Clamped-clamped Beam Research Articles

Theoretical and experimental analysis of the exact receptance function of a clamped-clamped beam with concentrated masses

This paper establishes the exact receptance function of a clamped-clamped beam carrying concentrated masses. The derivation of exact receptance and the numerical simulations are provided. The proposed receptance function can be used as a convenient tool for predicting the dynamic response at arbitrary point of the beam acted by a harmonic force applied at arbitrary point. The influence of the concentrated masses on the receptance is investigated. The numerical simulations show that peak in the receptance will decrease when there is a mass located close to that peak position. The numerical results have been compared to the experimental results to justify the theory.

A New Method of Resolution of the Bending of Thick FGM Beams Based on Refined Higher Order Shear Deformation Theory

The aim of this work is to study the static bending of functionally graded beams accounting higher order of shear deformation theory. The governing equations, derived from the virtual work principle, are a set of ordinary differential equations describing a static bending of a thick beam. Thus, this paper presents the differential transform method used to solve the previous system of equations. The results obtained lay the foundation to determine the exact analytical solution for different boundary conditions and external loadings. The axial displacement and the bending and shear displacements, in the exact analytical form, of a thick clamped-clamped beam with functionally graded material under a uniform load will be fully developed. Moreover, normal and shear stresses will be analyzed. To confirm the efficiency of this work, a comparison with the numerical results provided by literature is performed. Through this work, the given analytical results provide engineers with an accurate tool to determine the analytical solution for the bending of plates and shells. In addition, the geometric and material parameters that appear clearly in the analytical results allow for a more optimized design of functionally graded material beams. This type of beams is frequently used in mechanical engineering fields such as aerospace engineering.

Distributed wireless-based control strategy through Selective Negative Derivative Feedback algorithm

In this study the use of many stand-alone devices is considered for active vibration control. A stand-alone device represents a control unit which is able to independently perform the vibration control task, since it is embedded with sensors, an inertial actuator and a microcontroller, in which the control algorithm is implemented. The aim of this architecture is to improve the performance of a decentralized control strategy through a partial sharing of data between devices, without the use of cables. The proposed algorithm also aims to avoid instability phenomena related to the dynamics of inertial actuators. First, the formulation of the modified Negative Derivative Feedback resonant controller is derived; then, the steps of the control strategy are explained, which are based on a preliminary modal identification of the structure and an automatic assignment of a compensator for each resonance frequency to be controlled. Finally, the proposed method is validated with experimental results from a clamped-clamped beam.

Changing the critical snap-through loads of post-buckled beams using piezoelectric actuation

Post-buckled and curved structures experience snap-through instabilities when external loads from mechanical, fluid, or thermal environments result in a loss of local stability and a dramatic jump to a remote stable equilibrium. Fatigue caused by snap-through is a concern in many engineered systems because of the large stress reversals involved. This paper studies the extent to which the strategic placement and actuation of piezoelectric materials bonded to clamped-clamped post-buckled beams can influence the loads at which snap-through occurs. The electromechanical system is modeled using elastica theory with extensions to account for the influence of piezoelectric actuation on the structure. Static equilibrium positions and their stability are computed across a large configuration space using numerical integration and a shooting method. The results indicate that the effect of piezoelectric actuation on critical snap-through load depends on the degree to which the beam is buckled, the location of the external load, the placement of the piezoelectric material, and the applied actuation voltage. Experiments are performed to validate the numerical results and provide a physical demonstration of changing snap-through loads with piezoelectric actuation. Experimental results demonstrate that critical snap-through loads can be altered by factors ranging from 0.4 to 2.0, and numerical results indicate that even larger changes to snap-through loads are physically realizable.

Characterisation of the internal resonances of a clamped-clamped beam MEMS resonator

Nonlinear coupling between modes through internal resonance has many applications in MEMS resonators. The primary objective of this work is to systematically study all possible internal resonance conditions between the first three modes and the feasibility of mode interaction in an electrostatically excited, straight, clamped-clamped beam. The static displacement and the first three natural frequencies of the beam are obtained for an applied DC voltage by using Galerkin based reduced order model and finite element method. All six possible commensurable relations between the first three frequencies of the beam are obtained by sweeping the non-dimensional parameters ($$\alpha _1$$ and $$\alpha _2 V_\mathrm{dc}^2$$) which depend on beam dimensions, material properties and external forcing. It is also demonstrated by a further examination of the dynamical equations that only one resonance condition is capable of exhibiting modal coupling when externally excited. A detailed analysis is carried out for this feasible resonance condition by altering $$\alpha _1$$ and $$\alpha _2 V_\mathrm{dc}^2$$ and solving the relevant nonlinear coupled equations by using both numerical time integration and the method of multiple scales. From this study, we observe that the lower mode is automatically excited after driving amplitude for the higher mode reaches a critical value—a sign of mode interaction initiation. We also find that amplitude of the higher mode is saturated after interaction with the lower mode for a combination of $$\alpha _1$$ and $$\alpha _2 V_\mathrm{dc}^2$$. Moreover, we see that the frequency bandwidth of modal interaction increases with excitation amplitude for all combinations of $$\alpha _1$$ and $$\alpha _2 V_\mathrm{dc}^2$$. Finally, the role of external damping on the amplitude-frequency response curve of both modes during the mode interaction is also investigated.

Manipulating elastic waves through piezoelectric metamaterial with nonlinear electrical switched Dual-connected topologies

This research proposes a new kind of piezoelectric metamaterials with nonlinear SSDI (Synchronized Switching Damping on Inductor) Dual-connected electronic networks, which provides a new electrical medium coupled with the piezoelectric beam metamaterial for elastic wave manipulation and vibration control. Specifically, the proposed topology considers several piezoelectric elements periodically and evenly bonded onto the substructure. The proposed structure is composed of several electromechanical (EM) periodic cells. All the piezoelectric elements in one EM periodic cell are connected to each other in a cyclic fashion by identical SSDI shunts, and there is no electrical connection between every two EM periodic cells, shaping the Dual-connected electrical connection. Based on the wave propagation theory and finite element modeling, band structure and eigenmode shapes of the proposed structure are investigated for elastic wave manipulation and low-frequency vibration control. Results show that the proposed structure with SSDI 2-order-Dual-connected electrical networks exhibits broad phononic band gaps. Especially, band gap hybridization with Bragg-type band gap generation mechanism in the proposed structure with SSDI 2-order-Dual-connected electrical networks outperforms that of the piezoelectric metamaterial with SSDI independent networks. Furthermore, the low frequency vibration reduction and wave attenuation performance of the proposed structure is experimentally validated by measuring the harmonic response of a periodic clamped-clamped piezoelectric beam structure.

Optimal position of piezoelectric actuators for active vibration reduction of beams

Abstract This paper investigates the optimal placement of piezoelectric actuators for the active vibration attenuation of beams. The governing equation of the beam is achieved by coupled first order shear deformation theory with two node element. The velocity feedback controller is designed and used to calculate the feedback gain and then apply to the beam. In order to search for the optimal placement of the piezoelectric actuators, a new optimization criterion is considered based on the use of genetic algorithm to reduce the displacement output of the beam. The proposed optimization technique has been tested for two boundary conditions configurations; clamped -free and clamped-clamped beam. Numerical examples have been provided to analyze the effectiveness of the proposed technic.

Exact receptance function and receptance curvature of a clamped-clamped continuous cracked beam

The receptance function has been studied and applied widely since it interrelates the harmonic excitation and the response of a structure in the frequency domain. This paper presents the derivation of the exact receptance function of continuous cracked beams and its application for crack detection. The receptance curvature is defined as the second derivative of the receptance. The influence of the crack on the receptance and receptance curvature is investigated. It is concluded that when there are cracks the receptance curvature has sharp changes at the crack positions. This can be applied for the crack detection purpose. In this paper, the numerical simulations are provided.

A chip-scale frequency down-conversion realized by MEMS-based filter and local oscillator

A wide-width clamped-clamped beam (CC-beam) resonator and a filter realized using mechanically-coupled resonator tanks have been extensively characterized to attain mixler (i.e., mixer-filter) functionality for a chip-scale frequency translation. Such a resonator has been driven into the nonlinear regime to explore its effect on the mixler operation as well as to obtain the best phase noise performance using the phase-feedback method. The proposed electrostatically transduced mixler operates by utilizing the local oscillator (LO) signal generated from the resonator itself, thus eliminating the need of a function generator or an external quartz crystal oscillator. This technique implemented with all the components on the same silicon substrate shows great potential for miniaturization in wireless communication while achieving 14.4 dB overall mixler loss.

A magnetically coupled nonlinear T-shaped piezoelectric energy harvester with internal resonance

This letter originally presents a magnetically coupled nonlinear T-shaped piezoelectric energy harvester with multiple resonance bands based on internal resonance principle. The proposed harvester consists of a clamped-clamped buckled beam and a vertical cantilever with tip magnetically coupled. The corresponding theoretical model is established. Two resonance bands with hardening nonlinear phenomenon are obtained. 1:2:3:4 internal resonance characteristics with the peak amplitude ratio of 1:1.81:0.56:0.31 are found for the generated voltage of the first resonance. The energy conversion efficiency can be enhanced by utilizing high frequency response under low-frequency excitations. The bandwidth of full width at half maximum output power reaches 13.8 Hz within the excitation between 5∼30 Hz. The response of random excitations is also investigated and an average power of 9.64 μW is achieved at the random excitation intensity of 0.001 g2Hz−1. Overall, the proposed harvester can achieve ultra-wide working frequency range under low-level excitations and is suitable for low-frequency random vibrations.

A CMOS-Integrated MEMS Platform for Frequency Stable Resonators—Part II: Design and Analysis

The Part II of this paper presents the numerical model and experimental validations of the CMOS-MEMS resonant transducers fabricated by a reliable TiN-composite (TiN-C) platform. To characterize the frequency behavior of the proposed resonator, an analytical model based on average displacement is used to determine the equivalent lumped parameters of the proposed devices. Considering the free-free beam (FFB) design elaborated in Part I, the oxide fin provides a very strong boundary restriction, which forces the vibration like a short clamped-clamped beam (CCB) combined with two additional tip cantilevers. This segmented analytical model shows a good agreement with our measurement results. In addition, the implementation of TiN-to-TiN transduction gap and oxide-rich composite structure has enabled a deep sub-micron ( $TC_{f}$ ) is achievable by taking temperature coefficient of Young’s modulus ( $TC_{E}$ ), the coefficient of thermal expansion (CTE), and residual stress of standard BEOL into consideration. As a result, the proposed TiN-C pseudo free-free beam (PFFB) resonator shows the lowest passively compensated $TC_{f}$ of 0.6 ppm/K in CMOS-MEMS technology to date, revealing the excellent frequency stability over temperature and time for practical applications in the future.

Stable transitions between remote equilibria of post-buckled beams using piezoelectric actuation

This paper considers the extent to which a clamped-clamped post-buckled beam bonded to two elongating piezoelectric actuators can be made to stably transition between remote equilibria. The electromechanical system is modeled using elastica theory with extensions to account for the influence of the actuators on the structure. It expresses this piezoelectric coupling effect in terms of a non-dimensional parameter, σ, that can be easily calculated for candidate substrate/actuator configurations. Parameter studies determining the threshold values of σ required to execute stable transitions under different actuation and loading situations are presented. The results indicate that the lowest threshold values of σ occur when the beam is actuated from approximately 15%–85% of its span. Experiments validate the numerical results and offer the first physical demonstrations of the use of piezoelectric actuators to achieve stable transitions between remote equilibria.

Modal Kinetic Energy Change Ratio-based Damage Assessment of Laminated Composite Beams using Noisy and Incomplete Measurements

Modal kinetic energy (MKE) feature has been mostly employed for optimal sensor layout strategies; nevertheless, little attention is paid to use the feature to the field of structural damage detection. The article presents the extensive applicability of MKE change ratio (MKECR), a good damage sensitive parameter, to damage localization and quantification of laminated composite beams. The formulation of the parameter is based on the closed-form of element MKE sensitivity. The performance of the offered damage detection method is numerically verified by a clamped-clamped composite beam and a two-span continuous composite beam with different hypothetical damage scenarios. The influence of incomplete mode shapes, various noise levels as well as damage magnitudes on damage prediction results are also investigated. The obtained results from these numerical examples indicate that the offered method reliably localize the actually damaged elements and approximately estimate their severities, even under incomplete measurements at a high noise level.This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium provided the original work is properly cited.

Design and experimental investigation of a T-shaped piezoelectric energy harvester

This paper presents a T-shaped piezoelectric energy harvester (TPEH) by utilizing multimodal techniques. The proposed TPEH is consisted of a clamped-clamped beam and a branched cantilever beam at the center. One end of the clamped-clamped beam could move freely along the axial direction and a piezoelectric wafer is glued at the end. Two operational bandwidths are shown in the experimental results. Moreover, various cantilever beam configurations were investigated. It was found that the first resonance band moves to the low frequency range as the tip mass of the cantilever beam increases. However, the second resonance band accesses to lower frequency range slightly. Finally, it was demonstrated that the presented TPEH can be used in low frequency and low amplitude vibration environment.

Bifurcation diagram and dynamic response of a MEMS resonator with a 1:3 internal resonance

The dynamic response of a nonlinear resonator in the presence of resonant mode coupling is studied experimentally and theoretically. For the case of a clamped-clamped beam resonator in the presence of a 1:3 internal resonance, we show that at the onset of internal resonance, steady state oscillations cannot be sustained. At higher drive levels, stable oscillations can be maintained but the resonator amplitude undergoes amplitude modulated responses. We use these dynamic responses to build a bifurcation diagram that can be described remarkably well with a simple model consisting of a Duffing resonator coupled to a linear one.

Precise measurement of a concentrated mass and its position by the vibration of a rectangular micro-diaphragm

Free vibration of a micro-diaphragm was firstly solved using the Rayleigh–Ritz method by constructing the mode shape as combinations of the clamped-clamped beam functions in width and length directions. An asymptotic solution to the inverse problem of the dynamic system of the diaphragm with a concentrated mass was then derived. It was found that the rectangular micro-diaphragm delivers a unique capability to simultaneously measure the concentrated mass and its position from the observed frequency changes. The proposed strategy in mass sensing was validated by extracting the concentrated mass and its position using simulated and experimental results on a micro-diaphragm resonator.

RF MEMS In-Line Type Phase Detector With Large Dynamic Range

An in-line type X-band phase detector is presented in this letter. It is constructed by a power combiner and a capacitive power sensor with clamped-clamped beam to obtain large dynamic range with respect to RF power. The fabrication process is based on MEMS technology. The experiments show that the return loss is smaller than −12.3 dB over X-band, the isolation is better than −9.1 dB and the insertion loss is around −3.5 dB. The conversion between phase, power, and capacitance is accomplished inside the detector for phase detection from 0 to 180°. The varying capacitance of the beam can sustain a dynamic range of 0.25 ~ 2 W with satisfactory detection sensitivity, 163.8aF/deg. at 2W. Third-order intermodulation distortion (IM3) of the detector is rather small and the IIP3 is about 37.4 dBm for 10-kHz frequency interval.

Thermal Analysis of 2D FGM Beam Subjected to Thermal Loading Using Meshless Weighted Least-Square Method

The paper analyzed the thermal problem of the 2D FGM beam using meshless weighted least-square (MWLS) method. The MWLS as a meshless method is fully independent of mesh, and an approximate function was used to construct a series of linear equations to solve the unknown field variable, which avoided the troublesome task of numerical integration. The effectiveness and accuracy of the approach were illustrated by a clamped-clamped FGM beam which was subjected with interior heat source. The volume fraction of FGM beam was assumed to be given by a simple power law distribution. The effective material properties of the FGM beam were assumed to be temperature independent and calculated by Mori-Tanaka method. The results showed that a good agreement was achieved between the proposed meshless method and commercial COMSOL Multiphysics.

Analytical and numerical modeling of a clamped-clamped micro beam under electric actuation

This paper studies the deflection of a clamped-clamped micro beam actuator subjected to an electric voltage. Two approaches were applied to model the micro beam under study: the first one is analytical and the second is numerical. A solution to the analytical model was developed to investigate the mechanical behaviour of the micro beam-based MEMS device under electric actuation and simulations was done using Wolfram Mathematica® software. On the other hand, a finite element model (FEM) simulation was performed using COMSOL Multiphysics® for data validation for the micro beam model, and the results showed very good agreement. The effect of changing beam length, thickness and height from the ground on both the maximum beam displacement and pull-in voltage was also analyzed.

Nonlinear vibrations of a beam with non-ideal boundary conditions and stochastic excitations - experiments, modeling and simulations

This paper presents experiments and numerical simulations of a nonlinear clamped-clamped beam subjected to stochastic broadband excitations. Broadband excitations are performed experimentally in order to show the hardening effect and the enlargement of the response peak in the vicinity of the primary resonance, and to detect the presence of secondary peaks resulting from the harmonics generated by the primary resonance.Using the decomposition of a random signal into a multi-harmonic periodic equivalent signal, the shooting method and the Harmonic Balance Method (HBM) are used to simulate the response of the nonlinear clamped-clamped beam subjected to a stochastic excitation. The latter is modeled as a multi-harmonic periodic equivalent signal in order to be used by both the shooting method and the HBM. A periodogram strategy is used to ensure a good estimate of the resulting Power Spectral Density (PSD). The two nonlinear methods are compared in terms of computation time and precision. A specific attention is paid on correct use of these methods.Finally, comparison between experiments and numerical simulations are performed for different levels of stochastic excitations. Good correlations are observed thus validating the global nonlinear proposed strategy.