A novel numerical solution strategy for solving nonlinear free and forced vibration problems of cylindrical shells

Nonlinear vibration analysis of circular cylindrical shells has received considerable attention from researchers for many decades. Analytical approaches developed to solve such problem, even not involved simplifying assumptions, are still far from sufficiency, and an efficient numerical scheme capable of solving the problem is worthy of development. The present article aims at devising a novel numerical solution strategy to describe the nonlinear free and forced vibrations of cylindrical shells. For this purpose, the energy functional of the structure is derived based on the first-order shear deformation theory and the von–Kármán geometric nonlinearity. The governing equations are discretized employing the generalized differential quadrature (GDQ) method and periodic differential operators along axial and circumferential directions, respectively. Then, based on Hamilton's principle and by the use of variational differential quadrature (VDQ) method, the discretized nonlinear governing equations are obtained. Finally, a time periodic discretization is performed and the frequency response of the cylindrical shell with different boundary conditions is determined by applying the pseudo-arc length continuation method. After revealing the efficiency and accuracy of the proposed numerical approach, comprehensive results are presented to study the influences of the model parameters such as thickness-to-radius, length-to-radius ratios and boundary conditions on the nonlinear vibration behavior of the cylindrical shells. The results indicate that variation of fundamental vibrational mode shape significantly affects frequency response curves of cylindrical shells.     

A free boundary problem and stability for the nonlinear beam

The stability bound for the classical nonlinear Euler beam is determined in the case that its deflection is limited by an obstacle parallel to the plane of the beam. Let a clamped or simply supported beam be axially compressed by a force P > P₀, where P₀ denotes the critical load. So far only a linear theory has been applied to analyze the stability of the solutions in contact with the obstacle and the jumping to a different state. Utilizing a free boundary problem formulation we analytically as well as numerically answer these questions for the nonlinear beam.     

Active nonlinear saturation-based control for suppressing the free vibration of a self-excited plant

A nonlinear saturation-based control strategy for the suppression of the free vibration of a self-excited plant is presented. The self-excitation to have the classical form of that of the van der Pol oscillator is considered. The control technique is implemented by coupling the active absorber with the plant via a specific set of quadratic nonlinearities. The perturbation method of multiple scales is employed to find the first-order approximate solutions to the governing equations. Then a stability analysis is conducted for the response of the system and the performance of the control strategy is investigated. A parametric investigation is carried out to see the effects of changing the damping ratio of the absorber, and the value of the feedback gain on the responses of the plant and the absorber. Finally, the perturbation solutions are verified by numerical integration of the governing differential equations. It is demonstrated that the saturation-based control method is effective in reducing the vibration response of the self-excited plant when the absorber’s frequency is exactly tuned to one-half the natural frequency of the plant.     

Permanence and oscillation of a system of two nonlinear difference equations

In this paper we study the permanence and the oscillatory behavior of the solutions of a system of two nonlinear difference equations.     

A class of boundary-value problems for a system of nonlinear equations with a free boundary

The method of fixed domains is applied to derive conditions which ensure unique solvability of boundary-value problems for a system of nonlinear evolution equations with a partially free boundary.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 65, pp. 83–92, 1988.     

A new solution method for state equations of nonlinear system

Along with the computation and analysis for nonlinear system being more and more involved in the fields such as automation control, electronic technique and electrical power system, the nonlinear theory has become quite a attractive field for academic research. In this paper, we derives the solutions for state equation of nonlinear system by using the inverse operator method (IOM) for the first time. The corresponding algorithm and the operator expression of the solutions is obtained. An actual computation example is given, giving a comparison between IOM and Runge-kutta method. It has been proved by our investigation that IOM has some distinct advantages over usual approximation methods in that it is computationally convenient, rapidly convergent, provides accurate solutions not requiring perturbation, linearization, or the massive computations inherent in discrietization methods such as finite differences. So the IOM provides an effective method for the solution of nonlinear system, is of potential application valuable in nonlinear computation.     

Existence and stability of a synchronous oscillation in a neural system with delayed coupling

The article deals with a particular case of a two-neuron network with two discrete time delays. We analyse the linear stability for the particular equilibrium (0,0)$(0,0)$, and address the Hopf bifurcation which may occur as one of the delays crosses some critical values. The network under consideration has no self-connections, and thus only one time delay appears in the characteristic equation.     

A quasi-residual principle in regularization for a common solution of a system of nonlinear monotone ill-posed equations

In this paper we study the Browder–Tikhonov regularization method for finding a common solution for a system of nonlinear ill-posed equations with potential, hemicontinuous and monotone mappings in Banach spaces. We give a principle, named quasi-residual, to choose a value of the regularization parameter and an estimate of convergence rates for the regularized solutions.     

On the solution of the trajectory survival problem for a nonlinear dynamical system

Nonlinear control systems possessing the flatness property are encountered in many applied mathematical models. In this paper, a trajectory survival problem is considered for a specific nonlinear system that possesses the above property. A method based on the properties of the system is proposed for constructing a control that solves the trajectory survival problem when the controlled object moves to the goal set within a bounding set containing an obstacle. Results of numerical calculations of the control and the trajectory of a system with a given initial position are presented.     

一类非线性对偶系统的渐近解及其可解性条件

利用渐近理论,讨论了一类非线性对偶系统.在适当的条件下,得出了这一类非线性系统解的存在性条件及其渐近解.将此结果用于二自由度陀螺系统,较简捷地得到了该系统的具有小而有限振幅的渐近解.     

A multiple times scale solution for non-linear vibration of mass grounded system

In this paper, the vibration of a mass grounded system which includes two linear and non-linear springs in series has been considered. Since this system, depending on its parameters can oscillate symmetrically and asymmetrically, both cases have been solved using multiple times scales (MTS) method and some analytical relations have been obtained for natural frequency of oscillations. The results have been compared with previous work and good agreement has been obtained. Also forced vibrations of system in primary and secondary resonances have been studied for the first time and the effects of different parameters on the frequency-response have been investigated.     

A priori estimates for a solution to the first initial-boundary-value problem for a system of fully nonlinear parabolic equations

A priori estimates for a solution to a system of fully nonlinear parabolic equations are obtained in a bounded domain under the condition that the solution vanishes on the boundary of the domain. The method of obtaining a priori estimates is based on the possibility of reducing the problem under consideration to the Cauchy problem for a scalar equation on a manifold without boundary in some linear space. Bibliography: 9 titles. Translated fromProblemy Matematicheskogo Analiza, No. 17, 1997, pp. 46–71.     

A new approach for asymptotic stability system of the nonlinear ordinary differential equations

In this paper, we use measure theory for considering asymptotically stable of an autonomous system [1] of first order nonlinear ordinary differential equations(ODE’s). First, we define a nonlinear infinite-horizon optimal control problem related to the ODE. Then, by a suitable change of variable, we transform the problem to a finite-horizon nonlinear optimal control problem. Then, the problem is modified into one consisting of the minimization of a linear functional over a set of positive Radon measures. The optimal measure is approximated by a finite combination of atomic measures and the problem converted to a finite-dimensional linear programming problem. The solution to this linear programming problem is used to find a piecewise-constant control, and by using the approximated control signals, we obtain the approximate trajectories and the error functional related to it. Finally the approximated trajectories and error functional is used to for considering asymptotically stable of the original problem.     

一个耦合系统自由边界问题的整体古典解

§ 1　ＩｎｔｒｏｄｕｃｔｉｏｎＩｎｔｈｉｓｐａｐｅｒｗｅｄｉｓｃｕｓｓｔｈｅｇｌｏｂａｌｃｌａｓｓｉｃａｌｓｏｌｕｔｉｏｎｏｆａｍｕｌｔｉｄｉｍｅｎｓｉｏｎａｌｑｕａｓｉｓｔａｔｉｏｎａｒｙｐｒｏｂｌｅｍ .Ｔｈｅｐｒｏｂｌｅｍｃｏｍｅｓｆｒｏｍｔｈｅｄｉｓｃｕｓｓｉｏｎｏｆａｇｒｏｗｔｈｍｏｄｅｌｏｆｓｅｌｆｍａｉｎｔａｉｎｉｎｇｐｒｏｔｏｃｅｌｌ(ｓｅｅ [1— 3])ｉｎｍｕｌｔｉｄｉｍｅｎｓｉｏｎａｌｃａｓｅ .Ｔｈｅｐｒｏｔｏｃｅｌｌｃａｎｂｅｖｉｓｕａｌｉｚｅｄａｓｈａｖｉｎｇａｐｏｒｏｕｓｓｔ…     

Existence, stability and smoothness of a bounded solution for nonlinear time-varying thermoelastic plate equations

In this paper we study the existence, stability and the smoothness of a bounded solution of the following nonlinear time-varying thermoelastic plate equation with homogeneous Dirichlet boundary conditions