Nonlinear String Research Articles

Reconstructing the piano hammer force from measurements and filtering of the string velocity

A method is presented for reconstructing the piano hammer force from measurements and filtering of the string velocity. The construction of the filter is based on the propagation of the pulses generated by the hammer blow in two opposite directions along the string. For the first five octaves of most pianos, and for a measuring point situated between hammer and agraffe, the hammer force can be derived by considering the incoming wave from the hammer and its first reflection by the agraffe only. For the higher notes, 4- or 8-waves schemes must be considered in order to reconstruct the hammer force over the complete range of the piano. The theory is first validated on simulated string velocities, by comparing imposed and reconstructed hammer forces. The simulations are based on a nonlinear damped stiff string model (JASA 134, pp. 648–665). In a second step, the method is applied to string velocities measured of real pianos. Its efficiency is further validated through simulation of the string velocity with the reconstructed hammer force at the input, and comparison with the measured velocity. [Work supported by the Lise-Meitner-Fellowship M1653-N30 of the Austrian Science Fund (FWF).]

A physical model of a highly nonlinear string and its use in the music composition Quartet for Strings

A real-time virtual string model is created that supports two modes of haptic force-feedback interaction. Using one haptic controller, the string can be plucked, and using a second haptic controller, the virtual string's pitch can be continuously varied by applying pressure to the string. Via both controllers, the vibrations of the virtual string can be felt while it is played. These functionalities are achieved by modeling the string as a finite sequence of interleaved masses and stiffening springs. For artistic purposes, a nonlinear spring force characteristic function is selected that limits the effective stiffness of each spring to a range between k and (k + ks). This function is F(x) = kx + (ksx3)/(β2+x> + x2), where x is the displacement of the nonlinear spring and β approximately adjusts the displacement at which the stiffness becomes significantly nonlinear. For small displacements, the string's undamped fundamental frequency is tuned by k, and for sufficiently large displacements, the string's un...

On polynomials associated with an Uvarov modification of a quartic potential Freud-like weight

In this contribution we consider sequences of monic polynomials orthogonal with respect to the standard Freud-like inner product involving a quartic potential〈p,q〉=∫Rp(x)q(x)e−x4+2tx2dx+Mp(0)q(0).We analyze some properties of these polynomials, such as the ladder operators and the holonomic equation that they satisfy and, as an application, we give an electrostatic interpretation of their zero distribution in terms of a logarithmic potential interaction under the action of an external field. It is also shown that the coefficients of their three term recurrence relation satisfy a nonlinear difference string equation. Finally, an equation of motion for their zeros in terms of their dependence on t is given.

Nonlinear Evolution Equations with Infinitely Many Derivatives

We study the nonlinear evolutionary euclidean bosonic string equation $$\begin{aligned} u_t = \Delta e^{-c \Delta }\,u + U(t,u) , \quad c > 0 \; \end{aligned}$$ on the Euclidean space \({\mathbb {R}}^n\). We interpret the nonlocal operator \(\Delta e^{-c\,\Delta }\) using entire vectors of \(\Delta \) in \(L^2({\mathbb {R}}^n)\). We prove that it generates a bounded holomorphic \(C_0\)-semigroup on \(L^2({\mathbb {R}}^n)\) (so that it also satisfies maximal \(L^p\) regularity) and we show the well-posedness of the corresponding nonlinear Cauchy problem.

Three-dimensional dynamic behavior of suspended single wall carbon nanotubes

We investigate the nonlinear dynamics of high aspect ratio single wall carbon nanotubes (SWCNTs) both theoretically and experimentally. A nonlinear stretchable string model was used to describe the complicated dynamics of very slender SWCNTs with tension dominated stiffness and its transition from linear to nonlinear response. Finite difference spacial and time discretization were implemented and the resulting system of ordinary differential equations was solved numerically. A fabrication process developed to incorporate SWCNTs into patterned silicon structure was used to fabricate structures of taut, long SWCNTs suspended over trenches. Time depending electric fields were applied to the SWCNTs and their mechanical resonance responses were captured based on averaged electrical measurements (i.e., electrical resistance) of the oscillating SWCNTs. Measured resonance frequencies are reasonably consistent with those extracted from our model as resonance frequency of 24.3MHz was measured comparing to the calculated frequency of 20.4MHz. The transition from linear to nonlinear regime was demonstrated experimentally.

Boundary control of nonlinear elastic systems

This paper presents a design of boundary controllers for global stabilization of nonlinear elastic systems, which cover nonlinear elastic strings and membranes, under external bounded forces. The boundary controllers guarantee exponential convergence of the unique system solution to a ball centered at the origin. The Faedo–Galerkin approximation method is used to prove existence and uniqueness of the solution of the closed-loop system. The control design is based on the Lyapunov direct method, Gronwall’s, Poincare’s, and Holder’s inequalities, and Sobolev embedding theorems. Simulations illustrate the effectiveness of the proposed controllers.

A string reaction coordinate for the folding of a polymer chain

We investigate the crystallization mechanism of a single, flexible homopolymer chain with short range attractions. For a sufficiently narrow attractive well, the system undergoes a first-order like freezing transition from an expanded disordered coil to a compact crystalline state. Based on a maximum likelihood analysis of committor values computed for configurations obtained by Wang–Landau sampling, we construct a non-linear string reaction coordinate for the coil-to-crystal transition. In contrast to a linear reaction coordinate, the string reaction coordinate captures the effect of different degrees of freedom controlling different stages of the transition. Our analysis indicates that a combination of the energy and the global crystallinity parameter Q6 provide the most accurate measure for the progress of the transition. While the crystallinity paramter Q6 is most relevant in the initial stages of the crystallization, the later stages are dominated by a decrease in the potential energy.

Exact travelling wave solutions of a variety of Boussinesq-like equations

In this paper, we obtain exact traveling wave solutions of a variety of Boussinesq-like equations by using two distinct methods with symbolic computation. The Boussinesq equations play an important role in physical applications, such as in nonlinear lattice waves, acoustic waves, iron sound waves in a plasma, and vibrations in a nonlinear string. More precisely, the modified tanh-coth method is employed to obtain single soliton solutions, and the extended Jacobi elliptic function method is applied to derive doubly periodic wave solutions. Further, it is shown that soliton solutions and triangular solutions can be established as the limits of the Jacobi doubly periodic wave solutions. The employed approaches are quite efficient for the determination of the solutions, and are practically well suited for solving nonlinear evolution equations arising in physics.

Nonlinear energy sink to control elastic strings: the internal resonance case

The use of nonlinear energy sink as a passive control device is extended here to a nonlinear elastic string, in internal resonance conditions, excited by an external harmonic force. The Multiple Scale/Harmonic Balance Method is directly applied to the partial differential equations ruling the dynamics of the system. The internal resonance condition of the string involves a rich response containing essentially both the resonant, directly excited, mode and a superharmonic one. Numerical results on a case study are presented.

Nonlinear energy sink to control vibrations of an internally nonresonant elastic string

A nonlinear elastic string is considered here as a main structure to be passively controlled using a Nonlinear Energy Sink (NES). The string is internally nonresonant due to a point mass and an elastic spring applied at a free tip, and a distributed force with harmonic time-law is assumed. The Multiple Scale/Harmonic Balance method, already introduced for finite degree of freedom systems, is extended here in direct approach, being applied to the partial differential equations ruling the dynamics of the system. Amplitude modulation equations are obtained and discussion of some solutions, where the beneficial effect of the NES is evident, are made.

Adaptive Boundary Control of a Nonlinear Flexible String System

In this brief, the vibration control problem is investigated for a flexible string system in both transverse and longitudinal directions. The vibrating string is nonlinear due to the coupling between transverse and longitudinal displacements. Using the Hamilton's principle, the dynamics of the nonlinear string are presented by two partial and four ordinary differential equations. With the Lyapunov's direct method, adaptive boundary control is developed to suppress the string's vibration and the adaptive law is designed to compensate for the system parametric uncertainties. With the proposed control, the states of the system eventually converge to a compact set. Numerical simulations are carried out to verify the effectiveness of the proposed control.

Finite mechanical proxies for a class of reducible continuum systems

We present the exact finite reduction of a class ofnonlinearly perturbed wave equations --typically, a non-linear elastic string-- based on the Amann--Conley--Zehnder paradigm.By solving an inverse eigenvalue problem,we establish an equivalence between the spectral finite description derived from A--C--Z and a discrete mechanical model, a well definite finite spring--mass system. By doing so, we decrypt the abstract information encoded in the finite reduction and obtain a physically sound proxy for the continuous problem.

Global and Blow-Up Solutions for Nonlinear Hyperbolic Equations with Initial-Boundary Conditions

We consider an initial-boundary value problem to a nonlinear string equations with linear damping term. It is proved that under suitable conditions the solution is global in time and the solution with a negative initial energy blows up in finite time.

Constraints in structural and rigid body mechanics: a frictional contact problem

In this work, we propose some basic approaches towards a unification of the theories for deformable and rigid bodies. This unification process is based on two fundamental mechanical concepts, which are the principle of virtual work and the principle of d’Alembert–Lagrange. The basic idea is to initially look upon structural elements as general continua, and to endow them later with specific properties like rigidity, imposed by perfect bilateral constraints. It is shown by the example of a simple flexible multibody system how this unifying and systematic approach has to be carried out. The system under consideration consists of a rigid disk and a nonlinear elastic string, which may come into contact with each other. The contact is modeled as a hard unilateral geometric constraint combined with a one-dimensional Coulomb friction element. The contact interactions are formulated as set-valued force laws and impact laws, and the system is consequently treated within the framework of nonsmooth dynamics. The model of the string allows for large deformations in time and for a nonlinear elastic material response. By constraining the kinematics of the string to finite dimensions, a nonlinear finite element formulation is achieved in a very natural way.

The analysis on coupling vibration of drill string and marine riser in deep-water drilling

During drilling in the deep water, the system including marine riser and drill string over the wellhead exits strong dynamic response because of the motion of platform. The contact-collision problem between the drill string and marine riser under platform swaying is a highly non-linear dynamics problem, drill string and marine riser contact-collision position along the axis of marine riser is randomly distributed, and that there exits friction. In this paper, by using the method of finite element, the model of dynamical behavior of the system is set up, the state of contact between marine riser and drill string shows randomness. The rule of free lateral coupling vibration of drill string and marine riser has been obtained; the result shows the situation of random contact between drill string and marine rising. By case analysis the result shows that the platform swaying has great effect to the system of marine riser and drill string.

Fiber Bragg grating strain modulation based on nonlinear string transverse-force amplifier

The only effective method of fiber Bragg grating (FBG) strain modulation has been by changing the distance between its two fixed ends. We demonstrate an alternative that is more sensitive to force based on the nonlinear amplification relationship between a transverse force applied to a stretched string and its induced axial force. It may improve the sensitivity and size of an FBG force sensor, reduce the number of FBGs needed for multiaxial force monitoring, and control the resonant frequency of an FBG accelerometer.

Optimizing parametric oscillators with tunable boundary conditions

Parametric excitation or pumping is an effective method to create large oscillations by periodically altering a physical parameter of the governing dynamics. Precisely tuned pumping frequencies can lead to exponentially growing oscillations limited only by nonlinear effects like axial stretching of transversely vibrating string. It is demonstrated that a tuned passive dynamical system amplifies the otherwise limited transverse vibrations amplitudes of a nonlinear string considerably and thus increasing the selectivity of the system. This effect becomes more noticeable for shorter wavelengths where nonlinear stretching limits the obtainable vibration amplitudes severely. The present work analyses a passive dynamical system connected to one end of a taught string which parametrically couples its axial motion to transverse vibration. Analysis shows that a specific selection of parameters can reduce the limiting effect of nonlinear stretching thus allowing one to excite high-order modes with small external forces. The result can possibly affect other disciplines where effective parametric amplification is necessary.

The acoustics of pianos

The manufacturing of pianos remains largely empirical, with numerous trial-and-error procedures and fine adjustments at each step of the building process. The “skeleton” of the instrument obeys fundamental principles of vibrations, acoustics, and material science. An abundance of literature is available on its different constitutive parts. However, scientific studies based on a global model of the instrument that connects all of these constitutive parts together are more recent. Such modeling sheds useful light on the essential coupling properties between elements and, in particular, on the string-soundboard coupling at the bridge, and on the radiation of the soundboard. Fine analysis of piano tones also shows that in most cases, a nonlinear model of the strings is necessary to account for perceptually significant features such as precursors in the time-domain and the so-called “phantom partials” in the spectrum. This nonlinearity is based on the coupling between transverse and longitudinal waves in the string. In this lecture, a time-domain model of a complete piano is presented that couples together nonlinear strings, soundboard vibrations, and radiation in air. It highlights the transmission of both transverse and longitudinal string forces to the soundboard, and the influence of rib design and bridge on soundboard mobility and radiation patterns. Comparisons between the results of the model and measurements made on real pianos will be discussed.

Existence and Stability of Almost Periodic Solutions of Nonlinear Damped Equations of a Suspended String

In this paper we shall show the existence and the stability of almost periodic solutions of the boundary value problem to a nonlinear suspended string equation with a linear damping term and an almost periodic weakly nonlinear forcing term. We treat both weak solutions and strong solutions. Also we show the existence of time global solutions of the initial boundary value problem to the equation.

On exact controllability of networks of nonlinear elastic strings in 3-dimensional space

This paper concerns a system of nonlinear wave equations describing the vibrations of a 3-dimensional network of elastic strings. The authors derive the equations and appropriate nodal conditions, determine equilibrium solutions, and, by using the methods of quasilinear hyperbolic systems, prove that for tree networks the natural initial, boundary value problem has classical solutions existing in neighborhoods of the “stretched” equilibrium solutions. Then the local controllability of such networks near such equilibrium configurations in a certain specified time interval is proved. Finally, it is proved that, given two different equilibrium states satisfying certain conditions, it is possible to control the network from states in a small enough neighborhood of one equilibrium to any state in a suitable neighborhood of the second equilibrium over a sufficiently large time interval.