Frequencies Of Longitudinal Vibrations Research Articles

Design and optimization of stepped compound horn based on finite element method

A stepped composite horn structure with column holes at the front end is proposed for use in power ultrasonic fields. Based on the finite element method, the vibration characteristics and amplitude magnification of the horn at resonance frequency are calculated, Analyzed the influence of different structural parameters on its performance parameters, analyzed the influence of the aperture size on the resonance frequency of the horn, Optimize the design of the proposed composite horn to obtain the maximum amplitude magnification, Perform simulation analysis on the horn through ANSYS Workbench, extract the seventh-order modal results, and perform modal analysis. The simulation results show that: compared with the traditional horn, the maximum amplitude of the step compound-shaped horn is increased by 72%, The frequency of longitudinal vibration of this open-hole horn is 26.92KHZ. The performance parameters of the optimized horn have been significantly improved.

Vibratory Conveying by Harmonic Longitudinal and Polyharmonic Normal Vibrations of Inclined Conveying Track

Abstract Vibratory conveying of a material point by harmonic longitudinal and polyharmonic normal vibrations of an inclined conveying surface is considered. The dependence of dimensionless conveying velocity—a ratio of velocity to the product of frequency and amplitude of longitudinal vibration—on several dimensionless parameters is investigated in the moving modes without hopping. Maximal conveying velocity is achieved at the certain values of normal vibration amplitudes and phase difference angle between the longitudinal and normal vibrations, which are called optimal. Their values are dependent on two dimensionless parameters: the inclination angle parameter—a ratio of an inclination angle tangent to a frictional coefficient, the intensive vibration coefficient—a ratio of the longitudinal amplitude of vibration to the amplitude of the first harmonic of normal vibration and frictional coefficient. In a condition of the intensive longitudinal vibration, when its amplitude significantly greater than amplitudes of normal vibration, dimensionless velocity is almost independent of the intensive vibration parameter and it depends only on inclination angle parameter, i.e., on inclination angle and frictional coefficient. The optimal values of harmonics’ amplitudes of polyharmonic normal vibration are determined in dependence of inclination angle parameter with the number of harmonics from 2 to 7. The graphs of considered dependencies are presented and the most important values of parameters are presented in the table. Conclusions are made to determine the optimal vibration parameters and the problems of further research are indicated. The considered vibrations can be used in different vibratory conveying devices with electromagnetic drives.

Nonlinear interaction of longitudinal and transverse vibrations of a rod at an internal combinational resonance in view of opto-thermal excitation of N/MEMS

In this paper, the character of coupled longitudinal-flexural vibrations of a rod is investigated in the case of an internal combinational resonance between two flexural and lower longitudinal vibration modes. An approximate analytical solution is obtained by a multiple time-scales method applied directly to a system of nonlinear partial differential equations. It is shown that the initial perturbation in the longitudinal direction can effectively excite bending vibration modes, which, on the whole, leads to a longitudinal-transverse beating regime with noticeable amplitudes. It is noted that the period and amplitude of these beats substantially depend on the frequency detuning parameter between the sum of the bending vibration frequencies and the longitudinal vibration frequency. This effect can be used as a new principle of operation of opto-thermally excited N/MEMS resonant sensors.

Numerical and experimental verification of a method of identification of localized damages in a rod by natural frequencies of longitudinal vibration

A method for identification of multiple transverse cracks and other localized defects in a rod by means of two spectra of longitudinal vibrations that correspond to free–free and fixed–free end conditions is presented. A numerical algorithm is developed that implements the method. The algorithm was tested on experimental data obtained on a cylindrical sample of aluminum alloy D16 with the created localized damages. The experiments were carried out on a rod with free ends. The created damages were ring grooves symmetrically located relative to the middle of the rod. With the help of such experiments, natural frequencies were obtained that correspond to two types of boundary conditions for a half-length rod. The experimental data were processed using the developed algorithm. The results showed that the model on which the developed algorithm is based describes well the longitudinal vibrations of the rod with localized damages in a fairly wide frequency range, and the algorithm enables to reconstruct multiple damages accurately enough.

Acoustic control of the strengthening of building ceramics obtained by extrusion through an ultrasonic die

Extrusion through a die, vibrating with an ultrasonic frequency is a promising method for hardening ceramic materials (bricks, tiles, pipes, etc.). We present the results of studying low-frequency acoustic spectra of cylindrical samples of face brick ceramics reinforced by extrusion and addition of fiberglass waste. Correlations between structure changes in the obtained ceramics, parameters of the acoustic spectrum of the samples, and the destructive forces upon their compression (bending) in a press were analyzed. It is shown that ultrasonic extrusion increases the destruction force of the cylinders by 1 – 7 kN (~30 %) and thus shifts the frequency of the fundamental tones of transverse and longitudinal oscillations (6 and 17 kHz) with a coefficient of ~0.1 kHz/kN. The fiberglass particles strengthen the ceramics due to the formation of a needle-like crystalline bond during firing. At the same time, an increased addition of the fiberglass particles reduces the plasticity of the batch thus resulting in formation of the microcracks, cavities and internal stresses in the samples. The occurrence of internal defects shifts the frequency of longitudinal and transverse vibrations to a different extend, which can be used as an indicator of the defects present in the samples and provide the possible of their objective sorting and rejection. The vibration relaxation time (about 0.1 sec) does not change significantly under the described strengthening effects and being characterized by a wide scatter in each series of the samples cannot be considered an informative parameter. Thus, the effectiveness of the presented combined method of ceramics hardening can be accurately tested by the shift in the frequencies of longitudinal and transverse free vibrations of cylindrical specimens. The presented approach facilitates the process of developing new ceramic compositions and extrusion technology in conditions of depletion of sources of high-quality clay raw materials.

Design and dynamic analysis of low-frequency mounting system for marine thrust bearing

Longitudinal vibration reduction of marine propulsion shafting has drawn great attention of research. In this paper, a low-frequency mounting system (LFMS) is designed. LFMS consists of three sub-systems: floatable self-alignment thrust bearing, floating raft, and air spring mounting system. Floatable self-alignment thrust bearing is a new type of thrust bearing, which can provide small displacement compensation for shafting. In order to maximize the overall vibration behavior of the control system, thrust bearing and marine propulsion unit are integrated on the floating raft air spring mounting system can adjust the floating raft attitude to meet the shafting alignment requirements. A theoretical model is presented to estimate the isolation effect of LFMS. The first-order longitudinal vibration frequency of shafting is 9.8 Hz. Experiments are carried out to verify the isolation effect of LFMS. Experiments show that LFMS has an obvious vibration isolation effect above 10 Hz.

Mathematical Model of Root Crop Digging with Longitudinal Vibrations

The problem how to reduce damage to tubers when they are dug up is urgent. For the new design of a vibrating digging working body for root crops the mathematical model of longitudinal vibrations of a root crop in the soil is developed as an elastic body in an elastically damped medium. The Ostrogradsky-Hamilton variational principle is applied for the analytical description of the process. The Ritz method was applied to find the frequencies of natural vibrations, the amplitudes of forced vibrations of a root crop as a solid elastic body when it is captured by a vibrating digging body. The frequency equation for the discussed vibrational process was obtained. The values of the first proper frequency of longitudinal vibrations of the considered elastic body of the root crop with specific geometric physical parameters are found. Graphs of the dependence of the first natural frequency upon the elastic deformation coefficient, the damping coefficient of the soil as an elastic damping medium are obtained. When the soil damping coefficient changes within 0 to 10 N∙s²∙m^–3, the first proper frequency changes within 500 to 750 s^-1 (80 to 119 Hz) at soil elastic deformation coefficient 2∙10⁵ N∙m^–3. Dependence of the elastic body forced vibration amplitude upon the change in the amplitude of the disturbing force have been obtained. When the amplitude of the disturbing force changes within 100 to 600 N, the amplitude of forced vibrations of the root crop body changes within 0.30 to 0.68 mm.

The inverse problem of the oscillation of a rod with a variable cross section

Variable cross-section rods are used in many parts and mechanisms. For example, conical rods are widely used in percussion mechanisms. The strength of such parts directly depends on the natural frequencies of vibrations. Oscillations and vibration are also used in diagnostics, for example in acoustic diagnostics of defects. The paper presents a method that allows numerically finding a variable cross section of an elastic rod from the natural frequencies of longitudinal vibrations. It is assumed that the cross-sectional area varies along the axis and is described by an exponential function of a polynomial of degree n. The boundary condition on the left end is hard, on the right-elastic. It is shown that to determine n unknown coefficients of the cross section function, n natural frequencies are required.

Determination of Elastic Properties of Beech Plywood by Analytical, Experimental and Numerical Methods

This research article examines the application of various methods to determine the effective elastic properties of beech veneer-wood composites. Using laminate theory, the theoretically calculated effective values of the in-plane and out-of-plane modulus of elasticity as well as shear modulus are compared with the values determined from the natural frequencies of flexural, torsional and longitudinal vibrations of samples having different orientations and numbers of composite layers. The samples are also modelled using the finite element method, and their natural frequencies are calculated by the modal analysis. Research has shown that the laminate theory, which is well established and applied in the world of synthetic composites, can also be applied to beech plywood composites, where the theoretically calculated effective values can be up to 15% higher. Similarly, due to the higher calculated effective elastic properties, higher natural frequencies of flexural, torsional and longitudinal vibrations are also calculated by the finite element method.

Analysis of Vibration Monitoring Data of Flexible Suspension Lifting Structure Based on Time-Varying Theory

An elevator is a typical flexible lifting machine. In order to monitor the vibration of elevator structure, the vibration characteristics of an elevator with a traction ratio of 1:1 has been tested experimentally. Sensors were arranged on the platform frame, car roof, and hoist rope to test the vibrations of the elevator in both ascending and descending conditions. The transverse, longitudinal, and coupled transverse-longitudinal vibrations were compared and analyzed. Further, the short-time Fourier transform (STFT) method was used to examine the power spectral density (PSD) of the test results, and the main frequency distribution and influencing factors of the vibration of elevator components were investigated. The results revealed that the transverse and longitudinal vibrations of the platform frame were low-frequency vibrations, which was attributed to the elastic interaction between the platform frame and the car frame. The form and amplitude of longitudinal vibration of the car frame were basically consistent with those of the platform frame, but PSD of the transverse vibration had an obvious peak in the high-frequency region. The transverse and longitudinal vibration frequencies of the hoist rope were higher. Furthermore, the peak PSD value of transverse vibration of the hoist rope was 421 times larger than that of the car frame, so a small disturbance at the end of the rope could lead to a huge disturbance in the center of the rope. Overall, this study provides useful insights on designing an elevator monitoring sensor and relevant data processing.

Relaxation of atomic temperature anisotropy in a one-dimensional optical lattice enhanced by dynamic control of the aspect ratio

We study the relaxation dynamics of temperature anisotropy between longitudinal and transverse motions of $^{87}\mathrm{Rb}$ atoms in a one-dimensional optical lattice. The temperature imbalance is established by Raman sideband cooling applied along the lattice axis. At low temperature, a large difference between the longitudinal and transverse vibration frequencies limits the relaxation. We transform the trap potential by phase modulating the lattice beam to reduce the anisotropy and thereby enhance cross-dimensional sympathetic cooling. An extension of classical theory, which incorporates the quantization of the atomic motion, explains the experimental results well. We also report that effects of trap anharmonicity and atomic vector polarizability on broadening of the Raman sideband cancel each other at a particular lattice depth, which greatly facilitates sideband cooling.

Spectral Diagnostics of the Dynamics of the Formation of a Homoconjugated Complex [HCN.H.NCH

A quantum-mechanical analysis of the manifestations of the NHN hydrogen bond in the vibrational spectra of the [HCN.H.NCH]+ linear complex along the profile of the proton transition reaction path is given, and the laws governing the diagnostic parameters, that is, potential descriptors of the dynamics of this process, are determined. We calculated the surface of the potential energy and harmonic frequencies of normal vibrations along the profile of the reaction path of the proton transition in the system under study. When the fragments [HCNH]+ and NCH approach each other, a noticeable distortion of the forms of their skeletal vibrations occurs, up to complete mixing into the symmetric and antisymmetric forms. The frequency of the longitudinal vibration of the central proton ν(NH) varies along the reaction path from ~3600 to ~500 cm–1. An abrupt frequency change is observed in the region of intersection of the terms ν(NH) and ν(CN).

Dynamic analysis of the longitudinal vibration on bottom drilling tools

With extreme complexity, the drilling process is a dynamic process which is severely influenced by longitudinal vibration. Longitudinal vibration, as one of the most important reason, is directly generated by the fatigue failure of the bottom hole assembly. In this paper, the natural frequencies of longitudinal vibration along the drillstring are analyzed by the finite element method. The deformed plot, stress nephogram, and displacement contour map under 1 to 4 ordered the natural frequency of the longitudinal vibration are obtained. The analysis results show that the maximum deformation always appears in the central part of the string so that some technological process on these positions is required to reduce the collision between the string and wellbore wall. Additionally, a time series of longitudinal vibration of a bottom rotating drillstring is extracted from real-time field data, which is measured while drilling near the drill bit. Then the time-frequency and energy spectrum analysis of the longitudinal vibration is carried out. The results of the statistical analysis show that, when the drillstring uniformly rotates, the longitudinal vibration can be considered as a kind of random vibration. However, if the stick-slip phenomenon occurs during the drilling process, the energy concentration will appear in the time series spectrum of the longitudinal vibration, by which means the vibration could be regarded as random no longer.

Optimization of vibratory conveying upward by inclined track with polyharmonic normal vibrations

The paper is devoted to the research of vibratory conveying of piece goods along an inclined track, performing harmonic longitudinal and polyharmonic normal vibrations. It is considered the conditions of reaching maximum conveying velocity at specified values of frequency and amplitude of longitudinal vibrations – the conditions of maximum dimensionless conveying velocity, depending on several dimensionless parameters in the moving modes without hopping. These dimensionless parameters are the inclination angle parameter – a ratio of an inclination angle tangent to a frictional coefficient, the intensive vibration coefficient – a ratio of the longitudinal amplitude of vibration to the amplitude of the first harmonic of normal vibration and frictional coefficient. Maximal conveying velocity is achieved at the certain values of normal vibration amplitudes and values of phase difference angles between longitudinal and normal vibrations, which are called optimal, and their values are dependent on these two dimensionless parameters, while maximum normal vibration acceleration should be equal to the gravitational acceleration. The research was made by approximate harmonic balance method and by numerical step-by-step integration method, which allows performing calculations with any given accuracy. The results obtained by the two methods are compared. To determine the maximal and optimal values of elevation angles, there are calculated the maximal value of the inclination angle parameter at which the value of dimensionless velocity is equal to zero, and the value of the inclination angle parameter at which a particle moves to a specified height in the minimum time. The optimal values of amplitudes of harmonics of polyharmonic normal vibration are determined in dependence on the inclination angle parameter with the number of harmonics from four to seven. The graphs of these dependencies are presented and the most important values of dimensionless parameters are presented in the table.

The influence of specimen size on the dynamic properties of lightweight concrete determined by the resonance method

The paper examines the way specimen size influences measurements by the resonance method. The experiment was performed with a lightweight concrete with a porous aggregate (Liapor). Several sets of specimens were made; they differed both in size and type (drilled cylindrical cores, cast prisms). All the specimens were tested for the natural frequency of longitudinal, transverse, and torsional vibration, which were then used to calculate values of the dynamic modulus of elasticity. The outcome of this experiment is a statistical analysis of the test results.

Longitudinal and Torsional Vibration Characteristics of Boron Nitride Nanotubes

Boron nitride nanotubes (BNNTs) are new cousins of carbon nanotubes (CNTs), which alternately substitute boron and nitrogen atoms for carbon atoms entirely. Its excellent physical and chemical properties, such as excellent thermal stability, chemical inertness and strong acid corrosion resistance, make BNNTs products be favorable to machine, thus scientific activities focused on them have profound practical significance and great application prospect. However, the vibration characteristics of BNNTs is still the least understood aspects. Longitudinal and torsional free vibration of single-walled boron nitride nanotubes are investigated through a coupling atomistic-continuum model with Euler beam model and atomistic simulations. Euler beam model with the involved atomistic-continuum multiscale model is capable of capturing the size effect since physical meanings of the integration for involved higher order gradient tensors are corresponding to size effect. The analytical solutions obtained by the coupling method are compared with atomistic simulation and are found to be in a good agreement. Computational results demonstrate that fundamental frequencies are in an order of 1 THz for both cantilevered and clamped boron nitride nanotubes, independent of tubular chirality. The effect of tubular radius on fundamental frequencies of both longitudinal and torsional free vibrations are large when it is small, especially the latter. The error between coupling method and atomistic simulation generally decreases as the length/diameter ratio increases. For a given length/diameter ratio, tubular radius have a distinctly decreasing effect on the error no matter what the kind of constraint is. However, a small difference observed in the study of torsional free vibration is that the clamped constraint gives rise to a slight increasing effect on the error as the tubular radius increases, but limited to a small degree. The vibrational modes confirm that the difference is originated from the involved breathing vibration in both torsional and longitudinal free vibration.

Longitudinal oscillation of a rod with a variable cross section

Variable cross-section rods are used in many parts and mechanisms. For example, conical rods are widely used in percussion mechanisms. The strength of such parts directly depends on the natural frequencies of longitudinal vibrations. The paper presents a method that allows numerically finding the natural frequencies of longitudinal vibrations of an elastic rod with a variable cross section. This method is based on representing the cross-sectional area as an exponential function of a polynomial of degree n. Based on this idea, it was possible to formulate the Sturm-Liouville problem with boundary conditions of the third kind. The linearly independent functions of the general solution have the form of a power series in the variables x and λ, as a result of which the order of the characteristic equation depends on the choice of the number of terms in the series. The presented approach differs from the works of other authors both in the formulation and in the solution method. In the work, a rod with a rigidly fixed left end is considered, fixing on the right end can be either free, or elastic or rigid. The first three natural frequencies for various cross-sectional profiles are given. From the analysis of the numerical results it follows that in a rigidly fixed rod with thinning in the middle part, the first natural frequency is noticeably higher than that of a conical rod. It is shown that with an increase in the rigidity of fixation at the right end, the natural frequencies increase for all cross section profiles. The results of the study can be used to solve inverse problems of restoring the cross-sectional profile from a finite set of natural frequencies.

Comparison of various multivariate models to estimate structural properties by means of non-destructive techniques (NDTs) in Pinus sylvestris L. timber

Abstract The predictability of modulus of elasticity (MOE), modulus of rupture (MOR) and density of 120 samples of Scots pine (Pinus sylvestris L.) were investigated using various non-destructive variables (such as time of flight of stress wave, natural frequency of longitudinal vibration, penetration depth, pullout resistance, visual grading and concentrated knot diameter ratio), and based on multivariate algorithms, applying WEKA as machine learning software. The algorithms used were: multivariate linear regression (MLR), Gaussian, Lazy, artificial neural network (ANN), Rules and decision Tree. The models were quantified based on the root-mean-square error (RMSE) and the coefficient of determination (R2). To avoid model overfitting, the modeling was built and the results validated via the so-called 10-fold cross-validation. MLR with the “greedy method” for variable selection based on the Akaike information metric (MLRak) significantly reduced the RMSE of MOR and MOE compared to univariate linear regressions (ULR). However, this reduction was not significant for density prediction. The predictability of MLRak was not improved by any other of the tested algorithms. Specifically, non-linear models, such as multilayer perceptron, did not contribute any significant improvements over linear models. Finally, MLRak models were simplified by discarding the variables that produce the lowest RMSE increment. The resulted models could be even further simplified without significant RMSE increment.

Development of ultrasonic vibration assisted uniaxial tension device and utilization in evaluation of micro tension deformation behavior of copper foil

ABSTRACTFor lots of advantages e.g. softening effect, ultrasonic vibration was widely used in industry. However, investigations were mostly focused on acoustic softening effect of flow stress, and other effects were seldom studied, which were very important for micro sheet forming. In this paper, a vibrator with sandwich structure was developed using piezoelectric ceramics PZT-8 as the ultrasonic transducer. A groove structure was designed at the end of vibrator to clamp the copper foil in the tension test. Dimensions of vibrator were determined by calculation and FE-simulation. The longitudinal vibration frequency and amplitude of manufactured vibrator were measured by laser vibrometer Polytec OFV-5000 as 35.21 kHz and 0.735 μm, respectively. With the developed device, a serial of tension tests were carried out using copper foil with different grain sizes. The results show that yield stress decreases when ultrasonic vibration is applied, which can be attributed to the acoustic softening effect. It is interesting to find that hardening index increases with ultrasonic vibration, and this means that limit formability can be improved in bulging forming, etc. Extensibility becomes smaller for the transverse vibration and fatigue damage. These findings are very helpful for understanding effect of ultrasonic vibration in micro sheet forming processes.

Evolution of Natural Frequencies of Longitudinal Vibrations of a Bar as Its Cross-Section Defect Increases

We study the evolution of characteristics of natural longitudinal vibrations of a circular bar in the case of increasing defect in its cross-section. It is shown that, in the limit case where the defect separates the bar into two independent fragments, the natural frequencies of the initially defect-free bar pass into the natural frequencies of its separate parts. The respective evolution of the natural modes of vibrations is observed. The evolution predicted by the theoretical analysis can be observed experimentally by using the resonance method and constantly increasing the defect till the final separation of the bar into two parts.