Clamped-clamped Beam Research Articles

Thermal-mechanical coupling buckling analysis of porous functionally graded sandwich beams based on physical neutral plane

Porosity of functionally graded materials (FGMs) is usually aroused by fabrication defects. It had been proven that the porosity has a significant influence on the static responses of their structures, but the effects of porosity on buckling behaviors are still worth investigating. To reveal these effects, the thermal-mechanical coupling buckling issue of a clamped-clamped porous FGM sandwich beam is investigated in this paper by employing the high-order sinusoidal shear deformation theory. The modified Voigt mixture rule is used to approximate the temperature-dependent material properties of porous FGMs. The physical neutral plane of FGM sandwich beams is taken into account to reflect the actual condition of the structures and simplify the calculation. The thermal environments are considered as uniform, linear and nonlinear temperature rises, and both the temperature-independent and temperature-dependent material properties are discussed in order to justify the importance of the thermal-mechanical coupling effect. An iterative algorithm is used to solve the thermal-mechanical coupling critical buckling temperature. The present theoretical results are verified by comparing with the literature and ABAQUS results, and the effects of porosity, the physical neutral plane, gradient index, material temperature dependence, sandwich structural parameters are discussed. Results show that for buckling issue excluding the pre-buckling deformation effect, considering either the physical neutral plane or the geometrical middle plane of FGM beams would produce alike critical buckling temperatures. With the rise of porosity, the critical temperature increases greatly, which is quite different from the changing rule observed in the buckling issue of inplane-loaded porous FGM plates in literature. The beam with a smaller face-to-core ratio is more sensitive to the change in porosity. Moreover, to improve the thermal buckling load of FGM beams, ceramic constituents with the lower thermal expansion coefficient would be preferred.

Axisymmetric vibrations of the composite orthotropic cylindrical shell with rigid weightless end disks

A solution of the vibration problem for a composite orthotropic cylindrical shell with rigid weightless disks attached to its ends is presented in the paper. Using the Ritz method, the frequencies of the axisymmetric vibrations of the shell are determined. The clamped-clamped beam functions and their third derivatives are adopted as approximating functions for the shell deflection and axial displacement. Such an approximation satisfies the boundary conditions, according to which the shell deflections, angles of rotation, and axial reactive forces exerted on the end disks are zero. Based on this solution an analytical formula is derived for the calculations of the frequencies of axisymmetric vibrations. Using this formula, the vibration frequencies have been calculated for composite orthotropic shells with different lengths. The results were successfully verified by the finite element analyses. The effect of the fibre orientation on the vibration frequencies of the shell made of unidirectional carbon fibre reinforced plastic is investigated. Based on this analysis, the angles of the reinforcement orientation delivering the maximum and minimum values of the frequencies are determined. It is found that irrespective of the shell length, the high mode frequencies reach their maximum values for the shells reinforced in the hoop direction.

Semi-analytic solution of Eringen’s two-phase local/nonlocal model for Euler-Bernoulli beam with axial force

Eringen’s two-phase local/nonlocal model is applied to an Euler-Bernoulli nanobeam considering the bending-induced axial force, where the contribution of the axial force to bending moment is calculated on the deformed state. Basic equations for the corresponding one-dimensional beam problem are obtained by degenerating from the three-dimensional nonlocal elastic equations. Semi-analytic solutions are then presented for a clamped-clamped beam subject to a concentrated force and a uniformly distributed load, respectively. Except for the traditional essential boundary conditions and those required to be satisfied by transferring an integral equation to its equivalent differential form, additional boundary conditions are needed and should be chosen with great caution, since numerical results reveal that non-unique solutions might exist for a nonlinear problem if inappropriate boundary conditions are used. The validity of the solutions is examined by plotting both sides of the original integro-differential governing equation of deflection and studying the error between both sides. Besides, an increase in the internal characteristic length would cause an increase in the deflection and axial force of the beam.

Free vibration of bi-directional functionally graded material circular beams using shear deformation theory employing logarithmic function of radius

Curved beams such as arches find ubiquitous applications in civil, mechanical and aerospace engineering, e.g., stiffened floors, fuselage, railway compartments, and wind turbine blades. The analysis of free vibrations of curved structures plays a critical role in their design to avoid transient loads with dominant frequencies close to their natural frequencies. One way to increase their applications and possibly make them lighter without sacrificing strength is to comprise them of Functionally Graded Materials (FGMs) that are composites with continuously varying material properties in one or more directions. Here, we study free vibrations of FGM circular beams by using a shear deformation theory that incorporates through-the-thickness logarithmic variation of the circumferential displacement, does not require a shear correction factor, and has a parabolic through-the-thickness distribution of the shear strain. The radial displacement of a point is assumed to depend only upon its angular position. Thus the beam theory generalizes the Timoshenko beam theory. Equations governing transient deformations of the beam are derived by using Hamilton’s principle. Assuming a time harmonic variation of displacements, and by utilizing a generalized differential quadrature method (GDQM), the free vibration problem is reduced to solving an algebraic eigenvalue problem whose solution provides frequencies and corresponding mode shapes. Results are presented for different spatial variations of the material properties, boundary conditions, and the beam aspect ratio. It is found that frequencies of the FGM beam are bounded by those of two geometrically identical homogeneous beams composed of the two constituents of the beam. Keeping other variables fixed, the change in the beam opening angle results in very close frequencies of the first two modes of vibration at a critical value of the opening angle, a phenomenon usually called mode transition. The critical opening angle is essentially the same for radially graded, bidirectionally graded and monolithic beams. It equals about 80° (60°) for clamped-clamped (hinged-hinged) beams.

Dynamic buckling of smart sandwich beam subjected to electric field based on hyperbolic piezoelasticity theory

In this paper, dynamic buckling of the smart subjected to blast load subjected to electric field is studied. The sandwich structure is rested on Pasternak foundation with springs and shear elements. Applying piezoelasticity theory and hyperbolic shear deformation beam theory (HSDBT), the motion equations are derived by energy method. For calculating the dynamic instability region (DIR) of the sandwich structure, differential quadrature method (DQM) along with Bolotin method is used. The aim of this study is to investigate the effects of applied voltage, geometrical parameters of structure and boundary conditions on the DIR of the structure. The results show that applying negative voltage, the DIR will be happened at higher excitation frequencies. In addition, the clamped-clamped beam leads to higher excitation frequency with respect to simply supported boundary condition.

Large Amplitude Free Vibration Analysis of Tapered Timoshenko Beams Using Coupled Displacement Field Method

Tapered beams are more efficient compared to uniform beams as they provide a better distribution of mass and strength and also meet special functional requirements in many engineering applications. In this paper, the linear and non-linear fundamental frequency parameter values of the tapered Timoshenko beams are evaluated by using the coupled displacement field (CDF) method and closed form expressions are derived in terms of frequency ratio as a function of slenderness ratio, taper ratio and maximum amplitude ratio for hinged-hinged and clamped-clamped beam boundary conditions. The effectiveness of the CDF method is brought out through the solution of the large amplitude free vibrations, in terms of fundamental frequency of tapered Timoshenko beams with axially immovable ends. The results obtained by the present CDF method are validated with the existing literature wherever possible.

Primary High Acceleration Calibration System at NIM

Currently the maximum acceleration range of primary vibration calibration cannot meet the requirement of secondary vibratoin calibration and vibration testing field. In some special vibration test field, the accelerometers are required to be calibrated up to 4900 m/s2 or more, but the maximum acceleration of primary vibratoin calibration capability is normlly 100 m/s2. In order to meet the high acceleration traceability requirement, a primary high acceleration calibration system has been set up in National Institute of Metrology. Two high acceleration exciters based on resonance amplification are used to produce high acceleration vibration. In order to impove the maximum acceleration of the resonator, a simple clamped-clamped beam is used, only the clamped ends are excited by a shaker to reduce the damping. The phase control method are used to track the resoant frequency. A primary vibration measuring system was set up according to ISO 16063-11. The experimental results show that the maximum acceleration of this vibratoin calibration system can reach more than 10000 m/s2.

High-Power Handling Analysis of a Capacitive MEMS Power Sensor at <inline-formula> <tex-math notation="LaTeX">$X$ </tex-math> </inline-formula>-Band

The high-power handling capacity of a capacitive microelectromechanical systems (MEMS) power sensor is investigated in this paper. The power sensor is designed for a parallel-plate structure with clamped-clamped beam in X-band (8-12 GHz) applications. The fabrication of this device adopts GaAs monolithic microwave integrated circuit process and MEMS technology. Experiments show that the output response of this sensor is non-linear at high power levels. To analyze this characteristic, the relevant discussion and simulation are given. First, the structure of the sensor is simulated by ANSYS HFSS software to determine the effective sensing region. Then, the mechanical behavior of the sensor analyzed by using the finite element software ANSYS. Through reasonable hypothesis, an electro-mechanical model is proposed. The model considers two effects: 1) the mode shape of the beam and 2) the signal reflection. The calculated results using this model are in good agreement with the measured ones. It indicates the impedance mismatching of the device mainly causes the non-linear response of capacitive power sensor under high-power signal.

A hybrid piezoelectric and electromagnetic energy harvester for scavenging low frequency ambient vibrations

This work presents a hybrid type piezoelectric and electromagnetic multi-degree of freedom (MDOF) energy harvester for low-frequency ambient vibrations. The proposed device is designed with unique MDOF mechanical structure that helps to obtain multiple resonant modes operation of the harvester at low frequency range. The mechanical design comprises of a clamped-clamped beam as primary beam and four secondary beams as cantilever array. During external excitation, the secondary cantilever beams vibrate simultaneously along with the magnetic proof mass that causes mechanical strain to the piezoelectric elements as well as relative motion between magnets and coils placed underneath of it. A prototype of the hybrid MDOF energy harvester has been fabricated, simulated, and experimentally verified. The frequency response of the prototype implies that the proposed hybrid MDOF harvester has four resonant modes which concentrate around 12, 16, 17, and 21 Hz, respectively. On the other hand, a maximum of 218μW and 204µW peak powers have been obtained across the respective optimal load resistances from a single piezoelectric and electromagnetic generator of the array at 3rd resonance (17Hz) under 0.1g base acceleration.

Improving energy harvesting from random excitation by nonlinear flexible bi-stable energy harvester with a variable potential energy function

Abstract In this paper, aim at improving the performance of bi-stable energy harvester (BEH), we proposed a flexible bi-stable energy harvester (FBEH). The FBEH is composed of two elastic beams: one is the piezoelectric cantilever beam with a tip magnet and the other is the clamped-clamped beam with a mid-magnet. The dynamic behavior of FBEH was studied, and the results show that the FBEH owning a variable potential energy function is beneficial for snap-through. The governing equations of FBEH are derived by energy principles. Then its dynamic response under random excitation was studied. The results prove that the FBEH not only has a smaller threshold for snap-through, but also can generate a larger power output for the excitation intensity larger than the critical one. Validation experiments were designed and carried out. The experimental results are in good agreement with the simulation ones.

Pull-in behavior of electrostatically actuated stepped microbeams

The stepped microbeams are typical low stiffness structures widely used in MEMS devices. Approximated solutions of the natural frequency and the pull-in voltage of stepped microbeams, including clamped-free (CF) and clamped–clamped (CC) boundary conditions are developed, and the unique pull-in behavior of the stepped microbeams is investigated. The stepped microbeam is viewed as a beam with a rectangular electrode pad at its tip for CF beam and at its center for CC beam. The motion equations are deduced based on the Euler–Bernoulli beam and the modified couple stress theory. The natural frequency and the pull-in voltage are extracted with a one-degree-of-freedom model. The present model correctness is validated by comparing with the finite element results, and the effect of the length ratio of the electrode pad to the beam and the width ratio of the beam to the electrode pad are discussed. The results show that both the natural frequency and the pull-in voltage monotonously increase with the increase of the width ratio, and first decrease and then increase with the increase of the length ratio. The minimum value of them does not appear at the same time, but is determined by the width ratio. The results can be used to design and improve the performance of MEMS devices.

Fundamental frequency of a composite anisogrid lattice cylindrical panel with clamped edges

A derivation and validation of an analytical formula for the calculation of the fundamental frequency of a composite anisogrid lattice cylindrical panel with clamped edges is presented in this paper. Free vibration analysis is performed based on the continuous model of a lattice structure using the equations of engineering theory of orthotropic cylindrical shells. The problem was solved using the Galerkin method in which the displacements of the panel were approximated by the clamped-clamped beam functions. The analytical formula derived from this solution was employed to study the effects of the structural parameters of composite lattice panels on their fundamental frequencies. The results of these parametric analyses were successfully verified by comparisons with the finite-element solutions. It is shown that the analytical model that only takes into account the inertia of the transverse motion of the panel in the direction normal to its surface provides a reasonable estimate of the value of fundamental frequency. It is also demonstrated how the formula works in the calculations delivering the required fundamental frequency when designing the composite lattice panels.

Elastic moduli of cellulose nanofibers isolated from various cellulose resources by using aqueous counter collision

An atomic force microscopy (AFM) three-points bending (TPB) test is very useful to measure the mechanical properties of cellulose nanofibers (CNFs). This study investigates the Young’s moduli of various CNFs by using AFM TPB test. CNFs were isolated from hardwood, softwood, bamboo and cotton by using an aqueous counter collision (ACC) method. Single CNF was transferred to the AFM calibration chip by using contact transfer for the AFM TPB test and the CNF was considered as a clamped-clamped beam. An Euler equation was used to calculate Young’s modulus under linear deformation assumption. The cross section of single CNF is assumed to be circular shape. Young’s moduli of hardwood, softwood, bamboo and cotton CNFs isolated by ACC method are in the range of 88–110 GPa, which are comparable with the previously reported values.

Fabrication and characterization of a piezoelectric energy harvester with clamped-clamped beams

This work presents a piezoelectric energy harvester with clamped-clamped beams, and it is fabricated with MEMS process. When excited by sinusoidal vibration, the energy harvester has a sharp jumping down phenomenon and the measured frequency responses of the clamped-clamped beams structure show a larger bandwidth which is about 56Hz, more efficient than that with cantilever beams. When the exciting acceleration ac is 12m/s2, the energy harvester achieves to a maximum open-circuit voltage of 94mV on one beam. The load voltage is proportional to the load resistance, and it increased with the increase of load resistance. Connected four beams in series, the output power reaches the maximum value of 730 nW and the optimal load is 15KΩ to one beam.

Actuation of Piezoelectric Layered Beams With and Coupling.

In this paper, we derive and compare the linear static bending of piezoelectric actuators with transversal ( ) and longitudinal ( ) coupling. The transducers are, respectively, structures utilizing top and bottom electrodes (TBEs) and interdigitated electrodes (IDEs). While the theory is well developed for the TBE beam, governing equations for the bending of the piezoelectric beams with IDEs are far less developed. We improve on this by deriving the governing equation for the IDE beam with an arbitrary number of layers and with coupling consistently included. In addition, we introduce a phenomenological quadratic form for the nonuniform field that lets us derive a deflection formula with nontrivial effects of the field accounted for. The theory is applied to derive deflection formulas for both cantilever and clamped-clamped beams. All analytic results are validated with numerical simulations. From the analytic models, two different figures of merit (FOMs) are derived. We show that these FOMs are the same for cantilevers and doubly clamped beams. The analysis indicates the optimal transducer length for clamped-clamped beams and gives a criterion that can be used to determine which design concept ( or ) gives the largest deflection.

Frequency equations of nonlocal elastic micro/nanobeams with the consideration of the surface effects

A nonlocal elastic micro/nanobeam is theoretically modeled with the consideration of the surface elasticity, the residual surface stress, and the rotatory inertia, in which the nonlocal and surface effects are considered. Three types of boundary conditions, i.e., hinged-hinged, clamped-clamped, and clamped-hinged ends, are examined. For a hinged-hinged beam, an exact and explicit natural frequency equation is derived based on the established mathematical model. The Fredholm integral equation is adopted to deduce the approximate fundamental frequency equations for the clamped-clamped and clamped-hinged beams. In sum, the explicit frequency equations for the micro/nanobeam under three types of boundary conditions are proposed to reveal the dependence of the natural frequency on the effects of the nonlocal elasticity, the surface elasticity, the residual surface stress, and the rotatory inertia, providing a more convenient means in comparison with numerical computations.

Non-linear vibrations of a beam with non-ideal boundary conditions and uncertainties – Modeling, numerical simulations and experiments

Abstract This paper presents experiments and numerical simulations of a nonlinear clamped-clamped beam subjected to Harmonic excitations and epistemic uncertainties. These uncertainties are propagated in order to calculate the dynamic response of the nonlinear structure via a coupling between the Harmonic Balance Method (HBM) and a non-intrusive Polynomial Chaos Expansion (PCE). The system studied is a clamped-clamped steel beam. First of all, increasing and decreasing swept sine experiments are performed in order to show the hardening effect in the vicinity of the primary resonance, and to extract the experimental multi-Harmonic frequency response of the structure. Secondly, the Harmonic Balance Method (HBM) is used alongside a continuation process to simulate the deterministic response of the nonlinear clamped-clamped beam. Good correlations were observed with the experiments, on the condition of updating the model for each excitation level. Finally, the effects of the epistemic uncertainties on the variability of the nonlinear response are investigated using a non-intrusive Polynomial Chaos Expansion (PCE) alongside the Harmonic Balance Method (HBM). A new methodology based on a phase criterion was developed in order to allow the PCE analysis to be performed despite the presence of bifurcations in the nonlinear response. The efficiency and robustness of the proposed methodology is demonstrated by comparison with Monte Carlo simulations. Then, the stochastic numerical results are shown to envelope the experimental responses for each excitation level without the need for model updating, validating the nonlinear stochastic methodology as a whole.

Dynamic model updating based on strain mode shape and natural frequency using hybrid pattern search technique

Aiming at providing a precise dynamic structural finite element (FE) model for dynamic strength evaluation in addition to dynamic analysis. A dynamic FE model updating method is presented to correct the uncertain parameters of the FE model of a structure using strain mode shapes and natural frequencies. The strain mode shape, which is sensitive to local changes in structure, is used instead of the displacement mode for enhancing model updating. The coordinate strain modal assurance criterion is developed to evaluate the correlation level at each coordinate over the experimental and the analytical strain mode shapes. Moreover, the natural frequencies which provide the global information of the structure are used to guarantee the accuracy of modal properties of the global model. Then, the weighted summation of the natural frequency residual and the coordinate strain modal assurance criterion residual is used as the objective function in the proposed dynamic FE model updating procedure. The hybrid genetic/pattern-search optimization algorithm is adopted to perform the dynamic FE model updating procedure. Numerical simulation and model updating experiment for a clamped-clamped beam are performed to validate the feasibility and effectiveness of the present method. The results show that the proposed method can be used to update the uncertain parameters with good robustness. And the updated dynamic FE model of the beam structure, which can correctly predict both the natural frequencies and the local dynamic strains, is reliable for the following dynamic analysis and dynamic strength evaluation.

Nonlinear cc-beam microresonator model for system level electrical simulations: Application to bistable behavior analysis

Microelectromechanical resonators nonlinearities can be exploited in many ways to obtain a set of diverse new applications. In particular, some applications of bistable behavior includes threshold mechanical switches, memory cells, energy harvesting and chaotic signal generators. A key step for practical and efficient design for bistability behavior involves accounting for accurate and efficient models. In this paper we present a nonlinear electromechanical model for capacitive clamped-clamped beam resonators implemented in an analog hardware description language (AHDL) enabling system level electrical simulations. The model accounts for nonlinearities from variable resonator-electrode gap, residual fabrication stress, fringing field contributions as well as an accurate resonator deflection profile in contrast to parallel plate approximations. The model has been applied to derive for first time accurate analytical expressions for bistability design conditions. The work includes FEM analysis and experimental data that corroborates the correctness of the model in describing the required bias voltage conditions for bistability.

Linear flexural natural frequencies and stability analysis of spinning Rayleigh beams: application to clamped-clamped beams.

In the present paper 3D bending linear free vibrations of spinning Rayleigh beams are investigated. Four linear models, that differ in the linearization process, are studied. A focus on analytical computation of natural frequencies for a broad range of boundary conditions is highlighted. Then, the conditions of occurrence of divergence and flutter instabilities are determined. Finally, a case study consisting of a clamped-clamped Rayleigh beam is studied. It is found that the free vibrations destabilization process depends on the used linearization approach.