Vibration of scanning near-field optical microscope probe with laser-induced thermal effect using Timoshenko beam theory

The Timoshenko beam theory, including the effects of rotary inertia and shear deformation, is used to analyze the resonant frequency of lateral vibration of scanning near-field optical microscope (SNOM) tapered probe with a laser-induced thermal effect. In the analysis, the thermal effect can be considered as an axial force and is dependent of temperature distribution of the probe. The Rayleigh–Ritz method is used to solve the vibration problem of the probe. According to the analysis, the frequencies of the first three vibration modes increase when the thermal effect is taken into account. The effects of shear deformation and rotary inertia on the frequency ratio of a Timoshenko beam to an Euler beam increase when the mode number increases and the contact stiffness decreases. In addition, the frequency of mode 1 increases with increasing taper angle and coating thickness of the probe. Comparison of the frequency of a SNOM probe coated with Al, Ag, or Au, the highest is with Al coating, and the lowest is with Au coating.     

Free vibration of a single-walled carbon nanotube containing a fluid flow using the Timoshenko beam model

The flexural vibration of the fluid-conveying single-walled carbon nanotube (SWCNT) is derived by the Timoshenko beam model, including rotary inertia and transverse shear deformation. The effects of the flow velocity and the aspect ratio of length to diameter on the vibration frequency and mode shape of the SWCNT are analyzed. Results show that the effects of rotary inertia and transverse shear deformation result in a reduction of the vibration frequencies, especially for higher modes of vibration and short nanotubes. The frequency is also compared with the previous study based on Euler beam model. In addition, if the ratio of length to diameter increased to 60, the influence of the shear deformation and rotary inertia on the mode shape and the resonant frequencies can be neglected. However, the influence is very obvious when the ratio decreased to 20. As the flow velocity of the fluid increases in the vicinity of 2π, the SWCNT reveals the divergence instability. It regains stability when the flow velocity reaches about 9. As the velocity increases further, the SWCNT undergoes a coupled-mode flutter and results in a larger amplitude.     

Resonance frequency of chiral single-walled carbon nanotubes using Timoshenko beam theory

This Letter develops a model that analyzes the resonant frequency of the chiral single-walled carbon nanotubes (SWCNTs) subjected to a thermal vibration by using Timoshenko beam model, including the effect of rotary inertia and shear deformation. The analytical solution is derived and the frequency equation is obtained. The results based on the beam model show that the frequency increases with decreasing the nanotube aspect ratio of length to diameter. In addition, the frequency obtained by Timoshenko beam model is lower than that calculated by Euler beam model. As the nanotube aspect ratio of length to diameter decreased, the discrepancy is more obvious. Furthermore, as the effect of thermal vibration increases, the frequency for chiral SWCNTs decreases.     

Scale effects on thermal buckling properties of carbon nanotube

In this Letter, the thermal buckling properties of carbon nanotube with small scale effects are studied. Based on the nonlocal continuum theory and the Timoshenko beam model, the governing equation is derived and the nondimensional critical buckling temperature is presented. The influences of the scale coefficients, the ratio of the length to the diameter, the transverse shear deformation and rotary inertia are discussed. It can be observed that the small scale effects are significant and should be considered for thermal analysis of carbon nanotube. The nondimensional critical buckling temperature becomes higher with the ratio of length to diameter increasing. Furthermore, for smaller ratios of the length to the diameter and higher mode numbers, the transverse shear deformation and rotary inertia have remarkable influences on the thermal buckling behaviors.     

Free vibration analysis of Mindlin plates with linearly varying thickness

A method based on the variational principles in conjunction with the finite difference technique is applied to examine the free vibration characteristics of isotropic rectangular plates of linearly varying thickness by including the effects of transverse shear deformation and rotary inertia. The validity of the present approach is demonstrated by comparing the results with other solutions proposed for plates with uniform and linearly varying thickness. Natural frequencies and mode shapes of Mindlin plates with simply supported and clamped edges are determined for various values of relative thickness ratio and the taper thickness constant.     

Free vibration and parametric resonance of shear deformable functionally graded cylindrical panels

This paper investigates free vibration and dynamic instability of functionally graded cylindrical panels subjected to combined static and periodic axial forces and in thermal environment. Theoretical formulations are based on Reddy's higher order shear deformation shell theory to account for rotary inertia and the parabolic distribution of the transverse shear strains through the panel thickness. Thermal effects due to steady temperature change are included in the analysis. Material properties are assumed to be temperature dependent and graded in the thickness direction according to a power-law distribution in terms of the volume fractions of the constituents. The panel under current consideration is clamped or simply supported on two straight edges and may be either free, simply supported or clamped on the curved edges. A semi-analytical approach, which takes the advantages of one-dimensional differential quadrature approximation, Galerkin technique and Bolotin's method, is employed to determine the natural frequencies and the unstable regions of the panel. Numerical results for silicon nitride/stainless-steel cylindrical panels are given in both dimensionless tabular and graphical forms. Effects of material composition, temperature rise, panel geometry parameters, and boundary conditions on free vibration and the parametric resonance are also studied.     

Free vibration analysis of functionally graded curved panels using a higher-order finite element formulation

Free vibration analysis of functionally graded curved panels is carried out using a higher-order formulation. A C⁰ finite element formulation is used to carry out the analysis. The element consists of nine degrees of freedom per node with higher-order terms in the Taylor's-series expansion, which represents the higher-order transverse cross-sectional deformation modes. The formulation includes Sanders’ approximation for doubly curved shells considering the effects of rotary inertia and transverse shear. A realistic parabolic distribution of transverse shear strains through the shell thickness is assumed and the use of shear correction factor is avoided. Material properties are assumed to be temperature independent and graded in the thickness direction according to a simple power-law distribution in terms of the volume fractions of the constituents. Heat conduction between ceramic and metal constituents is neglected. The accuracy of the formulation is validated by comparing the results with those available in the literature. Effects of panel geometry parameters and boundary conditions are studied.     

A variational approach to free vibration analysis of shear deformable polygonal plates with variable thickness

This paper presents a simple and general variational approach for the study of the free vibration behaviour of polygonal isotropic plates with variable thickness. The Reissner-Mindlin plate theory is used to take into account the effects of shear deformation and rotary inertia in the analysis. Moreover, this theory allows obtaining greater accuracy of frequency coefficients corresponding to vibration higher modes, even for the thin plates.The governing eigenvalue equation is obtained employing the Ritz method. The plate geometry is approximated by using non-orthogonal triangular co-ordinates, while sets of independent polynomials, expressed in these co-ordinates, are employed to approximate the displacement and rotation fields. The developed algorithm allows obtaining approximated analytical solutions for plates with different aspect ratios, thickness variation and boundary conditions, including edges elastically restrained by both translational and rotational springs. Therefore, a unified program has been easily implemented. Convergence and comparison analyzes are carried out to verify the reliability and accuracy of the numerical solutions. Finally, sets of parametric studies are performed and the results are given in graphical and tabular form.     

Large amplitude vibration of an initially stressed cross ply laminated plates

In this paper, nonlinear equations of large amplitude vibration for a laminated plate in a general state of nonuniform initial stress are derived. The equations include the effects of transverse shear and rotary inertia. Using these derived governing equations, the large amplitude vibration behaviour of an initially stressed cross-ply laminated plate is studied. The initial stress is taken to be a combination of pure bending stress plus an extensional stress in the plane of the plate. The Galerkin method is used to reduce the governing nonlinear partial differential equations to ordinary nonlinear differential equations and the Runge-Kutta method is used to obtain the nonlinear to linear frequencies. The frequency responses of nonlinear vibration are sensitive of the vibration amplitude, aspect ratio, thickness ratio, modulus ratio, stack sequence, layer number and state of initial stresses. The effects of various parameters on the large amplitude free vibrations are presented.     

Wave propagation of functionally graded material plates in thermal environments

The wave propagation of an infinite functionally graded plate in thermal environments is studied using the higher-order shear deformation plate theory. The thermal effects and temperature-dependent material properties are both taken into account. The temperature field considered is assumed to be a uniform distribution over the plate surface and varied in the thickness direction only. Material properties are assumed to be temperature-dependent, and graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents. Considering the effects of transverse shear deformation and rotary inertia, the governing equations of the wave propagation in the functionally graded plate are derived by using the Hamilton’s principle. The analytic dispersion relation of the functionally graded plate is obtained by solving an eigenvalue problem. Numerical examples show that the characteristics of wave propagation in the functionally graded plate are relates to the volume fraction index and thermal environment of the functionally graded plate. The influences of the volume fraction distributions and temperature on wave propagation of functionally graded plate are discussed in detail. The results carried out can be used in the ultrasonic inspection techniques and structural health monitoring.     

Effects of mode coupling on the admittance of an AT-cut quartz thickness-shear resonator

We study the effects of couplings to flexure and face-shear modes on the admittance of an AT-cut quartz plate thickness-shear mode resonator. Mindlin’s two-dimensional equations for piezoelectric plates are employed. Electrically forced vibration solutions are obtained for three cases: pure thickness-shear mode alone; two coupled modes of thickness shear and flexure; and three coupled modes of thickness shear, flexure, and face shear. Admittance is calculated and its dependence on the driving frequency and the length/thickness ratio of the resonator is examined. Results show that near the thickness-shear resonance, admittance assumes maxima, and that for certain values of the length/thickness ratio, the coupling to flexure causes severe admittance drops, while the coupling to the face-shear mode causes additional admittance changes that were previously unknown and hence are not considered in current resonator design practice.     

Coupled bending and twisting of a timoshenko beam

Allowance is made for shear deflection and for rotary inertia of a non-uniform beam that executes coupled bending and twisting vibration. Principal modes are found, orthogonality conditions established and modal equations of forced motion derived.     

Dynamic characteristics of curved nanobeams using nonlocal higher-order curved beam theory

Here, an analytical approach for the dynamic analysis, viz., free and forced vibrations, of curved nanobeams using nonlocal elasticity beam theory based on Eringen formulation coupled with a higher-order shear deformation accounting for through thickness stretching is investigated. The formulation is general in the sense that it can be deduced to analyse the effect of various structural theories pertaining to curved nanobeams. It also includes inplane, rotary and coupling inertia terms. The governing equations derived, using Hamiltons principle, are solved in conjunction with Naviers solutions. The free vibration results are obtained employing the standard eigenvalue analysis whereas the displacement/stress responses in time domain for the curved nanobeams subjected to rectangular pulse loading are evaluated based on Newmarks time integration scheme. The formulation is validated considering problems for which solutions are available. A comparative study is done here by different theories obtained through the formulation. The effects of various structural parameters such as thickness ratio, beam length, rise of the curved beam, loading pulse duration, and nonlocal scale parameter are brought out on the dynamic behaviours of curved nanobeams.     

Stability and vibration of isotropic,orthotropic and laminated plates according to a higher-order shear deformation theory

A higher-order shear deformation theory is used to determine the natural frequencies and buckling loads of elastic plates. The theory accounts for parabolic distribution of the transverse shear strains through the thickness of the plate and rotary inertia. Exact solutions of simply supported plates are obtained and the results are compared with the exact solutions of three-dimensional elasticity theory, the first-order shear deformation theory, and the classical plate theory. The present theory predicts the frequencies and buckling loads more accurately when compared to the first-order and classical plate theories.     

Rayleigh-Ritz vibration analysis of Mindlin plates

The Rayleigh-Ritz method is applied to the prediction of the natural frequencies of flexural vibration of square plates having general boundary conditions. The analysis is based on the use of Mindlin plate theory so that the effects of shear deformation and rotary inertia are included. The spatial variations of the plate deflection and the two rotations over the plate middle surface are assumed to be series of products of appropriate Timoshenko beam functions. Results are presented for a number of types of plate and these demonstrate the manner of convergence of the method as the number of terms in the assumed series increases.     

Vibrations of annular plates including the effects of rotatory inertia and transverse shear deformation

An investigation of the natural vibrations of isotropic annular plates of uniform thickness has been made by considering the effects of rotatory inertia and shear deformation. The frequency determinantal equations are derived in explicit form for nine sets of common boundary conditions. Numerical results for the frequency parameters of annular plates having various thickness ratios and inner to outer radii ratios have been obtained. The results are compared with those given by the classical plate theory wherever possible. Among the effects of transverse shear deformation and rotatory inertia, the effect of shear deformation has been found to be more prominent.     

Elastic vibrations of a cylindrical nanotube with the effect of surface stress and surface inertia

Vibration frequency analysis of nanostructures may be essential for study of their thermal conductivity and mechanical characterization. Given the high surface-to-volume ratio, the elastic vibrations of an infinitely long cylindrical nanotube have been studied by considering both the effects of surface stress and that of surface inertia within the framework of surface elasticity. The phonon dispersion and the resonant frequencies for the specific vibration modes have been calculated. Numerical results have indicated that the surface stress and the surface inertia have equally important effect on the vibration behavior of the nanotube that may depend on the vibration modes as well. Due to the surface effect, the vibration modes of lower order by the classical elasticity may be indeed the modes of higher order. The surface effect on the low-frequency Raman shift has also been found.     

Buckling and free vibration analysis of functionally graded cylindrical shells subjected to a temperature-specified boundary condition

Linear thermal buckling and free vibration analysis are presented for functionally graded cylindrical shells with clamped-clamped boundary condition based on temperature-dependent material properties. The material properties of functionally graded materials (FGM) shell are assumed to vary smoothly and continuously across the thickness. With high-temperature specified on the inner surface of the FGM shell and outer surface at ambient temperature, 1D heat conduction equation along the thickness of the shell is applied to determine the temperature distribution; thereby, the material properties based on temperature distribution are made available for thermal buckling and free vibration analysis. First-order shear deformation theory along with Fourier series expansion of the displacement variables in the circumferential direction are used to model the FGM shell. Numerical studies involved the understanding of the influence of the power-law index, r/h and l/r ratios on the critical buckling temperature. Free vibration studies of FGM shells under elevated temperature show that the fall in natural frequency is very drastic for the mode corresponding to the lowest natural frequency when compared to the lowest buckling temperature mode.     

Analysis of a disk-type stator for the piezoelectric ultrasonic motor using impedance matrix

Impedance and admittance matrices of a piezoelectric annular actuator with segmented electrodes are presented for the analysis of the disk-type piezoelectric ultrasonic motors (USM). Equations of motion and the conjugate parameters for the impedance and admittance matrices are derived using the variational principle. In the derivation, the electric field in the piezoelectric layer is assumed to be constant over the area covered by a particular electrode, and the effects of both shear deformation and rotary inertia are taken into account. The resonance and antiresonance frequencies and the vibrating modes are calculated for the various resonance modes and boundary conditions, and the results are compared with those by the three-dimensional finite element methods. They are in excellent agreement with each other. It is expected that the derived impedance matrix can be effectively applied to the analysis and the design of the USM.     

Free vibration analysis of functionally graded plates using the element-free kp-Ritz method

A free vibration analysis of metal and ceramic functionally graded plates that uses the element-free kp-Ritz method is presented. The material properties of the plates are assumed to vary continuously through their thickness according to a power-law distribution of the volume fractions of the plate constituents. The first-order shear deformation plate theory is employed to account for the transverse shear strain and rotary inertia, and mesh-free kernel particle functions are used to approximate the two-dimensional displacement fields. The eigen-equation is obtained by applying the Ritz procedure to the energy functional of the system. Convergence studies are performed to examine the stability of the proposed method, and comparisons of the solutions derived with those reported in the literature are provided to verify its accuracy. Four types of functionally graded rectangular and skew plates—Al/Al₂O₃, Al/ZrO₂, Ti–6Al–4V/Aluminum oxide, and SUS304/Si₃N₄—are included in the study, and the effects of the volume fraction, boundary conditions, and length-to-thickness ratio on their frequency characteristics are discussed in detail.