Vibrations Of Micro Research Articles

On nonlinear forced vibration of micro scaled panels

The main target of this paper is to evaluate the geometrically nonlinear and size-dependent response of shallow sandwich curved micro-panels (SCMPs). In this regard, the governing equations are developed by using modified couple stress theory (MCST) based on the first-order shear deformation theory (FSDT). The proposed sandwich micro panels are assumed to be made of three layers so that the middle layer is a porous core and the other ones are the nanocomposite enriched face-sheets. The extended nonlinear differential equations are analytically solved by using the main concepts of the Galerkin method which results a closed-form formulation for the nonlinear equation of motion of the proposed micro-panels. Then, the obtained equation of motion is solved numerically using the forth-order Runge-Kutta method. Accuracy of the present method is validated by comparing the results of some examples with those are available in the other studies. Using an extensive parametric study, the effects of various mechanical and geometrical properties on the size-dependent forced vibration of proposed micro-panels are evaluated.

Effect of two-temperature parameter on thermoelastic vibration in micro and nano beam resonator

Thermoelastic vibration of micro and nano-beam resonators is described by the Euler-Bernoulli beam theory in context with two-temperature generalized thermoelasticity theory. The amplitude of deflection and thermal moment in the case of simply supported and the isothermal beam are obtained by the Laplace transform method along with the finite Fourier sine transform method and the vibration frequency is examined by the normal mode analysis when the beam-ends are clamped and isothermal. The effects of the two-temperature parameter and micro and nano-beam thickness on the amplitude of deflection, thermal moment, and thermoelastic vibration frequency of the micro and nano-beam resonator have been studied and shown with the numerical results. A comparison of the results with the corresponding results of the two-temperature generalized thermoelasticity (2TLS) and the Lord-Shulman (LS) theories is also presented. Besides, the size dependency nature of the micro and nano-beam is analyzed by analytical and numerical results.

Analytical solutions of nonlocal coupled thermoelastic forced vibrations of micro-/nano-beams by means of Green's functions

Thermoelastic dynamic analysis of micro-/nano-beams is essential in the field of micro-/nano-electro-mechanical systems (MEMSs/NEMSs). However, the classical coupled thermoelastic theory is not suitable for the microscopic/nanoscopic case. As elementary parts of MEMSs/NEMSs, the size effect of micro-/nano-beams must be considered in their thermoelastic vibrations. This work originally explores analytical solutions of coupled thermoelastic forced vibrations of micro-/nano-beams based on Rayleigh beam theory and Eringen nonlocal elasticity theory. The heat conductivity equation is obtained by using the type Ⅲ Green-Naghdi theory. Coupled thermoelastic dynamic equations are decoupled by use of Green's function method and expressions of displacements, temperature fields, and thermal moments of micro-/nano-beams are derived. Furthermore, movement behaviors of an “ice core” of the temperature field are used to explain dynamic phenomena of the coupled system. Results from finite element analysis are compared with analytical solutions for verification purposes. This work primarily discusses influences of a small-scale parameter on coupled displacement and temperature fields and those of other important physical parameters, such as heating positions and height-to-length ratios of micro-/nano-beams, on system responses.

Forced axial vibration of micro and nanobeam under axial harmonic moving and constant distributed forces via nonlocal strain gradient theory

In this paper the forced axial vibration of a nanobeam by deploying the nonlocal strain gradient theory exposed to a distributed moving force, is investigated. The assumed nanobeam in this paper is under the axial constant distributed load. The governing equations of the vibration of the nanobeam are obtained via the classic beam theory and the Hamilton principle. The determined partial differential equations are converted to differential equations by applying the assumed modes method. The dynamic axial displacement of the nanobeam along its length, is obtained by solving the differential equations via the convolution integral. The effect of length scale parameter, nonlocal parameter, velocity parameter, the constant distributed load and the excitation frequency on the maximum nondimensional dynamic axial displacement of the nanobeam are analyzed. Accordingly, the effect of these two parameters on the maximum nondimensional axial displacement of the nanobeam under the harmonic moving load is of significance importance.

Piezo electric bone surgery – Review

Piezoelectric surgery was first explained by French Jacques and Pierre Curie in 1880. Piezoelectric bone surgery is a novel osteotomy and osteoplasty technique invented by Professor Vercellotti in 1988 to overcome the limitations of traditional instrumentation in oral bone surgery by modifying and improving conventional ultrasound technology. The cutting action is the result of linear microvibrations, in amplitude between 30 and 200 μm that allow to obtain a micrometer precision in the range of 25-29 KHz modulated ultrasonic resonant frequency. Nerves, vessels, and soft tissue are preserved by the microvibrations (60 to 200 mm/sec), which are optimally adjusted to target only mineralized tissue. It is a minimally invasive osteotomy device to achieve unparallel precision, patient comfort, intra-operative sensitivity, intra-operative visibility, blood free surgical site and reduced damage to the surrounding soft tissues, osteocytes and important structures like nerves, muscles and mucosa. Also Piezosurgery is very useful in harvesting autogenous bone for regenerating periodontal defects. Various treatment modalities are based on removal of etiologic factors and preserving bone architecture. Recently this novel surgical approach has gained popularity in various fields of dental implantology, oral and maxillofacial surgery, endodontics, and orthopedics. This article aims to review the clinical calibre of piezosurgery in the field of periodontology and implant dentistry. Key Words : LESS INVASIVE, PRECISION CUTTING, MICRO VIBRATIONS, FASTER RECOVERY, PIEZO TECHNIQUE

Free vibration analysis of micro and nano fiber-metal laminates circular cylindrical shells based on modified couple stress theory

ABSTRACTIn this manuscript the frequencies of micro and nano fiber-metal laminate (FML) cylindrical shells based on modified couple stress theory (MCST) have been obtained. Love's first approximation shell theory has been applied to consider the thin cylindrical shell and beam modal function model used to satisfy the governing equations of motion subjected to different boundary conditions. Vibration of micro and nano FML cylindrical shells has been considered for carbon/epoxy, glass/epoxy and aramid/epoxy as composites and for aluminum as metal. Also, the frequencies of the shells have been calculated for different volume fractions of composite section, lay-ups, material length scale parameters, length to radius ratio, axial and circumferential wave numbers. The results have been compared to the other researches and showed excellent agreement with them. Also, the frequencies of aramid reinforced aluminum laminate micro and nano cylinder are more than the other ones.

Nonlinear dynamic behavior of small-scale shell-type structures considering surface stress effects: An isogeometric analysis

Surface influences on the nonlinear vibrations of micro- and nano-shells are investigated by an efficient numerical approach. The seven-parameter geometrically nonlinear first-order shear deformation shell theory in Lagrangian description is formulated for the bulk part of structure. To consider surface stress effects, the Gurtin–Murdoch surface elasticity theory with considerations proposed by Ru (2016) and Shaat et al. (2013) is employed. In this regard, two thin inner and outer surface layers are considered, and the corresponding constitutive relations are incorporated into the shell formulations. The stress–strain and strain–displacement relations are represented in a novel matrix–vector form by which the governing equations of motion are derived based on Hamilton’s principle. The isogeometric analysis (IGA) is then utilized due to having the capability to construct exact geometries of shells and the associated powerful features. The obtained ordinary differential equations from IGA are finally solved by the periodic grid approach which can be considered as a suitable solution strategy for the analysis of free and harmonically forced vibrations of different structures. The present work contributes to the literature with developing the isogeometric model of size-dependent geometrically nonlinear shells subjected to large-amplitude vibrations.

Linear free vibration of micro-/nano-plates with cut-out in thermal environment via modified couple stress theory and Ritz method

This article aims to study the natural frequency of defective micro-/nano-plates because not only the existence of cut-outs in plates may be essential on the basis of their desired functionality but also the resonant frequency can be changed by cut-outs. In this study, the modified couple stress theory is combined with Kirchhoff thin plate hypothesis to incorporate size effect to classical plate theory. The Rayleigh-Ritz method is employed to derive discrete equations of motion which yields eigenvalue problem. In the ‘numerical results’ section, the effects of different boundary conditions, location of the cut-out, side-length to thickness ratio and a rise in the temperature on dimensionless fundamental frequency of polysilicon micro-plates are studied. Results demonstrate that the natural frequency of micro-plate, the cut out of which is far from the plate center is more than that of micro-plate with a square hole close to the plate center.

Soft-mode dynamics in micrograin and nanograin ceramics of strontium titanate observed by hyper-Raman scattering

The low-frequency vibrations of micro- and nanograin ceramics of strontium titanate are investigated by hyper-Raman spectroscopy. The combination of a confocal optical microscope with an ultranarrow Notch filter enabled an accurate spectroscopy on both the Stokes and anti-Stokes side down to about 8 cm−1. A splitting of the soft mode into E-u- and A-2u-symmetry vibrations is clearly observed in the micrograin sample. The peculiar temperature dependence of ω^2_A2u−ω^2_Eu emphasizes a nonlinear coupling between the square of the TiO6 tilting angle and the spontaneous strain c/a − 1. In addition, the very different temperature behavior of the linewidth of the two soft components is assigned to the growth of a disorder anisotropy on cooling, probably on the titanium site.

Free Vibration of Micro- and Nano-Shells Based on Modified Couple Stress Theory

Free Vibration of Micro- and Nano-Shells Based on Modified Couple Stress Theory

DYNAMICAL INSTABILITY ANALYSIS OF NANOTUBES USING NONLOCAL SHEAR BEAM THEORY

The dynamics stability of distributed systems (continua) has been an object of considerable attention over the past half century. Numerous papers are available on isotropic and laminated beams, shafts, plates and shells under periodic and random forces. Most papers have applied finite dimensional or modal approximations in the analysis of vibration and stability. This paper focuses on the stochastic parametric vibrations of micro- and nanorods based on Eringens nonlocal elasticity theory and shear beam theory. The almost sure asymptotic instability criteria involving a damping coefficient, structure and loading parameters are derived using Liapunov's direct method. Using the appropriate energy-like Liapunov functional, sufficient conditions for the almost sure asymptotic instability of undeflected form of beam are derived. The nonlocal shear beam accounts for the scale effect, which becomes significant when dealing with short micro- and nanorods. From the obtained analytical formulas it is clearly seen that the small scale effect increases the dynamic instability region. Instability regions are functions of the axial force variance, the constant component of axial force and the damping coefficient.

Surface effects on nonlinear free vibration of nanobeams

The nonlinear flexural vibrations of micro- and nanobeams in presence of surface effects are studied within the framework of Euler–Bernoulli beam theory including the von Kármán geometric nonlinearity. Exact solution is obtained for the natural frequencies of a simply-supported nanobeam in terms of the Jacobi elliptic functions by using the free vibration modes of the corresponding linear problem. Numerical results include the normalized natural frequencies of vibration as functions of mode number, vibration amplitude, and nanobeam length and thickness. Also, the influence of surface effects on the system phase trajectory is considered.

Computational and quasi-analytical models for non-linear vibrations of resonant MEMS and NEMS sensors

Large-amplitude non-linear vibrations of micro- and nano-electromechanical resonant sensors around their primary resonance are investigated. A comprehensive multiphysics model based on the Galerkin decomposition method coupled with the averaging method is developed in the case of electrostatically actuated clamped–clamped resonators. The model is purely analytical and includes the main sources of non-linearities as well as fringing field effects. The influence of the higher modes and the validation of the model is demonstrated with respect to the shooting method as well as the harmonic balance coupled with the asymptotic numerical method. This model allows designers to investigate the sensitivity variation of resonant sensors in the non-linear regime with respect to the electrostatic forcing.

A Model for Bending, Torsional, and Axial Vibrations of Micro- and Macro-Drills Including Actual Drill Geometry—Part I: Model Development and Numerical Solution

Part I of this work presents a combined one-dimensional/three-dimensional approach for obtaining a unified model for the dynamics of micro- and macro-drills. To increase the numerical efficiency of the model, portions of the drill with circular cross-section (shank, extension, and tapered sections) are modeled using one-dimensional beam models without compromising model accuracy. A three-dimensional model is used for an accurate representation of the fluted section, considering the actual geometry with the pretwisted shape and axially varying (nonaxisymmetric) cross-section. The actual cross-section of the drills is incorporated to the model through a polynomial mapping while the pretwist effect is captured by defining a rotating reference frame. The boundary-value problem for both one- and three-dimensional models are derived using a variational approach, based on the extended Hamilton’s principle, and are subsequently solved by applying the spectral-Tchebychev technique. A component-mode synthesis is used for connecting the individual sections to obtain the dynamic model for the entire drill. Convergence of the model is studied by varying the number of polynomials for each section. The experimental validation of the model is included in Part II for both macro- and micro-drills. Also included in Part II is an analysis of drill dynamics for varying drill-geometry parameters and axial (thrust) force.

A Model for Bending, Torsional, and Axial Vibrations of Micro- and Macro-Drills Including Actual Drill Geometry—Part II: Model Validation and Application

In Part II of this work, an experimental study is conducted to validate the three-dimensional (3D-ST) drill dynamics model. Modal experiments on macro- and micro-drills are performed by exciting the drills with small piezoelectric elements directly attached to the drill body. The response measurements are conducted in a noncontact manner using a laser Doppler vibrometer system. In addition, to perform the comparison on a complete frequency response function, rather than on only natural frequencies and mode shapes, an impact hammer test with a miniature hammer and a small accelerometer was conducted on one of the macro-drills. In the validation study, five macro-drills and three micro-drills with different geometric parameters are used. It was concluded that the 3D-ST model can capture both bending and torsional-axial natural frequencies and mode shapes of macro-drills (up to 15 kHz) and micro-drills (up to 90 kHz) with better than 4.5% accuracy, and with an average absolute error of 1.5%. For each case, the natural frequencies are also compared with those from detailed solid-element finite-elements (FEs) model to gain further insight about the 3D-ST model. The natural frequencies from the FE and 3D-ST models are seen to match with better than 1.5% accuracy. Subsequently, the effects of tool geometry (diameter, aspect ratio, helix angle, and web-taper) and axial (thrust) force on dynamics of macro- and micro-drills are analyzed.

Finite Element Analysis of Thermoelastic Damping in Contour-Mode Vibrations of Micro- and Nanoscale Ring, Disk, and Elliptical Plate Resonators

Thermoelastic damping in contour-mode in-plane vibrations of rings, disks, and elliptical plates is investigated on various size scales, using a reduced finite element formulation. The Fourier scheme is applied to the axisymmetric geometries including circular rings and disks, and is found to be remarkably efficient in searching solutions. The numerical accuracy is further improved by the implementation of quadratic interpolation functions. The computational results are validated by comparing with the commercial software packages as well as the existing analytical solutions in literature. For resonators of elliptical shapes, the dominant frequency has a weak dependence on the geometric aspect ratio γ, whereas the effect of γ on the quality factor (Q value) is much stronger and the peak Q value of the leading mode consistently occurs in the vicinity of γ=1.42.

Stochastic instability via nonlocal continuum mechanics

The dynamic stability problem is solved for one-dimensional structures subjected to time-dependent deterministic or stochastic axial forces. The stability analysis of structures under time-dependent forces strongly depends on dissipation energy. The paper is concerned with the stochastic parametric vibrations of micro- and nano-rods based on Eringen’s nonlocal elasticity theory and Euler–Bernoulli beam theory. The asymptotic instability, and almost sure asymptotic instability criteria involving a damping coefficient, structure and loading parameters are derived using Liapunov’s direct method. Using the appropriate energy-like Liapunov functional sufficient conditions for the asymptotic instability, and the almost sure asymptotic instability of undeflected form of beam are derived. The nonlocal Euler–Bernoulli beam accounts for the scale effect, which becomes significant when dealing with short micro- and nano- rods. From obtained analytical formulas it is clearly seen that the small scale effect increases the dynamic instability region. Instability regions are functions of the axial force variance, the constant component of axial force and the damping coefficient.

Free vibration of microscaled Timoshenko beams

In this paper, a comprehensive model is presented to investigate the influence of surface elasticity and residual surface tension on the natural frequency of flexural vibrations of microbeams in the presence of rotary inertia and shear deformation effects. An explicit solution is derived for the natural oscillations of microscaled Timoshenko beams considering surface effects. The analytical results are illustrated with numerical examples in which two types of microbeams are configured based on Euler–Bernoulli and Timoshenko beam theory considering surface elasticity and residual surface tension. The natural frequencies of vibration are calculated for selected beam length on the order of nanometer to microns and the results are compared with those corresponding to the classical beam models, emphasizing the differences occurring when the surface effects are significant. It is found that the nondimensional natural frequency of the vibration of micro and nanoscaled beams is size dependent and for limiting case in which the beam length increases, the results tends to the results obtained by classical beam models. This study might be helpful for the design of high-precision measurement devices such as chemical and biological sensors.

Mechanical detection and mode shape imaging of vibrational modes of micro and nanomechanical resonators by dynamic force microscopy

We describe a method based on the use of higher order bending modes of the cantilever of a dynamic force microscope to characterize vibrations of micro and nanomechanical resonators at arbitrarily large resonance frequencies. Our method consists on using a particular cantilever eigenmode for standard feedback control in amplitude modulation operation while another mode is used for detecting and imaging the resonator vibration. In addition, the resonating sample device is driven at or near its resonance frequency with a signal modulated in amplitude at a frequency that matches the resonance of the cantilever eigenmode used for vibration detection. In consequence, this cantilever mode is excited with an amplitude proportional to the resonator vibration, which is detected with an external lock-in amplifier. We show two different application examples of this method. In the first one, acoustic wave vibrations of a film bulk acoustic resonator around 1.6 GHz are imaged. In the second example, bending modes of carbon nanotube resonators up to 3.1 GHz are characterized. In both cases, the method provides subnanometer-scale sensitivity and the capability of providing otherwise inaccessible information about mechanical resonance frequencies, vibration amplitude values and mode shapes.

Thermoelastic damping in the contour-mode vibrations of micro- and nano-electromechanical circular thin-plate resonators

This paper presents an analytical study on thermoelastic damping (TED) in the contour-mode vibrations of micro- and nano-electromechanical circular thin-plate resonators. Instead of expressing TED in terms of a commonly used complex frequency value, this work calculates TED by using a thermal-energy approach in which the generation of thermal energy per cycle of vibration is considered. To demonstrate its validity, this thermal-energy approach is first utilized to tackle the well-known TED in a flexural-mode beam resonator. Then, it is extended to analyzing TED in the contour-mode vibrations of a circular thin-plate micro-/nano-resonator. Consequently, the behavior of TED versus the key design parameters, namely thin-plate radius and resonant frequency, is predicted, and the attainable quality factors of such type of resonators are defined. From this work, it is found that the Q TED of the contour-mode vibrations of a circular thin-plate resonator is well above 1×10 6 when its resonant frequency is below 1 GHz and TED becomes a significant source of dissipation for circular thin-plate resonators at the nanometer scale.