共查询到20条相似文献,搜索用时 31 毫秒
1.
Maria V. Demina 《Studies in Applied Mathematics》2023,150(3):755-817
We provide the necessary and sufficient conditions of Liouvillian integrability for Liénard differential systems describing nonlinear oscillators with a polynomial damping and a polynomial restoring force. We prove that Liénard differential systems are not Darboux integrable excluding subfamilies with certain restrictions on the degrees of the polynomials arising in the systems. We demonstrate that if the degree of a polynomial responsible for the restoring force is greater than the degree of a polynomial producing the damping, then a generic Liénard differential system is not Liouvillian integrable with the exception of linear Liénard systems. However, for any fixed degrees of the polynomials describing the damping and the restoring force we present subfamilies possessing Liouvillian first integrals. As a by-product of our results, we find a number of novel Liouvillian integrable subfamilies. In addition, we study the existence of nonautonomous Darboux first integrals and nonautonomous Jacobi last multipliers with a time-dependent exponential factor. 相似文献
2.
Fatiha Alabau-Boussouira 《NoDEA : Nonlinear Differential Equations and Applications》2007,14(5-6):643-669
We consider systems of Timoshenko type in a one-dimensional bounded domain. The physical system is damped by a single feedback
force, only in the equation for the rotation angle, no direct damping is applied on the equation for the transverse displacement
of the beam. Moreover the damping is assumed to be nonlinear with no growth assumption at the origin, which allows very weak
damping. We establish a general semi-explicit formula for the decay rate of the energy at infinity in the case of the same
speed of propagation in the two equations of the system. We prove polynomial decay in the case of different speed of propagation
for both linear and nonlinear globally Lipschitz feedbacks.
相似文献
3.
We consider the decay rate of energy of the 1D damped original nonlinear wave equation. We first construct a new energy function. Then, employing the perturbed energy method and the generalized Young’s inequality, we prove that, with a general growth assumption on the nonlinear damping force near the origin, the decay rate of energy is governed by a dissipative ordinary differential equation. This allows us to recover the classical exponential, polynomial, or logarithmic decay rate for the linear, polynomial or exponentially degenerating damping force near the origin, respectively. Unlike the linear wave equation, the exponential decay rate constant depends on the initial data, due to the nonlinearity. 相似文献
4.
This article is concerned with a damped critical non-linear Schrödinger equation with fourth-order dispersion which models propagation of fibre arrays in non-linear Kerr media. We analyse the effect of the damping force on the solution for the system and prove that there exists a threshold value of the damping parameter for global existence and blowup of the system. Furthermore, we estimate the threshold value. 相似文献
5.
Alexander P. Seyranian Alexei A. Mailybaev 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2011,27(1):539-548
The paper presents a general approach to the paradox of Nicolai and related effects analyzed as a singularity of the stability
boundary. We study potential systems with arbitrary degrees of freedom and two coincident eigenfrequencies disturbed by small
non-conservative positional and damping forces. The instability region is obtained in the form of a cone having a finite discontinuous
increase in the general case when arbitrarily small damping is introduced. This is a new destabilization phenomenon, which
is similar to well-known Ziegler’s paradox or the effect of the discontinuous increase of the combination resonance region
due to addition of infinitesimal damping. It is shown that only for specific ratios of damping coefficients, the system is
stabilized due to presence of small damping. Then, we consider the paradox of Nicolai: the instability of a uniform axisymmetric
elastic column loaded by axial force and a tangential torque of arbitrarily small magnitude. We extend the results of Nicolai
showing that the column is stabilized by general small geometric imperfections and internal and external damping forces. It
is shown that the paradox of Nicolai is related to the conical singularity of the stability boundary which transforms to a
hyperboloid with the addition of small dissipation. As a specific example of imperfections, we study the case when cross-section
of the column is changed from a circular to elliptic form. 相似文献
6.
Alexander P. Seyranian Alexei A. Mailybaev 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2011,62(3):539
The paper presents a general approach to the paradox of Nicolai and related effects analyzed as a singularity of the stability boundary. We study potential systems with arbitrary degrees of freedom and two coincident eigenfrequencies disturbed by small non-conservative positional and damping forces. The instability region is obtained in the form of a cone having a finite discontinuous increase in the general case when arbitrarily small damping is introduced. This is a new destabilization phenomenon, which is similar to well-known Ziegler’s paradox or the effect of the discontinuous increase of the combination resonance region due to addition of infinitesimal damping. It is shown that only for specific ratios of damping coefficients, the system is stabilized due to presence of small damping. Then, we consider the paradox of Nicolai: the instability of a uniform axisymmetric elastic column loaded by axial force and a tangential torque of arbitrarily small magnitude. We extend the results of Nicolai showing that the column is stabilized by general small geometric imperfections and internal and external damping forces. It is shown that the paradox of Nicolai is related to the conical singularity of the stability boundary which transforms to a hyperboloid with the addition of small dissipation. As a specific example of imperfections, we study the case when cross-section of the column is changed from a circular to elliptic form. 相似文献
7.
We study a class of elastic systems described by a (hyperbolic) second-order partial differential equation. Our working example is the equation of a vibrating string subject to a destabilizing linear disturbance. Our main goal is to establish conditions for stabilization and asymptotic stabilization of the equilibrium configuration of the string by applying to it fast oscillating controlled force. In the first situation studied we assume that the string is subject to damping; after that we consider the same system without damping. We extend the tools of high-order averaging and of chronological calculus for studying the stability of this distributed parameter system. 相似文献
8.
The damping of laminar fluid transients in piping systems is studied numerically using a two-dimensional water hammer model. The numerical scheme is based on the classical fourth order Runge–Kutta method for time integration and central difference expressions for the spatial terms. The results of the present method show that the damping of transients in piping systems is governed by a non-dimensional parameter representing the ratio of the Joukowsky pressure force to the viscous force. In terms of time scales, this non-dimensional parameter represents the ratio of the viscous diffusion time scale to the pipe period. For small values of this parameter, the damping of the fluid transient becomes more pronounced while for large values, the fluid transient is subjected to insignificant damping. Moreover, the non-dimensional parameter is shown to influence other important transient phenomena such as line packing, instantaneous wall shear stress values and the Richardson annular effect. 相似文献
9.
《Communications in Nonlinear Science & Numerical Simulation》2011,16(3):1640-1649
A generalized van der Pol oscillator is considered, with positive real power nonlinearities in the restoring and damping force, including fractional powers. An analytical approach based on the Krylov–Bogoliubov method is adjusted to derive analytical expressions for the amplitude of a limit cycle for small values of the damping coefficient. These expressions are also derived for some integer power nonlinearities in the equation of motion and the results obtained compared with the existing results from the literature. Relaxation oscillations are studied for larger values of the damping coefficient. Matched asymptotic expansions are used and the influence of the powers of the restoring and damping force on the period of these oscillations is investigated. It is shown that not only can the period increase with the damping power, but it can also have a decreasing trend for some cases and the condition for this to hold is obtained. 相似文献
10.
11.
S. L. Mathur 《Proceedings Mathematical Sciences》1973,77(6):293-304
The aim of the present paper is to study displacement due to an impulsive torsional body force, within a Semi-infinite isotropic elastic solid body in the presence of viscous damping as well as solid damping. Certain numerical results of the displacement of the stress-free solid have also been given in graphical form. 相似文献
12.
In this paper, a linear viscoelastic system is considered where the viscoelastic force depends on the past history of motion via a convolution integral over an exponentially decaying kernel function. The free‐motion equation of this nonviscous system yields a nonlinear eigenvalue problem that has a certain number of real eigenvalues corresponding to the nonoscillatory nature. The quality of the current numerical methods for deriving those eigenvalues is directly related to damping properties of the viscoelastic system. The main contribution of this paper is to explore the structure of the set of nonviscous eigenvalues of the system while the damping coefficient matrices are rank deficient and the damping level is changing. This problem will be investigated in the cases of low and high levels of damping, and a theorem that summarizes the possible distribution of real eigenvalues will be proved. Moreover, upper and lower bounds are provided for some of the eigenvalues regarding the damping properties of the system. Some physically realistic examples are provided, which give us insight into the behavior of the real eigenvalues while the damping level is changing. 相似文献
13.
Innovative stiffness and damping models for oil films are developed to account for the impacts in both normal and tangential directions. Given that these models are applied to a gear drive in line contact elastohydrodynamic lubrication (EHL), the combined stiffness is derived from the stiffness of both the oil film and gear tooth while the combined damping is established from the damping of these parts. The effects of three fundamental parameters (contact force, rotation speed, and tooth numbers) of the gear drive in line contact EHL on the combined stiffness and damping are then investigated. The results reveal that the small normal and tangential stiffness of the lubricant can alleviate meshing impact and shear vibration, while the impact and friction heat can be reduced by using an oil film with either a large normal damping or small tangential damping. Given that its amplitude and fluctuation are closely related to shear rate, effective viscosity, entrainment velocity, and curvature radii, the improved combined stiffness and damping can be obtained by rationally matching the geometric and operating parameters. 相似文献
14.
15.
《Mathematical Modelling》1984,5(6):363-381
A mathematical model of the dynamic response of atrioventricular valve leaflets to the systolic pressure pulse is used to calculate the resultant energy dissipation under the action of viscous damping. The energy dissipated from mitral and tricuspid leaflets is compared, and the effects of leaflet size, closing velocity, the magnitude of the damping force, and the rate of change of the atrioventricular pressure gradient are investigated. The results are interpreted in relation to experimentally documented first heart sound determinants. 相似文献
16.
17.
In this contribution we analyze the stabilizing and destabilizing effect of small damping for rather general class of continuous non-conservative systems, described by the non-self-adjoint boundary eigenvalue problems. Explicit asymptotic expressions obtained for the stability domain allow us to predict when a given combination of the damping parameters leads to increase or to decrease in the critical non-conservative load. The results obtained explain why different types of internal and external damping so surprisingly influence on the stability of non-conservative systems. As a mechanical example the stability of a viscoelastic rod with small internal and external damping, loaded by tangential follower force, is studied in detail. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
18.
19.
We developed the control technique for (non)linear oscillators when repellors are stabilized by adjusting the system to energy levels corresponding to their stable counterparts. The technique does not require knowledge of the system equations. Two control strategies are possible. Following the first one, we simply test the systems by changing the feedback strength. This strategy does not require any computation of the control signal, and, hence, can be useful for control as well as identification of unknown systems. If the desired target can be identified (say, from the system time series), one can use another strategy based on goal-oriented control of the desired target. We analyze how the perturbation shape can be tuned so as to preserve the system natural response and discuss how to calculate the minimal strength of the perturbation required for stabilization of a priori chosen orbit. Generally, the control represents addition of an extra nonlinear damping to the system. In two limits of the perturbation slope, it manifests itself in (i) changing the oscillator natural damping; (ii) suppressing (enhancing) the external driving force. In the case of large deviations between phases of the chaotic oscillator and the driving force, only first scenario holds. Generalization of the technique to the case of oscillator networks and 3D autonomous dynamical systems is considered. 相似文献
20.
Diego Chamorro Oscar Jarrín Pierre-Gilles Lemarié-Rieusset 《Comptes Rendus Mathematique》2019,357(2):175-179
We consider stationary Navier–Stokes equations in with a regular external force and we prove the exponential frequency decay of the solutions. Moreover, if the external force is small enough, we give a pointwise exponential frequency decay for such solutions. If a damping term is added to the equation, a pointwise decay is obtained without the smallness condition over the force. 相似文献