Vibration Of String Research Articles

Skills of Music Creation Based on Homogeneous First-Order Linear Partial Differential Equations

Abstract The research of differential equation models in music creation and audio is a current research hotspot. This paper uses homogeneous first-order linear partial differential equations to analyze the relevant music signals in music creation. This paper focuses on analyzing the expression of music signals based on time series. This paper discusses its characteristic function and conducts partial differential equation modeling of music characteristic signals. The method successfully established the string vibration model and the time-series autoregressive model of the music signal. This paper becomes a new differential equation to solve the music signal processing method in music creation.

Transverse vibration simulation of sucker rod string under normally distributed load caused by viscoelastic fluid in a curved wellbore

The sucker rod string is eccentric due to the effect of wellbore bending. When the sucker rod string undergoes an eccentric motion within the wellbore containing viscoelastic fluid in the polymer flooding pumping wells, it is subjected to an alternating normally distributed load caused by viscoelastic fluid (NDLVF). In this paper, it is pointed out that the alternating NDLVF acting on the sucker rod string is the excitation of the transverse vibration of the sucker rod string. Moreover, the coupling relationship between the NDLVF and the transverse vibration displacement of the sucker rod string is considered. Based on the elastic vibration theory, the mechanical and mathematical model of the transverse vibration of a sucker rod string under the NDLVF in the curved wellbore is established. Then, the Runge-Kutta method is used to solve the simulation model, and parameter sensitivity analysis is carried out. The simulation results show that: (1) the NDLVF can significantly affect the transverse displacement of the sucker rod string and the rod-tubing contact force, which is an essential excitation for the transverse vibration of the sucker rod string in the polymer flooding pumping wells. (2) Due to the NDLVF presence, the contact collision between the rod string and tubing string occurs more frequently, and the rod-tubing contact force is increased. (3) The NDLVF effect on the transverse vibration of the sucker rod string is most noticeable near the neutral point of the sucker rod string. (4) With an increase in the polymer concentration, stroke length, and stroke frequency, the transverse vibration of the rod string becomes stronger. The transverse vibration simulation model of the sucker rod string under NDLVF in a curved wellbore plays a guiding role when analyzing rod-tubing eccentric wear and optimal allocation of a centralizer in the polymer flooding pumping wells.

Nonlinear vibrations of composite drill strings considering drill string–wellbore contact and bit–rock interaction

Nonlinear vibrations of composite drill strings considering drill string–wellbore contact and bit–rock interaction

Application of the lumped-parameter method for modeling nonlinear vibrations of drill strings with stabilizers in a supersonic gas flow

In this paper, we study the nonlinear dynamics of a drill string under the effect of external loads in a supersonic gas flow, taking into account supporting stabilizers. The case of the drill string spatial lateral vibrations, which are the main cause of the drilling equipment failures, is considered. To account for nonlinear effects, a mathematical model based on the general relations of Novozhilov's nonlinear theory of elasticity is developed. The discretization of the model is conducted using the lumped-parameter method. The explicit fourth order Runge-Kutta method is utilized for obtaining a numerical solution for the studied system. The effect of damping elements (supporting stabilizers) on the drill string lateral vibrations is analyzed. Numerical illustrations are presented, and the corresponding recommendations for deep and shallow drilling are given.

Modelling and controlling of drill string stick slip vibrations in an oil well drilling rig

Mitigating of drill string vibrations, plays a distinctive role in increase of rate of penetration and consequently decrease of the operation costs, prevent of severe damages to drill string and bit, increase of drilling operation quality and safety, and enhance the performances of the controllers. This study deals with modelling and controlling of drill string vibrations with focus on stick slip vibrations in vertical wells. The approach taken to sub-divide the drill string into smaller torsional sections and model the complete system dynamically and employ the extracted torsional model to model the stick slip vibrations mathematically. To mitigate and or active control of stick slip vibrations, three architectures are developed and proposed, manipulation of the rotational speed, manipulation of the weight on bit, and increasing of the damping in the bottom of the drill string. The performance of each strategy is analysed individually as well as relative to each other, using a mathematical measure when the drill string length is divided into smaller torsional sections. By the aid of the simulations and mathematical measures, it is shown that manipulation of rotational speeds at the surface may not be an effective solution to reduce the stick slip vibrations, however manipulation of weight on bit and increasing of damping at the bottom of string can be two effective solutions to mitigate these kinds of vibrations. As a general conclusion, it is proved that the manipulation of bottom side drilling parameters is more effective than the manipulation of surface drilling parameters.

Bifurcation Diagram and Dynamic Response of a Drill String Applied in NGH Drilling

Natural gas hydrate (NGH) is a kind of unique resource among hydrocarbon fuels. It is stable within its sedimentary hosts and will only convert to its constituent gas and water phases relatively quickly if reservoir pressure-temperature conditions are artificially disturbed. The unwanted vibrations from the drill string with the complex external environment and excitation usually cause instability in the NGH. In this paper, a nonlinear mathematical model is presented taking into account the fluid–solid coupling vibration caused by the drilling fluid inside and outside the drill string, the non-smooth rub-impact between the drill string and borehole wall, and the influence of centralizer setting interval on nonlinear resonant vibration of the drill string. The relationship between the variable parameters of the drill string and the complex dynamics of the drill string is analyzed. Finally, the bifurcation diagram is given. The results show that the system response exhibits complex dynamic behavior due to rub-impact and nonlinear effects. However, there is always a suitable parameter in the bifurcation diagram of each parameter change, so that the amplitude of the system is small and the failure probability of the drill string is reduced. The research results have reference significance for exploring the nonlinear dynamic mechanism of a drill string under complex working conditions and reasonably optimizing drilling parameters.

Nonlinear analysis of axial-torsional vibration of drill string based on a 3 DOF model

Based on a model with three-degree-of-freedom, the nonlinear coupled axial-torsional vibration of drill string is numerically studied considering a modified contact model of bit-rock interaction. The effects of driving angular velocities, nominal drilling pressures, and formation stiffness on response are discussed. The simulation results show that response can be periodic, quasi-periodic or chaotic motion with the variation of system parameters. Increasing driving angular velocity is helpful to reduce the duration of sticking and even make the system stick-slip free. However, bit-bounce may occur if driving angular velocity is too large. Increasing nominal drilling pressure can cause the system to be more prone to stick-slip, but it has dampening effect on bit-bounce. The larger the formation stiffness is, the smaller the amplitude of the axial vibration response of the system is, and the driving angular velocity corresponding to the maximum amplitude of the axial vibration response is also larger, whereas the variation of the amplitude of torsional vibration is smaller. Compared with the results from two-degree-of-freedom model, the amplitude of torsional vibration of the present model is remarkable greater. Meanwhile, the present model is more prone to stick-slip than the one with two-degree-of-freedom, and it is not easy to bit bounce.

Simulation of the geometrically exact nonlinear string via energy quadratisation

String vibration represents an active field of research in acoustics. Small-amplitude vibration is often assumed, leading to simplified physical models that can be simulated efficiently. However, the inclusion of nonlinear phenomena due to larger string stretchings is necessary to capture important features, and efficient numerical algorithms are currently lacking in this context. Of the available techniques, many lead to schemes which may only be solved iteratively, resulting in high computational cost, and the additional concerns of existence and uniqueness of solutions. Slow and fast waves are present concurrently in the transverse and longitudinal directions of motion, adding further complications concerning numerical dispersion. This work presents a linearly-implicit scheme for the simulation of the geometrically exact nonlinear string model. The scheme conserves a numerical energy, expressed as the sum of quadratic terms only, and including an auxiliary state variable yielding the nonlinear effects. This scheme allows to treat the transverse and longitudinal waves separately, using a mixed finite difference/modal scheme for the two directions of motion. A matrix decomposition algorithm is presented, so to treat the sparse and full parts of the update matrix separately. Numerical experiments are presented throughout.

Combined effect of silica nanofluid and wettability on enhanced oil recovery process

The objectives of this study are to characterize the properties of water-SiO2 nanofluid and quantify its flooding efficiency in an initially oil-filled micro-porous medium of two different surface wettability conditions, i.e. water-wet and neutral-wet. The average nanoparticle size (as a result of aggregation) in water-SiO2 nanofluid is characterized using dynamic light scattering method. Viscosity and surface tension of water-SiO2 nanofluid are measured respectively by vibration string and drop volume methods. For the flooding experiments, a transparent PDMS microchannel with internal square obstacles mimicking a porous medium is designed and fabricated. The wettability of PDMS over time, as a result of aging, is investigated. The effect of SiO2 nanoparticles on 3-phases contact angle (nanofluid, crude oil and PDMS) is reported. The flooding efficiency of water-SiO2 nanofluid is then quantified. The results reveal that the best oil recovery process occur when the water-SiO2 nanofluid flooding is used in a neutral-wet surface condition. As the injection rate increases, the oil displacement efficiency decreases. This oil recovery behavior implies the dominant nature of viscous forces at low flow rate. At high flow rate, the effect of viscous force is not obvious.

Non-Linear Vibrations of Composite Drill Strings–Wellbore Contact Considering Dynamics of Them at the Contact Zones

Non-Linear Vibrations of Composite Drill Strings–Wellbore Contact Considering Dynamics of Them at the Contact Zones

A Finite Difference Method for Positive Definite Fractional Damped String Vibration Equations

In this paper, the finite difference method is used to solve the positive fractional derivative damped string vibration equations, and the vibration attenuation phenomenon of the model is described by numerical simulation. In numerical examples, the effects of the order of the positive definite fractional derivative and the damping coefficient on vibration are studied and compared respectively. The results show that, on the one hand, when the damping coefficient c is fixed, the closer the order p(0 < p < 1) is to 1, the faster the attenuation is. On the other hand, when the order p is fixed, the larger the damping coefficient c is, the faster the attenuation is.

Free Response and Musical Pitch of Shape Memory Alloy Wires Under Voltage Loading

An experimental setup, on which the shape memory effect is based, is developed to measure the free response frequency and musical pitches of SMA wires. In this paper, the voltage, force, and sound signals of SMA wires are determined simultaneously under two kinds of experiments, constant temperature uniaxial tensile tests and fixed-length thermal cycle tests. Based on constitutive equations and thermodynamic equations, an empirical formula for the driving voltage and the steady-state stress of SMA wires is proposed, and the undetermined coefficients at radii of 0.050 mm and 0.070 mm are determined by experiments. The musical pitches of SMA wires under a constant voltage are obtained by combining the empirical formula with the string vibration equation. The theoretical results are compared with the actual musical pitches measured by the sound signals, and the two have a high degree of coincidence. Just for an SMA wire with a radius of 0.070 mm, the free response frequency range reaches 640 Hz to 1120 Hz, which covers 6 whole musical tones in total. Thus the SMA wire can be used as a string to play simple music.

Metode Backward Time Central Space dalam Penyelesaian Model Matematika Vibrasi Dawai pada Alat Musik Petik

This research was conducted to obtain a numerical solution of the mathematical model of string vibration on stringed instruments. This mathematical model is a representation of the phenomenon of string vibration on a stringed instrument subject to deviation. The model was constructed by Kusumastuti, et al (2017) and is in the form of a second-order partial differential equation. The method used in completing this research is the BTCS (Backward Time Central Space) method. The numerical solution is obtained by the following steps, 1). Discretize mathematical models, as well as discretize initial conditions and boundary conditions. 2). Performing stability analysis of numerical solutions to determine the terms of solution stability and conducting consistency analysis as a condition of the convergence of the obtained numerical solutions. 3). Simulate numerical solutions and perform graph interpretations. The results show that the numerical solution of the mathematical model of string vibration on stringed instruments is unconditionally stable and has an error order (〖∆x〗^2,〖∆t〗^3).

Model Matematika Vibrasi Dawai Dikenai Massa yang Berjalan di Atasnya

This study discusses the formulation of a mathematical model of string vibration when subjected to mass. There are two variables resulting from the formulation of the model, namely the deflection of the string and the angle of the string . The deflection of the string and the angle of the string . are affected by the friction force, tension force, spring force, and gravity. Then the identification of the working potential and kinetic energy is carried out to obtain the Lagrange equation of the deflection and angle of the string. Based on the formulation steps that have been described, the mathematical model obtained is an ordinary differential equation of the order of one to the power of two. Furthermore, the model is calculated numerically by assigning values to the parameters involved. So it is known that with a string mass of 0.005 kg, 0.05 kg, 0.5 kg, a string mass of 0.075 kg, and an object radius of 0.07 m, it is known that the deflection of the string is greater if the object's mass is greater. While the angle of the string is in a state of equilibrium before being subjected to a mass and experiencing vibration after being subjected to a mass.

Experimental study on flow-induced vibration response characteristics of leaching tubing in salt cavern gas storage

To address flow-induced vibration failure problems of water solution leaching tubing string in salt cavern gas storage, a laboratory experimental device for flow-induced vibration of leaching tubing string was developed based on the similarity principle, and the corresponding experimental bench was built, according to the on-site parameters of the M gas storage well in Jianghan Oilfield, China. The vibration simulation experiments were carried out, and the displacement and amplitude-frequency response data at the bottom end of the tubing string were measured. The influences of different operation parameters and structure modes, such as water injection flow rate, two-port distance, screen structure and circulation mode, on the vibration characteristics of the tubing string were explored, the failure mechanism of the leaching tubing string was revealed, and the corresponding vibration failure control methods were proposed. The results obtained demonstrate that, first, at the moment of pump start-up (corresponding to well opening on-site), the tubing string produced obvious water hammer phenomenon. With the increase of the water injection flow rate, the vibration displacement and frequency of the center tube first increase and then decrease, when the water injection flow rate is less than 1.55 L/min (corresponding to the on-site flow rate of 40 m3/h), the risk of vibration failure of leaching tubing can be reduced. Second, when the two-port distance is greater than 0.16 m (corresponding to the on-site two-port distance of 45 m), the amplitude and frequency of lateral vibration of the center tube increase significantly, and the risk of lateral vibration deformation of leaching tubing increases. Third, adding a symmetrical hole screen at the bottom of the center tube can significantly reduce the vibration displacement and frequency of leaching tubing, and improve the safety of the operation tubing string. Fourth, the vibration displacement and amplitude of the leaching tubing in the reverse circulation mode are smaller than those in the positive circulation mode. When the cavern shape requirements are met on-site, the vibration of the tubing string can be effectively reduced by using the reverse circulation cavern leaching.

Helmholtz vibrations in bowed strings.

For almost 160 years, it has been known that Helmholtz oscillations, unique to vibrating strings in bowed instruments (violin, cello, etc.), have two distinct regimes: "slip" and "stick." During the slip regime, the force at the bow-string interaction is attributed to friction between the sliding bow hair and the vibrating string, with a friction coefficient that decreases with increasing relative velocity. Yet the hair-string interaction during the stick regime is less understood. We propose that the interaction force during the stick regime is proportional to the product of the longitudinal acoustic impedance of the bow hair to the relative bow-string velocity. We validate this hypothesis by solving the string's differential equation of motion, including an enhanced formulation to avoid parasitic high-frequency oscillations. This physical model enables us to analyze, in real time, the characteristics of the Helmholtz oscillations, including the string shape, excitation of harmonics, Schelleng ripples, and string energy, showing that the bowed string gains energy during the stick regime and loses energy during the slip regime.

Control of String Vibrations by Displacement of One End with the Other End Fixed, Given the Deflection Form at an Intermediate Moment of Time

We consider a boundary control problem for the equation of string vibration with given initial and final conditions, given the deflection form at an intermediate moment of time. The control is carried out by displacement of one end with the other end fixed. The problem is reduced to the problem of a distributed action control with zero boundary conditions. We propose a constructive approach to constructing a boundary control action by the separation of variables and methods of the theory of control of finite-dimensional systems. The approach is applied to given functions. A computational experiment was carried out with the construction of the corresponding graphs and their comparative analysis. They confirm the obtained results.

Study on contactless measurement method of dropper load

The fatigue fracture of dropper will threaten the safe operation of high-speed railway. The accurate measurement of dropper load plays an important role in dropper fatigue test and service life prediction. In this work, according to the structural characteristics of catenary suspension and based on string vibration theory, a contactless method for measuring dropper load is proposed. In this method, the edge recognition technology is used to obtain the vibration displacement time history data of the dropper clamp position on the messenger wire, the random vibration is transformed into the linear superposition of sinusoidal vibration through Fourier transform to solve the vibration differential equation. Through the relationship between the shape change of messenger wire and concentrated force, the change of dropper load is analyzed. The measurement accuracy is tested on the test platform built in the laboratory, and the results show the measurement accuracy can reach 0.50%.

Investigation of the cutting force response to a PDC cutter in rock using the discrete element method

In the drilling industry, Polycrystalline Diamond Compact (PDC) bits with multiple fixed PDC cutters are widely applied to cut rock under high bottom hole pressure and high temperature. The bit-rock interaction is the main excitation source of drill string vibrations and needs to be understood and controlled better to improve the performance of the drilling process. Rock cutting mechanisms can be examined best for a single PDC cutter: Limitations of dynamic control and available sensors in experimental setups are eliminated in numerical simulations. Using the Discrete Element Method (DEM), deformation and strength of rock can be modeled through its full degradation in the PDC cutting process under typical environmental conditions. Stresses and strains in the rock material and on the cutter faces become available for examination.In this paper, numerical simulations of the cutting process are conducted using a modified bonded-particle model (specialized DEM), which is calibrated to a pressure series of high-strain triaxial laboratory tests. The simulation results are interpreted from the perspective of continuum mechanics providing stress and strain rate fields within the rock. These serve to identify two primary deformation modes predominant in two distinct zones moving with the PDC cutter: A crush zone below the chamfer and a shear zone ahead of the cutter face. The influences of two cutter design parameters, the back rake angle and the chamfer size, on these zones and the cutting force, are investigated. The shear angle is found to decrease with increasing back rake angle, resulting in a wider shear band. Moreover, the back rake angle has a considerable influence on the cutting forces both in the normal and tangential direction, whereas the chamfer size affects the normal cutting force predominantly. These results are in good accordance with experimental observations in the literature. Furthermore, the relationship between the cutting force and underlying rock failure mechanisms is clarified.This study enhances the understanding of the rock cutting process and gives first indications for dynamic cutting force control through cutter design.

Dynamic analysis and control of a string-stiffened single-link flexible manipulator with flexible joint

In this paper, an innovative method of controlling the endpoint precision of a single-link flexible manipulator has been presented. The flexibility of the link has been stiffened by using a pair of strings while retaining its advantage of being lightweight. The flexible link is modeled as an Euler-Bernoulli beam. Dynamic analyses of the string-stiffened single-link flexible manipulator (SSSLFM) with flexible joint have been carried out using Euler-Lagrange’s formulation with the assumed mode method. The effects of string-stiffening on tip deflection of the link have been found. Coupling effects of string vibration on tip deflection of the link as well as on the joint of the manipulator have also been investigated. It has been observed that by the stiffening of the flexible link with a pair of strings, tip deflection of the link has been reduced significantly. Residual vibrations have been controlled by applying robust proportional-integral and derivative (PID) control.