Transverse Vibration Modes Research Articles

Localized vibration of a microtubule surrounded by randomly distributed cross linkers.

Based on finite element simulation, the present work studies free vibration of a microtubule surrounded by 3D randomly distributed cross linkers in living cells. A basic result of the present work is that transverse vibration modes associated with the lowest frequencies are highly localized, in sharp contrast to the through-length modes predicted by the commonly used classic elastic foundation model. Our simulations show that the deflected length of localized modes increases with increasing frequency and approaches the entire length of microtubule when frequency approaches the minimum classic frequency given by the elastic foundation model. In particular, unlike the length-sensitive classic frequencies predicted by the elastic foundation model, the lowest frequencies of localized modes predicted by the present model are insensitive to the length of microtubules and are at least 50% lower than the minimum classic frequency for infinitely long microtubules and could be one order of magnitude lower than the minimum classic frequency for shorter microtubules (only a few microns in length). These results suggest that the existing elastic foundation model may have overestimated the lowest frequencies of microtubules in vivo. Finally, based on our simulation results, some empirical relations are proposed for the critical (lowest) frequency of localized modes and the associated wave length. Compared to the classic elastic foundation model, the localized vibration modes and the associated wave lengths predicted by the present model are in better agreement with some known experimental observations.

Exact closed form solution for the analysis of the transverse vibration modes of a Timoshenko beam with multiple concentrated masses

Concentrated masses on the beams have many industrial applications such as gears on a gearbox shafts, blades and disks on gas and steam turbine shafts, and mounting engines and motors on structures. Transverse vibration of the beam carrying a point mass was studied in many cases by both Euler–Bernoulli and Timoshenko beam theory for a limited number of concentrated masses mounted on a specific place on the beam. This was also investigated for a beam carrying multiple concentrated masses, yet they were solved by numerical methods such as Differential Quadrature (DQ) method. The present study investigated an exact solution for free transverse vibrations of a Timoshenko beam carrying multiple arbitrary concentrated masses anywhere on the beam with various boundary conditions. Using Dirac’s delta in governing equations, the effects of concentrated masses were imposed. After extracting a closed form solution, basic functions were used to reduce the amount of computations. Standard symmetric and asymmetric boundary conditions were enforced for beam; in addition, the effects of value, position, and number of concentrated masses were examined. Generally, while the existence of concentrated masses reduces the natural frequencies, the reduction depends on the parameters of concentrated masses. Finally, there were acquired mode shapes for different boundary conditions and different value, position, and number of concentrated masses.

Evaluation of Mode Dependent Fluid Damping in a High Frequency Drumhead Microresonator

Design of high quality factor (Q) micromechanical resonators depends critically on our understanding of energy losses in their oscillations. The Q of such structures depends on process induced prestress in the structural geometry, interaction with the external environment, and the encapsulation method. We study the dominant fluid interaction related losses, namely, the squeeze film damping and acoustic radiation losses in a drumhead microresonator subjected to different prestress levels, operated in air, to predict its Q in various modes of oscillation. We present a detailed research of the acoustic radiation losses, associated with the 15 transverse vibration modes of the resonator using a hybrid analytical-computational approach. The prestressed squeeze film computation is based on the standard established numerical procedure. Our technique of computing acoustic damping based quality factor Q <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ac</sub> includes calculation of the exact prestressed modes. We find that acoustic losses result in a non-monotonic variation of Q <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ac</sub> in lower unstressed modes. Such non-monotonic variation disappears with the increase in the prestress levels. Although squeeze film damping dominates the net Q at lower frequencies, acoustic radiation losses dominate at higher frequencies. The combined computed losses correctly predict the experimentally measured Q of the resonator over a large range of resonant frequencies.

Mobility Matrix of a Thin Circular Plate Carrying Concentrated Masses Based on Transverse Vibration Solution

When calculating the vibration or sound power of a vibration source, it is necessary to know the point mobility of the supporting structure. A new method is presented for the calculation of point mobility matrix of a thin circular plate with concentrated masses in this paper. Transverse vibration mode functions are worked out by utilizing the structural circumferential periodicity of the inertia excitation produced by the concentrated masses. The numerical vibratory results, taking the clamped case as an instance, are compared to the published ones to validate the method for ensuring the correctness of mobility solution. Point mobility matrix, including the driving and transfer point mobility, of the titled structure is computed based on the transverse vibration solution. After that, effect of the concentrated masses on the mechanical point mobility characteristics is analyzed.

Seismic Response on the Mangjiedu Long-Span PC Continuous Rigid Frame Bridge with High Piers

The selected high-span continuous rigid frame bridge for studying is Mangjiedu Bridge in this paper. Based on the principle of structure dynamics and the method of seismic response analysis, the finite element model of the main bridge in Mangjiedu Bridge under Midas-civil was built according to the structural features, site conditions and seismic fortification intensity of the bridge. Then the characteristics of main bridge structure were studied using Lanzcos modal analysis method, and the seismic responses of main bridge are studied by earthquake response spectrum analysis and time history analysis respectively. The results show that the fundamental frequency of the main bridge is 0.1943Hz, and the transverse vibration mode shapes are most remarkable in the former 10 ordered types of vibration of main bridge. The main bridge is in the conditions of elastic range and does not failure under E1 earthquake. Both response spectrum analysis and time history analysis show that the largest displacement along the main bridge appears at the top of pier and the largest transverse displacement appears in the central position of main span. It also shows that the maximum shear and maximum moment occur at the bottom of main pier.

Seismic Response on the Mangjiedu Long-Span PC Continuous Rigid Frame Bridge with High Piers

The selected high-span continuous rigid frame bridge for studying is Mangjiedu Bridge in this paper. Based on the principle of structure dynamics and the method of seismic response analysis, the finite element model of the main bridge in Mangjiedu Bridge under Midas-civil was built according to the structural features, site conditions and seismic fortification intensity of the bridge. Then the characteristics of main bridge structure were studied using Lanzcos modal analysis method, and the seismic responses of main bridge are studied by earthquake response spectrum analysis and time history analysis respectively. The results show that the fundamental frequency of the main bridge is 0.1943Hz, and the transverse vibration mode shapes are most remarkable in the former 10 ordered types of vibration of main bridge. The main bridge is in the conditions of elastic range and does not failure under E1 earthquake. Both response spectrum analysis and time history analysis show that the largest displacement along the main bridge appears at the top of pier and the largest transverse displacement appears in the central position of main span. It also shows that the maximum shear and maximum moment occur at the bottom of main pier.

Dynamic approach to the Ishlinsky-Lavrent’ev problem

The dynamic stability problem for a hinged rod in the case of a stepwise axial load is considered. The method of expansions in series in natural vibration modes is systematically applied to the longitudinal as well as bending vibrations. The longitudinal vibrations result in the occurrence of longitudinal periodic forces which, in turn, cause unstable bending vibrations. An application of the Galerkin method leads to a system of ordinary differential equations with periodic coefficients, which can be reduced to Mathieu type equations. The instability domains, whose shape depends on the spectral properties of longitudinal and bending vibrations, the damping value, and the longitudinal force, are constructed. An example of nonstandard location of the instability domains is given; namely, the rod parameters are chosen so that the twelfth transverse vibration mode caused by the first longitudinal mode turns out to be unstable. The expression for the minimum value of a stepwise load leading to instability of this transverse vibration mode is obtained.

Stability of transverse vibration of rod under longitudinal step-wise loading

The problem of dynamic stability of a simply supported rod subjected to axial jump loading is considered. A systematic application of the method of expansion in terms of the normal axial and bending vibration modes is utilised. Longitudinal vibrations give rise to periodic longitudinal forces which in turn causes unstable bending vibrations. Application of the Galerkin approach results in a system of ordinary differential equations with periodic coefficients which are reduced to Mathieu equation. The instability regions whose form depends on the spectral properties of the longitudinal and flexural vibrations, damping values and longitudinal force are obtained. An example of unusual shapes of the instability regions is shown: the twelfth transverse mode caused by the first longitudinal mode turns out to be unstable for some parameters of the rod. The critical value of the jump load leading to instability of the considered transverse vibration modes is derived.

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FTIR-ATR spectroscopy in thin film studies: The importance of sampling depth and deposition substrate

Fourier transform infrared (FTIR) spectroscopy in the attenuated total reflectance mode (ATR) was used to characterise SiOxHy thin films deposited on either polypropylene foil or silicon wafers through a cold atmospheric plasma discharge. Compared to a classical transmission spectrum with transverse (TO) vibrational modes, the FTIR-ATR spectra revealed modified and/or exhibited additional features caused by either the non-orthogonal angle of incidence of the infrared radiation with respect to the sample normal or the partial light reflection on the deposition substrate. On one hand, recording the infrared spectra with an angle of incidence other than 90° produced a longitudinal (perpendicular to the sample normal) component in the electric field of the incident light, which enabled the detection of longitudinal (LO) vibrational modes. On the other hand, the transverse vibrational modes of thin films deposited on silicon were slightly extinguished with a concomitant increase of the spectral intensity of the LO features, due to both the partial withdrawing between the incident and reflected electric fields of the infrared light lying in the sample plane and, the addition of those perpendicular to this sample plane. These data thus clearly show the enormous potential of FTIR-ATR to characterise thin film molecular order, provided that a prior comprehensive analysis is performed on the sampling depth and the light reflection on the deposition substrate.

Dynamic Modeling and Characteristics Analysis of Rolls along Axial Direction for Four High Mill Based on Timoshenko Theory

The dynamic model of rolls along axial direction for four high mill was established, and the characteristics of rolls was investigated in this paper. The work roll and backup roll were considered as elastic continuous bodies, and usually they were regarded as rigid body in previous studies. Considering the effect of shear and moment of inertia, the coupled equations of work roll and backup roll were solved based on Timoshenko theory. The vibration characteristic, natural frequency and principal mode were determined. The transverse vibration modes include two forms which are synchronous vibration in low frequencies and asynchronous vibration in high frequencies. The free transverse vibrations corresponding to the first two modals are investigated. Numerical results reveal that the work roll vibrates more strenuously than that of backup roll and the high order modal can be neglected. The rolls parameters have great influences on the natural character of rolls. The transverse dynamic characters of the rolls are investigated clearly and therefore serve as a useful source of reference for engineers in designing rolling mill and controlling strip shape.

Coupling between Transverse Vibrations and Instability Phenomena of Plates Subjected to In-Plane Loading

As it is known, the problems of free transverse vibrations and instability under in-plane loads of a plate are two different technological situations that have similarities in their approach to elastic solution. In fact, they are two eigenvalue problems in which we analyze the equilibrium situation of the plate in configurations which differ very slightly from the original, undeformed configuration. They are coupled in the event where in-plane forces are applied to the edges of the transversely vibrating plate. The presence of forces can have a significant effect on structural and mechanical performance and should be taken into account in the formulation of the dynamic problem. In this study, distributed forces of linear variation are considered and their influence on the natural frequencies and corresponding normal modes of transverse vibration is analyzed. It also analyzes their impact for the case of vibration control. The forces' magnitude is varied and the first natural frequencies of transverse vibration of rectangular thin plates with different combinations of edge conditions are obtained. The critical values of the forces which cause instability are also obtained. Due to the analytical complexity of the problem under study, the Ritz method is employed. Some numerical examples are presented.

Cantilever Beam Static and Dynamic Response Comparison with Mid-Point Bending for Thin MDF Composite Panels

A new cantilever beam apparatus has been developed to measure static and vibrational properties of small and thin samples of wood or composite panels. The apparatus applies a known displacement to a cantilever beam, measures its static load, then releases it into its natural first mode of transverse vibration. Free vibrational tip displacements as a function of time were recorded. This paper compares the test results from the cantilever beam static bending and vibration with standard mid-point simply supported bending samples. Medium density fiberboard panels were obtained from four different commercial sources. Comparisons were made using a set of fiberboard panels with thicknesses of 8.1, 4.5, 3.7, and 2.6 mm and nominal densities of 700, 770, 780, and 830 kg/m3, respectively. Cantilever beam static modulus and dynamic modulus of elasticity linearly correlated well but were consistently higher than standard mid-point bending modulus of elasticity having linear correlations of 1.12:1 and 1.26:1, respectively. The higher strain rates of both the static and vibrating cantilever beam could be the primary reason for the slightly higher dynamic modulus values. The log decrement of the displacement was also used to calculate the damping ratio for the cantilever beam. As expected, damping ratio had a slightly decreasing slope as density increased. This paper discusses the new apparatus and initial results.

Spectral analysis of the sound produced during femoral broaching and implant insertion in uncemented total hip arthroplasty

Preparation of the proximal femur using incremental broaches to create the ideal cancellous bone envelope is an important technique to perfect in uncemented hip arthroplasty. To guide broaching adequacy and final implant position, the surgeon can use audible pitch changes produced by the femoral broach and definitive implant. The aim of this study was to characterise these pitch changes by analysing the sound spectra created by the first broach, last broach and implant using spectral analysis software. The last broach and implant introduction spectra demonstrated low-frequency (400-1200 Hz) spectral peaks that were not detected when using the first broach. These frequencies corresponded to the natural resonant frequency of a standing sound wave within the femoral bone canal (approximately 894 Hz) that was estimated using acoustic physics theory. The remaining spectral peaks were associated with transverse vibration modes produced by striking the metal broach handle and implant introducer and were a function of the constructs geometry and material properties.

Coupled transverse vibration modeling of drillstrings subjected to torque and spatially varying axial load

Predicting and mitigating unwanted vibration of drillstrings is an important subject for oil drilling companies. Uncontrolled vibrations cause premature failure of the drillstring and associated components. The drillstring is a long slender structure that vibrates in three primary coupled modes: torsional, axial and transverse. Among these coupled modes, the transverse mode is the major cause of drillstring failures and wellbore washout. Modal analysis of drillstrings reveals critical frequencies and helps drillers to avoid running the bit near critical modes. In this article, the coupled orthogonal modes of transverse vibration of a drillstring in the presence of torque and spatially varying axial force (due to mud hydrostatic effect, self-weight and hook load) are derived and the mode shapes and natural frequencies are determined through the expanded Galerkin method. The results are verified by the nonlinear finite element method. Modal mass participation factor, which represents how strongly a specific mode contributes to the motion in a certain direction, is used to determine the appropriate number of modes to retain so that computational efficiency can be maximized.

Free transverse vibration analysis of a toothed gear

Analysis of transverse vibration of the gear found in a high-speed gearbox considered as an annular plate reflecting gear geometry is the subject of this paper. How gear angular velocity affects the deformation of normal modes of transverse vibration of the system under consideration is analysed. Models considered were discretized by the finite elements method. Numerical computations have been performed in the ANSYS environment. The algorithm to identify the proper distorted mode shapes is presented. The Campbell diagram for the system under consideration is elaborated. The problems discussed here can be useful for engineers dealing with dynamics of rotating machine systems.

The tones of the kalimba (African thumb piano)

The acoustic spectrum of the kalimba (African thumb piano) is measured and analyzed for tonal structure. The frequency f(1) of the fundamental tone of each tine (key) is investigated in relation to the frequencies of its two dominant overtones, f(2) and f(3). These frequencies are identified as the first three modes of transverse vibration of a beam of rectangular cross section. As is typical for vibrating-beam instruments, the overtone sequence is inharmonic, that is, the sequence f(1), f(2), f(3),[ellipsis (horizontal)] is unevenly spaced and the frequency ratios f(2)/f(1) and f(3)/f(1) are not integers. The kalimba tines are modeled by applying the Euler-Bernoulli beam equation with one end clamped, the other end free, and an intermediate point (the bridge) simply supported. Unlike the cases of free-free and clamped-free beams, it is found that the clamped-supported-free frequency ratios f(2)/f(1) and f(3)/f(1) are not fixed values, but depend uniquely upon where the bridge supports and subdivides the tine. The model solution is more thoroughly investigated analytically for the special case in which the beam segment ratio is unity, which has some analytic solutions. Numerically computed mode frequencies agree well with acoustic measurements, validating the model. Mode shapes are computed for the first three modes of a typical tine.

Structural, optical and electrical properties of cubic AlN films deposited by laser molecular beam epitaxy

Cubic AlN films were successfully deposited on TiN buffered Si (100) substrates by a laser molecular beam epitaxy (LMBE) technique, and their crystal structure and optical and electrical properties were studied. The results indicate that cubic AlN films show the NaCl-type structure with a (200) preferred orientation, and the lattice parameter is determined to be 0.4027 nm. The Fourier transform infrared (FTIR) pattern of the cubic AlN film displays sharp absorption peaks at 668 cm−1 and 951 cm−1, corresponding to the transverse and longitudinal optical vibration modes. Ellipsometric measurements evidence a refractive index of 1.66–1.71 and an extinction coefficient of about zero for the cubic AlN film in the visible range. Capacitance–voltage (C–V) traces of the metal–insulator–semiconductor (MIS) device exhibit that the cubic AlN film has a dielectric constant of 8.1, and hysteresis in the C–V traces indicates a significant number of charge traps in the film.

Nonlinear transverse steady-state periodic forced vibration of 2-dof discrete systems with cubic nonlinearities

A method based on Hamilton’s principle and spectral analysis has been applied recently to nonlinear transverse vibrations of discrete systems with cubic nonlinearities, leading to calculation of the nonlinear free modes of transverse vibration and their associated nonlinear frequencies. The objective of the present work was the extension of this method to the nonlinear forced transverse steady-state periodic response of 2-dof system leading to nonlinear frequency response function in the neighbourhood of the two modes

Analysis of Dynamic Stability of Space Launch Vehicles under Aerodynamic Forces Using CFD Derived Data

A computational framework is developed to investigate the dynamic stability of space launch vehicles subjected to aerodynamic forces. A detailed mechanics based mathematical model of a moving flexible vehicle is used. The aerodynamic forces on the vehicle are obtained from simulation using Computational Fluid Dynamics (CFD) package. The objective behind this investigation is to analyze the problem of aeroelastic instability in blunt/conical nose slender space launch vehicles. Coupling among the rigid-body modes, the longitudinal vibration modes, and the transverse vibrational modes are considered. The effect of propulsive thrust as a follower force is also considered. A one-dimensional finite element model is developed to investigate the occurrence of aeroelastic instabilities of various types. Eigenvalues of the vehicle are determined in order to analyze the stable regimes. As a special case, we show numerical simulations by considering a typical vehicle configuration, for a vehicle Mach number of 0.8. Phenomenon of flutter is observed at this Mach number. The proposed analysis is suitable for different launch events such as vehicle take-off, maximum dynamic pressure regime, thrust transients, stage separation etc. The approach developed in this paper can be utilized for preliminary design of launch vehicles and establishing the stability boundaries for different trajectory parameters.