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Deep brain stimulation (DBS) is an effective treatment for Parkinson's disease. The cannula insertion process plays an important role in DBS. The friction force during needle insertion influences the precision of the insertion and the degree of damage to the brain tissue. This paper proposes a method of longitudinal vibration assisted insertion to reduce the friction during insertion and improve the effects of the insertion. Cannulas were inserted into twenty eight pig brains at multiple frequencies and fixed amplitudes, and the resulting friction force was measured. On this basis, the LuGre model was used to analyze the friction force trend under vibration-assisted conditions. The frictional forces of vibration-assisted insertion with frequencies ranging from 200–1200 Hz and an amplitude of 1 μm were measured. The results show that the friction between the needle shaft and the tissue is smaller with vibration than without vibration. In this experiment, the friction is reduced by up to 24.43%. The friction force trend of vibration-assisted insertion conforms to the simulation results of the LuGre model.

In this study, a 55.5 kHz frequency ultrasonic transducer is designed for an electrosurgical electrode to satisfy the requirement of minimally invasive surgery. The structure of the ultrasonic transducer is modified to induce longitudinal vibration at 55.5 kHz. Simulation results demonstrate that the ultrasonic transducer would longitudinally vibrate at frequencies of 48.01 kHz and 56.035 kHz. An ultrasonic transducer is fabricated to experimentally test the simulation results; the results demonstrate that the resonant frequency is approximately 55.5 kHz, and the output amplitude is satisfactory for cutting soft tissues.

In this paper a new non-iterative computational technique referred to as the symbolic-numeric method of initial parameters (SNMIP) is proposed. The SNMIP represents a modification of the iterative numeric method of initial parameters in differential form known in the literature. The SNMIP is applied to study free vibrations of Euler-Bernoulli axially functionally graded tapered, stepped, and continuously segmented rods and beams with elastically restrained end with attached masses. Both the longitudinal vibration of rods and transverse vibration of beams are considered. The influence of the attached masses and springs on the natural frequencies of vibration of axially functionally graded rods and beams is examined. The validity and accuracy of the method are proven through the comparison with the known results in the available literature.

The article deals with the problem of computing efficiently the nonlinear response of a rod involving a fractional constitutive model, and exposed to random excitation. The constitutive model is a three-parameter model comprising an instant elasticity modulus, a prolonged elasticity modulus, and a relaxation parameter. The nonlinear term is a linear-plus-cubic force of the Winkler kind. The resulting nonlinear fractional partial differential equation governing the rod displacement has no known exact solution. Thus, the article proposes an approximate analytical solution by relying on the statistical linearization technique. Further, it develops a Boundary Element Method (BEM)-based approach to estimate numerically the rod response statistics. The statistical linearization solution is obtained by representing the rod displacement as the superposition of linear modes of vibration having time-dependent coefficients. In this context, it is shown that the equation governing the time variation of the mode coefficients is a nonlinear fractional ordinary differential equation, whose solution is computed by a surrogate linear system identified by minimizing the response error between the linear system and the nonlinear one in a mean square sense. Relevant Monte Carlo studies pertaining to rods with fixed-fixed, and fixed-free ends show that the proposed analytical solution is in a good agreement with data obtained by the numerical (BEM) approach.

An inverse spectral problem for a rod of variable cross-section containing a finite number of transverse cracks is considered. The cracks are simulated by translational springs. It is proved that the number of springs, their locations and flexibilities are uniquely reconstructed by two spectra corresponding to longitudinally vibrating rod with free-free and free-fixed end conditions. A constructive procedure for reconstructing unknown damage parameters is developed. Numerical examples are considered.

Titanium has widely been used in chemical and aerospace industries. In order to overcome the drawbacks of cold compaction of titanium, the process was assisted by an ultrasonic vibration system. For this purpose, a uniaxial ultrasonic assisted cold powder compaction system was designed and fabricated. The process variables were powder size, compaction pressure and initial powder compact thickness. Density, friction force, ejection force and spring back of the fabricated samples were measured and studied. The density was observed to improve under the action of ultrasonic vibration. Fine size powders showed better results of consolidation while using ultrasonic vibration. Under the ultrasonic action, it is thought that the friction forces between the die walls and the particles and those friction forces among the powder particles are reduced. Spring back and ejection force didn’t considerably change when using ultrasonic vibration.

This paper examines the effectiveness of a MR damping system with lumped mass for a longitudinal vibration control of two-span bridge, preventing spans from pounding each other and mitigating its seismic responses when earthquake-induced load is inflicted. The MR damping system with lumped mass proposed in this study it was constituted by three components: a MR damper (30 kN) which connected two spans of a bridge and two spans themselves made different mass and stiffness. Its function was designed to impose an effective check on relative displacement and pounding. In order to prove its effectiveness, a two-span bridge (8.3 m long) was constructed with a MR damper installed beneath the two spans. First, a mathematical model of the bridge was derived; also, a model of the MR damper was verified through a separate test on its performance. Next, shaking table tests were carried out by inflicting 150% of El-centro earthquake and 60% of Kobe earthquake to see how effective the system would be in mitigating seismic responses. All the tests were done under various conditions (un-control, passive off, passive on, Lyapunov control, and Clipped-optional control) in order to closely examine and analyze under which condition mitigation could be maximized and how mitigation could be effected. As a result of the tests, it was found that the MR damping system with lumped mass was effective in preventing two spans from pounding each other and its active performance decreased the relative displacement of the bridge structure. Therefore, it was proven to be effective in attenuating the seismic responses of two-span bridge in longitudinal direction.

Longitudinal vibration of a cracked beam subjected to the transverse magnetic field is addressed. One dimensional wave theory is used and the crack part of the beam is modelled as by a linear spring, in this analysis. The obtained explicit solution, for clamped-free and clamped–clamped boundary conditions, is illustrated as graphically. It is generally seen that the longitudinal frequencies increase with increase of the intensity of magnetic field. However, the results of the analysis reveal that the transverse magnetic field has a more significant effect on the variation of longitudinal vibration frequency under the clamped–clamped boundary conditions.

The longitudinal vibration of a sucker rod may change plunger stroke, thus affecting the pumping efficiency. In the study, a simplified thermo-solid coupling model of the longitudinal vibration of a sucker rod is established based on the consideration of the effect of the temperature along the well depth. Then, with the finite element method, the influence of longitudinal vibration of a continuous steel sucker rod on plunger stroke is studied. The calculation model involves two stages. In the first stage, the polished rod position remains stationary, whereas the sucker rod vibrates only under its gravity and liquid column gravity. The final stable state of the sucker rod is the initial state of the second stage. This stage aims to eliminate the effect of the initial tensile vibration. The longitudinal vibration characteristics of the sucker rod in the second stage are calculated based on the consideration of the excitations of both the polished rod displacement and the liquid column load in order to ensure the accuracy of calculation results. The influences of the frequency of stroke, stroke, and the length of sucker rod on plunger displacement are studied. The plunger displacement is slightly lagged behind the excitation on the polished rod due to the joint action of elasticity and load. The frequency of stroke, stroke, and the length of sucker rod affect the overtravel. When the sucker rod length is 2000 m, the overtravel reaches 0.12 m. Other parameters can also be optimized with this model.

A novel approach is presented to describe the dynamic interaction system of a large-diameter floating pipe pile and surrounding soils, taking the three-dimensional wave effects into account. The corresponding analytical solutions for longitudinal complex impedance are obtained and subsequently validated via comparisons with existing solutions. Comparative analyses are also performed to illustrate the difference between the present and previous solutions, concerning the wave propagation effect in the radial direction on the longitudinal dynamic vibration of pile shaft. Furthermore, the effects of Poisson’s ratio and visco-elastic support beneath the pile toe, on the longitudinal dynamic vibration of pile shaft, are investigated. It is indicated that the presented approach and corresponding solutions provide a more wide-ranging application for longitudinal vibration analysis of a large-diameter floating pipe pile, which can also be reduced to analyze the longitudinal vibration problems of large-diameter floating solid pile and fixed- end pipe pile.

Classical nonlinear vibration methods used for structural damage detection are often based on higher- and sub-harmonic generation. However, nonlinearities arising from sources other than damage – e.g. boundary conditions or a measurement chain – are a primary concern in these methods. This paper focuses on localisation of damage-related nonlinearities based on higher harmonic generation. Numerical and experimental investigations in longitudinal vibration of beams with breathing cracks are presented. Numerical modelling is performed using a two-dimensional finite element approach. Different crack depths, locations and boundary conditions are investigated. The results demonstrate that nonlinearities in cracked beams are particularly strong in the vicinity of damage, allowing not only for damage localisation but also for separation of crack induced nonlinearity from other sources of nonlinearities.

81 timber beams (14 of Spruce and 67 of Silver fir) were salvaged from different historical buildings. All the timber were visually graded, dynamically (vibration and acoustic methods were used) and statically (four point bending test) tested, to calculate the actual MOE/MOR. The results of the visual strength grading and dynamic tests were combined, in order to predict MOE and MOR on-site, or on reclaimed timber after their dismantling. For salvaged timbers, comparable results to those reported in literature for new solid timbers were obtained on MOE/MOR prediction, combining longitudinal vibration tests and wane extension (r 2 = 0.71 and 0.43 for MOE and MOR prediction respectively). Within this research, alternative simplified methods, for the MOE/MOR prediction, which are based on the stress wave velocity and wane extension, applicable on site, were proposed (r 2 = 0.57 and 0.29 for MOE and MOR prediction respectively). Since the results were obtained by testing a sample with large variability, adopting some simplifications, they can be used as a starting point for the assessment of MOE/MOR of timber directly on-site, or to evaluate the mechanical properties of reclaimed timber after their dismantling, to assess their potential reuse in timber structures.

A problem of identification of a finite number of small cracks in a rod by means of natural frequencies corresponding to longitudinal vibration is considered. The cracks are simulated by massless translational springs. It is known, that generally for unique identification of multiple cracks in a rod it is necessary to know two spectra. An algorithm is developed enabling in case of small cracks to extract an important information about the possible number of cracks, their positions and the flexibilities of the corresponding springs by means of only one spectrum. The developed method is extended for identification small multiple cracks in a simply supported beam.

Accurate evaluation of dynamic resistance provided by the surrounding soil is the key to determining the dynamic performance of the pile foundation, especially for long piles. A new method is proposed to evaluate the dynamic pile-side soil resistance (DPSR). A dynamic coupled model of the pile-soil system is established considering the three-dimensional nature of the surrounding soil. Analytical solutions are then deduced for the dynamic response of each point along the pile shaft. The derived analytical solutions are then used to investigate several characteristics of the DPSR by analyzing the attenuation of the incident wave propagating along the pile shaft. The results show that the effect of DPSR decreases gradually from the pile top to tip under low-strain dynamic loading conditions, and it increases as the shear wave velocity of the surrounding soil increases between points under consideration along the pile shaft. It is also found that energy dissipation is the most important mechanism of soil resistance during longitudinal vibration associated with low strain axial loading, and thus damping should be evaluated carefully. The findings from this study aid in understanding the dynamic pile-soil interaction and provide practical guidance for analyzing the effect of the DPSR in engineering practice.

An analytical friction model considering material properties and surface topography is established to predict the friction reduction performance of longitudinal vibration, and a set of closed-form solutions are derived and verified. A parameter study is conducted based on the model. Results show that the vibration parameters, normal load, surface morphology, and material properties have significant influences on the friction reduction effect. The friction reduction mechanism is mainly attributed to the changes in the magnitude and direction of asperities tangential deformation. The analytical model can efficiently and accurately calculate the contact parameters and instantaneous friction force of rough contact surfaces, and provide theoretical guidance for the rational utilization of longitudinal vibration to achieve better friction reduction performance in mechanical systems.

This paper discusses the accuracy of the fundamental governing equation based on Rayleigh rod theory and the nonlocal integral model, for the longitudinal vibration derived in [6]. This discussion shows that; when shear presence is neglected, the nonlocal Rayleigh-Bishop wave solution existing in the literature cannot be reduced to the present nonlocal wave solution presented by these authors. Since the nonlocal Love wave solution must be recovered from the nonlocal Rayleigh-Bishop wave solution, hence, this case is an unexpected result.

The analytic modeling and one experimental assess of the ultrasonic surface mechanical attrition treatment are presented. The bombarding ball speed, induced energy, hardness and grain size are incorporated into this model, based on harmonic longitudinal vibration motion of ultrasonic-wave-driven impact. There appear some optimum attrition working parameters for the best attrition effect; beyond which, the sample surface would be subject to bombarding microcracking and the grain size would not be further reduced. The benefits from attrition would become lower.

A problem of identification of multiple cracks in a rod using spectral data is considered. The cracks are simulated by translational springs. A method for determination of the number of springs, their locations and flexibilities is developed. The problem is reduced to the problem which can be solved by Krein׳s method. The Krein method enables to reconstruct the damage parameters by means of the use of natural frequencies for longitudinal vibration of the rod with free–free and fixed–free end conditions.

The ultrasonic machining is a processing method using both the ultrasonic vibration of the tool horn and abrasive slurry. We studied a new ultrasonic machining method using ultrasonic complex vibration caused by the longitudinal and torsional vibration. In previous studies, we found that the machining speed and the machining accuracy when using a complex vibration are improved as compared with that using conventional ultrasonic machining method. However, the mechanism of ultrasonic machining using longitudinal-torsional vibration has not been clarified. In this presentation, we study that the observation of machining marks of soda-lime glass caused by ultrasonic machining using complex vibration.

Surface adhesion between wet wafers poses great challenges for silicon wafer handling. It has been shown that both the shear and normal handling forces of the solar silicon wafers can be dramatically reduced by using the ultrasound energy. Approximately 20 and 5 times reduction in horizontal and vertical forces were achieved by as low power as 10 W, and a good agreement was found between the measured values and the predictions of a simple model for the effect of longitudinal vibration we developed.

In recent years, there have been many modern engineering materials introduced recently which are hard, brittle and difficult to machine. These materials are very difficult to machine with conventional turning methods. Ultrasonic assisted turning (UAT) is a new non-conventional technique, used to remove an unwanted material to produce a desired product. Ultrasonically assisted turning (UAT) is an advanced machining technique, where a high frequency of vibration (frequency f ≈ 20±0.5 kHz, amplitude around a ≈ 20 μm) is superimposed on the movement of a cutting tool. In this method, high frequency electrical energy is converted into mechanical vibrations via a transducer or booster combination, which are transmitted through a horn or tool assembly. An ultrasonic vibratory tool (UVT) designed and analyzed using ANSYS® (one of the FE code) for the calculation of its natural frequency and working amplitude of vibration. Taguchi with TOPSIS method is used to optimize both cutting force and surface roughness to find the best possible machining parameters under the used experimental working condition in UAT.

An ultrasonic vibration assisted micro-milling (UVAM) with longitudinal vibration of workpiece is investigated in this paper. The most essential difference between ordinary micro-milling and UVAM lies in the trajectory of cutter tooth. A mathematical model has been established to simulate the trajectory of the cutter tooth. A more complex and ergodic trajectory has been found in UVAM. Series of experiments have been performed to investigate the surface roughness of Al6061 processed by micro-milling with and without ultrasonic vibration. It is found that the surface of Al6061 processed by UVAM has a smaller roughness value compared to that processed by ordinary micro-milling when choose an appropriate ultrasonic vibration amplitude. The influence of different ultrasonic vibration amplitude on surface roughness of Al6061 processed by UVAM has been analyzed. It is found that ultrasonic vibration amplitude has an optimum value with regard to the surface roughness of Al6061 processed by UVAM and is not the bigger the better.

This study is to investigate the wire rope governing equation in order to establish the longitudinal vibration characteristics of bonded PSC strands. For this purpose, the longitudinal vibration tests have been conducted for six bonded PSC specimens with the different stress levels. Subsequently, the wave velocities of the strands have been estimated from the test results. Then, a sensitivity-based system identification algorithm has been applied to evaluate the constitutive constants of the strands using the measured wave velocities. The present study demonstrates that the longitudinal dynamic behaviour of the strands is governed by coupled extensional-torsional oscillations. The study also reveals that longitudinal elastic wave velocity of not only the bonded strands but also unbounded strands is nonlinearly increased as the applied tensile stress level increases.

The propulsion shafting is one typically continuous, elastic and stepped body. To design an active controller for propulsion shafting by applying the modern control theory, an accurate reduced-order model with lumped parameters is required. This research concentrated on the reduced-order modelling method for longitudinal vibration control of a direct transfer marine propulsion shafting. The longitudinal vibration modes of the propulsion shafting were determined with one-dimension elastic wave approach based on its continuous model in advance. Taking into account of the unobservability and uncontrollability of mode nodes, the nodes of the neglected low highest mode were selected as locations of the reduced-order model mass points. By modifying mode shapes of the reduced-order model with sensitivity analysis, the reduced-order model can accurately represent the first low order modes of the continuous propeller-shafting system, enabling the effectiveness of controller design in the physical state space.

The aim of this paper is to study the longitudinal frequency of a cracked nanobeam. The frequency equation of the nanobeam with clamped–clamped and clamped–free boundary conditions is derived based on the nonlocal elasticity theory. According to the equation, it can be found that the effects of the crack parameter, crack location, and nonlocal parameter on the longitudinal frequency of the cracked nanobeam are significant. The frequency decreases with an increase of the crack parameter. However, the increasing nonlocal parameter results in a decrease of the crack effect on the frequency. In addition, when the crack location is near the support, a larger decrease in the frequency can be observed.

The problem of detecting localized large-scale internal damage in structures with imperfect bolted joints is considered. The proposed damage detection strategy utilizes the structural damping and an equivalent linearization of the bolted lap joint response to separate the combined boundary damage from localized large-scale internal damage. The frequencies are found approximately using asymptotic analysis and a perturbation technique. The proposed approach is illustrated on an example of longitudinal vibrations in a slender elastic bar with both ends clamped by bolted lap joints with different levels of damage. It is found that while the proposed method allows for the estimation of internal damage severity once the crack location is known, it gives multiple possible crack locations so that other methods (e.g., mode shapes) are required to obtain a unique crack location.

The modulus of elasticity (MOE) of fiber-reinforced plastic (FRP) reinforced fast-growing poplar glulam was investigated and predicted in this study. Firstly, MOE of FRP reinforced glulam with different FRP lengths were measured by longitudinal vibration and static bending tests and the effect of FRP length on MOE was investigated. Secondly, MOE of FRP globally reinforced (GR) poplar glulam and non-reinforced (NR) poplar glulam was predicted by improved transformed section method based on dynamic MOE of poplar laminas. At last, the deflection curve equation of FRP locally reinforced (LR) poplar glulam was obtained by singularity function method. Then MOE was predicted based on deflection curve equation of LR glulam. The results indicated that the reinforced effect of FRP on MOE of reinforced glulam increased as the FRP length increased. The predicted MOE had a good agreement with measured static MOE of glulam, and the accuracy of MOE prediction of LR glulam decreased as the FRP length decreased.

In this paper, evaluation of tensile force on bonded seven-wire strand embedded in nuclear reactor containments has been nondestructively investigated. For this purpose, a total of eight prestressed concrete beam specimens with different tension force levels have been fabricated and tested in terms of using an impact pulse at one end of the strand. Subsequently, various longitudinal vibration characteristics excited by the pulse have been analyzed in detailed. In short, the present study has demonstrated that longitudinal frequency, elastic wave velocity, and elastic modulus are nonlinearly increased as the tensile force level is increased. It is thus postulated that the longitudinal vibration characteristics of the existing bonded tendons can be of promising indicators for the inspection of a tensile force loss.

This paper presents a new approach to solve dynamic bimodular problems. By utilizing a smoothing technique, the constitutive discontinuity can be avoided, and a FE-based precise incremental dynamic equation is derived. In order to secure the dynamic analysis credible, both a step-by-step linear acceleration algorithm and Wilson-θ method are employed in the solution process. Numerical verification gives satisfactory results.

Ultrasonically assisted drilling (UAD) has received great interest in the past few years by both academia and industry. The technology has already demonstrated a multitude of advantages over conventional drilling technology although its use has been mainly thwarted by inconsistent results. In order to aid the further development of UAD, a better understanding of the underlying dynamical process is required. In this work, an investigation into UAD is performed. Longitudinal vibration is used to excite standard 8 mm high-speed steel drill bits and mild steel samples are used for cutting tests. A swept sine wave is used to excite the system whilst the cutting forces that result are acquired. Clearly optimal regions are revealed and these are discussed. Subsequent to the experimental investigation, three-dimensional finite element drill bit models are employed to further understand the drill bit's vibrational characteristic. The simulation results show new areas of interest in which further work is necessitated.

The problem of synthesising discrete physical, mechanical, electrical and electronic systems is widely developed in scientific research. However, it is the problem to find examples of methods of synthesising discrete–continuous mechanical systems. In the paper a foundation of the synthesis such systems has been presented, and its exact formulation and solution is still to deal with.

In this paper we consider the combined bending and axial vibration of beams and we analyze the magnitudes of the different effects that contribute to the motion. Specifically, we consider the transverse elasticity, the axial elasticity, and the shortening of the projection. The goal of our study is to ascertain the circumstances under which the axial elasticity is larger than the shortening of the projection and vice versa. We show that for the majority of cases the shortening of the projection has larger amplitude than the axial stretch. We also develop a general closed-form solution for the response when a force is applied at a free end.

Although there are solutions for the axial vibration of compound bars, i.e., bars of different cross-section connected in mechanical series, this note is to illustrate another solution. The primary objective is to analyze this problem by a relatively new technique originated in 1986 and known as the differential transformation (DT) method. The numerical solutions are compared with classical exact solution and other studies in the literature. The accuracy, simplicity, and effectiveness of the DT approach are demonstrated.

The nonlinear dynamic responses including chaotic oscillations caused by a fatigue crack growth are presented. Fatigue tests have been conducted on a novel fatigue-testing rig, where the loading is generated from inertial forces. The nonlinearity is in the form of discontinuous stiffness caused by the opening and closing of a growing crack. Nonlinear dynamic tools such as Poincare maps and bifurcation diagrams are used to unveil the global dynamics of the system. The results obtained indicate that fatigue crack growth strongly influences the dynamic response of the system leading to chaos.

Longitudinal vibrations of rheological rod with variable cross section are examined. Particular solutions and eigenfunction are accomplished for natural vibrations of the rod with hereditary material of standard hereditary body. Some examples are given.

This note is concerned with the natural vibration problem of a mechanical system, consisting of a fixed-free axially vibrating elastic rod which is restrained by a linear spring in-span. The frequency equation of the system is derived first. Then the mode shapes are given and finally a sensitivity formula is established

In the first step, the characteristic equation of the continuous system is derived. In the second, the rod is modelled as a uniform oscillator consisting of n equal masses and springs. The main purpose of the work was to study the dependence of the convergence of the uniform oscillator model eigencharacteristics towards those obtained from the continuous system, on the number of discrete masses

It is shown that the analysis can be separated into three cases depending on the range in which the natural frequency lies

The first alternative enables one to determine the eigenfrequency parameters via the eignenvalues of a special matrix, whereas the second alternative yields the eigenfrequency parameters as the roots of a simple non linear equation

The Johnson-Thorn bag model which explains the rotational motion of the elongated MIT bag is extended so as to describe the longitudinal vibration along the elongated bag. Our model is a kind of string model for baryons but it reduces to the ordinary spherical MIT bag model in the ground states. It is shown that the mass spectra of the P11 and P33 resonances of baryons can be reproduced by the present model as well as the Regge trajectories.

Referred to by R.I. Sujith; Exact solutions for the longitudinal vibration of non-uniform rods: Author's reply Journal of Sound and Vibration, Volume 329, Issue 19, 13 September 2010, Page 4108