Transverse Vibration Modes Research Articles

Vibration Behavior Analysis for Simply Supported Beam under Multiple Moving Vibrating Systems

In this paper, the transverse vibration mode of simply supported beam subjected to moving mass and moving spring-mass system has been studied considering the gravity of the beam and the inertia of the moving mass. A coupled vibration model for multiple moving spring-mass systems (MSMSs) moving on the simply supported beam is established. The numerical simulations of lateral transient response of simply supported beams subjected to four different moving systems are carried out. It is found that the vibration frequency of the midpoint of the beam will be changed once the moving mass system (MMS) is changed into the MSMS. This phenomenon is more pronounced under the action of a single moving system. In addition, be different from the single MMS, the results show that the lateral displacement curves of the positions near the supports will occur obvious bimodal characteristics when a single MSMS pass through the simply supported beam.

Free vibration analysis of cracked Timoshenko beams carrying spring-mass systems

In this paper, an analytical approach is proposed for determining vibration characteristics of cracked non-uniform continuous Timoshenko beam carrying an arbitrary number of spring-mass systems. This method is based on the Timoshenko beam theory, transfer matrix method and numerical assembly method to obtain natural frequencies and mode shapes. Firstly, the beam is considered to be divided into several segments by spring-mass systems and support points, and four undetermined coefficients of vibration modal function are contained in each sub-segment. The undetermined coefficient matrices at springmass systems and pinned supports are obtained by using equilibrium and continuity conditions. Then, the overall matrix of undetermined coefficients for the whole vibration system is obtained by the numerical assembly technique. The natural frequencies and mode shapes of a cracked non-uniform continuous Timoshenko beam carrying an arbitrary number of springmass systems are obtained from the overall matrix combined with half-interval method and Runge-Kutta method. Finally, two numerical examples are used to verify the validity and reliability of this method, and the effects of cracks on the transverse vibration mode shapes and the rotational mode shapes are compared. The influences of the crack location, depth, position of spring-mass system and other parameters on natural frequencies of non-uniform continuous Timoshenko beam are discussed.

Intermolecular interaction effect on the inelastic electron tunneling spectroscopy of bi-octane-monothiol junctions

The inelastic electron tunneling spectroscopy (IETS) of bi-octane-monothiol junctions is theoretically studied based on first-principles calculations. The results reveal that IETS is very sensitive to the vertical and lateral distance of the two molecules in the bimolecular junctions owing to the changes of interaction between the two molecules. It is further demonstrated that the transverse vibrational modes ν(C-H) around 0.38V will be triggered when the two molecules are close to each other and open a new path for electron tunneling. Our theoretical results provide new insight into understanding the origin of the IETS peaks around 0.38V.

Dynamic performance of existing double track railway bridges at resonance with the increase of the operational line speed

This article addresses the dynamic behaviour of double track simply supported bridges of short to medium span lengths (10m<L<25m) belonging to conventional railway lines. These structures are susceptible to experience inadmissible levels of vertical vibrations when traversed by trains at high speeds, and in certain cases their dynamic performance may require to be re-evaluated in case of an increase of the traffic velocity above 200km/h. In engineering consultancies, these structures have been traditionally analysed under the passage of trains at different speeds using planar models, neglecting the contribution of transverse vibration modes and also the flexibility of the elastomeric bearings. The study presented herein endeavours to evaluate the influence of these two aspects in the verification of the Serviceability Limit State of vertical accelerations, which is of great interest in order to guarantee a conservative prediction of the dynamic behaviour. In the present study, the dynamic response of representative slab and girder bridges has been evaluated using an orthotropic plate finite element model, leading to practical conclusions regarding the circumstances under which the above mentioned factors should be considered in order to adequately evaluate the transverse vibration levels of the deck.

Semi-analytical solution for free transverse vibrations of Euler–Bernoulli nanobeams with manifold concentrated masses

ABSTRACTIn the present study, transverse vibrations of nanobeams with manifold concentrated masses, resting on Winkler elastic foundations, are investigated. The model is based on the theory of nonlocal elasticity in the presence of concentrated masses applied to Euler–Bernoulli beams. A closed-form expression for the transverse vibration modes of Euler–Bernoulli beams is presented. The proposed expressions are provided explicitly as the function of two integrated constants, which are determined by the standard boundary conditions. The utilization of the boundary conditions leads to definite terms of natural frequency equations. The natural frequencies and vibration modes of the concerned nanobeams with different numbers of concentrated masses in different positions under some typical boundary conditions (simply supported, cantilevered, and clamped–clamped) have been analyzed by means of the proposed closed–form expressions in order to show their efficiency. It is worth mentioning that the effect of various nonlocal length parameters and Winkler modulus on natural frequencies and vibration modes are also discussed. Finally, the results are compared with those corresponding to a classical local model.

A new tangent stiffness-based formulation to study the free vibration behavior of a transversely loaded Timoshenko beam with geometric nonlinearity

A new formulation is introduced to study the free vibration behavior of a statically loaded beam with geometric nonlinearity. The tangent stiffness of the statically loaded beam is used to investigate the free vibration behavior of the beam about its loaded configuration. The problem is formulated for a linearly tapered beam, and a uniform beam is obtained as a special case. Energy principles based on the variational approach are used to derive the governing equations for the static and dynamic problems. The Ritz method of approximate displacement field is followed to solve the governing equations. The Ritz coefficients are used to derive the tangent stiffness of the loaded beam. Components of the tangent stiffness matrix are derived for a Timoshenko beam with von Kármán-type nonlinearity. Illustrative results are presented for four different classical boundary conditions having in-plane restraint. Results for the first two modes of transverse vibration are presented in the nondimensional deflection-frequency plane. Validation of the work is carried out using finite element software ANSYS. The formulation is new of its kind and can be used for any displacement-based problem following the Ritz method.

Emergence of linear elasticity from the atomistic description of matter.

We investigate the emergence of the continuum elastic limit from the atomistic description of matter at zero temperature considering how locally defined elastic quantities depend on the coarse graining length scale. Results obtained numerically investigating different model systems are rationalized in a unifying picture according to which the continuum elastic limit emerges through a process determined by two system properties, the degree of disorder, and a length scale associated to the transverse low-frequency vibrational modes. The degree of disorder controls the emergence of long-range local shear stress and shear strain correlations, while the length scale influences the amplitude of the fluctuations of the local elastic constants close to the jamming transition.

Equilibrium distribution of the wave energy in a carbyne chain

The steady-state energy distribution of thermal vibrations at a given ambient temperature has been investigated based on a simple mathematical model that takes into account central and noncentral interactions between carbon atoms in a one-dimensional carbyne chain. The investigation has been performed using standard asymptotic methods of nonlinear dynamics in terms of the classical mechanics. In the first-order nonlinear approximation, there have been revealed resonant wave triads that are formed at a typical nonlinearity of the system under phase matching conditions. Each resonant triad consists of one longitudinal and two transverse vibration modes. In the general case, the chain is characterized by a superposition of similar resonant triplets of different spectral scales. It has been found that the energy equipartition of nonlinear stationary waves in the carbyne chain at a given temperature completely obeys the standard Rayleigh–Jeans law due to the proportional amplitude dispersion. The possibility of spontaneous formation of three-frequency envelope solitons in carbyne has been demonstrated. Heat in the form of such solitons can propagate in a chain of carbon atoms without diffusion, like localized waves.

An analytical method for crack detection of Timoshenko beams with multiple open cracks using a test mass

A crack identification approach in uniform simply supported beams with open cracks, using a test mass, is developed, based on changes in natural frequencies of cracked beams due to the test mass. Test mass is defined as a stationary mass, which is located in different places of the beam modelled by Timoshenko beam theory. The beam with an arbitrary number of cracks is modelled as segments connected by elastic springs illustrating open edge cracked cross-sectional flexibilities. The stiffness of the springs is determined using fracture mechanics theory. Solving the differential equations, the eigenfunctions of the problem are explicitly derived. Rayleigh’s quotient method is then used to consider the effect of the test mass. Consequently, natural frequencies of the cracked beam with a test mass are derived for each mode of transverse vibration. Knowing the natural frequencies of the beam carrying a test mass at some different locations, it is possible to identify cracks’ parameters (i.e. location and depth) by solving the system of equations. The proposed closed-form approach is also validated using numerical studies on multiple-cracked beam examples. There is quite encouraging agreement between the results of the present approach and those numerically computed using the finite element method.

Damage-Caused Frequency Shifts in Beams and their Use in Structural Health Assessment

Transverse cracks affect the stiffness and consequently the dynamic behavior of beams, by altering the natural frequencies. Actual cracked beam models are not able to explain in-deep the vibration mechanism and, therefore, no mathematical relation able to predict frequency changes due to damage exist. This paper proposes a new damage model which better explain the dynamic behavior of beams with open and closed cracks and a mathematical relation able to predict the frequency changes due to damage. In contrast to actual models, a global approach is used, by considering the influence of the stored energy distribution for each transversal vibration mode. Since the energy loss globally quantifies the state change due to damage, it was possible to replace the cracked beam with an equivalent one having constant but reduced stiffness. Based on it, a relation indicating the frequency drop damage was contrived and tested, by means of the finite element analysis, for various damage types and locations along the beam. Simulations have shown that accurate prediction about the natural frequency shifts can be made with the proposed relation.

Natural frequency changes due to damage in composite beams

Transversal cracks in structures affect their stiffness as well as the natural frequency values. This paper presents a research performed to find the way how frequencies of sandwich beams change by the occurrence of damage. The influence of the locally stored energy, for ten transverse vibration modes, on the frequency shifts is derived from a study regarding the effect of stiffness decrease, realized by means of the finite element analysis. The relation between the local value of the bending moment and the frequency drop is exemplified by a concrete case. It is demonstrated that a reference curve representing the damage severity exists whence any frequency shift is derivable in respect to damage depth and location. This curve is obtained, for isotropic and multi-layer beams as well, from the stored energy (i.e. stiffness decrease), and is similar to that attained using the stress intensity factor in fracture mechanics. Also, it is proved that, for a given crack, irrespective to its depth, the frequency drop ratio of any two transverse modes is similar. This permitted separating the effect of damage location from that of its severity and to define a Damage Location Indicator as a sequence of squared of the normalized mode shape curvatures.

Vibration analysis of a tapered laminated thick composite plate with ply drop-offs

In this study, vibration characteristics of a tapered laminated thick composite plate have been investigated using finite element method by including the shear deformation and rotary inertia effects. The governing differential equations of motion of a tapered laminated thick composite plate are presented in the finite element formulation based on first-order shear deformation theory for three types of taper configurations. The effectiveness of the developed finite element formulation in identifying the various dynamic properties of a tapered laminated thick composite plate is demonstrated by comparing natural frequencies evaluated using the present FEM with those obtained from the experimental measurements and presented in the available literature. Various parametric studies are also performed to investigate the effect of taper configurations, aspect ratio, taper angle, angle ply orientation and boundary conditions on free and forced vibration responses of the structures. The comparison of the transverse free vibration mode shapes of the uniform and tapered composite plates under various boundary conditions is also presented. The forced vibration response of a composite plate is investigated to study the dynamic response of tapered composite plate under the harmonic force excitation in various tapered configurations. It is concluded that the dynamic properties of laminated thick composite plates could be tailored by dropping off the plies to yield various tapered composite plate.

Thermoelastic coupling effect analysis for gyroscope resonator from longitudinal and flexural vibrations

The paper illustrates thermoelastic modeling, analysis and simulation solutions of gyroscope resonator from the longitudinal and transverse vibration modes. In the study, the sensitive components of gyroscope are the cantilever beams within the drive mode and the detection mode. Then, the effect of thermoelastic coupling, coupling strength on gyroscope performance is analyzed by two different numerical calculation methods, and the results are validated by FEM simulation solutions. The corresponding parameters which are analyzed in the study for thermoelatic coupling are temperature, thermoelastic damping (TED) and frequency shift ratio. It is found that the thermoelastic damping has the order of 10?4 at both transverse and longitudinal vibrations. And the shift frequency sharply increased and then gradually approaches the horizontal line at the longitudinal and flexural vibrations. Then the comparison of thermoelastic damping is studied at two vibration modes. Compared with longitudinal vibration, the thermoelastic damping for flexural vibration has a similar pattern, while the former's peak value is twice lower than that of the latter.

Study of the Vibratory Stress Relief for Large-Scale Parallel Welded Steel Truss

Parallel welded steel truss component is widely used in construction, tower, bridges, aerospace and other fields. And the welding residual stress is distributed complicatedly in many areas and directions, which will definitely deform the welded steel truss or even crack it. So, it is necessary to do aging treatment to avoid such thing happened, as well as improve the accuracy and reliability. Vibratory stress relief (VSR) is a new kind of technology which may eliminate the residual stress in metal component with high efficiency, energy-saving, environment protection and low cost compared with thermal stress relief (TSR). But its technical effects vary with the vibration parameters. To the parallel steel truss structures, the excitation parameter of VSR is determined by welding residual stress distribution and the natural mode of vibration. The residual stress concentration areas can get enough vibration, if the parallel steel truss component vibrates with different kinds of natural mode of vibration, which will obtain good effect of eliminating stress. Firstly, the inherent vibration mode of parallel welded steel truss component is measured by the modal analysis technology. Secondly, multiple intrinsic modes are chosen according to principle of making all the welding residual stress concentration regions and directions get enough vibration and then main technical parameters are determined such as excitation frequency, excitation point and support point. In this paper, two examples are given out: the specimen 1is treated with the transverse bending vibration mode and torsion vibration mode respectively according to the above method. As for specimen 2, it is treated with 3 exciting frequencies which are located automatically by the vibration aging machine. The residual stresses of the two specimens are measured before and after their vibrations with blind holes method. Results show that after vibration aging, the welding residual stress peak figure of the Specimen 1 is reduced by an average of 48.65% and the mean value of that dropped by an average of 32.50%. Correspondingly, the residual stress peak value of the Specimen 2 shows a decline of 3.38%, its average fell by an average of 2.55%.

Analytic model of vibrations of a carbon dioxide molecule. Fermi resonance

We use the invariant normalization method to study nonlinear autonomous vibrations of a CO2 molecule near its stable configuration. If the frequencies of symmetric and deformation vibrations are related as 2: 1, then a third-order resonance occurs in the molecule. The simulation discovered the following two nonlinear effects: the energy transfer between modes of longitudinal and transverse vibration modes which participate in the resonance and the frequency splitting in the molecule spectrum; namely, instead of one line of symmetric vibration, there is a group of four closely located lines. These effects are known as the Fermi resonance phenomenon.

Simulation of crack growth in the compressor blade subjected to resonant vibration using hybrid method

This paper presents results of crack growth analysis of an aero-engine compressor blade subjected to resonant vibration. In the work an original hybrid method were used for crack dynamic estimation. Proposed procedure connects three methods: analytical, numerical and experimental fatigue analysis of the blade tested in high cycle fatigue conditions. During investigations only first mode of transverse vibration was considered. In first step of analysis the stress intensity factor for the simplified model of blade with half-elliptical crack was calculated using Raju–Newman method. Crack dimensions and crack location were determined on the base of experimental investigations. The bending stress used in Raju–Newman solution was computed for the real blade geometry using the finite element method. Stress intensity factor values were next used as an input data into Paris–Erdogan equation. As a result proposed hybrid analysis, the plot of crack length in function of number of load cycles for the compressor blade vibrating at constant amplitude was determined. Obtained results of simulation were finally compared with the results of the experimental crack growth analysis performed for the first stage compressor blade of helicopter turbo-engine.

Understanding the atomic-level Green-Kubo stress correlation function for a liquid through phonons in a model crystal

In order to gain insight into the connection between the vibrational dynamics and the atomic-level Green-Kubo stress correlation function in liquids, we consider this connection in a model crystal instead. Of course, vibrational dynamics in liquids and crystals are quite different and it is not expected that the results obtained on a model crystal should be valid for liquids. However, these considerations provide a benchmark to which the results of the previous molecular dynamics simulations can be compared. Thus, assuming that vibrations are plane waves, we derive analytical expressions for the atomic-level stress correlation functions in the classical limit and analyze them. These results provide, in particular, a recipe for analysis of the atomic-level stress correlation functions in Fourier space and extraction of the wave-vector and frequency-dependent information. We also evaluate the energies of the atomic-level stresses. The energies obtained are significantly smaller than the energies previously determined in molecular dynamics simulations of several model liquids. This result suggests that the average energies of the atomic-level stresses in liquids and glasses are largely determined by the structural disorder. We discuss this result in the context of equipartition of the atomic-level stress energies. Analysis of the previously published data suggests that it is possible to speak about configurational and vibrational contributions to the average energies of the atomic-level stresses in a glass state. However, this separation in a liquid state is problematic. We also introduce and briefly consider the atomic-level transverse current correlation function. Finally, we address the broadening of the peaks in the pair distribution function with increase of distance. We find that the peaks' broadening (by $\ensuremath{\approx}40%$) occurs due to the transverse vibrational modes, while contribution from the longitudinal modes does not change with distance.

Vibration Analysis for Piezoceramic Circular Plates with V-Notches. Part 1: Theory

AbstractIn this paper the transverse vibration characteristics of piezoceramic circular plates with V-notches are investigated theoretically through use of the Ritz's method incorporated with the defined equivalent constants. The Ritz's method is employed with two sets of admissible displacement functions, algebraic-trigonometric polynomials and corner functions, to guarantee convergence sufficiently and represent the stress singularity, respectively. Moreover, the equivalent constants derived by comparing the characteristic equations of transverse vibration between isotropic and piezoceramic disks are applied to suspend the electrical field consideration regarding the piezoelectricity. With the aid of theoretical analysis, the non-dimensional frequency parameters of transverse vibration modes for completely free V-notching circular plates are exhibited; in addition, the frequency variations depending on various notch angles and depths are explored. Numerical calculations using the finite element method (FEM) are performed and the results are compared with the theoretical analysis. It is shown that the resonant frequencies predicted by theoretical analysis and calculated by FEM are in good agreement.

Analysis of Transverse Vibration Modes of a Gun Barrel Subjected to Recoil

This paper focuses on the calculation of modal frequencies and corresponding mode shapes of Gun Barrel by analytical method and the Finite Elements Method. Modal analysis is used to find out natural and succeeding two modes frequencies for transverse vibration. Barrel is considered as a cantilever beam. Recoil of Gun Barrel is incorporated as variation in the length cantilever beam. Increase in modal frequencies due to gun barrel recoil is observed. It is concluded that the transverse vibration of gun barrel coupled with the recoil produces a band of frequencies in each modes. The analysis and results discussed can be used for design and performance evaluation, selection of accessories and attachments of gun systems.

The quenched state with dominant shear vibration mode originated from domain reconfiguration in [001]-oriented Pb(Mg1/3Nb2/3)O3-PbTiO3 single crystals

We present results of investigations into a low-frequency vibration mode (LVM) in [001] poled Pb(Mg1/3Nb2/3)O3-PbTiO3 crystal. Through a series of impedance measurement and polarization microscopy observations of domain structure, we find that the LVM is a shear mode arising from a polarization component lying in the (001) plane. Through quenching, a metastable state can be reached where the LVM is greatly enhanced and the thickness vibration mode of the typical poled state is highly suppressed. Measurements on the transverse vibration modes of differently oriented crystal samples show that the LVM state possesses a macroscopic orthorhombic symmetry (mm2) with its polarization vector along the [110] direction. The structural differences between the LVM-enhanced state and the usual poled state are discussed in detail. The two states have the same crystalline phase but different ferroelectric domain structures. When heated above the rhombohedral-to-tetragonal transition temperature, Trt, a tetragonal phase with (110) domain walls is formed. This domain structure can be frozen by the rapid quenching through the Trt. Based on compatibility of ferroelectric polarizations at domain walls, and observed polarization microscopy results, we are able to infer the possible arrangement of domain structures in rhombohedral and tetragonal phases. The [110] polarization component associated with LVM originates from the domain structure with well-organized (110) domain walls.