Thermoelastic vibrations in micro-/nano-scale beam resonators with voids

In this paper the closed form expressions for the transverse vibrations of a homogenous isotropic, thermoelastic thin beam with voids, based on Euler-Bernoulli theory have been derived. The effects of voids, relaxation times, thermomechanical coupling, surface conditions and beam dimensions on energy dissipation induced by thermoelastic damping in (micro-electromechanical systems) MEMS/(nano-electromechanical systems) NEMS resonators are investigated for beams under clamped and simply supported conditions. Analytical expressions for deflection, temperature change, frequency shifts and thermoelastic damping in the beam have been derived. Some numerical results with the help of MATLAB programming software in case of magnesium like material have also been presented. The computer simulated results in respect of damping factor and frequency shift have been presented graphically.     

Application of the Green functions to the problem of the thermally induced vibration of a beam

This paper considers the problem of the transverse vibrations of a beam induced by a mobile heat source. The formulation of the problem is based on the differential equations of heat conduction and transverse vibrations of the beam, which are complemented by suitable initial and boundary conditions. The effect of internal and external damping on the vibrations of the beam is considered. The solution to the problem in analytical form is obtained by using the properties of the Green functions. A time partitioning method has been used to avoid the difficulties associated with the slow convergence of the series occurring in the solution to the heat conduction problem. The numerical results of the thermally induced vibration of the beam are presented.     

Thermoelastic damping in torsion microresonators with coupling effect between torsion and bending

Predicting thermoelastic damping (TED) is crucial in the design of high Q MEMS resonators. In the past, there have been few works on analytical modeling of thermoelastic damping in torsion microresonators. This could be related to the assumption of pure torsional mode for the supporting beams in the torsion devices. The pure torsional modes of rectangular supporting beams involve no local volume change, and therefore, they do not suffer any thermoelastic loss. However, the coupled motion of torsion and bending usually exists in the torsion microresonator when it is not excited by pure torque. The bending component of the coupled motion causes flexural vibrations of supporting beams which may result in significant thermoelastic damping for the microresonator. This paper presents an analytical model for thermoelastic damping in torsion microresonators with the coupling effect between torsion and bending. The theory derives a dynamic model for torsion microresonators considering the coupling effect, and approximates the thermoelastic damping by assuming the energy loss to occur only in supporting beams of flexural vibrations. The thermoelastic damping obtained by the present model is compared to the measured internal friction of single paddle oscillators. It is found that thermoelastic damping contributes significantly to internal friction for the case of the higher modes at room temperature. The present model is validated by comparing its results with the finite-element method (FEM) solutions. The effects of structural dimensions and other parameters on thermoelastic damping are investigated for the representative case of torsion microresonators.     

Damping in micro-scale generalized thermoelastic circular plate resonators

The out-of-plane vibrations of a generalized thermoelastic circular plate are studied under different environmental temperature, plate dimensions and boundary conditions. The analytical expressions for thermoelastic damping of vibration and phase velocity of circumferential surface wave modes are obtained. It is noticed that the damping of vibrations and phase velocities of circumferential surface wave modes significantly depend on thermal relaxation time in addition to thermoelastic coupling in circular plates under resonance conditions. The surface conditions also impose significant effects on the vibrations of such resonators. The expressions for displacement and temperature fields in the plate resonator are also derived and obtained. Some numerical results have also been presented for illustration purpose in case of silicon material plate.     

Rod Vibrations Caused by Axial Impact

The axial impact by an elastic body on an elastic-rod end with a fixed opposite end is considered. The propagation of elastic waves in the rod and the local deformations in the contact zone are taken into account. After recoil of the body, the rod performs free longitudinal vibrations, which can under certain conditions cause parametric transverse vibrations having the character of beats. Depending on the parameters of the problem, the collision time, the shock-pulse shape, and the greatest amplitude of the transverse vibrations under parametric resonance are determined.     

Thermoelastic vibrations of a semi-infinite strip

The forced vibrations of a thermoelastic strip (x₁ϵ [0, l], x₂ ⩾ 0) produced by a prescribed heating at the boundary x₂ = 0 are considered. The analysis is based on classical coupled thermoelastic theory (plane strain) and finite Fourier transforms. The solution of the problem under discussion is expressed in terms of the Lamé scalar and vector potentials.     

Dynamic Interaction of Longitudinal and Transverse Laser-Induced Oscillations of a Rod

Doklady Physics - The problem of the interaction of longitudinal and transverse vibrations of a rod under thermal actions using laser radiation is considered. Simultaneously taking into account the...     

The coupled transverse vibrations of a spinning membrane disk with a central hub

The results of an investigation of the effect of hub coupling on the transverse free vibrations of a spinning membrane disk are presented. The equations of motion are formulated by an energy method and Rayleigh-Ritz procedure. It is found by using assumed modal distributions consisting of the normal modes for the spinning membrane with a rigidly clamped hub that the problem partitions into sub-problems associated with (i) symmetric vibrations involving only the generalized coordinates from the symmetric input modes, (ii) unsymmetric vibrations involving only the input modes with one nodal diameter, and (iii) unsymmetric vibrations involving input modes with more than one nodal diameter. The degree of hub coupling is found to depend primarily on the ratio of the mass of the membrane to the mass of the disk and on the ratio of the radius of the hub to the radius of the disk.     

Free vibrations of a generally restrained rectangular plate with an internal line hinge

This paper deals with the study of free transverse vibrations of rectangular plates with an internal line hinge and elastically restrained boundaries. The equations of motion and its associated boundary and transition conditions are rigorously derived using Hamilton’s principle. The governing eigenvalue equation is solved employing a combination of the Ritz method and the Lagrange multipliers method. The deflections of the plate and the Lagrange multipliers are approximated by polynomials as coordinate functions. The developed algorithm allows obtaining approximate solutions for plates with different aspect ratios, boundary conditions, including edges elastically restrained by both translational and rotational springs, and arbitrary locations of the line hinge. Therefore, a unified algorithm has been implemented. Sets of parametric studies are performed and the results are given in graphical and tabular form.     

Possible nature of oscillations in the time dependences of the internal friction and effective shear modulus in Al-Cu alloys

The conditions of the appearance of in-phase oscillations in the time dependences of the low-frequency internal friction and effective shear modulus G _eff in Al-0.01 wt% Cu and Al-0.001 wt % Cu alloys are studied. The in-phase oscillations are shown to appear in the alloys only when their dislocation-impurity structure is disturbed from equilibrium and the impurity concentration near a dislocation is optimum. This fact suggests that the in-phase oscillations are due to a collective character of dislocation-impurity interactions, which leads to a transfer of energy of translational dislocation motion to transverse dislocation vibrations in the main slip plane.     

Identification of laser-generated ultrasounds in the response of the cylinder over time and space

A theoretical solution is presented to identify laser-generated ultrasounds in the transient response over time and space of a cylinder impacted by a laser line pulse. Theoretical radial displacements at various observation angles are obtained for an aluminum cylinder under thermoelastic regime. The corresponding displacements are observed experimentally by the laser ultrasonic technique. Good agreement is found in the time arrival, shape and relative amplitude of surface waves and various longitudinal and transverse bulk waves. These laser-generated ultrasounds are further identified by the ray trajectory analysis. This work will be helpful for the inverse problem of the nondestructive evaluation of cylinder parts.     

Roper Resonances as Vibrating Flux Tubes Between Quarks

The Roper resonances of the nucleon are described as transverse vibrations of astretched flux tube between the three quarks. The proton is modeled using currentmass quarks interacting with a confining linear flux tube potential plus thespin-dependent parts of a one-gluon exchange potential. The proton ground state hasno vibrations and the confining flux tube has the minimum length required toconnect the three quarks. The flux tube has a V or Y shape of two or threesegments, depending on the locations of the quarks. The vibrations of the fluxtube have nodes at the quarks and at the apex of the Y-shaped configuration andprovide the vibrational excitation energy to describe the proton excitations. Theamplitude of the transverse vibrations is found from a geometric analysis, anddepends on the string constant of the flux tube potential. The Roper 1. 440-GeVresonance energy is very nearly reproduced by the vibration with mode number1 acting in only one segment of the flux tube. The vibration with mode number2 in one segment of the flux tube closely reproduces the second proton excitationat 1.710 GeV. The D excited states are also well reproduced by these modes ofa vibrating flux tube.     

Transient analysis of nonlinear Euler–Bernoulli micro-beam with thermoelastic damping,via nonlinear normal modes

In this paper an Euler–Bernoulli model has been used for vibration analysis of micro-beams with large transverse deflection. Thermoelastic damping is considered to be the dominant damping mechanism and introduced as imaginary stiffness into the equation of motion by evaluating temperature profile as a function of lateral displacement. The obtained equation of motion is analyzed in the case of pure single mode motion by two methods; nonlinear normal mode theory and the Galerkin procedure. In contrast with the Galerkin procedure, nonlinear normal mode analysis introduces a nonconventional nonlinear damping term in modal oscillator which results in strong damping in case of large amplitude vibrations. Evaluated modal oscillators are solved using harmonic balance method and tackling damping terms introduced as an imaginary stiffness is discussed. It has been shown also that nonlinear modal analysis of micro-beam with thermoelastic damping predicts parameters such as inverse quality factor, and frequency shift, to have an extrema point at certain amplitude during transient response due to the mentioned nonlinear damping term; and the effect of system?s characteristics on this critical amplitude has also been discussed.     

Calculation of bulkhead vibrations in a supported shell simulating a plane fuselage

Noise from external sources penetrates a plane cabin through the board construction in several ways. They include direct penetration through loose-fiber layers and indirect penetration through the attachment points of interior panels to transverse ribs (bulkheads). The analytical method of calculating vibrations of an orthogonally supported shell (developed by us earlier) makes it possible to correctly calculate bulkhead vibrations. As a result, noise penetration into the cabin through the attachment points of interior panels can be determined analytically. The first part of the solution to this problem is presented (i.e., the relations and examples of calculating bulkhead vibrations upon point excitation of the shell and excitation by pressure fluctuations of the turbulent boundary layer are given).     

Transverse rod vibrations under a short-term longitudinal impact

Transverse vibrations of a thin rod caused by a short-term longitudinal impact are considered. After the impact, a system of compression-tension waves, which causes transverse vibrations, appears in the rod. Parametric resonance, which leads to an unbounded rise in the amplitude of transverse vibrations, is investigated in the linear approximation. With a nonlinear approach, beats appear in the resonance vicinity, in which the mutual exchange of the energy of longitudinal and transverse vibrations occurs. The influence of viscoelastic resistance forces is investigated.     

Forced transverse vibrations of an elastically connected complex simply supported double-beam system

The present paper is devoted to analyzing undamped forced transverse vibrations of an elastically connected complex double-beam system. The problem is formulated and solved in the case of simply supported beams. The classical modal expansion method is applied to ascertain dynamic responses of beams due to arbitrarily distributed continuous loads. Several cases of particularly interesting excitation loadings are investigated. The action of stationary harmonic loads and moving forces is considered. In discussing vibrations caused by exciting harmonic forces, conditions of resonance and dynamic vibration absorption are determined. The beam-type dynamic absorber is a new concept of a continuous dynamic vibration absorber (CDVA), which can be applied to suppress excessive vibrations of corresponding beam systems. A numerical example is presented to illustrate the theoretical analysis.     

Flexural and transversal wave motion in homogeneous isotropic thermoelastic plates by using asymptotic method

The present investigation is concerned with the flexural and transversal wave motion in an infinite, transversely isotropic, thermoelastic plate by asymptotic method. The governing equations for the flexural and transversal motions have been derived from the system of three-dimensional dynamical equations of linear theory of coupled thermoelasticity. The asymptotic operator plate model for free vibrations; both flexural and transversal, in a homogenous thermoelastic plate leads to fifth degree and cubic polynomial secular equations, respectively, that governs frequency and phase velocity of various possible modes of wave propagation at all wavelengths. All the coefficients of differential operator have been expressed as explicit functions of the material parameters. The velocity dispersion equations for the flexural and transversal wave motion have been deduced from the three-dimensional analog of Rayleigh-Lamb frequency equation for thermoelastic plate waves. The approximations for long and short waves and expression for group velocity have also been derived. The thermoelastic Rayleigh-Lamb frequency equations for the considered plate are expanded in power series in order to obtain polynomial frequency and velocity dispersion relations whose equivalence is established with that of asymptotic method. The dispersion curves for phase velocity, group velocity and attenuation coefficient of various flexural and transversal wave modes are shown graphically for aluminum-epoxy material elastic and thermoelastic plates.     

Acoustic waves generated by a laser line pulse in a hollow cylinder

A theoretical solution is proposed to predict acoustic waves generated in a homogeneous and isotropic hollow cylinder by a laser line source under either ablation or thermoelastic regime. The Fourier series expansion is introduced for one spatial coordinate to solve this transient response problem. Theoretical displacements are obtained in both regimes for aluminum hollow cylinders with various thickness including a rod of the same size. The corresponding displacements are observed experimentally by the laser ultrasonic technique. Agreement has been found in the time arrival, shape and relative amplitude of surface waves and various longitudinal and transverse bulk waves. These acoustic waves are further identified by the ray trajectory analysis. This work will be helpful when dealing with the inverse problem of the nondestructive evaluation of hollow cylindrical parts.     

High-speed Monte Carlo calculations of crystal blocking dips

Summary A new method for the calculation of blocking dips in single-crystals is described. It makes use of the continuum model to generate the distribution of initial transverse energies and of a Monte Carlo sampling for the exit angular distribution of ions emitted with a given transverse energy. The results are very similar to traditional Monte Carlo simulations and are obtained with a factor of 100 smaller CPU time consumption. Compensating shoulders are absent, or appear insuficient if use is made of the half-way-plane model. This suggests that correlated thermal vibrations might be important in blocking phenomena. Work partly supported by MPI founds (60%).