Fixed-fixed Beam Research Articles

Use of resonance and antiresonance frequencies for better matching of frequency response function

Most of the finite element model updating method uses resonance frequencies and mode shapes. The mode shapes are not very accurate as suggested by many researchers. Whereas, antiresonance frequencies can be identified accurately from measured frequency response functions (FRFs). In this paper, a new model updating method is proposed, which uses resonance and antiresonance frequencies for better FRF matching of updated FRF and experimental FRF. The effectiveness of the proposed finite element updating method is demonstrated by measured data of fixed-fixed beam structure. The updated results have shown that the use of resonance along with antiresonance frequencies for finite element model updating can be used to derive accurate model of the system. This is illustrated by matching of the FRFs obtained from the updated model with that of experimental data.

Design, analysis and testing of a parallel-kinematic high-bandwidth XY nanopositioning stage

This paper presents the design, analysis, and testing of a parallel-kinematic high-bandwidth XY nanopositioning stage driven by piezoelectric stack actuators. The stage is designed with two kinematic chains. In each kinematic chain, the end-effector of the stage is connected to the base by two symmetrically distributed flexure modules, respectively. Each flexure module comprises a fixed-fixed beam and a parallelogram flexure serving as two orthogonal prismatic joints. With the purpose to achieve high resonance frequencies of the stage, a novel center-thickened beam which has large stiffness is proposed to act as the fixed-fixed beam. The center-thickened beam also contributes to reducing cross-coupling and restricting parasitic motion. To decouple the motion in two axes totally, a symmetric configuration is adopted for the parallelogram flexures. Based on the analytical models established in static and dynamic analysis, the dimensions of the stage are optimized in order to maximize the first resonance frequency. Then finite element analysis is utilized to validate the design and a prototype of the stage is fabricated for performance tests. According to the results of static and dynamic tests, the resonance frequencies of the developed stage are over 13.6 kHz and the workspace is 11.2 μm × 11.6 μm with the cross-coupling between two axes less than 0.52%. It is clearly demonstrated that the developed stage has high resonance frequencies, a relatively large travel range, and nearly decoupled performance between two axes. For high-speed tracking performance tests, an inversion-based feedforward controller is implemented for the stage to compensate for the positioning errors caused by mechanical vibration. The experimental results show that good tracking performance at high speed is achieved, which validates the effectiveness of the developed stage.

Vibration Analysis of a Beam Structure Attached with a Dynamic Vibration Absorber

In this paper the transverse vibration of a fixed-fixed end beam will be investigated by experimental measurement. This paper is concerned with the classical theory formula as well as the analysis and design of the dynamic vibration absorber (DVA) which is composed by a flexible beam with two masses symmetrically mounted at its both sides. The fixed end beam clamped to a static structure where dynamic vibration absorber then being attached onto it. One side of end of the beam is harmonically excited in transversal direction by an electric shaker. The structure equipped with accelerometer sensor to measure its vibration response amplitudes and natural frequencies. The dynamic vibration absorber arrangement located under motor near the beam end was then being vibrated and done in two conditions; before and after mounting dynamic vibration absorber. The comparison amplitudes before and after equipped dynamic vibration absorber were compared and discussed. From the experimental results, proved that the DVA has successfully absorbed the beam vibration hence reduced the vibration amplitude of the beam structure. The knowledge and result obtained from this study can help engineers control the vibration level of beam structure.

A Tunable Miniaturized RF MEMS Resonator With Simultaneous High <formula formulatype="inline"><tex Notation="TeX">$Q$</tex></formula> (500–735) and Fast Response Speed <formula formulatype="inline"><tex Notation="TeX">$(<\hbox{10}-\hbox{60}\ \mu\hbox{s})$</tex></formula>

This paper reports on the design, fabrication, and measurement of a novel radio frequency (RF) microelectromechanical systems (MEMS) tunable all-silicon evanescent-mode cavity-based resonator that simultaneously achieves high quality factor and fast response speed. The resonator is based on a 1.5-mm-deep silicon-etched cavity attached to a gold-coated silicon substrate with an array of 75 185-μm-long 20-μm-wide 1- μm-thick gold beams. The 54- mm <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> resonator is tunable from 15.2 GHz up to 16.5 GHz (analog tuning range) and up to 17.8-GHz range (digital tuning range) with an array of 75 MEMS fixed-fixed beams. The MEMS beams are biased against their own silicon substrate. This helps keep RF leakage at a minimum and permits high quality factors of 500-735 for the all-silicon configuration. By applying dynamic biasing waveforms, the MEMS tuners respond within 9 μs (actuation time) and 60 μs (release including settling time). To the best of the authors' knowledge, the presented resonator is more than 3 × smaller, achieves nearly 30% higher average quality factor, and is at least 10-100 × faster than state-of-the-art resonators based on similar technology, implemented in similar frequency ranges.

The Vibration Problem Studying in Micro Actuator System Using the Differential Transformation Method

The aim of this study is to derive the governing equation of an electrostatically actuated micro system by use of the Hamilton principle, and then the natural frequencies of a micro fixed-fixed beam are derived as the solutions to a boundary value problem with prescribed boundary conditions through the differential transformation method (D.T.M.). The differential transformation employed is a transformed function based on the Taylor series that is effective in solving nonlinear problems with fast convergence. The numerical results of the calculated natural frequencies are compared with the analytical data and were found to be in good agreement. Hence, the differential transformation method is one of the most efficient methods of simulating the electrostatic behavior of a micro-structure system, and it has a great potential for use in the analysis of the micro fixed-fixed beam.

Design, Electrostatic and Eigen Frequency Analysis of Fixed– Fixed Beam MEMS Resonator

The objective of this paper is to design of fixed-fixed MEMS Resonator. The beam of resonator is made up of poly-silicon. This paper also presents the electrostatic and Eigen frequency analysis of MEMS resonator. The resonator used in oscillator which can be fabricated on chip. It can also be used as Switch. The size of crystal oscillator is in order of centimeter but the size of MEMS Resonator is in order of microns, so by replace a crystal oscillator by MEMS Resonator, we can reduce the size of the system. The dc voltage is given to the beam of resonator for electrostatic analysis & Eigen frequency analysis in different modes. Keywords - MEMS Resonator, fixed-fixed beam, mechanical restoring force;

Transverse impact characteristics of a rubber pipe expansion joint

The transverse impact characteristics of a rubber pipe expansion joint are studied. A pair of joints assembled end to end with an inserted middle mass is tested on a drop shock testing machine. Based on the test results, an equivalent fixed-fixed beam model with polynomial stiffness and damping is applied to predict the transverse impact response and identify the nonlinear impact parameters. The least square residual between the computed and test results is defined to drive the identification optimization. The response surface methodology in combination with the generalized reduced gradient method is used to search the best matching coefficients. Final results show that the equivalent bending stiffness of the tested rubber expansion joint gradually decreases with the transverse deformation and is greatly influenced by its internal working pressure.

A Laboratory Project on the Theory, Fabrication, and Characterization of a Silicon-On-Insulator Micro-Comb Drive Actuator With Fixed-Fixed Beams

A laboratory course on the theory, fabrication, and characterization of microelectromechanical systems (MEMS) devices for a multidisciplinary audience of graduate students at the University of New Mexico, Albuquerque, has been developed. Hands-on experience in the cleanroom has attracted graduate students from across the university's engineering and science campuses, including students from the Nano-Science and Micro Systems program, itself an interdisciplinary program. This course has been offered yearly for the last four years (since 2007) at the University of New Mexico. In one project from the class, a MEMS actuator is fabricated. This paper details the theory, fabrication steps, and characterization of a silicon-on-insulator (SOI) comb drive actuator that relies on fixed-fixed beams for its restoring force. Discussions include force generation by comb drives and a linear and nonlinear description of the restoring force applied by the fixed-fixed beams. Course assessment data is given from 79 students from three semesters, based on which the effectiveness of the laboratory project is evaluated.

Fabrication and characterization of a thermal tunable bulk-micromachined optical filter

This paper presents and demonstrates a thermal tunable bulk-micromachined optical filter. The filter is based on a Fabry–Perot (F–P) interferometer which is realized by changing the refractive index of bulk silicon F–P cavity. Dense boron doped silicon resistors are used as the heater to achieve the function of wavelength tunable. Device silicon layer of SOI wafer is utilized for the parallel and high-quality bulk F–P cavity with a large dimension mirror structure in size of more than 700μm. Four fixed-fixed beams are designed to suspend the F–P cavity and provide a good thermal isolation. The tunable optical filter has been successfully fabricated and packaged. A pair of optical fiber collimators is used to couple the optical signal due to the large size of the mirror. The measured wavelength tuning range is about 17nm from 1565.32nm to 1582.36nm with driving voltage from 0V to 13.55V. The maximum power consumption is less than 130mW. The 3-dB bandwidth of the successfully fabricated filter is achieved in value of 0.29nm while the Free Spectral Range (FSR) is 21.9nm. The heating and cooling responding time is less than 50ms.

An Unstructured Finite-Volume Method for Structure–Electrostatics Interactions in MEMS

Radio-frequency microelectromechanical systems (RF MEMS) are widely used for contact actuators and capacitive switches, and involve metal–dielectric contact. In these devices, the structure is activated by an electrostatic force, whose magnitude changes as the gap closes. It is advantageous to model fluid and structural mechanics and electrostatics within a single comprehensive numerical framework to facilitate coupling between them. In this article, we extend a cell-based finite-volume approach popularly used to simulate fluid flow to characterize structure–electrostatics interactions. The method employs fully implicit second-order finite-volume discretization of the integral conservation equations governing elastic solid mechanics and electrostatics, and uses arbitrary convex polyhedral meshes. Results are presented in this article for a fixed-fixed beam under electrostatic actuation.

On the tunability of a MEMS based variable capacitor with a novel structure

In this work a novel MEMS based variable capacitor has been presented. To increase the tunability and decrease the applied voltage, the conventional fixed-fixed beam used in CPW lines has been changed to a fixed-simple supported beam. The proposed structure is a simple cantilever micro-beam in the first step of deflection and is changed to a fixed-simple supported micro-beam in the second step of motion. In the capacitive micro-structures increasing the applied voltage decreases the equivalent stiffness of the structure and leads the system to an unstable condition by undergoing to a saddle node bifurcation. In the proposed structure to avoid pull-in instability and increase the capacitance tuning range, mechanical stiffness of the structure is increased by changing boundary conditions by locating a pedestal in the end of the cantilever beam. The governing nonlinear equation for static deflection of the micro-beam, based on Euler–Bernoulli micro-beam theory has been presented. The results show that the proposed structure increases the capacitance tuning range and decreases the applied voltage. The results also show that the position of the pedestal affects the tunability and the threshold voltage of the structure.

Developing Ni–Al and Ru–Al intermetallic films for use in microelectromechanical systems

Ordered intermetallic films have a favorable combination of properties such as high strength, metallic electrical conductivity, good oxidation and corrosion resistance, and a high melting temperature and thermal stability that make them suitable for microelectromechanical systems (MEMS). One potential drawback to intermetallics is a lack of ductility at room temperature; however, the B2 compounds NiAl and RuAl show some ductility at room temperature, which has been shown to increase as the grain size decreases. Additionally, the fracture toughness of both materials is higher than those of Si and SiGe. It is also possible to deposit these materials at temperatures that make them compatible with complementary metal oxide semiconductor processing. The authors have shown that by controlling the Ar pressure during cosputtering, NiAl and RuAl thin films can be deposited near room temperature with stresses ranging from compressive to tensile, possibly eliminating the need for annealing. This article examines Ni–Al and Ru–Al intermetallic thin films with an overall composition of 50% Al and 50% Ni (or Ru) deposited by cosputtering or the annealing of multilayer stacks for use in MEMS by first establishing an appropriate combination of etchant/sacrificial layer that results in crack-free, freestanding, fixed-fixed beams. Finally, two different MEMS devices were successfully fabricated using a XeF2 etch and actuated: electrostatically actuated resonators and bent beam thermal actuators.

Vibration Suppression of Symmetric Cross-Ply Laminated Composite Beam

A procedure using the dynamic Green's functions was proposed to suppress the transverse vibration of a symmetric cross-ply laminated composite beam excited by an external harmonic force. This was achieved by attaching a set of properly chosen masses and/or translational and rotational grounded springs at arbitrary selected locations along the beam. A scheme was proposed for choosing the optimal values of the added masses and the stiffness of the springs and their locations along the beam to confine the vibration in a certain chosen region. Beams with different boundary conditions, such as pinned-pinned, fixed-fixed, cantilever, free-free, simply-clamped, and simply-free beams, are considered. Other unconventional boundary conditions can be treated as well.

Comparative study of finite element model updating methods

The effects of vibrations present major hazards and operating limitations ranging from discomfort (including noise), malfunctioning, reduced performance, early breakdown and structural failure which, in the worst case can be catastrophic. Hence, accurate mathematical models are required to describe the vibration characteristics of structures, which subsequently can be used for design purposes to limit the negative effects of vibrations. Finite element (FE) predictions are often called into question when they are in conflict with test results. Inaccuracies in FE models and errors in results predicted by them can arise due to the use of incorrect modeling of boundary conditions, incorrect modeling of joints, and difficulties in modeling of damping. This has led to the development of model updating techniques, which aim at reducing the inaccuracies present in an analytical model in the light of measured dynamic test data. In this paper, a detailed comparison of two approaches of obtaining updated FE models are evaluated with the objective that the frequency response functions (FRFs) obtained from updated FE models are able to predict the measured FRFs accurately. In the first method, the updated FE model is obtained by a direct method, which uses modal data. In the second method, the updated model is obtained by an iterative method, which uses FRF data and is also a parameter-based method. The effectiveness of both methods is evaluated by numerical examples, as well as by actual experimental data. Firstly, a study is performed using a numerical simulation based on fixed-fixed beam structure. The numerical study is followed by a case involving actual measured data for the case of an F-shaped test structure. The updated results have shown that the iterative method gives 20% better matching of FRFs with the experimental data and also the predictions of the iterative method is better than the direct method beyond the considered frequency range. The updated results have shown that the FE model obtained using the response function method, an iterative method, can be used to derive accurate model of the system. Updated models obtained by both methods are subsequently evaluated for its application in dynamic design.

Design of a contact probe with high positioning accuracy for plasmonic lithography

Plasmonic lithography with a contact probe records nano-meter scale features and has high-throughput owing to its capability to scan in contact mode. The probe is commonly based on a micrometer-scale cantilever, which leads to the tip-positioning problem due to force-deflection that induces lateral tip displacement. We propose a geometrically modified probe to achieve high positioning accuracy. Contrary to a conventional cantilever-tip probe, we designed a "circular probe" with arc-shaped arms that hold the tip in the center. The mechanism is based on the "fixed-fixed beam" concept in material mechanics. To confirm its positioning accuracy, we used a finite element method (FEM) to calculate the tip displacement for a circular probe and compared the results with those using a conventional cantilever-tip probe. The probe was designed considering a silicon-based micro-fabrication process. The designed probe has a square outline boundary with a length of 50 µm, four arms, and a pyramidal tip with a height of 5 µm. The ratio of the lateral tip displacement to the vertical deflection was evaluated to indicate the accuracy of the probe. The probe has higher positioning accuracy by a factor of 10(3) and 10 in its approach mode and scan mode, respectively, compared with a cantilever-tip probe. We expect that the probe is suitable for the applications that require high positioning accuracy, such as nanolithography in contact mode and applications based on multiple-probe arrays.

Reducing Pull-In Voltage by Adjusting Gap Shape in Electrostatically Actuated Cantilever and Fixed-Fixed Beams

A gap with variable geometry is presented for both cantilever beam and fixed-fixed beam actuators as a method to reduce the pull-in voltage while maintaining a required displacement. The method is applicable to beams oriented either in a plane parallel to or perpendicular to a substrate, but is most suitable for vertically oriented (lateral) beams fabricated with a high aspect ratio process where variable gap geometry can be implemented directly in the layout. Finite element simulations are used to determine the pull-in voltages of these modified structures. The simulator is verified against theoretical pull-in voltage equations as well as previously published finite element simulations. By simply varying the gap in a linear fashion the pull-in voltage can be reduced by 37.2% in the cantilever beam case and 29.6% in the fixed-fixed beam case over a structure with a constant gap. This can be reduced a further 4.8% by using a polynomial gap shape (n = 4/3) for the cantilever beam and 1.2% for the fixed-fixed beam by flattening the bottom of the linearly varying gap.

Random Eigenvalue Problems in Structural Dynamics: Experimental Investigations

An experimental investigation of uncertainty in natural frequencies of linear structures estimated from measured frequency response function (FRF) under dynamic loading is presented. Experiments were conducted on one hundred nominally identical realizations of a fixed-fixed beam; each realisation was obtained by placin g twelve identical masses at random spatial locations (generated by a computer) along the length of the beam. The total random mass is about 2% of the total mass of the beam. These experiments contain parametric uncertainty in the mass matrix, and hence may be useful for the validation of many random eigenvalue analysis and prediction methods currently available to structural dynamicist. Predictions from Monte Carlo simulation of deterministic finite elements model are compared with experiments. It is co ncluded that the method of estimation of natural frequencies from FRFs and the spatial location of the measurements has significant influence upon the first two moments (mean and standard deviation) of the natural frequency ensemble. Furthermore, whilst the Monte Carlo simulation estimates of the mean and standard deviation are in reasonable agreement with experiments at higher frequencies, the probability density function differ appreciably, within the limits of the sample size investigated in this study.

Mechanics of energy transfer and failure of ductile microscale beams subjected to dynamic loading

The mechanical response of microelectromechanical systems (MEMS) under impulse loading conditions has not been thoroughly studied to date, partially because of the lack of means to provide such extreme loading rates to miniature devices. However, the increasing use of MEMS-based sensors and actuators in adverse environments, which include extreme strain rate loading, has motivated the investigation of the response of MEMS components under these conditions. In this work, basic and mostly commonly employed Au MEMS components were subjected to impulse loads of 40 ns in duration, which were generated by a high power pulsed laser in order to achieve acceleration levels on the order of 10 9 g. This allowed for the microdevice mechanical/structural response to be investigated at time scales that were of the order of wave transit times in the substrate and the devices. Basic microscale structures, such as cantilevers and fixed-fixed beams of uniform cross-section, were employed to facilitate comparisons with companion finite element simulations in order to gain insight into the mechanisms responsible for impulsive deformation at the microscale. The simulations investigated the effect of loading rate, boundary conditions, beam length, material constitutive response, and damping on the final deformed shapes of the beams. It was found that contact and momentum transfer mechanisms were responsible for the large permanent beam deflections which were measured postmortem. Additionally, the effects of both damping and material property rate dependence were found to be dominant in determining the final deformed shape of the beams. In fact, our observations suggest that the contributions of material rate dependence and damping are not simply additive, but rather involve a coupling between them that affects the final structure response.

Free transverse vibrations of an elastically connected simply supported twin pipe system

In this paper, free vibration analyses of a parallel placed twin pipe system simulated by simply supported-simply supported and fixed-fixed Euler-Bernoulli beams resting on Winkler elastic soil are presented. The motion of the system is described by a homogenous set of two partial differential equations, which is solved by a simulation method called the Differential Transform Method (DTM). Free vibrations of an elastically connected twin pipe system are realized by synchronous and asynchronous deflections. The results of the presented theoretical analyses for simply supported Euler-Bernoulli beams are compared with existing ones in open literature and very good agreement is demonstrated.

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