Vibration Of String Research Articles

Problems of Boundary Control and Optimal Control of String Vibrations with Multipoint Intermediate Conditions on the State Functions

Problems of Boundary Control and Optimal Control of String Vibrations with Multipoint Intermediate Conditions on the State Functions

On the Optimal Control Problem for Vibrations of the Rod/String Consisting of Two Non-Homogeneous Sections with the Condition at an Intermediate Time

We consider an optimal boundary control problem for a one-dimensional wave equation consisting of two non-homogenous segments with piecewise constant characteristics. The wave equation describes the longitudinal vibrations of a non-homogeneous rod or the transverse vibrations of a non-homogeneous string with given initial, intermediate, and final conditions. We assume that wave travel time for each of the sections is the same. The control is carried out by shifting one end with the other end fixed. The quality criterion is set on the entire time interval. A constructive approach to building an optimal boundary control is proposed. The results obtained are illustrated with an analytical example.

Nonlinear dynamic analysis of liquid-solid coupling and rubbing on drill string

Natural deposit of natural gas hydrate (NGH) exists in the colder regions on the earth, which is an unconventional energy and has great potential. About 97% of the NGH is located in the offshore and only 3% is located on the land. Due to its sensitivity to the external environment and easy decomposition, it is necessary to suppress the external environment changes caused by the complex vibration of the drill string during its exploitation. In this paper, the fluid-solid coupling nonlinear vibration of drill string is studied. The nonlinear dynamic model of drill string with drilling fluid is established. The relationship between variable parameters of drill string and complex dynamics of drill string is analyzed. Furthermore, the bifurcation diagram is given. The results have certain significance for discussing the non-linear dynamic mechanism of drill string under complex conditions and reasonably selecting the drilling process parameters of NGH.

Simulasi Numerik Model Matematika Vibrasi Dawai Flying Fox Menggunakan Metode Adams-Bashforth-Moulton

This study discusses numerical simulation using the Adams-Bashforth-Moulton (ABM) method of order 4 in the flying fox string mathematical model which is in the form of ordinary differential equations depending on time, consisting of two equations, namely the equation of the flying fox string y(t) and the angular equation of the flying fox string θ(t). This mathematical model is a model that has been constructed by Kusumastuti, et al (2017) and has been validated by comparing analytical solutions to its numerical solutions by Sari (2018). The analysis of the behavior of the Kusumastuti 2017 model conducted by Makfiroh (2020) shows that the phase portrait graph is in the form of a spiral with eigenvectors pointing towards the equilibrium point so that the mathematical model of the flying fox string vibration can be concluded as a valid mathematical model that is close to the actual situation. This study attempts to determine the numerical simulation of the deflection of the flying fox string y(t) and the numerical simulation of the angle of the flying fox string θ(t). The Runge-Kutta method of order 4 was used to generate 3 initial values for order 4 ABM. Next, a comparison of the y(t) and θ(t) solution graphs of order 4 ABM with the solution graph with Runge-Kutta of order 4 was performed in Sari 2018. The first simulation was carried out when h=1, the difference in the value of y(t) of order 4 ABM and Runge-Kutta order 4 fluctuated in the range of [0,0.09] with almost the same graphic profile, and the difference in the value of θ(t) ABM of order 4, and Runge-Kuta order 4 which is quite large with different graphic profiles. The second simulation was carried out when h=0.01, the difference in the value of y(t) of order 4 ABM and Runge-Kutta order 4 was fluctuating which also ranged from [0.0.09] with the same graphic profile, and the difference in the values of θ(t) ABM of order 4 and Runge -Kutta order 4 fluctuates in the range of [0,1] with the same graphic profile. So concluded that when h=0.01 comparison of ABM of order 4 and Runge-Kutta of order 4 is the best for displaying the graph profiles of y(t) and θ(t). Further research can explore numerical solutions using other methods.

Simulasi Numerik Model Matematika Vibrasi Dawai Flying Fox Menggunakan Metode Adams-Bashforth-Moulton

This study discusses numerical simulation using the Adams-Bashforth-Moulton (ABM) method of order 4 in the flying fox string mathematical model which is in the form of ordinary differential equations depending on time, consisting of two equations, namely the equation of the flying fox string y(t) and the angular equation of the flying fox string θ(t). This mathematical model is a model that has been constructed by Kusumastuti, et al (2017) and has been validated by comparing analytical solutions to its numerical solutions by Sari (2018). The analysis of the behavior of the Kusumastuti 2017 model conducted by Makfiroh (2020) shows that the phase portrait graph is in the form of a spiral with eigenvectors pointing towards the equilibrium point so that the mathematical model of the flying fox string vibration can be concluded as a valid mathematical model that is close to the actual situation. This study attempts to determine the numerical simulation of the deflection of the flying fox string y(t) and the numerical simulation of the angle of the flying fox string θ(t). The Runge-Kutta method of order 4 was used to generate 3 initial values for order 4 ABM. Next, a comparison of the y(t) and θ(t) solution graphs of order 4 ABM with the solution graph with Runge-Kutta of order 4 was performed in Sari 2018. The first simulation was carried out when h=1, the difference in the value of y(t) of order 4 ABM and Runge-Kutta order 4 fluctuated in the range of [0,0.09] with almost the same graphic profile, and the difference in the value of θ(t) ABM of order 4, and Runge-Kuta order 4 which is quite large with different graphic profiles. The second simulation was carried out when h=0.01, the difference in the value of y(t) of order 4 ABM and Runge-Kutta order 4 was fluctuating which also ranged from [0.0.09] with the same graphic profile, and the difference in the values of θ(t) ABM of order 4 and Runge -Kutta order 4 fluctuates in the range of [0,1] with the same graphic profile. So concluded that when h=0.01 comparison of ABM of order 4 and Runge-Kutta of order 4 is the best for displaying the graph profiles of y(t) and θ(t). Further research can explore numerical solutions using other methods.

Theoretical study on fatigue damage of sonic standing wave resonant drill-string

To achieve high-speed and undisturbed core drilling, the standing wave vibration of the drill string in a sonic drill is excited by a high-frequency inertial vibrator; the resulting high alternating stress cycle in the drill string can easily cause fatigue damage. In order to minimize the fatigue failure of drill-string at the stage of its design, it is necessary to assess the fatigue damage caused by alternating stress to guide engineering practice. In this paper, based on one-dimensional wave theory, we analyse the standing wave vibration in a drill-string excited by a sonic vibrator, and theoretically prove that the dynamic resonant stress of a drill-string is the key factor influencing the fatigue damage. By using the Palmgren–Miner fatigue damage rule, we establish a theoretical formula for the cumulative fatigue damage of a variable-length standing wave vibration drill string and reveal the fatigue damage mechanism of the variable-length resonant drill string. Furthermore, the effects of sonic drill systems and process parameters on the damage are quantified. It was found that by an appropriate choice of a drill-pipe length, the fatigue damage can be reduced whilst the axial stress concentration factor (aSCF) kσ on threaded connections can significantly increase it. At the fundamental frequency of the resonant sonic drilling, the maximum fatigue damage point, xf, is located approximately la/2 above the drill bit, not exceeding the theoretical sonic standing wave starting length, la, and unrelated to the hole depth. This study promotes the theoretical understanding and exploration of variable-length standing wave oscillators.

A Nonlinear Vibration Control of a String Using the Method Based on Its Time-Varying Length

Strings are common components in various mechanical engineering applications, such as transmission lines, infusion pipes, stay cables in bridges, and wire rope of elevators. The string vibrations can affect the stability and accuracy of systems. In this paper, a time-varying string length method is studied for string vibration suppression. A dynamic model of a string with the time-varying length is formulated. The dimensionless variables are introduced into the nonlinear dynamic model to realize the separation of time and space variables. The finite difference method is used to solve the differential equations of time functions. The vibration characteristics of time-varying length string are analyzed, such as the free vibrations, forced vibrations and damping effect. The influences of the length time-varying frequency and length time-varying range on the suppression performances are discussed. The results show that the time-varying string length method can effectively disperse the resonance peak energy and suppress multimodal resonance at the same time. The suppression performance is better for the time-varying length string with a higher time-varying frequency and a higher time-varying range.

In-plane free vibrations of small-sag inclined cables considering bending stiffness with applications to cable tension identification

This study proposed an analytical model to characterize in-plane free vibration of a cable considering sags, inclined angles, and non-negligible bending stiffness to enhance the accuracy and efficiency of vibration analysis and health monitoring of cable structures under complex engineering environments. The static equilibrium configuration of the cable is derived, and the singularity at boundary regions is elucidated in detail. The analytical solution of the dynamical control equation is obtained using the successive approximation method. The influences of the sags, inclined angles, and bending stiffness on eigenfrequencies and mode shapes are individualy studied via introducing three non-dimensional parameters. The analytical model is further examined through parametric anaysis. As a typical extension, the model is used to establish a cable tension identification method, in which an inverse problem is solved by the trust-region-dogleg technique. The identification method is verified through comparisons with existing methods established based on classical string vibration theory and various numerical methods, e.g., finite element method and finite difference method. Validations are made by using experimental outcomes, on which the data collected from both an experiment in laboratory and a field experiment in the Poyanghu bridge are used for comparison. It is found that the proposed model along with the method is able to capture precise vibration features and identify cable tensions accurately in favor of the improvement of cable condition monitoring and health management in fields such as bridge engineering.

Explicit exactly energy-conserving methods for Hamiltonian systems

For Hamiltonian systems, simulation algorithms that exactly conserve numerical energy or pseudo-energy have seen extensive investigation. Most available methods either require the iterative solution of nonlinear algebraic equations at each time step, or are explicit, but where the exact conservation property depends on the exact evaluation of an integral in continuous time. Under further restrictions, namely that the potential energy contribution to the Hamiltonian is non-negative, newer techniques based on invariant energy quadratisation allow for exact numerical energy conservation and yield linearly implicit updates, requiring only the solution of a linear system at each time step. In this article, it is shown that, for a general class of Hamiltonian systems, and under the non-negativity condition on potential energy, it is possible to arrive at a fully explicit method that exactly conserves numerical energy. Furthermore, such methods are unconditionally stable, and are of comparable computational cost to the very simplest integration methods (such as Störmer-Verlet). A variant of this scheme leading to a conditionally-stable method is also presented, and follows from a splitting of the potential energy. Various numerical results are presented, in the case of the classic test problem of Fermi, Pasta and Ulam and for nonlinear systems of partial differential equations, including those describing high amplitude vibration of strings and plates.

The Review and Development of Devices with an Increasing Rate of Penetration (ROP) in Deep Formation Drilling Based on Drill String Vibration

The oil and gas resources stored in deep strata are an important replacement field of the petroleum industry. Accelerating the exploration and development of deep oil and gas is of great significance to the security of energy strategy. Drilling is the primary link and necessary means of deep oil and gas exploration and development. Slow drilling speed is one of the key problems restricting the exploration and development of deep oil and gas. The research and development of down-hole equipment with an increasing ROP provides a technical means for increasing the ROP. However, the energy of existing down-hole equipment with an increasing ROP comes from the drilling circulation medium, and the ROP increase effect of such equipment is relatively obvious in shallow and middle formations. However, with the increase in well depth, energy in the circulation medium increasingly struggles to reach deeper formations, and the ROP increase effect is not good at the later stage of drilling. In the drilling process, drill string vibration is a frequently encountered complex situation, but it also contains sufficient energy, and the energy of drill string vibration will increase with an increase in the well depth, which can meet the energy demand of increasing the ROP in deep oil and gas exploration. This paper analyzes the characteristics of drill string vibration, and introduces six kinds of devices that take drill string vibration as energy and realize drill string vibration reduction, bottom-hole pressurization, and high-pressure pulse jet, providing a new idea for the development of deep down-hole speed-increasing devices.

Pattern recognition of stick-slip vibration in combined signals of DrillString vibration

Vibration signal measurement is one of the important methods to study the movement mode of the drill string. Through the interpretation of the vibration signal, the vibration pattern of the drill string can be accurately identified. MWD is a measurement method which can more directly reflect the vibration information of the drill string. In this paper, firstly, the drill string vibration signal is obtained by the sensors with a three-axis accelerometer and a gyroscope, and the sensor’s response has been solved and analyzed, which proves the effectiveness of this method for expressing drill string vibration information and the correlation between the measured signals. Then, the correlation coefficient, as the characteristic information basing on signal combination, is proposed; since the correlation coefficients of different patterns have different distribution characteristics, supervised machine learning is used to establish a classification model of vibration patterns. Finally, through the application of classification model in the actual drill data, the specific vibration types of various modes are clarified, which proves that the classification model can effectively distinguish different vibration modes and its accuracy can reach up to 95%.

MODELLING OF HORIZONTAL DRILL STRING MOTION BY THE LUMPED-PARAMETER METHOD

The motion of drill strings is modeled in the drilling of geotechnological wells in the mining industry by the Lumped-Parameter Method (LPM). This method is widely used in structural mechanics and is most justified in modeling dynamic systems with a variable structure. On the example of horizontal drilling of geotechnological wells, longitudinal vibrations of a drill string with a static compressive load at the left end are considered [1]. The contact interaction of the drill string with the borehole walls and the inertia force of the bit on the destructible rock at the right end of the string are taken into account. The analysis of the column splits number, which specifies the dimension of the system of discrete equations, is carried out by verifying the obtained results with the previously known data [1]. For verification, the developed C# software was used, allowed to determine the error of the column splits in comparison with the test data. The optimal number of the drill string splits in terms of "implementation time - calculation error" by the LPM was identified. The numerical implementation of the model is conducted by the fourth-order Runge-Kutta method. In connection with the increase in the implementation time of the program code due to the increase in the dimension of the system, the numerical algorithm is optimized using the parallel programming tools. The expediency of this optimization is analyzed.

Investigation into the effect of friction decay factor on the modelling and attenuation of stick-slip vibrations of oilwell drilling systems

The self-excited stick-slip oscillations of oilwell drillstrings are attributed to the nonlinear interaction between the drill-bit and the rock formation. Development of more accurate models will lead to improved predictions allowing more potential for successful suppression of the drillstring vibrations, thus reducing damage to the drilling system, prevention of expensive failures and increased output from the oilwell. In this paper, the effect of the transition from static friction to Coulomb friction on modelling of stick-slip phenomenon of oil well drill string is investigated through an analysis of the so called ‘decay factor’. Based on a distributed-lumped parameter model (DLPM) of the drilling system, the governing equations of motion for the system are obtained. By using different values of decay factor (low, high and medium), the stick-slip vibrations of the drill string are validated against published data from full-scale drill strings. The results from the simulation show that lowering the decay factor increases the critical speed and thus reduces the propensity for stick slip motion. However, a reduction in the decay factor also has the effect of inducing worse stick-slip motion once the critical speed has been reached. The results indicate the wider impact of both correct modelling of the decay factor, but also the importance of correct characterisation of the mud viscosity and drill/well contact for more accurate selection of drilling parameters in the field.

Model of a Drive System for Low-Floor Trams with a Four-Linkage Coupling

The article describes the structure and principles of operation of a drive system used in some low-floor trams. The system consists of electric motors, gear transmissions as well as hollow shaft s and four-linkage couplings causing the wheels to move. During the study, a dynamic model was built and the parameters needed to carry out a simulation were determined. NGT6 low-floor tram wagons operated by MPK S.A. Cracow were used as examples of existing vehicles. Due to the lack of available data, part of the work was devoted to determining mass moments of inertia of drive system components using an experimental method of torsional vibrations of a string. In subsequent chapters, a mathematical model was developed and a tram start-up simulation was performed based on specific parameters of the individual system components. Some of the results were presented in the form of graphs. Keywords: tram riding simulation, mathematical model of dynamics, start-up, mass moments of inertia, method of torsional vibration of string

Time periodicity from randomness in quantum systems

Many complex systems can spontaneously oscillate under non-periodic forcing. Such self-oscillators are commonplace in biological and technological assemblies where temporal periodicity is needed, such as the beating of a human heart or the vibration of a cello string. While self-oscillation is well understood in classical non-linear systems and their quantized counterparts, the spontaneous emergence of periodicity in quantum systems without a semi-classical limit is more elusive. Here, we show that this behavior can emerge within the repeated-interaction description of open quantum systems. Specifically, we consider a many-body quantum system that undergoes dissipation due to sequential coupling with auxiliary systems at random times. We develop dynamical symmetry conditions that guarantee an oscillatory long-time state in this setting. Our rigorous results are illustrated with specific spin models, which could be implemented in trapped-ion quantum simulators.

Analysis of Drill-String Nonlinear Dynamics Using the Lumped-Parameter Method

This work aims at studying the nonlinear dynamics of drill strings using the lumped-parameter method (LPM). The study is based on the good consistency of the results of the test problem where the model of the longitudinal vibrations of a horizontal drill string with a static compressive load at the left end is considered. In this paper, this method is applied to discretize linear and nonlinear models of the lateral vibrations of a vertical drill string under the effect of a supersonic gas flow. The obtained results are verified with the previously published data. The optimal number of the drill-string partitions is determined using the developed application, which allows us to estimate the accuracy of the loaded data. The numerical solution of the model is obtained using the fourth-order Runge–Kutta method. The optimization of the numerical algorithm using parallel-programming tools is carried out, and the efficiency of the method is analyzed.

Optimal Research in Piano Shape Sound and Sound Propagation Model Based on Nonlinear Differential Equations

Abstract Nonlinear differential equations are being used in many applied science fields such as engineering, there are more and more applications, such as musical instrumentation, chemistry, aerodynamics, cybernetics, signal and image processing, biophysics, complex media dynamics, etc. Mainly based on the analysis of the vibration and attenuation characteristics of piano strings, and on the theoretical basis of the discussion on the resonance effect of the resonance system, the model of the piano music simulation system is constructed, investigated a class of fractional nonlinear differential (integrative-differential) equations and systems of equations, fractional nonlinear differential equations on unbounded domains, and with multiple different fractional derivatives, and the existence of solutions to integral nonlinear impulse differential equations, uniqueness, multiple solutions, eigenvalue interval, as well as monotonic iterative sequences and error estimates that converge to the solutions of the corresponding equations. Finally, from the pitch frequency detection, formant point frequency estimation, and three aspects of spectrogram analysis, subjective evaluation experiments are carried out on the algorithm, and the experimental results verify the effectiveness of musical simulation.

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