String Oscillations Research Articles

Light-Powered Self-Translation of an Asymmetric Friction Slider Using a Liquid Crystal Elastomer String Oscillator

In recent years, there have been many studies focused on improving the performance of active materials; however, applying these materials to active machines still presents significant challenges. In this study, we introduce a light-powered self-translation system for an asymmetric friction slider using a liquid crystal elastomer (LCE) string oscillator. The self-translation system was composed of a hollow slide, two LCE fibers, and a mass ball. Through the evolution of photothermal-induced contraction, we derived the governing equations for the system. Numerical simulations revealed two distinct motion modes: the static mode and the self-translation mode. As the mass ball moved, the LCE fibers alternated between illuminated and non-illuminated states, allowing them to effectively harvest light energy to compensate for the energy dissipation within the system. Unlike traditional self-oscillating systems that oscillate around a fixed position, the asymmetric friction enabled the slider to advance continuously through the oscillator’s symmetric self-sustained oscillation. Furthermore, we explored the critical conditions necessary for initiating self-translation as well as key system parameters that influence the frequency and amplitude of the oscillator and average speed of the slider. This self-translation system, with its simple design and ease of control, holds promising potential for applications in various fields including soft robotics, energy harvesting, and active machinery.

Nodes and energy flow in the transverse oscillations of resonantly end-driven stretched strings with viscous damping

Abstract Most of the works that analyze the forced oscillations of strings assume that, although they can support a tension T, they are perfectly elastic/flexible entities that do not have losses. In this work we will show that the incorporation of losses, even maintaining the analysis in a linear regime, allows us to clarify some details of the behavior that is observed in the forced oscillations of a real string. By introducing a dissipative force into the analysis, it is possible to predict not only the resonance frequencies but also how the oscillation amplitudes depend on the strength of the driving force. We can also analyze the variation of the amplitude of the antinodes when the excitation frequency approaches the resonance frequency. This amplitude variation has a behavior very similar to that of the resonance amplitude of a spring-mass oscillator. It is generally recognized that the nodes of a string, in forced oscillation, are not points of absolute rest because the energy of vibration must be transmitted through these points. However, it is difficult to find works that analyze the peculiar properties of this small movement. In this work we calculate the amplitudes of the nodes and deduce the peculiar properties of their movement. We also calculate the time averaged energy that flows along the string, from its excitation point to its fixed end.

Towards reproducing bowed-string instrument transients using physical modeling

Physical models of bowed-string instruments can forecast various aspects of bow-string interaction. However, accurately replicating transient waveforms of string oscillations remains a challenge. In this work, measured transient behavior of the bowed string is compared with predictions from a simulation model. Fixed boundary conditions are assumed and a finite-width bow, bow-hair compliance, torsional string motion, and an elasto-plastic friction model are incorporated. The latter links relative velocity and friction force through a differential equation, and therefore, exhibits hysteresis behavior. Simulated Guettler diagrams are then compared to measured ones obtained from a robot bowing a monochord. The general shape of the playability region is successfully predicted, but some details in the measured time-domain waveforms are not reproduced by the model. Employing inverse modeling, parameter values for the elasto-plastic friction model are derived, demonstrating the ability to generate reliable reconstructions of the measured transient signals. These results emphasize the need to account for variable friction coefficients in order to reproduce measured signals. This observation aligns with prior findings indicating that static and dynamic friction coefficients vary with bow force and bow acceleration. [Work funded in whole or in part by the Austrian Science Fund (FWF) (P34852-N).]

Problem of the Boundary Control of Oscillations of a Sample of a Layered Two-Phase Composite Material

We consider the problem of the boundary control of one-dimensional oscillations of an effective (averaged) medium corresponding to a two-phase medium consisting of periodically alternating layers of elastic and viscoelastic materials with long-term memory or various viscoelastic materials with Kelvin–Voigt friction and long-term memory. The averaged model is described by a boundary value problem for an integrodifferential equation. It is shown that for this model, it is impossible to bring oscillations to a state of rest in finite time (in contrast to the equation of string oscillations) by a force acting at one end of the band. A hypothesis is formulated on the possibility of bringing the specified object to a state of rest with the help of force effects distributed along the entire length of the object.

On Some Non-local Boundary Value and Internal Boundary Value Problems for the String Oscillation Equation

The work is devoted to the problem of setting new boundary and internal boundary value problems for hyperbolic equations. The consideration of these settings is given on the example of a wave equation. The research involves the d’Alembert method, the mean value theorem and the method of successive approximations. The paper formulates and studies a number of non-local problems summarizing the classical Goursat and Dardu tasks. Some of them are marginal, and the other part is internal-marginal, and in both cases both characteristic and uncharacteristic displacements are considered. It should also be noted that a number of problems discussed below arose as a special case in the construction of the theory of correct problems for the model loaded equation of string oscillation.

EFT strings and emergence

We revisit the Emergence Proposal in 4d mathcal{N} = 2 vector multiplet sectors that arise from type II string Calabi-Yau compactifications, with emphasis on the role of axionic fundamental strings, or EFT strings. We focus on large-volume type IIA compactifications, where EFT strings arise from NS5-branes wrapping internal four-cycles, and consider a set of infinite-distance moduli-space limits that can be classified in terms of a scaling weight w = 1, 2, 3. It has been shown before how one-loop threshold effects of an infinite tower of BPS particles made up of D2/D0-branes generate the asymptotic behaviour of the gauge kinetic functions along limits with w = 3. We extend this result to w = 2 limits, by taking into account D2-brane multi-wrapping numbers. In w = 1 limits the leading tower involves EFT string oscillations, and one can reproduce the behaviour of both weakly and strongly-coupled U(1)’s independently on whether the EFT string is critical or not, by assuming that charged modes dominate the light spectrum.

Impact of Damping on Oscillation Patterns on the Plain Piano Soundboard

The influence of internal damping on the vibration of a piano soundboard is investigated using a Finite-Difference Time Domain (FDTD) physical model and experimental measurements. The damping constant of the model is varied according to a range similar to those found with measurements on a real grand piano at different production stages. With strong damping, a clear driving-point dependency of the forced string oscillation on the oscillation pattern of the soundboard is found. When decreasing the damping, this driving-point dependency is decreasing, nevertheless, it is still present. High damping, therefore, decreases soundboard vibration when strings drive the soundboard at the soundboard’s eigenfrequencies. However, such large damping increases soundboard vibrations when strings drive the soundboard at frequencies which are not eigenfrequencies. Therefore, strong damping smooths out the frequency response spectrum of an instrument. Extreme damping without any presence of distinct eigenmodes leads to a radiation of the strings sound spectrum without soundboard filtering. Low damping leads to a strong influence of the soundboard on the string’s radiated sound. Therefore, the amount of soundboard characteristics can be designed to alter internal damping process by choice of materials, including wood or varnish, and geometry. Additionally, damping reduces the presence of ’dead spots’, notes which are considerably lower in volume compared to other notes.

Boundary Value Problems for a Mixed Fourth-order Parabolic-Hyperbolic Equation With Discontinuous Gluing Conditions

The theorem of the existence and uniqueness of the solution of the boundary value problem for the equation in partial derivatives of the fourth order with variable coefficients containing the product of the mixed parabolic-hyperbolic operator and the differential operator of the oscillation string with discontinuous conditions of gluing in the pentagon to the plane is proved. By the method of reducing the order of equations, the solvability of the boundary value problem is reduced to the solution of the Tricomi problem for the mixed parabola-hyperbolic equation with variable coefficients and discontinuous gluing conditions. Solving this problem is reduced to the solution of Fredholm’s integral equation of the second order relative to the trace of the derivative function on y along the line of variation of the equation type. In the hyperbolic part of the domain, the representation of the solution of the problem for the hyperbolic equation with the smallest terms was obtained by using the Riemann function method. In the parabolic part of the domain, the solution of the first boundary value problem for the parabolic equation with the smallest terms is obtained by the method of successive approximations and the Green’s function. As a result, the solution of the problem is realized by the method of solving the Gursa problem and the first boundary value problem for the equation of string oscillation.

Optimal boundary control of string oscillations by displacement at two ends with specified values of deflection function at intermediate moments of time

We consider the problem of optimal boundary control for the equation of string vibrations with given initial, final conditions and given values of the string deflection function at intermediate moments of time and with a quality criterion specified over the entire time interval. Using the method of separation of variables and methods of optimal control theory with multipoint intermediate conditions, optimal boundary controls are constructed for arbitrary numbers of the first harmonics. As an application of the proposed constructive approach, a boundary optimal control is built with a given string deflection function at an intermediate moment of time.

The experimental study of the influence of the string length and the position of the electromagnetic excitation coil on the metrological characteristics of the strain sensors of the intelligent monitoring system

The article is devoted to the study of the influence of changes in the string length of string converters and the position of the electromagnetic excitation coil relative to the points of attachment of the string on the amplitude of its vibrations. The re-search was carried out on a prototype of a string converter made under the patent for the invention of the Russian Federation No. 2685803 and having a constant string tension force. The results of theoretical and experimental studies showing the influence of the highest harmonics on the amplitude of string vibrations are presented. The factors determining the composition of harmonics are considered. The contribution of the harmonics to the resulting string oscillation amplitude and the measurement error of the controlled parameter of the stress-strain state is considered. The ranges of the measurements with the help of string converters with a constant string tension force and the ways of their expansion are justified.

Geometric Comments on Music

In fact degrees of the middle chromatic scales are related with the logarithm (base 2). With relying on this fact, In this study by using logarithm and string oscillation properties, we show that music notes (A − B − · · ·) are related with the interval (0, a] (a is constant) and then we define a topology on this interval. Finally Lie algebra for the mentioned interval of music will be characterized.

String excitation by initial singularity of inflation

We discuss excitation of string oscillation modes by an initial singularity of inflation. The initial singularity of inflation is known to occur with a finite Hubble parameter, which is generally lower than the string scale, and hence it is not clear that stringy effects become significant around it. With the help of Penrose limit, we find that infinitely heavy oscillation modes get excited when a singularity is strong in the sense of Krolak’s classification. We demonstrate that the initial singularities of Starobinsky and hill top inflation, assuming the slow roll inflation to the past infinity, are strong. Hence stringy corrections are inevitable in the very early stage of these inflation models. We also find that the initial singularity of the hill top inflation could be weak for non-slow roll case.

Nambu dynamics and hydrodynamics of granular material

Abstract On the basis of the intimate relation between Nambu dynamics and hydrodynamics, hydrodynamics on a non-commutative space (obtained by the quantization of space), proposed by Nambu in his last work, is formulated as hydrodynamics of granular material. In Sect. 2, the quantization of space is done using a Moyal product, and the hydrodynamic simulation is performed for the thus-obtained 2D fluid, which flows inside a channel with an obstacle. The obtained results differ between two cases in which the size of a fluid particle is zero and finite. The difference seems to come from the behavior of vortices generated by an obstacle. In Sect. 3, considering a vortex as a string, two models are examined; one is the hybrid model in which vortices interact with each other by exchanging Kalb–Ramond fields (a generalization of stream functions), and the other is the more general string field theory in which the Kalb–Ramond field is one of the excitation modes of string oscillations. In the string field theory, an Altarelli–Parisi-type evolution equation is introduced. This is expected to describe the response of the distribution function of a vortex inside turbulence, when the energy scale is changed. The behavior of viscosity differs in string theory compared with particle theory, so that the Landau theory of fluids to introduce viscosity may be modified. In conclusion, hydrodynamics and string theory are almost identical theories. It should be noted, however, that the string theory needed to reproduce a given hydrodynamics is not the usual string theory.

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).

Cusp properties of high harmonic loops

In determining the gravitational signal of cusps from a network of cosmic string loops, a number of key parameters have to be assumed. These include the typical number of cusps per period of string oscillation and the typical values of the sharpness parameters of left- and right-moving waves on the string, evaluated at the cusp event. Both of these are important, as the power stored in the gravitational waves emitted from the loops of string is proportional to the number of cusps per period and inversely proportional to the product of the sharpness parameters associated with the left- and right-moving modes on the string. In suitable units, both of these quantities are usually thought to be of order unity. To try and place these parameters on a more robust footing, we analyze in detail a large number of randomly chosen loops of string that can have high harmonics associated with them, such as one might expect to form by chopping off an infinite string in the early Universe. This allows us to analyze tens of thousands of loops and obtain detailed statistics on these crucial parameters. While we find in general the sharpness parameters are indeed close to unity, as assumed in previous work [with occasional exceptions where they can become $O({10}^{\ensuremath{-}2})$], the cusp number per period scales directly with the number of harmonics on the loop and can be significantly larger than unity. This opens up the possibility of larger signals than would have otherwise been expected, potentially leading to tighter bounds on the dimensionless cosmic string tension $G\ensuremath{\mu}$.

Напрямки наукових дослiджень Ю.В. Козаченка: дослiдження розв’язкiв задач математичної фiзики з випадковими факторами

Одним з напрямкiв наукових дослiджень Ю. В. Козаченка є рiвняння математичної фiзики з випадковими факторами. Цi фактори можуть мати рiзну природу: випадковi початковi умови, випадковi крайовi умови, випадкова права частина, випадковi коефiцiєнти i т. д. Умови та оцiнки збiжностi за ймовiрнiстю випадкових рядiв знаходять широке застосування при розв’язаннi задач математичної фiзики з випадковими умовами. Фiзичнi постановки таких задач розглядав Кампе де Фер’є . Вiн розглядав крайову задачу для рiвняння коливання струни з випадковими початковими умовами. У роботах В. В. Булдигiна показано, що вимога, щоб майже всi реалiзацiї випадкової початкової функцiї задовольняли умови, при яких є розв’язуваною детермiнована задача, значно звужує клас випадкових умов, за яких розв’язок iснує в класичному розумiннi. Є багато робiт, в яких вивчались задачi математичної фiзики з випадковими умовами, якi базуються на дослiдженнi збiжностi за ймовiрностю в функцiональних просторах послiдовностi випадкових функцiй, що апроксимують розв’язки крайових задач. Зауважимо, що у бiльшостi з цих робiт, для знаходження умов рiвномiрної збiжностi випадкових рядiв застосовується метод, що ґрунтується на iдеї Ж.Канаха. Булдигiним В.В. та Козаченком Ю.В. був запропонований метод, який дозволяє обґрунтовувати застосування методу Фур’є до задач математичної фiзики у багатовимiрному випадку. Метод, що ґрунтується на iдеї Кахана для цього випадку не пiдходить. У роботах Козаченка Ю.В. та його учнiв дослiджувалися рiвняння гiперпболiчного та параболiчного типiв математичної фiзики з випадковими факторами. Зокрема, вивчалися властивостi класичних та узагальнених розв’язкiв таких задач, було обґрунтувано застосування методу Фур’є, знайдено оцiнок для розподiлу супремуму розв’язкiв, та побудовано моделi розв’язкiв деяких задач, що наближають розв’язок iз заданою надiйнiстю та точнiстю в рiвномiрнiй метрицi. Всi цi результати мають не лише теоретичне, але й практичне застосування для подальшого вивчення та розвинення теорiї гiперболiчних i параболiчних рiвнянь математичної фiзики з випадковими факторами. Крiм того, цi результати дозволяють моделювати розв’язки крайових задач математичної фiзики iз заданою надiйнiстю та точнiстю в рiвномiрнiй метрицi, що може застосовуватися в наукових дослiдженнях в галузi радiотехнiки, фiзики, геофiзики, фiнансової математики, математичної економiки, в технiчних науках та в механiцi, зокрема, де використовуються методи комп’ютерного моделювання випадкових процесiв.

Diffraction of nonstationary waves on wedges and half‐planes under mixed boundary conditions: Exact solutions and weakening of blast waves by wedge‐shaped barriers

AbstractThe problem of diffraction of nonstationary plane waves on a wedge under mixed boundary conditions is considered when the Neumann condition is set on one side of the wedge and the Dirichlet condition is set on other side. For the case of an incident plane wave in the form of a step function, an exact solution of the problem (expressed through elementary functions) is obtained by the direct method based on self‐similarity of the problem, i.e., using the fact that its solution is a homogeneous function of space coordinates and time. This method reduces the dimension of the problem by unity and leads it to the initial‐boundary value problem for the equation of string oscillations in the hyperbolic region and to the mixed boundary problem for the two‐dimensional Laplace equation in the elliptic region. The exact solution is given by reducing the mixed problem for an elliptic domain to a problem with boundary conditions of one type on the whole boundary by means of analytic continuation of unknown function. The results for a half‐plane are significantly simplified in comparison with the solution for a wedge. The behaviour of the obtained solution at the tip of the wedge and near the diffraction front is investigated and qualitative differences from the results for diffraction problems with one‐type boundary conditions on the whole boundary are noted. The numerical comparison is given of the near‐front asymptotic solution of mixed diffraction problem and the similar asymptotic solution by Friedlander for a wedge with acoustically rigid walls. This comparison shows that in the case of a wedge with mixed boundary conditions there is a much greater weakening (attenuation) of the blast waves in the shadow region.

Examination of polarization coupling in a plucked musical instrument string via experiments and simulations

In this article, the transient motion of a realistically plucked guitar string is studied experimentally and numerically in both transversal polarizations. The frequency dependent damping and suitable initial conditions are identified in the experiment and used in a simulation. For this reason an experimental set-up consisting of a string, an excitation mechanism and two laser Doppler vibrometers is developed. The excitation mechanism performs a realistic and reproducible plucking motion with a plectrum. Two laser Doppler vibrometers are used to measure the string oscillation transversally in two polarizations. The experimental set-up makes it possible to measure the string’s motion under reproducible conditions and, hence, at different positions for the same oscillation. This capability renders the identification of suitable initial conditions, i.e., initial displacement and velocity as well as the pre-tension, for a string model possible. Furthermore, a finite element model for the string is developed that takes into account the oscillation in both transversal planes of polarization and the coupling between them. Finally, the model results are in good agreement with the measurements. With help of the numerical model it can be vividly shown that the coupling between the polarizations of the oscillation is due to a torsional movement of the string on the saddle.

Influence of the Gravitational Field of a Null String Domain on the Dynamics of a Test Null String

The paper studies the influence of periodically changing boundary conditions on the motion of a test null string “inside” an axially symmetric null string domain that has a layered structure and preserves its size. It is shown that the action of the gravitational field of the null string domain on a test null string whose size (radius) is smaller than the size of the domain, under any initial conditions, leads to oscillations of the test null string inside a limited spatial region. The motion (drift) of the region where the test null string oscillations take place, depends on the relationship between the initial parameters characterizing the test null string and the null string domain. These regions can be considered as particles with an effective nonzero rest mass, localized in space. It is noted that the properties of these particles should be determined by the trajectories of null strings moving within a limited spatial volume, and under the influence of changing external conditions, one sort of particles can pass into another one.

Non-stationary localized oscillations of an infinite string, with time-varying tension, lying on the Winkler foundation with a point elastic inhomogeneity

We consider non-stationary oscillations of an infinite string with time-varying tension. The string lies on the Winkler foundation with a point inhomogeneity (a concentrated spring of negative stiffness). In such a system with constant parameters (the string tension), under certain conditions a trapped mode of oscillation exists and is unique. Therefore, applying a non-stationary external excitation to this system can lead to the emergence of the string oscillations localized near the inhomogeneity. We provide an analytical description of non-stationary localized oscillations of the string with slowly time-varying tension using the asymptotic procedure based on successive application of two asymptotic methods, namely the method of stationary phase and the method of multiple scales. The obtained analytical results were verified by independent numerical calculations based on the finite difference method. The applicability of the analytical formulas was demonstrated for various types of external excitation and laws governing the varying tension. In particular, we have shown that in the case when the trapped mode frequency approaches zero, localized low-frequency oscillations with increasing amplitude precede the localized string buckling. The dependence of the amplitude of such oscillations on its frequency is more complicated in comparison with the case of a one degree of freedom system with time-varying stiffness.