Kirchhoff String Research Articles

On the convergence of a three-layer semi-discrete scheme for the nonlinear dynamic Kirchhoff string equation

Abstract In this work, the initial-boundary value problem is considered for the dynamic Kirchhoff string equation u t ⁢ t - ( α ⁢ ( t ) + β ⁢ ∫ - 1 1 u x 2 ⁢ d x ) ⁢ u x ⁢ x = f u_{tt}-\bigl{(}\alpha(t)+\beta\int_{-1}^{1}u_{x}^{2}\,\mathrm{d}x\bigr{)}u_{xx}=f . Here α ⁢ ( t ) \alpha(t) is a continuously differentiable function, α ⁢ ( t ) ≥ c 0 > 0 \alpha(t)\geq\mathrm{c}_{0}>0 and 𝛽 is a positive constant. For solving this problem approximately, a symmetric three-layer semi-discrete scheme with respect to the temporal variable is applied, in which the value of a nonlinear term is taken at the middle point. This approach allows us to find numerical solutions per temporal steps by inverting the linear operators. In other words, applying this scheme, a system of linear ordinary differential equations is obtained. The local convergence of the scheme is proved. The results of numerical computations using this scheme for different test problems are given for which the Legendre–Galerkin spectral approximation is applied with respect to the spatial variable.

Boundary stabilization for axially moving Kirchhoff string under fractional PI control

AbstractIn this paper, we investigate vibration control for an axially moving Kirchhoff string. A fractional PI boundary control is applied transversally to the right boundary and the transverse vibration of the system is shown to be suppressed. By constructing an auxiliary equivalent system (PDE‐ODE) and a novel energy like function, the well‐posedness and exponential stability of the closed‐loop system are established by means of the Faedo‐Galerkin method and the Lyapunov method. The effectiveness of the proposed controller is verified by a numerical simulation using finite element method.

A Kirchhoff type equation in a nonlinear model of shell vibration

The paper deals with the study of the initial boundary value problem for the Donnell‐Mushtari‐Vlasov nonlinear system of differential equations, which describes large deflections of a rectangular shallow shell. The sought functions and are respectively longitudinal and transverse displacements of points of the shell midsurface. It is assumed that Dirichlet and Neumann type homogeneous conditions are fulfilled for functions u1 and u2. After expressing u1 and u2 through w, i.e. after constructing the relations with a help of Fourier series method, we come to a nonlinear integro‐differential hyperbolic equation for the transverse displacement function w, which by its structure is analogous to Kirchhoff string, Woinowsky‐Krieger beam and Berger plate equations. These equations are united under the common name – Kirchhoff type equations. The advantage of the described technique consists in splitting the initial boundary value problem at the end into two subproblems. From the integro‐differential equation we first find w and using the explicit relations we construct u1 and u2. The issue of the use of this approach in the general case not restricted by the rectangular domain and Dirichlet and Neumann conditions on the boundary is discussed. It is shown that in order to obtain formulas we have to solve a linear plane problem of elasticity with respect to the functions u1 and u2.

Uniform Stabilization of an Axially Moving Kirchhoff String by a Boundary Control of Memory Type

In this paper, we study the stabilization of solutions of an axially moving string of kirchhoff type by a viscoelastic boundary control. We prove that the dissipation produced by the viscoelastic term is sufficient to suppress the transversal vibrations that occur during the axial motion of the string, and we also show that the string displacement decays in an arbitrary rate. When comparing with immobile strings, we conclude that the movement of the string itself produces enough extra damping ensuring the stabilization.

Absolute stability of the axially moving Kirchhoff string with a sector boundary feedback control

In this study, the absolute stability problem for the axially moving Kirchhoff string with nonlinear boundary feedback control has been investigated. Based on the generalized Hamilton’s variational principle, the nonlinear governing equations of the motion and boundary conditions have been deduced. The proposed boundary control, which satisfies a sector constraint condition, is a negative feedback of the transversal velocity at the right end of the string. Applying the integral-type multiplier method, the absolute stability of the axially moving Kirchhoff system is established. To validate the proposed theoretical results, numerical simulations are expressed by the finite element method.

Absolute stability of the Kirchhoff string with sector boundary control

This paper addresses the stability problem of the nonlinear Kirchhoff string with nonlinear boundary control. The nonlinear boundary control is the negative feedback of the transverse velocity of the string at one end, which satisfies a sector constraint. Employing the integral type multiplier method, we establish explicit absolute exponential stability of the Kirchhoff string system.

Introducing a Green–Volterra series formalism to solve weakly nonlinear boundary problems: Application to Kirchhoff's string

This paper introduces a formalism which extends that of “Green's function” and that of “the Volterra series”. These formalisms are typically used to solve, respectively, linear inhomogeneous space–time differential equations in physics and weakly nonlinear time-differential input-to-output systems in automatic control. While Green's function is a space–time integral kernel which fully characterizes a linear problem, the Volterra series expansions involve a sequence of multi-variate time integral kernels (of convolution type for time-invariant systems). The extension proposed here consists in combining the two approaches, by introducing a series expansion based on multi-variate space–time integral kernels. This series allows the representation of the space–time solution of weakly nonlinear boundary problems excited by an “input” which depends on space and time.

Exponential and polynomial stabilization of the Kirchhoff string by nonlinear boundary control

This paper addresses the stabilization problem of the nonlinear Kirchhoff string using nonlinear boundary control. Nonlinear boundary control is the negative feedback of the transverse velocity of the string at one end, which satisfies a polynomial-type constraint. Employing the multiplier method, we establish explicit exponential and polynomial stability for the Kirchhoff string. The theoretical results are assured by numerical results of the asymptotic behavior for the system.

Free vibrations in space of the single mode for the Kirchhoff string

We study a single mode for the Kirchhoff string vibrating in space. In 3D a single mode is generally almost periodic in contrast to the 2D periodic case. In order to show a complete geometrical description of a single mode we prove some monotonicity properties of the almost periods of the solution, with respect to the mechanical energy and the momentum. As a consequence of these properties, we observe that a planar single mode in 3D is always unstable, while it is known that a single mode in 2D is stable (under a suitable definition of stability), if the energy is small.

Stabilization of a nonlinear Kirchhoff equation by boundary feedback control

This paper is concerned with stabilization of an axially moving Kirchhoff string by boundary feedback control. We derive a nonlinear system describing transverse oscillation of the string using Hamilton’s principle and prove exponential stabilization of the system by the direct Lyapunov method. The result shows that oscillation of a nonlinear axially moving Kirchhoff string can be suppressed by a new type of boundary feedback control.

On local convergence of a symmetric semi-discrete scheme for an abstract analogue of the Kirchhoff equation

The present work considers a nonlinear abstract hyperbolic equation with a self-adjoint positive definite operator, which represents a generalization of the Kirchhoff string equation. A symmetric three-layer semi-discrete scheme is constructed for an approximate solution of a Cauchy problem for this equation. Value of the gradient in the nonlinear term of the scheme is taken at the middle point. It makes possible to find an approximate solution at each time step by inverting the linear operator. Local convergence of the constructed scheme is proved. Numerical calculations for different model problems are carried out using this scheme.

Compact invariant sets for some quasilinear nonlocal Kirchhoff strings on ℝ N

We consider the quasilinear nonlocal dissipative Kirchhoff String problem with the initial conditions u(x, 0) = u 0(x) and u t (x, 0) = u 1(x), in the case where N ≥ 3, δ ≥ 0, f(u) = |u| a u for example, and (φ(x))−1 ∈ L N/2(ℝ N ) ∩ L ∞(ℝ N ) is a positive function. The purpose of our work is to study the long-time behaviour of the solution of this equation. The compactness of the embeddings , is widely applied.

Global solutions for dissipative Kirchhoff strings with non-Lipschitz nonlinear term

We investigate the evolution problem { u ″ + δ u ′ + m ( | A 1 / 2 u | H 2 ) A u = 0 , u ( 0 ) = u 0 , u ′ ( 0 ) = u 1 , where H is a Hilbert space, A is a self-adjoint nonnegative operator on H with domain D ( A ) , δ > 0 is a parameter, and m ( r ) is a nonnegative function such that m ( 0 ) = 0 and m is nonnecessarily Lipschitz continuous in a neighborhood of 0. We prove that this problem has a unique global solution for positive times, provided that the initial data ( u 0 , u 1 ) ∈ D ( A ) × D ( A 1 / 2 ) satisfy a suitable smallness assumption and the nondegeneracy condition m ( | A 1 / 2 u 0 | H 2 ) > 0 . Moreover, we study the decay of the solution as t → + ∞ . These results apply to degenerate hyperbolic PDEs with nonlocal nonlinearities.

Central manifold theory for the generalized equation of Kirchhoff strings on [formula omitted

We consider the generalized quasilinear dissipative Kirchhoff's String problem u tt = - ∥ A 1 / 2 u ∥ H 2 Au - δ Au t + f ( u ) , x ∈ R N , t ⩾ 0 with the initial conditions u ( x , 0 ) = u 0 ( x ) and u t ( x , 0 ) = u 1 ( x ) , in the case where N ⩾ 3 , δ > 0 . The purpose of our work is to study the stability of the initial solution u = 0 for this equation using central manifold theory.

The energetics and the stability of axially moving Kirchhoff strings (L)

The energetics and the stability of free transverse vibration of an axially moving string are investigated based on the Kirchhoff nonlinear model. The model is derived in a physically meaningful manner. The time-rate of the total mechanical energy associated with the vibration is calculated to show that the energy is not conserved. It is proved that there exists a conserved quantity that remains a constant during the nonlinear vibration. The conserved quantity is applied to verify the Lyapunov stability of the axially moving Kirchhoff string.

Energy bounds for some nonstandard problems in partial differential equations

We consider problems of the form d 2u dt 2 +Au=F, αu(0)+u(T)=g, β du dt (0)+ du dt (T)=h, for t∈(0, T), where A is a densely defined, linear, time independent, positive definite symmetric operator and α and β are constants. Although most of our results would hold for more general operators A, we restrict attention to the case in which A is a differential operator and determine ranges of values of α and β for which it is possible to obtain energy bounds, uniqueness results, and, in a special case, pointwise bounds. Some extensions which include a damping term or a term which arises in a generalization of the Kirchhoff string model are also discussed.

Global Solutions for Dissipative Kirchhoff Strings with m(r) = rp (p < 1)

We investigate the evolution problemu″+δu′+m|A1/2u|2HAu=0,u0=u0,u′0=u1,where H is a Hilbert space, A is a self-adjoint non-negative operator on H with domain D(A), δ>0 is a parameter, and m(r)=rp with p<1. We prove that this problem has a unique global solution for positive times, provided that the initial data (u0,u1)∈D(Aαi/2)×D(A(αi−1)/2) satisfy a suitable smallness assumption and the non-degeneracy condition m(|A1/2u0|2H)>0 (where p≥2−i and αi=2i+1). Moreover, we prove for this solution decay with a polynomial rate as t→+∞. These results apply to degenerate hyperbolic PDEs with non-local non-linearities.

Boundary Control of the Axially Moving Kirchhoff String

In this note, the axially moving Kirchhoff string is considered. It is proved that this non-linear string can be stabilized by the linear boundary control, which is the negative feedback of the transversal velocity of the string at one end. Furthermore, it is concluded that the axially moving Kirchhoff string is stable without an active control when one of its ends is free to move transversally.

SUPPRESSION OF VIBRATION IN THE AXIALLY MOVING KIRCHHOFF STRING BY BOUNDARY CONTROL

The Lyapunov technique has been used to prove that the non linear axially moving Kirchhoff string can be stabilized by the linear boundary control. The boundary control is the negative feedback of the transversal velocity of the string at one end

On Global Solutions and Blow-up Solutions of Nonlinear Kirchhoff Strings with Nonlinear Dissipation

On Global Solutions and Blow-up Solutions of Nonlinear Kirchhoff Strings with Nonlinear Dissipation