Dynamic responses to sinusoidal excitations of beams with frictional joints

We consider the dynamic responses of a beam with a frictional joint. The frictional force at the joint is modeled using the Coulomb friction model. The frictional force at the joint makes the nature of the boundary conditions at the joint uncertain. Therefore, this problem represents a type of nonlinear problems where the boundary conditions are coupled to the solutions. Using numerical integration of the resulting differential equations obtained by combining the finite element method and the Lagrange equations, we study the steady-state solutions of the system to sinusoidal excitations. We explore the dependence of the system responses to various parameters including the frictional force, the forcing frequency and the forcing amplitude. A result of special interest is the existence of an optimum friction force if the frictional joint is used to control the system response amplitude. We also examine the ways that friction affects the resonance frequency of the structure. Experiments are carried out, which agree qualitatively with the numerical results.     

Uniqueness and non-uniqueness of bounded solutions to singular nonlinear parabolic equations

We investigate well-posedness of initial-boundary value problems for a class of nonlinear parabolic equations with variable density. At some part of the boundary, called singular boundary, the density can either vanish or diverge or not need to have a limit. We provide simple conditions for uniqueness or non-uniqueness of bounded solutions, depending on the behaviour of the density near the singular boundary.     

Multipoint boundary value problems for nonlinear ordinary differential equations

In this paper we provide sufficient conditions for the existence of solutions to multipoint boundary value problems for nonlinear ordinary differential equations. We consider the case where the solution space of the associated linear homogeneous boundary value problem is less than 2. When this solution space is trivial, we establish existence results via the Schauder Fixed Point Theorem. In the resonance case, we use a projection scheme to provide criteria for the solvability of our nonlinear boundary value problem. We accomplish this by analyzing a link between the behavior of the nonlinearity and the solution set of the associated linear homogeneous boundary value problem.     

On the solvability of multipoint boundary value problems for discrete systems at resonance

We study nonlinear discrete-time boundary value problems at resonance. Criteria for the existence of solutions are established via topological degree theory and the Lyapunov–Schmidt procedure.     

Robin-type boundary control problems for the nonlinear Boussinesq type equations

In this paper, we study a linear and a nonlinear boundary control problems arising from viscous flows. The equations are of nonlinear Navier-Stokes type for the velocity and pressure, of transport-diffusion type for the temperature and the salinity. The essential difficulties are due to the nonlinear nature of a part of the boundary conditions and to the nature of the equations: time-dependent, coupled and nonlinear. The existence and the conditions of the uniqueness of the solution, for the variational problem, are studied. The control is of linear or nonlinear Robin-type and acts on a part of the boundary during a time T. The cost function measures the distance between the observed and the computed vorticity. The existence of an optimal control in the admissible set of states and controls is proved. A first order necessary conditions of optimality are obtained.     

First-order three-point boundary value problems at resonance

We consider three-point boundary value problems for a system of first-order equations in perturbed systems of ordinary differential equations at resonance. We obtain new results for the above boundary value problems with nonlinear boundary conditions. The existence of solutions is established by applying a version of Brouwer’s Fixed Point Theorem which is due to Miranda.     

Long Nonlinear Water Waves in a Channel Near a Cut-off Frequency

A theoretical study is made of the free-surface flow induced by a wavemaker, performing torsional oscillations about a vertical axis, in a shallow rectangular channel near a cut-off frequency. Exactly at cut-off, linearized water-wave theory predicts a temporally unbounded response due to a resonance phenomenon. It is shown, through a perturbation analysis using characteristic variables, that the nonlinear response is governed by a forced Kadomtsev—Petviashvili (KP) equation with periodic boundary conditions across the channel. This nonlinear initial-boundary-value problem is investigated analytically and numerically. When surface-tension effects are negligible, the nonlinear response reaches a steady state and exhibits jump phenomena. On the other hand, in the high-surface-tension regime, no steady state is obtained. These results are discussed in connection with similar forced wave phenomena studied previously in a deepwater channel and related laboratory experiments.     

Existence of periodic solutions for first order differential equations

In this paper we study the periodic boundary value problem for first order differential equations by combining techniques of the theory of differential inequalities, namely the method of upper and lower solutions, and the alternative method for nonlinear problems at resonance. The results obtained are in terms of the behavior of the nonlinear part at infinity.     

Nonlinear dynamics and stability analysis of piezo-visco medium nanoshell resonator with electrostatic and harmonic actuation

In this paper, nonlinear dynamics, vibration and stability analysis of piezo-visco medium nanoshell resonator (PVM-NSR) based on functionally graded (FG) cylindrical nanoshell integrated with two piezoelectric layers subjected to visco-pasternak medium, electrostatic and harmonic excitations is investigated. Nonclassical method of the electro-elastic Gurtin–Murdoch surface/interface theory with von-Karman–Donnell's shell model as well as Hamilton's principle, the assumed mode method combined with Lagrange–Euler's are considered. Complex averaging method combined with arc-length continuation is used to achieve a numerical solution for the steady state vibrations of the system. The stability analysis of the steady state response is performed. The parametric studies such as the effects of different boundary conditions, different geometric ratios, structural parameters, electrostatic and harmonic excitation on the nonlinear frequency response and stability analysis are studied. The results indicate that near the natural frequency of the nanoshell, it will lead to resonance and will have large motion amplitude and near the resonant frequency, the nanoshell shows a softening type of nonlinear behavior, and the nanoshell bandwidth increases due to nonlinear factors. In this range, nanoshell has three different ranges of motion, of which two are stable and the other unstable, and so the jump phenomenon and saddle-node bifurcation are visible in the behavior of the system. Also piezoelectric voltage influences on static deformation and resonant frequency but has no significant effect on nonlinear behavior and bandwidth and also system very sensitive to the damping coefficient and due to decrease of nano shell stiffness, natural frequency decreases. And also, increasing or decreasing of some parameters lead to increasing or decreasing the resonance amplitude, resonant frequency, the system's instability, nonlinear behavior and bandwidth.     

一类四阶非线性奇摄动方程的边值问题

利用匹配渐近展开法,讨论了一类四阶非线性方程的具有两个边界层的奇摄动边值问题．引进伸长变量,根据边界条件与匹配原则,在一定的可解性条件下,给出了外部解和左右边界层附近的内层解,得到了该问题的二阶渐近解,并举例说明了这类非线性问题渐近解的存在性．     

High-order nonlinear boundary value problems admitting multiple exact solutions with application to the fluid flow over a sheet

We frame a hierarchy of nonlinear boundary value problems which are shown to admit exponentially decaying exact solutions. We are able to convert the question of the existence and uniqueness of a particular solution to this nonlinear boundary value problem into a question of whether a certain polynomial has positive real roots. Furthermore, if such a polynomial has at least two distinct positive roots, then the nonlinear boundary value problem will have multiple solutions. In certain special cases, these boundary value problems arise in the self-similar solutions for the flow of certain fluids over stretching or shrinking sheets; examples given include the flow of first and second grade fluids over such surfaces.     

Study of cross-diffusion induced Turing patterns in a ratio-dependent prey-predator model via amplitude equations

Cross-diffusion models the situation where the presence, absence or abundance of one species of population affects the movement of other species of population in the domain under consideration and vice versa. Inclusion of cross-diffusion terms makes the modeling approach more realistic and shows significant impact on the spatio-temporal pattern formation scenario. In this paper, cross-diffusion is considered in a prey-predator model with ratio-dependent functional response, in addition to self-diffusion. Weakly nonlinear analysis is used near the Turing bifurcation boundary to derive the amplitude equations. From the stability analysis of the amplitude equations, conditions for emergence of Turing patterns such as cold spot, hot spot, mixture of spots and stripes and labyrinthine are identified. The analytical results are then verified with the help of numerical simulations. Results are general in nature and can be used to study the effect of cross-diffusion on other prey predator models both analytically and numerically.     

Corner singularities between free surfaces and open boundaries

We consider the solution of the Stokes problem at a corner between a free surface and an inflow or outflow boundary. A formal asymptotic solution for the dominant contribution to the streamfunction near the corner is derived. We give a heuristic discussion of the relevance of the nature of the corner singularity to the formulation of well-posed boundary value problems.     

Multiple solutions for semilinear elliptic boundary value problems with double resonance

In this paper we study the multiplicity of nontrivial solutions of semilinear elliptic boundary value problems which may be double resonance near infinity between two consecutive eigenvalues of −Δ with zero Dirichlet boundary data. The methods we use here are Morse theory, minimax methods and bifurcation theory.     

On numerical blow-up sets

In this paper we study numerical blow-up sets for semidicrete approximations of the heat equation with nonlinear boundary conditions. We prove that the blow-up set either concentrates near the boundary or is the whole domain.

     

Nonlinear vibrations of offshore structures under random wave excitation

An account is given of some mathematical methods which have been used to analyse the response of offshore structures to random wave excitation. The analysis of nonlinear phenomena and the assessment of related nongaussian probability distributions are emphasized. The following problems are analysed in some depth: probability distributions for Morison-type wave loading; response near resonance of nonlinearly damped systems to random excitation; parametric resonance and instability of nonlinearly damped systems with randomly fluctuating restoring force coefficient. Solutions to each of these problem areas are illustrated by practical examples.     

Uniqueness and support properties of solutions to singular quasilinear parabolic equations on surfaces of revolution

We study uniqueness, nonuniqueness and support properties of nonnegative bounded solutions of initial value problems on surfaces of revolution with boundary, for a class of quasilinear parabolic equations with variable density. At the boundary, the density can either vanish or diverge or need not to have a limit. In dependence of the behavior of the density near the boundary, we provide simple conditions for uniqueness or nonuniqueness of solutions; moreover, supposing that the initial datum does not intersect the boundary, we give criteria so that the support of any solution intersects the boundary at some positive time or it remains always away from it.     

A characterization of boundary conditions yielding maximal monotone operators

We provide a characterization for maximal monotone realizations for a certain class of (nonlinear) operators in terms of their corresponding boundary data spaces. The operators under consideration naturally arise in the study of evolutionary problems in mathematical physics. We apply our abstract characterization result to Port–Hamiltonian systems and a class of frictional boundary conditions in the theory of contact problems in visco-elasticity.     

An application of the Kantorovich theoremto nonlinear finite element analysis

Summary. Finite element solutions of strongly nonlinear elliptic boundary value problems are considered. In this paper, using the Kantorovich theorem, we show that, if the Fréchet derivative of the nonlinear operator defined by the boundary value problem is an isomorphism at an exact solution, then there exists a locally unique finite element solution near the exact solution. Moreover, several a priori error estimates are obtained. Received March 2, 1998 / Published online September 7, 1999