field nodes, with good convergence rates.

The Design of Piezoelectric Transducer for Structural Health Monitoring Using Longitudinal Vibration Modes

Longitudinal vibration mode is conveniently used for damage localization and detection in truss structure. However, a satisfactory piezoelectric impedance response due to longitudinal modes is not so easy to obtain in practical applications. The objective of this paper is to investigate how to obtain the impedance response of longitudinal modes with a piece of PZT transducer. In this case various vibration modes will be excited by unilaterally bonded PZT transducer, so the observation of longitudinal modes becomes complicated. In this paper, the effect of parameters of PZT transducer such as the length and attachment position is discussed in detail. Through selecting a suitable size and attachment position, the impedance response of expected longitudinal modes can be generated. An example for generating impedance peaks of specified longitudinal mode in fixed-fixed beam is given to demonstrate its validity.

Application of hybrid differential transformation/finite difference method to nonlinear analysis of micro fixed-fixed beam

Analyzing the dynamic response of electrostatic devices is problematic due to the complexity of the interactions between the electrostatic coupling effect, the fringing field effect and the nonlinear electrostatic force. To resolve this problem, this study presents an efficient computational scheme in which the nonlinear governing equation of the electrostatic device is obtained in accordance with Hamilton’s principle and is then solved using a hybrid differential transformation/finite difference method. The feasibility of the proposed approach is demonstrated by modeling the dynamic responses of two micro fixed-fixed beams with lengths of 250 and 350 μm, respectively. The numerical results show that the pull-in voltage reduces as the beam length increases due to a loss in the structural rigidity. Furthermore, it is shown that the present results for the pull-in voltage deviate by no more than 0.75% from those derived in the literature using a variety of different schemes. Overall, the results presented in this study demonstrate that the proposed hybrid method represents a computationally efficient and precise means of obtaining detailed insights into the nonlinear dynamic behavior of micro fixed-fixed beams and similar micro-electro-mechanical systems (MEMS)-based devices.

Smart Sand—a wide bandwidth vibration energy harvesting platform

We propose a concept for true wide bandwidth vibration energy harvesting. Our approach exploits nonlinear stretching of fixed-fixed beams in an off-resonance mode, effectively expanding the operational frequency range well beyond the narrow bandwidth of linear resonators. Our initial prototype demonstrates operation between 160–400 Hz, without the need for frequency tuning. A simple dynamic model shows good agreement with measurements. Optimized device geometry will allow for even lower frequency operation (starting at 60 Hz) at strain levels above 1e-3 (ideal for piezoelectric transduction).

Pull-in characterization of doubly-clamped composite beams

An electrostatic pull-in beam has been widely used to measure material properties. There has been an increasing need for MEMS devices with composite films in recent years. This paper presents an approach to characterize the material properties of the composite films using the electrostatic pull-in test structure which is composed of the composite fixed-fixed beam. An analytical model for the pull-in voltage of the composite doubly-clamped beam is presented, which takes into account the influence of lateral residual stress along width direction of the composite beam. Based on this improved model, a test structure is designed to avoid the influence of lateral residual stress along the width of the composite beam. An in situ method for simultaneously extracting material properties (Young's modulus and residual stress) of each layer for the composite films is reported. And doubly-clamped surface micromachined beams with multi-cup style anchors were fabricated to support the model of pull-in voltage. The validation and accuracy of the extracting method have been confirmed by FEM simulation and experiments.

Investigation of the Vibrational Mode of a Long Bone – The Ulna

An excised and dried ulna was vibrated at its balance pnoint about 42% (0.1m) along the length from the olecranon process, using an electromagnetic vibrator. The vibration was carried out in the anterior-posterior plane. The bone response to the mechanical vibration was picked up at equal intervals (0.02m) along the length of the bone in the plane of the vibration, using an accelerometer attached to the bone. From the response curve of the bone, the resonant frequencies at the various pick- up positions were identified. Analysis of the data shows that when the responses were picked up using the accelerometer at 0.02m proximal, and 0.1m distal, to the point of vibration, the excised ulna vibrated in the fundamental mode like a fixed-fixed beam at resonant frequencies of 340 Hz and 355Hz respectively. For resonant frequencies greater than 650 Hz, the ulna vibrates like a fixed-free beam and at modes greater than the fundamental. It is suggested from this study that the vibration of the long bone in the fundamental mode lends itself to detailed analysis in the use of resonant frequency method in the assessment of the bone quality.

Multimode thermoelastic dissipation

In this paper, we investigate thermoelastic dissipation (TED) in systems whose thermal response is characterized by multiple time constants. Zener [Phys. Rev. 52, 230 (1937)] analyzed TED in a cantilever with the assumption that heat transfer is one dimensional. He showed that a single thermal mode was dominant and arrived at a formula for quantifying the quality factor of a resonating cantilever. In this paper, we present a formulation of thermoelastic damping based on entropy generation that accounts for heat transfer in three dimensions and still enables analytical closed form solutions for energy loss estimation in a variety of resonating structures. We apply this solution technique for estimation of quality factor in bulk mode, torsional, and flexural resonators. We show that the thermoelastic damping limited quality factor in bulk mode resonators with resonator frequency much larger than the eigenfrequencies of the dominant thermal modes is inversely proportional to the frequency of the resonator unlike in flexural mode resonators where the quality factor is directly proportional to the resonant frequency. Purely torsional resonators are not limited by TED as the deformation is isochoric. We show that it is possible to express the quality factor obtained by full three-dimensional analyses as a weighted sum of Zener formula based modal quality factors. We analytically estimate the quality factor of a cantilever and a fixed-fixed beam and corroborate it with data to show that the assumption of a single dominant thermal mode, which is valid in one-dimensional analysis, is violated. The analytical formulation described in this paper permits estimation of energy lost due to heat transfer in orthogonal directions. It is found that the entropy generated due to heat transfer along the beam becomes significant in beams with aspect ratio (length/width) below 20.

Acoustic phonon characterisation of fixed-fixed beam MEMS switch

The reported work focuses on an acoustic phonon characterisation method for MEMS devices. This technique is uniquely suited to address challenges of MEMS device characterisation as a low cost and non-destructive method. The mechanical switching action of a fixed-fixed beam MEMS switch is characterised using this acoustic phonon characterisation technique and a correlation between phonon measurements and electrical waveforms is carried out. Insights on the mechanical state of the switch are also obtained using this acoustic phonon characterisation technique.

T0302-2-2 変断面両端固定梁を有する静電駆動MEMS共振器の簡易設計手法の構築(マイクロ・ナノ力学とシステム設計論(2))

T0302-2-2 変断面両端固定梁を有する静電駆動MEMS共振器の簡易設計手法の構築(マイクロ・ナノ力学とシステム設計論(2))

Development of a wide-tuning range and high Q variable capacitor using metal-based surface micromachining process

This study aims to improve the tuning range and quality-factor ( Q) of micro variable capacitors for wireless communication applications. A suspending 0.5 μm-thick gold thin-plate with two-gap structure in one-to-three ratio of spacing is designed for the maximization of tuning range. To enhance effectively the flexural rigidity of top metal-plate and improve further the tuning range of the varactor, a double-cross-type microstructure with two vertical fixed-fixed beam springs and four horizontal fixed-guided cantilever beams is introduced. Besides, a glass substrate (Corning 7740) was used to reduce substantially the power dissipation and improve the Q-factor of variable capacitor. The new glass-based double-cross-type micro variable capacitor has demonstrated many superior performances, including the wide-tuning range (2100%, at 1.0 MHz with 6.0 V), the moderate capacitance (0.56 pF, at 2.4 GHz and without DC bias), 6.5 V pull-in voltage, and the high Q-factor (40.6, at 2.4 GHz). These characteristics approximately match with the theoretical derivation or simulated results from Agilent-ADS, Ansoft-HFSS, and IntelliSuite software.

A Self-Aligned Fabrication Process for Capacitive Fixed-Fixed Beam RF MEMS Components

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> A self-aligned fabrication process for capacitive fixed-fixed beam RF microelectromechanical system (MEMS) components is disclosed. It enables the scaling of the critical dimensions and reduces the number of processing steps by 40% compared with a conventional RF MEMS fabrication process. RF MEMS varactors with beam lengths of 30 <formula formulatype="inline"><tex>${\rm \mu}\hbox{m}$</tex></formula> are demonstrated by using the self-aligned fabrication process, and the performance of a four-by-four RF MEMS varactor bank is discussed as well. At 20 GHz, the measured capacitance values range between 180.5 and 199.2 fF. The measured capacitance ratio is 1.15 when a driving voltage of 35 V is applied, and the measured loaded <formula formulatype="inline"> <tex>${Q}$</tex></formula> -factor ranges between 14.5 and 10.8. The measured cold-switched power handling is 200 mW. <formula formulatype="inline"><tex>$\hfill$</tex> </formula>[2008-0018] </para>

Multiscale electrostatic analysis of silicon nanoelectromechanical systems (NEMS) via heterogeneous quantum models

A multiscale method, seamlessly combining semiclassical, effective-mass Schr\odinger (EMS), and tight-binding (TB) theories, is proposed for electrostatic analysis of silicon nanoelectromechanical systems (NEMS). By using appropriate criteria, we identify the physical models that are accurate in each local region. If the local physical model is semiclassical, the charge density is directly computed by the semiclassical theory. If the local physical model is quantum mechanical (the EMS or TB model), the charge density is calculated by using the theory of local density of states (LDOS). The LDOS is efficiently calculated from the Green's function by using Haydock's recursion method where the Green's function is expressed as a continued fraction based on the local Hamiltonian. Once the charge density is determined, a Poisson equation is solved self-consistently to determine the electronic properties. The accuracy and efficiency of the multiscale method are demonstrated by considering two NEMS examples, namely, a silicon fixed-fixed beam with hydrogen termination surfaces and another silicon beam switch with 90\ifmmode^\circ\else\textdegree\fi{} single period partial dislocations. The accuracy and efficiency of the multiscale method are demonstrated.

Dynamic Analysis of A Symmetric Laminated Composite Beam Subjected to a Moving Load with Constant Velocity

This study deals with the dynamic behavior of a symmetric laminated composite beam subjected to a concentrated force traveling at a constant velocity. The dynamic analyses for a fixed-fixed composite beam under the action of a moving load are carried out by the finite element method. A three-dimensional finite element model based on the classical lamination theory is used. The Newmark integration method is employed in order to calculate the dynamic response. A computer code is developed using MATLAB to perform the finite element vibration analysis. The dynamic magnification, which is defined as the ratio between the dynamic and static displacements is obtained for different load velocities and ply orientations. The results reported in this paper show that load velocity and ply orientation have significant effect on the dynamic response, especially for those with [90]2s lay-up. For a fixed-fixed beam, the maximum mid-point deflection is observed when the total traveling time is equal to the first natural period of the beam.

Fourier Analysis of Mode Shapes of Damaged Beams

This paper investigates the effect of damage on beams with fixed boundary conditions using Fourier analysis of the mode shapes in spatial domain. A finite element model is used to obtain the mode shapes of a damaged fixed-fixed beam. Then the damaged beams are studied using a spatial Fourier analysis. This approach contrasts with the typical time domain application of Fourier analysis for vibration problems. It is found that damage causes considerable change in the Fourier coefficients of the mode shapes. The Fourier coefficients, especially the higher harmonics, are found to be sensitive to both damage size and location and amplify the changes in the mode shape due to the damage. Therefore, we formulate a damage index in the form of a vector of Fourier coefficients which is robust and unique for a given damage size and damage location. The effect of noise in the mode shape data is considered and it is found that Fourier coefficients provide a useful indication of damage even in the presence of noise. Various damage levels are considered and it is found that higher modes are needed to detect small amount of damage.

Thermal Buckling and Postbuckling of Euler-Bernoulli Beams Supported on Nonlinear Elastic Foundations

Deformations of homogeneous and isotropic pinned-pinned and fixed-fixed Euler-Bernoulli beams supported on nonlinear elastic foundations and heated uniformly into the postbuckling regime have been analyzed numerically. Geometric nonlinearities introduced by finite deflections and curvature of the deformed beams are incorporated in the problem formulation. First buckling due to the uniform temperature rise and buckling mode transitions are investigated analytically by analyzing the linear problem. Subsequently, the nonlinear boundary-value problems for postbuckling of beams are transformed into initial-value problems and analyzed by the shooting method. For different values of the elastic foundation parameters, postbuckled configurations are illustrated.

Vibration analysis of axially moving magnetic tape with comparisons to static and dynamic experimental results

The demand for greater storage capacity is prompting the development of data storage tape with greater volumetric and areal density. As the data tracks on these tapes become narrower, minimization of the lateral tape motion (LTM) becomes more important to prevent loss of data due to read/write errors. In order to identify and minimize sources of LTM, a vibration model is developed which simplifies the tape drive to a fixed-fixed Euler-Bernoulli beam model with axial velocity. The effects of varying axial velocity, tension, free span length, and tape thickness were investigated. The calculated natural frequencies are compared to those obtained by both static and dynamic experiments. Effects of varying the length, tape thickness, and tension were studied in the static experiment while the effects of changing speed and tension were studied in the dynamic experiment.