On the analytical approximation of damped oscillations of autonomous single degree of freedom oscillators

In this paper, we discuss the generalization of a recently developed perturbation method for conservative single degree of freedom systems to the cases of damping forces. We show that by explicitly considering the solution as a function of the amplitude parameter, frequency and phase of oscillation, one can merge the classical Krylov–Bogoliubov–Mitropolsky method and a modified Lindstedt–Poincaré method to derive better accurate slow-flow for damped oscillation of single degree of freedom systems.     

Analysis of a weakly non-linear autonomous oscillator by means of the field method

In this paper the field method is extended to the study of oscillatory systems with two degrees of freedom and weak quadratic non-linearity. The basic field method concept is combined with the technique of multiple time scales and the solution for both non-resonant case and the case out of the first resonance are found. The qualitative analysis of behavior in the resonant area is done by determining the values of the “adelphic” integral.     

基于微分求积法求解非粘滞阻尼系统的时程响应

研究一种含有指数型非粘滞阻尼线性多自由度振动系统的时程分析问题。该非粘滞阻尼模型假设阻尼力与质点速度的时间历程相关,数学表述为质点速度与核函数的卷积。由于阻尼模型的改变,常用的数值积分方法（如Newmark-β法、Wilson-θ法）不能直接应用于这种非粘滞阻尼系统。基于一种无条件稳定的微分求积方法,给出了这种非粘滞阻...     

Explicit computational method of dynamic response for non-viscously damped structure systems

Non-viscous damping models in which the damping forces depend on the past history of velocities via convolution integrals over some kernel functions have been raised in many engineering fields. This paper describes an explicit computational method of dynamic response for the non-viscously damped structure systems. The explicit formula is derived using the differential property of convolution and the central difference formula of acceleration. The explicit computational procedure of dynamic response is given in detail. Finally, the dynamic responses of MDOF structure system with double exponential model dampers and SDOF structure system with Gaussian model damper are computed using the proposed explicit method. The accuracy and efficiency are discussed by comparison with other two developed methods.     

A kinetic theory solution method for the Navier–Stokes equations

The kinetic-theory-based solution methods for the Euler equations proposed by Pullin and Reitz are here extended to provide new finite volume numerical methods for the solution of the unsteady Navier–Stokes equations. Two approaches have been taken. In the first, the equilibrium interface method (EIM), the forward- and backward-flowing molecular fluxes between two cells are assumed to come into kinetic equilibrium at the interface between the cells. Once the resulting equilibrium states at all cell interfaces are known, the evaluation of the Navier–Stokes fluxes is straightforward. In the second method, standard kinetic theory is used to evaluate the artificial dissipation terms which appear in Pullin's Euler solver. These terms are subtracted from the fluxes and the Navier–Stokes dissipative fluxes are added in. The new methods have been tested in a 1D steady flow to yield a solution for the interior structure of a shock wave and in a 2D unsteady boundary layer flow. The 1D solutions are shown to be remarkably accurate for cell sizes large compared to the length scale of the gradients in the flow and to converge to the exact solutions as the cell size is decreased. The steady-state solutions obtained with EIM agree with those of other methods, yet require a considerably reduced computational effort.     

Three-dimensional arbitrary Lagrangian–Eulerian numerical prediction method for non-linear free surface oscillation

A numerical prediction method has been proposed to predict non-linear free surface oscillation in an arbitrarily-shaped three-dimensional container. The liquid motions are described with Navier–Stokes equations rather than Laplace equations which are derived by assuming the velocity potential. The profile of a liquid surface is precisely represented with the three-dimensional curvilinear co-ordinates which are regenerated in each computational step on the basis of the arbitrary Lagrangian–Eulerian (ALE) formulation. In the transformed space, the governing equations are discretized on a Lagrangian scheme with sufficient numerical accuracy and the boundary conditions near the liquid surface are implemented in a complete manner. In order to confirm the applicability of the present computational technique, numerical simulations are conducted for the free oscillations of viscid and inviscid liquids and for highly non-linear oscillation. In addition, non-linear sloshing motions caused by horizontal and vertical excitations and a transition from non-linear sloshing to swirling are numerically predicted in three-dimensional cylindrical containers. Conclusively, it is shown that these sloshing motions associated with high non-linearity are reasonably predicted with the present numerical technique. © 1998 John Wiley & Sons, Ltd.     

Response probability density functions of strongly non-linear systems by the path integration method

The paper discusses challenges in numerical analysis and numerical/analytical results for strongly non-linear systems—systems with “signum”-type non-linearities. Such non-linearities are implemented for instantaneous variations of the systems’ parameters, to reduce their mean energy response when subjected to random excitations. Numerical results for displacement and velocity response probability density functions (PDFs), energy response PDFs and various order moments are obtained by the path integration technique. Attention is also given to evaluation of mean upcrossing rate, related to the system's half period, via Rice's formula informally applied to discontinuous response PDFs.     

Solution of the Navier–Stokes equations in velocity–vorticity form using a Eulerian–Lagrangian boundary element method

This paper describes the Eulerian–Lagrangian boundary element model for the solution of incompressible viscous flow problems using velocity–vorticity variables. A Eulerian–Lagrangian boundary element method (ELBEM) is proposed by the combination of the Eulerian–Lagrangian method and the boundary element method (BEM). ELBEM overcomes the limitation of the traditional BEM, which is incapable of dealing with the arbitrary velocity field in advection‐dominated flow problems. The present ELBEM model involves the solution of the vorticity transport equation for vorticity whose solenoidal vorticity components are obtained iteratively by solving velocity Poisson equations involving the velocity and vorticity components. The velocity Poisson equations are solved using a boundary integral scheme and the vorticity transport equation is solved using the ELBEM. Here the results of two‐dimensional Navier–Stokes problems with low–medium Reynolds numbers in a typical cavity flow are presented and compared with a series solution and other numerical models. The ELBEM model has been found to be feasible and satisfactory. Copyright © 2000 John Wiley & Sons, Ltd.     

An example of applications of the elementary catastrophe theory in Hamiltonian systems

The convection in atmosphere discussed in ref. [1] is rigorously treated by considering the variation of environmental temperature with the height. This represents an example of applications of the elementary catastrophe theory in Hamiltonian systems.     

Construction of the stationary probability density for a family of SDOF strongly non-linear stochastic second-order dynamical systems

In this paper, a new procedure is proposed to construct the stationary probability density for a family of the single-degree-of-freedom (SDOF) strongly non-linear stochastic second-order dynamical systems subjected to parametric and/or external Gaussian white noises. First of all, the Fokker-Planck-Kolmogorov (FPK) equation associated with the original Itô stochastic differential equation is replaced by the equivalent FPK equation by adding arbitrary anti-symmetric diffusion coefficient. Then, a family of invariant measures depending on the arbitrary anti-symmetric diffusion coefficient and another arbitrary function is constructed by vanishing the probability flows in two directions. Finally, the drift vector associated with a family of Itô stochastic differential equations is deduced by giving, a priori, these two arbitrary functions. It is shown that the known invariant measures dependent on energy are only the special cases of invariant measures presented in this paper, while some other classes of invariant measures are independent of energy. Thus, the invariant measures constructed in this paper are those belonging to the most general class of the SDOF strongly non-linear stochastic second-order dynamical systems so far.     

ON THE STABILITY BOUNDARY OF HAMILTONIAN SYSTEMS

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Solution of the planar Newtonian stick–slip problem with the singular function boundary integral method

A singular function boundary integral method (SFBIM) is proposed for solving biharmonic problems with boundary singularities. The method is applied to the Newtonian stick–slip flow problem. The streamfunction is approximated by the leading terms of the local asymptotic solution expansion which are also used to weight the governing biharmonic equation in the Galerkin sense. By means of the divergence theorem the discretized equations are reduced to boundary integrals. The Dirichlet boundary conditions are weakly enforced by means of Lagrange multipliers, the values of which are calculated together with the singular coefficients. The method converges very fast with the number of singular functions and the number of Lagrange multipliers, and accurate estimates of the leading singular coefficients are obtained. Comparisons with the analytical solution and results obtained with other numerical methods are also made. Copyright © 2005 John Wiley & Sons, Ltd.     

A Chimera method for the incompressible Navier–Stokes equations

The Chimera method was developed three decades ago as a meshing simplification tool. Different components are meshed independently and then glued together using a domain decomposition technique to couple the equations solved on each component. This coupling is achieved via transmission conditions (in the finite element context) or by imposing the continuity of fluxes (in the finite volume context). Historically, the method has then been used extensively to treat moving objects, as the independent meshes are free to move with respect to the others. At each time step, the main task consists in recomputing the interpolation of the transmission conditions or fluxes. This paper presents a Chimera method applied to the Navier–Stokes equations. After an introduction on the Chimera method, we describe in two different sections the two independent steps of the method: the hole cutting to create the interfaces of the subdomains and the coupling of the subdomains. Then, we present the Navier–Stokes solver considered in this work. Implementation aspects are then detailed in order to apply efficiently the method to this specific parallel Navier–Stokes solver. We conclude with some examples to demonstrate the reliability and application of the proposed method. Copyright © 2014 John Wiley & Sons, Ltd.     

Refining the submesh strategy in the two‐level finite element method: application to the advection–diffusion equation

We introduce a new submesh strategy for the two‐level finite element method. The numerical results show that the new submesh is able to better capture the boundary layer which is caused by the choice of bubble functions. The effect of an improved approximation of the residual free bubbles is studied for the advective–diffusive equation. Copyright © 2002 John Wiley & Sons, Ltd.     

Navier–Stokes computation of transonic vortices over a round leading edge delta wing

A 3D Navier–Stokes solver has been developed to simulate laminar compressible flow over quadrilateral wings. The finite volume technique is employed for spatial discretization with a novel variant for the viscous fluxes. An explicit three-stage Runge–Kutta scheme is used for time integration, taking local time steps according to the linear stability condition derived for application to the Navier–Stokes equations. The code is applied to compute primary and secondary separation vortices at transonic speeds over a 65° swept delta wing with round leading edges and cropped tips. The results are compared with experimental data and Euler solutions, and Reynolds number effects are investigated.     

A Chebyshev collocation method for the Navier–Stokes equations with application to double-diffusive convection

A Chebyshev collocation method for solving the unsteady two-dimensional Navier–Stokes equations in vorticity–streamfunction variables is presented and discussed. The discretization in time is obtained through a class of semi-implicit finite difference schemes. Thus at each time cycle the problem reduces to a Stokes-type problem which is solved by means of the influence matrix technique leading to the solution of Helmholtz-type equations with Dirichlet boundary conditions. Theoretical results on the stability of the method are given. Then a matrix diagonalization procedure for solving the algebraic system resulting from the Chebyshev collocation approximation of the Helmholtz equation is developed and its accuracy is tested. Numerical results are given for the Stokes and the Navier–Stokes equations. Finally the method is applied to a double-diffusive convection problem concerning the stability of a fluid stratified by salinity and heated from below.     

A new matrix perturbation method for analytical solution of the complex modal eigenvalue problem of viscously damped linear vibration systems

A new matrix perturbation analysis method is presented for efficient approximatesolution of the complex modal quadratic generalized eigenvalue problem of viscouslydamped linear vibration systems.First,the damping matrix is decomposed into the sum of aproportional-and a nonproportional-damping parts,and the solutions of the real modaleigenproblem with the proportional dampings are determined,which are a set of initialapproximate solutions of the complex modal eigenproblem.Second,by taking thenonproportional-damping part as a small modification to the proportional one and using thematrix perturbation analysis method,a set of approximate solutions of the complex modaleigenvalue problem can be obtained analytically.The result is quite simple.The new methodis applicable to the systems with viscous dampings-which do not deviate far away from theproportional-damping case.It is particularly important that the solution technique be alsoeffective to the systems with heavy,but not over,dampings.The solution formul     

Primary resonance of multiple degree-of-freedom dynamic systems with strong non-linearity using the homotopy analysis method

A homotopy analysis method（HAM）is presented for the primary resonance of multiple degree-of-freedom systems with strong non-linearity excited by harmonic forces.The validity of the HAM is independent of the existence of small parameters in the considered equation.The HAM provides a simple way to adjust and control the convergence region of the series solution by means of an auxiliary parameter.Two examples are presented to show that the HAM solutions agree well with the results of the modified Linstedt-Poincar＇e method and the incremental harmonic balance method.     

Validation of the Rayleigh–Ritz method for the postbuckling analysis of rectangular plates with application to delamination growth

Postbuckling solutions of laminated rectangular plates are obtained by the Rayleigh–Ritz method using von Karman’s nonlinear strain displacement relations and high-order polynomial expansions of the displacements. The potential energy function and the nonlinear algebraic equations governing the undetermined coefficients are obtained by Mathematica. Reasonably accurate solutions for the membrane forces, the bending and twisting moments and the pointwise energy-release rates generally require more undeterminated coefficients. Such refined postbuckling solutions show significant non-uniformity of the in-plane forces and strains and certain boundary effects characterized by concentration of the curvatures and the bending moment.