Vibration Of String Research Articles

Derivation and classical solutions for the Wave equation

The problem of string vibration prompted mathematicians to find a suitable equation to express this phenomenon, which led to the birth of the Fourier series and integrals. The physical law of string vibration can be expressed by wave equation and solved by Fourier series. The study on the wave equation is fraught with mathematical and physical controversy. D.Bernoulli based on results from physical experiments to make a bold conjecture that the shape of a vibrating string can be described as a combination of trigonmetric series, which was inspired and validated by work from Fourier on study on heat equation. This study reviews the wave equation using Fourier as a tool. This paper not only gives the derivation of the wave equation but also explores its solution and corresponding properties.

Elastic wave velocities in finitely pre-stretched soft fibers

Elastic wave velocity in a soft fiber that varies depending on material constitution and axial stress level is an essential measure of mechanical signals in many technical applications. In this work, based on the small-on-large theory, we establish a model of linear elastic wave propagation in a finitely pre-stretched soft fiber. The formulas of longitudinal (L-) and transverse (T-) wave velocities are provided and validated by numerical simulations as well as by experimental data on spider silk. The influences of material constitution, compressibility, and pre-stress on the wave propagation are investigated. We found that with increasing pre-stress, the variation of L-wave velocity highly relies on the concavity of the stress–strain curve. In contrast, an increase of T-wave velocity exhibits regardless of any constitutive model. For both L- and T-waves, the variation of the velocities is more significant in a compressible fiber than that in a nearly-incompressible one. Moreover, for minuscule pre-stress, we propose a modified formula for T-wave velocity based on the Rayleigh beam theory, which reveals the competition mechanism between “string vibration” and “beam vibration.” This may provide a reliable theoretical basis for precise mechanical characterization of soft fibers and open a route for lightweight, tunable wave manipulation devices.

The effect of bamboo clip dimension and position towards the frequency spectrum of a vibrating inhomogeneous bundengan string

Bundengan is a traditional musical instrument from Indonesia. One of its unique features is the ability to produce sound imitating the gamelan, a percussive metallophone. This is generated by plucking on the bundengan strings, which have small bamboo clips attached to them. In this work, the effect of the clip dimension and position on the frequency spectrum of the vibrating string is analysed by means of computer simulation and experiment. The string was modelled using Scilab, taking into account the transversal and rotational vibration of the string and bamboo clip, including air drag force. The height to diameter ratio of the clip can be varied in the model. Furthermore, we set up a bundengan string on a sonometer with no resonator, attached specially made bamboo clips on it, and measured the sound frequency spectrum of the vibrating string. The results showed that increasing the height to diameter ratio of the clip decreased the overtone frequencies of the string. It was also found that the fundamental frequency of the string decreased, but its overtones increased, when the clip is shifted towards the middle of the string. The frequency spectrum from the simulation corresponds well to that from the experiment.

On the stochastic bit–rock interaction disturbances and its effects on the performance of two commercial control strategies used in drill strings

The vibration of drill strings is a well-known source of undesirable disturbance in drilling operation, therefore it is important to design and evaluate active controllers to suppress it. The drill string dynamics is complex due to nonlinear interactions with the environment, and the uncertainties related to them. This paper aims to analyze two commercial control strategies acting at the top drive, namely stiff controller and SoftSpeed, applied to a pure torsional drill string model that takes into account uncertainties in the bit–rock interaction. The model is calibrated with field data and the numerical results present a good agreement with the measurements. Uncertainties in the torsional vibration severity are assessed through severity maps. Finally, the effect of the uncertainty level on the control strategies is analyzed. Although SoftSpeed gives better results, it is clear that the controller becomes less effective in suppressing vibrations as the uncertainties increases.

Energy transfer through parametric excitation to reduce self-excited drill string vibrations

Drilling a wellbore can result in several types of vibration that lead to inefficient drilling and premature failure of drill string components. These vibrations are subdivided based on their operating direction into lateral, torsional, and axial vibrations. Especially in hard and dense formations, high-frequency torsional oscillations are found in the bottom-hole assembly (BHA). These critical vibrations are induced by a self-excitation mechanism caused by the bit–rock interaction. Self-excitation mechanisms are regenerative effects, mode coupling, or a velocity-dependent torque characteristic at the drill bit. To increase drilling performance and reduce tool failure due to high-frequency torsional oscillations, the critical vibration amplitudes localized at the bottom-hole assembly need to be minimized. Increasing the damping of self-excited systems to affect the energy output during vibration is a common approach to mitigate self-excited vibrations. In drilling systems, the achievable damping is naturally limited by the small installation space due to the drilled borehole diameter. Therefore, alternative methods to influence vibrations are necessary. Applying parametric excitation in self-excited systems can result in a parametric anti-resonance and therefore in an energy transfer within different modes of the structure. This allows, among other benefits, improved utilization of the structural damping. In this article, the influence of additional stiffness–based parametric excitation on self-excited torsional vibration in downhole drilling systems is investigated. For this purpose, a finite element model of a drill string is reduced using the component mode synthesis and analyzed with the goal to mitigate torsional vibrations. The multiple degree of freedom drill string model is investigated regarding the additional energy transfer due to the parametric excitation. Robustness of various parameters, especially with regard to the positioning within the bottom-hole assembly, is analyzed and discussed. Additionally, the problem of multiple unstable self-excited modes due to the nonlinear velocity-dependent torque characteristic in drilling systems is addressed.

Dynamical characteristics analysis of a deepwater test string induced by high-pressure gas and its experimental verification

To explore the dynamical characteristics of the deepwater test string, a fluid–structure coupling vibration model of the deepwater test string was established, and the model was solved by using the method of characteristics. A similarity experiment of the deepwater test string was performed to verify the model. Combined with the related data of a well in the South China Sea, the dynamical characteristics of the test string were analyzed to obtain the vibration regulation of the test string. Results showed that the test string vibrated periodically when gas passed through it, and its lateral deformation in the seawater segment was evidently larger than that in the formation. The test string's largest deformation was in the middle of the seawater segment, and its deformation in the upper part of the formation segment was larger to the middle and lower parts. A large longitudinal amplitude occurred in the lower part of the formation segment. With the increasing of surface current velocity, the lateral and longitudinal displacements of the test string in seawater segment increased. The lateral and longitudinal displacements of the test string increased with gas production and decreased with test string's thickness. Besides, the interaction effect between the gas and the test string increased with the gas density increasing. These results are highly significant for reducing the vibration of the deepwater test string and protecting its safety.

Drill String Torsional Vibrations Modeling With Dynamic Drill Pipe Properties Measurement and Field Validation

Abstract This paper aims to present the drill string torsional dynamics through a lumped parameter modeling using the basic physical notions with continuous measurement of drill pipe mechanical properties (inertia, damping, and stiffness). The model represents the mechanical properties as a variable for each drilled stand. A rock bit interactions model is employed in the system considering the kinetic friction as variable and depends on surface drilling parameters and the well length. Field data, including surface and downhole recorded velocities, are used to validate the model by comparing both velocities and to confirm the existence of drill string vibrations together with the simulation results (bit velocity).

Dynamic characteristics of LIWELL flip-flow screen panel and particle movement

Flip-flow screen is an effective solution for screening high-viscosity and fine-grained materials, because of the large vibration intensity of the screen panel. Therefore, investigating the dynamic characteristics of the screen panel is the key to improving the flip-flow screen. In this paper, a new dynamic model is proposed to do dynamic analysis of screen panel. The vibration of the screen panel is simplified into a string vibration model, and the energy equation is established. Then, the Hamilton principle and the Galerkin method are employed to transform the energy equations into ordinary differential equations, with the numerical solutions obtained using MATLAB. The vibration model is established by ABAQUS to obtain the displacement data of each point on the screen panel. To verify the correctness and accuracy of the theory, the simulation results are compared with the theoretical calculation results, and the dynamic characteristics of the vibration of the screen panel are analyzed. Besides, displacement and velocity equations of particle on the screen panel based on the screen panel vibration theory are proposed. A new FEM-DEM (Finite Element Method-Discrete Element Method) coupling simulation model is established in ABAQUS, which is used to investigate the trajectory of one particle and the effect of screen panel’s acceleration on particles’ movement and screening percentage.

Research on the self-powered downhole vibration sensor based on triboelectric nanogenerator

Drill string vibration during the drilling must be measured in real-time as it will cause damage to the construction. This paper proposed a self-powered downhole vibration sensor based on the triboelectric nanogenerator. The downhole vibration sensor relies on the drill string vibration to induce triboelectric charge and electrostatic induction of nanomaterials, thereby realizing self-powered vibration measurement. Sensing performance test results show that the measurement range is between 0 to 5 Hz, the measurement error does not exceed 3.5%, and the output voltage amplitude with a range of 2 V to 5.5 V decreases with the increases of vibration frequency. Self-powered performance test results show that the output current can reach a maximum value of about 35 × E−8 A when a 50 ohm resistance is connected in series, the output power can reach a maximum value of about 924.5 × E−12 W when a 20k ohm resistance is connected in series at a vibration frequency of 0.8 Hz, and the output current and power all decrease with the increase of the vibration frequency.

Review on Stick-slip Vibration’s Mechanism Analysis and Its Control Strategy in Deep Drilling with Drag Bits

The stick-slip Vibration is one of the major causes for drilling quality reduction, shorter drilling tool lifespan, drilling efficiency decrease, it also affect the drilling cost and completion period seriously. The complex situations in the process of drilling result in the slow progress in this aspect, they rarely break through. The recent advances are outlined in stick-slip vibration of the drill string in three areas: theoretical analysis, experiment research and control strategy. The main research direction, methods and existing problem in the progress are introduced, which will provide reference and guidance for further research.

Investigation of the mitigation effect of an anti-stall tool on stick-slip vibrations of drill strings

ABSTRACT The anti-stall tool (AST) is a downhole tool used to mitigate stick-slip vibrations during drilling. It has the ability to adjust the cutting depth of the drill bit and is characterized by an axial spring and a screw pair. In this paper, the dynamic characteristics and working mechanism of the AST are studied by modeling and parametric analysis. Based on the lumped-mass model, torsional dynamic models of the drill string without and with the AST are built and simulated. The simulation results of the drill string model indicate that there is an upper limit on the weight on bit (WOB) (148 kN) and a lower limit on the rotary table speed (84 RPM) to avoid stick-slip vibration, and that a stick-slip vibration cycle can be divided into five phases. The simulation results of the model with the AST indicate that the AST can mitigate the stick-slip vibration and stabilize the bit angular velocity effectively for a large range of WOBs (120–200 kN) and rotary table speeds (40–120 RPM). The AST spring strength and helical angle have important influences on AST performance. Reasonable values for these two parameters should be determined to ensure AST effectiveness while avoiding other adverse effects. This work can provide guidance for AST applications and improve drilling efficiency and safety.

Orthogonal wear experiment of 13Cr-L80 tubing string in high-yield gas wells and analysis of its influencing factors

Due to wear failures caused by tubing string vibrations in high-yield gas wells, an orthogonal wear experiment was performed on the 13Cr-L80 tubing string. The influence of contact load, friction frequency, wear time, completion fluid density, and reciprocating stroke length on the wear characteristics of the tubing string were effectively analyzed using range analysis, variance analysis and the control variable methods. The results demonstrate that, the wear patterns of the tubing-casing were primarily abrasive and adhesive wears, with minimal corrosion wear. In-field operations, the service life of the tubing string in high-yield gas wells can be effectively augmented by reducing the contact load and longitudinal vibration displacement of the tubing-casing, maintaining the vibration frequency of the tubing string below 1.5 Hz, and utilizing a completion fluid with a higher density (approximately 1.2 g/cm3). The wear amount and friction coefficient of the tubing string are primarily influenced by the contact load and reciprocating stroke length, respectively. Moreover, the effect significances of the contact load and reciprocating stroke length were particularly significant and significant, respectively. These results provide useful guidance for designing and implementing approaches to improve the service life of tubing strings in high-yield gas wells.

Numerical Investigation of the Influence of the Drill String Vibration Cyclic Loads on the Development of the Wellbore Natural Fracture

Wellbore instability is one of the most serious issues faced in the drilling process. During drilling operations, the cyclic loads applied on the fractured formation progressively modify the initial parameters (i.e., length and width) of the fractured formation, thus resulting into undesirable wellbore instability. In this paper, using a nonlinear finite element software (ABAQUS) as the numerical simulator, a poro-elasto-plastic model has been established which aimed at analyzing the influence of drill string vibration cyclic loads on the development of the wellbore natural fracture. The numerical results showed that the fracture width as a function of time profiles followed a sinusoidal behavior similar to the drill string vibration cyclic load profiles. For different cyclic load magnitudes with constant number of cyclic loads, the highest percentage increase of the fracture width after integration of cyclic loads was 64.77%. Interestingly, the fracture width increased with the fracture length in the near wellbore region while it globally decreased in the region far away from the wellbore. But for constant cyclic load magnitude with different number of cyclic loads, the biggest percentage increase of the fracture width after integration of cyclic loads was slightly lower representing 63.12% while the oscillating period of the fracture width increased with the number of cyclic loads. The parametric study revealed that the drill string vibration cyclic loads were relatively independent of the fracture length and the bottom hole pressure. However, the fracture width and the loss circulation rates were considerably impacted by the drill string vibration and the highest percentage increase of the loss circulation rate after integration of cyclic loads was 14.3%. This study provides an insight into the coupling of the fracture rock development and the continuous cyclic loads generated by drill string vibrations which is an aspect that has been rarely discussed in the literature.

Numerical and experimental assessment of string gauge influence on guitar tone

String gauge is known for affecting a guitar's tone, playability and volume. It is often a subject of debate between younger players and becomes a staple and signature of older ones. The choice between the heavier or lighter strings is usually made based on personal taste and practical reasons. Although string gauge is intuitively associated with fatter or thinner sound, physics behind string vibration do not necessarily confirm this assumption. In the study, the impact of string gauge on the guitar's tone is assessed in several ways. A numerical study is carried out, introducing a time-domain dynamic analysis of a coupled string-structure system. A basic experimental study is also performed to validate the numerical results. The study is focused on estimation of the three aspects of sound—harmonic composition, sustain and volume—and how they are affected by the guitar's string gauge.

Wear Experiment and Influencing Factors Analysis of 13Cr-L80 Tubing String in High-Pressure, High-Temperature and High-Yield Gas Wells

Due to wear failures caused by tubing string vibrations in high-pressure, high-temperature and high-yield (3H) gas wells, a wear experiment was performed on the 13Cr-L80 tubing string. The influence of contact load, friction frequency, and reciprocating stroke length on the wear characteristics of the tubing string were effectively analyzed using the control variable method. The results demonstrate that, the wear patterns of the tubing-casing were primarily abrasive and adhesive wears, with minimal corrosion wear. The wear amount of tubing increases linearly with the increase of contact load and reciprocating stroke, but increases nonlinearly with the increase of friction frequency, and the friction coefficient of tubing string do not change with the increase of contact load, friction frequency and reciprocating stroke. In-field operations, the service life of the tubing string in 3H gas wells can be effectively augmented by reducing the contact load and longitudinal vibration displacement of the tubing-casing, maintaining the vibration frequency of the tubing string below 1.5 Hz. These results provide useful guidance for designing and implementing approaches to improve the service life of tubing strings in high-yield gas wells.

Mathematical model of the polymer concrete by fractional calculus with respect to a spatial variable

The article presents a solution of the vibration string equation containing the fractional derivative concerning the spatial variable, in which the fractional derivative is defined by Caputo. This model is utilized for characterizing the oscillation mechanical process in a viscoelastic medium. The theoretical solution is presented, and the fractional derivative parameter is determined. We compared the theoretical solution and the experimental data for polymer concrete samples. A sample of the problem of structural mechanics has been considered. This sample allows the demonstration of some advantages of the application of the suggested approach to solve the fractional vibration equation.

Effect of compact inclusion on the natural frequencies of vibrations of a pipe string in a well

The influence of small-size inclusion of pipes in a well column on the natural frequency of its longitudinal vibrations is investigated. Using the asymptotic expansion in a small parameter, an analytical relation is obtained that describes the change in the period of the column oscillations in the form of some additional small term to the period of the homogeneous column oscillations. Numerical calculations show that the obtained analytical relations almost accurately describe the oscillation period of a column with a massive compact inclusion, while its difference from the oscillation period of a homogeneous column is within ~20%. The results obtained can be useful for preventing resonant phenomena in the drill string when drilling wells, as well as for optimal use of the longitudinal vibrations of the tubing string to influence the bottom-hole zones of producing wells.

On Problem of Internal Boundary Control for String Vibration Equation

The article deals with the vibration control problem described by one dimensional wave equation with integral type boundary condition. As usual, the initial and final moments of time for arbitrary displacements and velocities of the wave are specified by points on a string (Cauchy data). It is shown that the minimum time for the realizable control is uniquely determined by the condition of correct solvability to the Cauchy problem involving data lying on disconnected manifold. This suggests that the internal boundary conditions does not affect the minimum time value. Necessary and sufficient conditions for the existence of the desired internal-boundary controls that move the process from the state initially specified to a predetermined final one are obtained and written out. The controls are presented in explicit analytical form. Moreover, it is shown that for the inner-boundary controls expressions, one should use not the representation of the solution to the Cauchy problem in the sought-for domain, but the formula for the general solution of the string oscillation equation (d’Alembert’s formula).

On one boundary control problem of string vibrations with given velocity of points at an intermediate moment of time

The boundary control problem for the string vibration equation with given initial and final conditions, with the given velocity of the string points at an intermediate moment of time is considered. It is controlled by displacement of one end while the other end is fixed. A constructive approach for constructing the boundary control action is proposed, which is applied for given functions.

Calculation of shock absorber parameters of torsional vibrations of screw pumps’ rod string

Torsional vibrations of the rod string occur during the operation of surface-driven screw pumping units, which reduces the reliability of the units. It is proposed to include a torsional vibration damper in the rod string to maintain stability. The calculation of the parameters of the shock absorber is presented.