Frequencies Of Longitudinal Vibrations Research Articles

Consideration of supports stiffness at torsional and longitudinal vibrations of the rod

The study of the impact of the stiffness of the supports on the value of the first natural frequency of torsional and longitudinal free vibrations of the spring-hinged rod was carried out. In both cases of vibrations the mathematical formulation of the problem of determining the first natural frequency of vibrations reduces to compiling and solving the corresponding differential equation under given boundary conditions. The solution of this complex mathematical problem is performed by the numerical method in a specially created program for many combinations of supports stiffness. To obtain an analytical solution, a quadratic approximation of the obtained values was performed, which made it possible to obtain simple dependencies convenient for engineering applied calculations. The quadratic approximation also allowed solving the inverse problem of determining the required stiffness of supports for the required value of the first natural frequency of vibrations. The obtained solutions were compared with known solutions, which showed that the approximation error is less than 5% if the stiffness of the rod supports of the same order. The obtained results can be used in engineering applied calculations of natural frequencies of torsional and longitudinal vibrations of rods with elastic supports.

A Kirchhoff Plate Model for Longitudinal Vibration Analysis of Restrained Nanoplate Including Thermal Effects

This work attempts to apply the Kirchhoff plate theory to find out the vibrational analyses of a nanoplate incorporating thermal effects. The effects of thermal environments on the natural frequency of longitudinal vibration of restrained nanomaterials, especially for restrained nanoplates, have not been investigated, and most of the previous research has been carried out for unrestrained nanoplates. Therefore, it must be emphasized that the vibrations of restrained nanoplate, including thermal effects, are novel and applicable to the nanodevices, in which nanoplates act as the main structure of the nanocomposite. A novel motion and frequency equation are derived using the Kirchhoff plate model. The present study illustrates that a nanoplate’s longitudinal vibration characteristics strongly depend on the temperature change and stiffness coefficients. The numerical results clearly show that the longitudinal natural frequencies of the nanoplate are less than unity for both cases of low and high temperatures. This means that applying the Kirchhoff plate model for restrained nanoplate analysis would lead to an over-prediction of the frequency if the small thermal stress effect is neglected. Finally, the investigation of the restrained and thermal impact on longitudinal vibration of nanoplates may be used as a valuable reference for the application and the design of nanoelectronics and nano-drives devices, nano-oscillators, and nano-sensors, in which nanoplates act as essential elements.

Effectiveness of Friction Force Reduction in Sliding Motion Depending on the Frequency of Longitudinal Tangential Vibrations, Sliding Velocity and Normal Pressure

Abstract The article presents the results of experimental research and simulation analyses of the influence of slip velocity, normal pressures and vibration frequency on the effectiveness of friction force reduction carried out in sliding motion in the presence of forced tangential vibrations. In experimental studies, changes in the driving force were measured during the slip of the upper body over the vibrating lower body. The direction of these vibrations was parallel both to the contact plane and to the direction of movement of the shifted body. The simulation tests were carried out in the Matlab/Simulink environment through the use of numerical procedures that were specially created for this purpose. Dynamic friction models considering the tangential compliance of contact and the phenomenon of pre-sliding displacement were used for calculations. The paper presents the designated values of the so-called coefficient of average friction force reduction in sliding motion for the following friction pairs: steel C45–steel C45, steel C45–cast iron GGG40 and steel C45–polytetrafluoroethy-lene PTFE (Teflon). The results of numerical analyses were in good agreement with those of experimental tests. A significant dependence of the level of average friction force reduction on the frequency of forced vibrations, sliding velocity as well as the kind of sliding pair material, and normal pressures was shown.

Modeling the deadwood bearings wear surface by optical scanning

The development of a methodology for modeling the wear surface of deadwood bearings based on optical scanning is considered in the paper. It is noted that the deadwood bearings of the ship’s shaft line wear out unevenly during operation, which affects the stress-strain state of the shaft and the system parameters responsible for the occurrence of resonance of transverse, torsional and longitudinal vibrations. The methodology and results of an experimental study of bearing wear surfaces are presented. The applied optical scanning scheme makes it possible to obtain high-precision digital models of objects due to synchronous shooting of an object with two cameras. The measurement results are obtained by processing with specialized software. The results obtained are an addition to the methodology for determining the elastic force acting on the shaft from the side of a bearing that is unevenly worn along the length with a non-constant gap. Based on the obtained wear values, a solid-state model is constructed and the natural frequencies of transverse, torsional and longitudinal vibrations are determined. For the constructed model, the forms of natural oscillations for practically significant frequencies are obtained. It is noted that bearing wear affects not only the displacement of the natural frequencies values, but also the oscillations shape. The impact of the wear of the aft bearing on the bearing capacity of the shaft line, taking into account the contact interaction, is evaluated. The simulation has shown that the bearing wear surface forms an area of increased contact pressures that arise from the impact of the propeller shaft. Analysis of the propeller shaft tip has revealed the presence of an area of increased concentration of equivalent stresses formed at the place of bearing wear formation. The proposed method of modeling the deadwood bearings wear allows us to evaluate its effect on the magnitude of the displacement of natural frequencies and the change in the bearing capacity of the shaft line. The results obtained in this work are an addition to the existing calculations in the design and evaluation of the ship’s power plant reliability.

Damage Detection in Free–Free Glass Fiber Fabric Composite Beams by measuring Flexural and Longitudinal Vibrations

This paper presents the experimental investigation results of the use of the vibration method by impulse excitation of free flexural and longitudinal vibrations of composite materials. The purpose of the study is to establish the sensitivity of the method used for defect detection and localization. To realize the objective, rectangular notch-type defects were simulated at different distances and depths. The influence of the location and depth of the artificial cracks on the dynamic properties of the beams was investigated by measuring the natural frequencies of flexural and longitudinal vibrations. The conducted experiments show a change in the dynamic characteristics of the beam depending on the dimensions and location of the defects.

First Frequency of Longitudinal Vibration of a Rod with Mass at one End and Partial Placement on Piecewise-Homogeneous Elastic Foundation

Statement of the problem. Despite the introduction of engineering practice of numerical methods for the calculation, in a case of engineering problems associated with extended structures, the well-known models «rod on an elastic foundation» and «beam on an elastic foundation» are widely used to describe the behavior of structures. Most of the known solutions were obtained in the conditions of an idealized setting and without differences in the properties of the foundation. This work is devoted to the study of the dynamics of a rod of finite length with a point mass at the end and partially placement on a piecewise-homogeneous elastic foundation, formed by two sections of different stiffness. The aim of this study is to obtain analytical dependences for the first frequency of the longitudinal vibrations of a rod with different boundary conditions at its ends: rigid support and free end. The available research results are obtained for special cases, which makes them difficult to apply in practice. To assess the influence of the inertial pro-perties of the foundation, the case with different linear masses of the rod in the pinched and cantilever sections is considered. Results. Using the Ritz method analytical dependences were obtained for the determination of the first frequency of the longitudinal vibrations of a fix and free rod with a point mass at the end and partially placement on a piecewise-homogeneous elastic foundation formed by two sections of different stiffness. Quantitative evaluation shows good agreement with those of other studies. The results for particular cases are obtained. Conclusions. It is shown that the first natural frequency of longitudinal oscillations is in a certain range and in the nonlinear ratios depending on the lengths of the sections various elastic. The inertial properties of the foundation leads to a decrease in the first frequency.

Identification of a longitudinal notch of a rod by natural vibration frequencies

Objectives. To study the direct and inverse problem of vibrations of a rectangular rod having a longitudinal notch, to analyze regularities of the behavior of natural frequencies and natural forms of longitudinal vibrations when changing the location and size of the notch, and to develop a method for uniquely identifying the parameters of the longitudinal notch using the natural frequencies of longitudinal vibrations of the rod.Methods. The rod with a longitudinal notch is modeled as two rods, where the first one does not have a notch, while the second one does. For connection, conjugation conditions are used, in which longitudinal vibrations and deformations are equated. The solution of the inverse problem is based on the construction of a frequency equation under the assumption that the desired parameters are included in the equation. Substituting natural frequencies into this equation, the nonlinear system with respect to unknown parameters is derived. The solution of the latter is the desired notch parameters.Results. Tables of eigenfrequencies and graphs of eigenforms are given for different notch parameters. The results for different boundary conditions are obtained and analyzed. A method for identifying notch parameters by a finite number of eigenfrequencies is presented. The inverse problem is shown to have two solutions, which are symmetrical about the center of the rod. The unambiguous solution requires eigenfrequencies of the same problem with different boundary conditions at the right end. By adding additional conditions at the ends of the rod, the inverse problem can be solved with new boundary conditions to construct the exact solution and develop an algorithm for checking the uniqueness of the solution.Conclusions. The developed method can be used to solve the problem of identification of geometric parameters of various parts and structures modeled by rods.

Design and optimization of a half-wavelength ultrasonic bone scalpel

Compared with traditional orthopedic surgical instruments, the ultrasonic bone scalpel (UBS) has obvious advantages in both safety and effectiveness. Based on the structure of the sandwich piezoelectric transducer, a design of half-wavelength UBS with a longitudinal vibration frequency of 22.5 kHz is proposed in this study. Firstly, the dimensions of each component of the UBS are calculated by the transmission matrix method, the initial design is verified by modal analysis, and the error is optimized by the response surface method. Then the harmonic response analysis is used to calculate the UBS’s tip amplitude, impedance and stress distribution. In addition, the influence of bolt pre-tightening force on the vibration characteristics of the UBS is analyzed. When the bolt pre-tightening force is 50-90MPa, the vibration performance of the UBS is stable, and the tip amplitude is large, which provides a reference for selecting appropriate tightening torque when the UBS is assembled. Finally, the bone-cutting experiment shows that the UBS has good cutting performance.

Design and test of a small high frequency ultrasonic cleaning transducer

In order to solve the problem of cleaning of complex surfaces and micro blind holes in industrial products, a small high frequency ultrasonic cleaning transducer was designed and tested. Firstly, the finite element model of the transducer was established, and the influence of structural parameters on the first order longitudinal vibration frequency was analyzed. By adjusting the structural parameters of the transducer, the first order longitudinal vibration frequency of the transducer was determined to be 58716hz.The harmonic response of the transducer was analyzed. The results showed that when the voltage amplitude applied on the piezoelectric ceramic sheet was 80V, the transducer produced resonance at 58716 Hz, and the end face of the front cover of the transducer produced an axial displacement of about 3.34 µm . The transducer was made according to the theoretical analysis results, and the frequency sweep test was carried out on the transducer. The theoretical analysis results were consistent with the test results, which showed that the designed small high frequency ultrasonic cleaning transducer could meet the requirements of high frequency cleaning.

Electromechanical coupling model of variable-section piezoelectric composite beams in longitudinal vibration

The traditional step-reduction method is popular for modeling mechanical vibration of variable-section homogeneous beams (VHBs). However, this method cannot transfer electrical information and cannot depict electromechanical coupling behaviors of variable-section piezoelectric composite beams (VPCBs) in longitudinal vibration. To solve this defect, considering the structural continuity and parallel circuit factors, we propose the electromechanical step-reduction method for VPCBs. Longitudinal vibration transfer equations, linking input and output state vectors, are first created for VPCBs according to the proposed method. Then, a piezoelectric actuator excited by a VPCB is designed as a case study. Its transfer matrix model, involving mechanical and electrical properties, is developed to prove the correctness of the proposed method and the created transfer equations. Besides, an actuator prototype was manufactured for experimental verifications. The calculation results show that the longitudinal vibration frequency and velocity tend to be stable with the increase of the division number n. Their velocity vibration shapes are also consistent with each other. The calculated and experimental impedance curves have the same trend. The minimum and maximum impedance differences between the calculated and experimental are 107 ohm and 127 ohm, respectively. Finally, the modeling differences between the traditional and electromechanical step-reduction methods are compared and discussed in detail. The proposed method is more applicable to the modeling of VPCBs, which provides a theoretical tool for optimizing the excitation of the longitudinal vibration.

Longitudinal Vibrations of Restrained Irregular Nanoplates

The main purpose of this work is to examine how surface irregularity affects the longitudinal vibration frequency of nanoplates. Based on the Kirchhoff plate model, the boundary conditions and governing equations are developed. The MATLAB R2013a Software is utilized to carry out the numerical solutions, and an accurate solution is shown. We derive a novel equation of motion as well as a new frequency equation. Examined is how several factors, such as thickness, stiffness, and surface irregularity, affect the longitudinal vibration of nanoplates. The studies revealed that from large to small irregularity parameters, the longitudinal natural frequencies of nanoplate increase. Additionally, it has been discovered that increasing the thickness of nanoplates is preferred for enhancing their vibration stability.

Longitudinal Vibrational Analysis of Restrained Irregular Single-Walled Carbon Nanotube Using Thin Cylindrical Shell Model

The main purpose of this work is to investigate the impacts of surface irregularity on the natural frequency of longitudinal vibration of the single-walled carbon nanotube (ISWCNT). The boundary conditions and governing equations are formulated based on thin cylindrical shell theory. An exact solution is presented and the MATLAB R2013a Software is used to carried out the numerical solutions. A novel equation of motion and frequency equation are derived. The impact of various parameters, including surface irregularity, stiffness and thickness, on longitudinal vibration of SWCNT is examined. It is concluded that, the longitudinal natural frequencies of SWCNT increase from large to small irregularity parameter. Additionally, it has been found that increasing the ration (h/R) of nanotube is preferable for improving the vibration stability of SWCNTs. Moreover, the findings of this work can be used to produce the best designs for nanodevices with SWCNT as a common structural component.

Metastructure with integrated internal oscillators of constant, linearly and nonlinearly varying natural frequency

This research focuses on the analysis of the model and performance of lightweight metastructures encompassing a distributed array of internal homogenous oscillators, integrated into the host structure to create a single-piece element. This metastructure performs longitudinal vibrations, whose axis is colinear with the direction of the transverse vibration of the internal oscillators. First, the mechanical models of the separate elements of the metastructure and the metastructure as a whole are created and considered. The first modal frequencies of longitudinal vibrations of the metastructure with blocked and free internal oscillators are tuned to the first modal frequency of transverse vibration of one internal oscillator, yielding the optimal number of internal oscillators for this to be achieved, which is a new result for the proposed design. This theoretical result is then checked experimentally with the metastructures produced by 3D printing technology, comprising a different number of internal oscillators, all of which have the same natural frequency. Besides validating the theoretical results, experimental investigations with blocked and freely vibrating internal oscillators of the constant natural frequency are used to explore other performance characteristics, such as the width of the regions where the reduced amplitude is achieved. Finally, based on the theoretical and additional numerical results, the internal oscillators are modified in two ways, which is an original approach: their natural frequency is increased linearly and nonlinearly along the metastructure in accordance with the previous new theoretical results. The benefits of such new redesigns for the multi-modal performance characteristics of the metastructure are discussed.

Modeling and optimization of longitudinal-torsional vibration horn with the large tool head

There are problems such as large deviations in resonant frequency and difficulties in selecting the structural parameters of helical slot when designing longitudinal-torsional vibration horn with large tool head (HWLTD) using traditional design theory of horn. Therefore, in this paper, HWLTD was designed integrally, the model of HWLTD was established, longitudinal vibration frequency equation of HWLTD was derived and consequently, the resonant frequency was predicted and optimized. Then, the principle of vibration mode transformation (POVMT) of helical slots was derived. Combined with POVMT and ANSYS simulation analysis, resonant frequency and torsional-longitudinal ratio of longitudinal-torsional vibration of HWLTD were optimized, thereby improving the vibration characteristics of HWLTD. The resonant frequency and vibration effect of HWLTD were tested, and the measurements were agreed with the theoretical design and simulation analysis, which validates the HWLTD model and POVMT. The research provides a reference for the efficient and reliable design of HWLTD in the field of ultrasonic machining.

A novel ultrasonic transducer with multiple vibrations in microelectronic packaging

Ultrasonic wire bonding is a typical packaging for semiconductor devices. Compared with single longitudinal transducer, we propose a novel ultrasonic transducer with multiple vibrations for the microelectronic packaging. The transducer is designed to excite the longitudinal and bending vibration individually or simultaneously. By simulation calculation, three longitudinal vibrations and a bending vibration is attained in the same node flange. The transducer structure is established by optimizing a balance and a Z-shaped flange, which can achieve the optimal modal shape. Under the 10 kg·cm torque for clamping the piezoelectrics stacked, the frequency of longitudinal vibration is 64.8 kHz 103.4 kHz 142.8 kHz and bending vibrations is 70.5 kHz. A dual channel driver with Field Programmable Gate Array (FPGA) and Digital Signal Processing (DSP) controller is self-developed to excite the multiple vibrations. In experiment, each vibration is excited individually, switched or simultaneously. It is found that all vibration for each operating modes is higher than 1 µm, and the vibration profile can be controlled as a rectangle or parallelogram, which has potential to be a new bonding process.

Investigation on Deformation Behavior in the Surface of Metal Foil with Ultrasonic Vibration-Assisted Micro-Forging.

Excitation of the acoustic field, leading to the Blaha effect, significantly affects the plasticity of a material. In the micro-forming field, the so-called impact effect is found to generate a larger amount of dislocation and produce greater plastic deformation than acoustic softening. In this study, the mechanism of deformation in the surface of the material with ultrasonic vibration assistance was investigated and compared with that in the bulk. Forging tests using a newly developed ultrasonic vibrator were carried out on pure Cu foils with various process conditions. The longitudinal vibration frequency of the ultrasonic transducer was 60 ± 2 kHz, and the vibration amplitude was in an adjustable range of 0~6 μm. Forging tests were carried out at different amplitudes. The result shows that acoustic softening and the impact effect could be separated by an oscilloscope in the micro-forging system. The difference in deformation on the surface asperity caused by acoustic softening and the impact effect is discussed. Compared to acoustic softening, which has a limited effect on the deformation of the surface asperity, the impact effect could create more plastic deformation on the surface asperity. Therefore, the reduction in the surface roughness would increase after the impact effect occurs. In addition, to confirm the mechanism of acoustic softening and the impact effect, the microstructural evolution of specimens, at the surface scale and inner scale, was investigated by electron backscatter diffraction (EBSD). It was found that acoustic softening could create more grain refinement, and with the amplitude increasing, the impact effect would oppositely cause the surface grains to grow. In this study, the mechanism of how the impact effect and acoustic softening affect the deformation behavior of the surface asperity was investigated.

Urban bridges studies on dynamic impacts from urban rolling stock

The article is dedicated to the actual dynamic impact study of rolling stock on urban bridges. The requirements for recording the dynamic impact specified in domestic and foreign regulatory documents are considered. Using a mobile measuring complex, the results of which determine the natural frequencies of vertical, horizontal transverse, and horizontal longitudinal vibrations. The technique used in the experiment allows us to obtain initial data for the dynamic calculation of structures, to clarify the real rolling stock dynamics depending on the state of the rail track on the bridge. Urban rolling stock is a relevant and necessary type of public transport that meets modern city requirements. The bridge crossings construction will ensure the connection of the urban areas separated by obstacles into a single network. When designing such structures, it is important to take into account the effects of actual dynamic coercitive forces. The dynamics study allows, on the one hand, to reasonably set the dynamic loads impact on bridges, on the other hand, to diagnose the structures state by dynamic parameters. In this work, using a specialized mobile measuring complex, the authors measured the oscillation parameters on moving vehicles. As a conducted experimental studies result, a methodology for determining the dynamic characteristics of a moving load and directly urban bridges has been demonstrated. As a result of the conducted experimental studies, it can be concluded that the proposed method allows obtaining initial data for the dynamic analysis of structures for the transport load. In addition, fixing the transport load dynamic impact during operation allows you to clarify the real rolling stock dynamics, depending on the state of the rail track on the bridge.

Influence of longitudinal vibration on the friction and wear characteristics of multi-layer winding hoisting wire rope

The longitudinal vibration of the multi-layer winding hoisting wire rope is inevitable due to the time-varying dead weight and the fluctuation of hoisting acceleration. Longitudinal vibration will lead to relative sliding between the hoisting wire rope and the rope groove, resulting in friction and wear, and thus threaten the service life of the wire rope. In order to investigate the influence of longitudinal vibration on the tribological characteristics of the wire rope and its wear mechanism, the friction and wear tests between the sliding wire rope and the fixed wire rope groove under different longitudinal vibration amplitudes and frequencies were carried out with the help of a self-developed test rig. Results show that the coefficient of friction (CoF) of the wire rope has experienced three stages of rapid growth, “concave” transition and relative stability with the increase of sliding distance, which increases first and then decreases with increasing the amplitude in the relatively stable stage and has no obvious change with an increase in the frequency. Additionally, the maximum temperature rises at the wind-out region of the sliding wire rope and the middle of the contact wire ropes increase first and then decrease with the increase of the amplitude in the relatively stable stage, but increase gradually with an increase in the frequency. The vibration intensities in the wear regions of the wire ropes have a great increase in the rapid wear stage. Furthermore, the main vibration wear mechanisms of the wire rope are abrasive wear and adhesive wear, the content of O element in the furrows and the abrasive adhesion regions are higher, and the frequency has less effect on the oxidation degree of the worn surface of the wire rope than the amplitude.

Determination of the variable density of the rod from natural frequencies of longitudinal vibrations

Rods of various configurations are elements of many structures and machines. Therefore, the acoustic and vibration diagnostics of such parts has been widely developed. The paper considers the problem of determining the variable density of the rod from the natural frequencies of longitudinal vibrations. It is assumed that the density changes along the axis and is described by a polynomial function. This approach allows one to determine the law of density variation from a finite set of eigenvalues. The results of the study can find applications for finding hidden defects in steel and composite rods, which arise during the production process or due to corrosion.

THE METHOD OF VARIATION OF ARBITRARY CONSTANTS IN THE CASE OF A SYSTEM OF LINEAR DIFFERENTIAL EQUATIONS OF DIFFERENT ORDERS

The paper considers structures consisting of rods connected in one node.Longitudinal and transverse vibrations of such a structure are described by systems of linear differential equations on star graphs.The noted system of equations consists of three linear differential equations of different orders.Two equations correspond to two transverse vibrations, and the third equation describes the longitudinal vibrations of the bar.Moreover, the system of three linear differential equations in the general case does not decompose. In this work, a fundamental system of solutions of a homogeneous system is constructed when the conjugation conditions are satisfied at the point of connection of the rods. Also, by the method of variation of arbitrary constants, a particular solution of an inhomogeneous system is constructed, which is subject to the conjugation conditions at the point of connection of the rods. In subsequent works, the authors intend to investigate the natural frequencies of longitudinal and transverse vibrations of a structure consisting of many rods