Evaporation of a two-phase drop in an immiscible liquid: A parametric study

This paper presents a theoretical analysis of the evaporation of a single two-phase droplet during its rise in a quiescent column of an immiscible liquid. Governing equations are derived and solved by a numerical method. Dimensionless parameters governing the phenomenon are identified and a parametric study is carried out for different non-dimensional parameters. The predicted results are compared with experimental results of previous workers.     

Dynamic stability in parametric resonance of axially accelerating viscoelastic Timoshenko beams

This paper investigates dynamic stability of an axially accelerating viscoelastic beam undergoing parametric resonance. The effects of shear deformation and rotary inertia are taken into account by the Timoshenko thick beam theory. The beam material obeys the Kelvin model in which the material time derivative is used. The axial speed is characterized as a simple harmonic variation about the constant mean speed. The governing partial-differential equations are derived from Newton's second law, Euler's angular momentum principle, and the constitutive relation. The method of multiple scales is applied to the equations to establish the solvability conditions in summation and principal parametric resonances. The sufficient and necessary condition of the stability is derived from the Routh-Hurvitz criterion. Some numerical examples are presented to demonstrate the effects of related parameters on the stability boundaries.     

Dynamic stability of pretwisted columns under periodic axial loads

This paper deals with the parametric instability of pretwisted columns subjected to static and periodic axial loads. The governing equations of the problem are reduced to time domain by applying Galerkin's method. The instability regions are determined with the procedure proposed by Hsu. The analysis includes free vibration and static buckling of the column as special cases. Although the method is quite general, numerical calculations are carried out only for a simply supported column. Free vibration frequencies, static buckling loads and the coefficients of the instability regions are obtained for various values of the pretwisting angle and those of the cross sectional rigidity ratio of the column.     

A 3D finite element model for the vibration analysis of asymmetric rotating machines

This paper suggests a 3D finite element method based on the modal theory in order to analyse linear periodically time-varying systems. Presentation of the method is given through the particular case of asymmetric rotating machines. First, Hill governing equations of asymmetric rotating oscillators with two degrees of freedom are investigated. These differential equations with periodic coefficients are solved with classic Floquet theory leading to parametric quasimodes. These mathematical entities are found to have the same fundamental properties as classic eigenmodes, but contain several harmonics possibly responsible for parametric instabilities. Extension to the vibration analysis (stability, frequency spectrum) of asymmetric rotating machines with multiple degrees of freedom is achieved with a fully 3D finite element model including stator and rotor coupling. Due to Hill expansion, the usual degrees of freedom are duplicated and associated with the relevant harmonic of the Floquet solutions in the frequency domain. Parametric quasimodes as well as steady-state response of the whole system are ingeniously computed with a component-mode synthesis method. Finally, experimental investigations are performed on a test rig composed of an asymmetric rotor running on nonisotropic supports. Numerical and experimental results are compared to highlight the potential of the numerical method.     

Convection-induced stabilization of optical dissipative solitons

In spatially extended convective systems, the reflection symmetry breaking induced by drift effects leads to a striking nonlinear effect that drastically affects the formation and stability of dissipative solitons in optical parametric oscillators. The phenomenon of nonlinear-induced convection dynamics is revealed using a model of the complex quintic Ginzburg-Landau equation with nonlinear gradient terms in it. Mechanisms leading to stabilization of dissipative solitons by convection are singled out. The predictions are in very good agreement with numerical solutions found from the governing equations of the optical parametric oscillators.     

A moment method analysis of the gain spectrum in bi-directionally pumped Raman amplifiers through continuous-spectrum radiation

A semi-analytical method is proposed to solve the equations governing a bi-directionally pumped Raman amplifier through continuous-spectrum radiation. The governing equations are systems of uncountable Nonlinear Ordinary Differential Equation (NODE). By applying the moment method, the uncountable system of NODE is reduced to a system of finite number of NODEs. This system of equations is solved numerically and the results are compared with that of the full numerical method. It was shown that the moment method is a powerful and efficient technique for the analysis of a bi-directionally pumped Raman amplifier through continuous-spectrum radiation.     

Aspect ratio effects on natural convection in a water-saturated porous cavity near its density maximum

In this study, two-dimensional steady-state solutions of buoyancy-driven convection in a water-saturated porous cavity is conducted numerically for a range of different aspect ratios. The left vertical wall is considered into a partially heating location. The Brinkman–Forchheimer extended Darcy model is used to investigate the average heat transfer rate. The governing equations are solved using a finite volume method. The results obtained for various values of parameters are presented graphically in the form of streamlines, isotherms and velocity at mid-plane of the cavity. In addition, numerical results for the average Nusselt number are presented for various parametric conditions.     

Numerical simulation for the flow inside V-shapes using grid generation

In this paper, we determined a numerical solution of the Navier-Stokes equations for the flow of incompressible fluid inside the contraction geometry. The governing equations are written in the vorticity-stream function formulations. The numerical solution is based on a technique of automatic numerical generation of a curvilinear coordinate system by transforming the governing equation into the computational plane. The transformed equations are approximated using central differences and solved simultaneously by the alternating direction implicit method and successive over relaxation iteration method.     

Reconstruction of non-linear time delay models from data by the use of optimal transformations

We propose a new technique for the numerical reconstruction of non-linear delay differential equations from time series by applying the method of optimal transformations, a concept of multiple non-linear regression analysis. By constructing a generalized correlation function, this method allows for testing time series for delay-induced dynamics. Also, the delay times and the governing differential equations are estimated with good numerical accuracy. We present several examples which show that this method is useful also for the analysis of short and noisy time series.     

Nonlinear dynamics of a pipe conveying pulsating fluid with combination,principal parametric and internal resonances

Nonlinear dynamics of a hinged–hinged pipe conveying pulsatile fluid subjected to combination and principal parametric resonance in the presence of internal resonance is investigated. The system has geometric cubic nonlinearity due to stretching effect out of immovable support conditions at both ends. The pipe conveys fluid at a velocity with a harmonically varying component over a constant mean velocity. For appropriate choice of system parameters, the natural frequency of the second mode is approximately three times that of the first mode for a range of mean flow velocity, activating a three-to-one internal resonance. The analysis is carried out using the method of multiple scales by directly attacking the governing nonlinear integro-partial-differential equations and the associated boundary conditions. The set of first-order ordinary differential equations governing the modulation of amplitude and phase is analyzed numerically for combination parametric resonance and principal parametric resonance. Stability, bifurcation and response behavior of the pipe are investigated. The amplitude and frequency detuning of the harmonic velocity perturbation are taken as the control parameters. The system exhibits response in the directly excited and indirectly excited modes due to modal interaction. Dynamic response of the system is presented in the form of phase plane trajectories, Poincare maps and time histories. A wide array of dynamical behavior is observed illustrating the influence of internal resonance.     

Analysis of transient responses in a laminated piezoelectric cylindrical shell

A hybrid numerical method is proposed for analysis of transient responses in a multilayered piezoelectric cylindrical shell.In the present method,the associated equations of the displacement field and the electro-potential field are developed using an analytical-numerical method.The piezoelectric cylindrical shell is discretized into layered annular elements along the wall thickness direction.The governing equations are determined by Hamilton's Principle considering the coupling between the elastic and elec...     

Nonlinear vibration of iced cable under wind excitation using three-degree-of-freedom model

High-voltage transmission line possesses a typical suspended cable structure that produces ice in harsh weather. Moreover, transversely galloping will be excited due to the irregular structure resulting from the alternation of lift force and drag force. In this paper, the nonlinear dynamics and internal resonance of an iced cable under wind excitation are investigated.Considering the excitation caused by pulsed wind and the movement of the support, the nonlinear governing equations of motion of the iced cable are established using a three-degree-of-freedom model based on Hamilton's principle. By the Galerkin method, the partial differential equations are then discretized into ordinary differential equations. The method of multiple scales is then used to obtain the averaged equations of the iced cable, and the principal parametric resonance-1/2 subharmonic resonance and the 2:1 internal resonance are considered. The numerical simulations are performed to investigate the dynamic response of the iced cable. It is found that there exist periodic, multi-periodic, and chaotic motions of the iced cable subjected to wind excitation.     

A moment method for analysis the gain spectrum in fiber Raman amplifier with broadband pumps

The governing equations of optical fiber Raman amplifier with broadband pumps in the steady state are systems of uncountable nonlinear ordinary differential equations (NODE). In this paper, the Moment method is used to reduce the uncountable system of NODE to a system of finite number of NODE. This system of equations is solved numerically. The results are compared with that of the full numerical method. It was shown that the Moment method is a precise and efficient technique for analysis of optical fiber Raman amplifier with broadband pumps.     

Multi-pulse chaotic dynamics in non-planar motion of parametrically excited viscoelastic moving belt

This paper investigates the multi-pulse global bifurcations and chaotic dynamics for the nonlinear, non-planar oscillations of the parametrically excited viscoelastic moving belt using an extended Melnikov method in the resonant case. Using the Kelvin-type viscoelastic constitutive law and Hamilton's principle, the equations of motion are derived for the viscoelastic moving belt with the external damping and parametric excitation. Applying the method of multiple scales and Galerkin's approach to the partial differential governing equation, the four-dimensional averaged equation is obtained for the case of 1:1 internal resonance and primary parametric resonance. From the averaged equations obtained, the theory of normal form is used to derive the explicit expressions of normal form with a double zero and a pair of pure imaginary eigenvalues. Based on the explicit expressions of normal form, the extended Melnikov method is used for the first time to investigate the Shilnikov-type multi-pulse homoclinic bifurcations and chaotic dynamics. The paper demonstrates how to employ the extended Melnikov method to analyze the Shilnikov-type multi-pulse homoclinic bifurcations and chaotic dynamics of high-dimensional nonlinear systems in engineering applications. Numerical simulations show that for the nonlinear non-planar oscillations of the viscoelastic moving belt, the Shilnikov-type multi-pulse chaotic motions can occur. Overall, both theoretical and numerical studies suggest that the chaos for the Smale horseshoe sense in motion exists.     

基于等几何分析方法求解任意截面波导本征问题

基于等几何分析方法具有自由度花费少、高精度、高阶连续性等特点,通过加权余量法对椭圆波导本征问题的亥姆霍兹方程等几何离散得出等几何分析方程.解决了传统数值方法的求解域与几何模型的非一致性问题,实现了将问题的分析计算构架于精确几何模型基础之上.分析任意截面波导的本征问题,对不同偏心率的椭圆波导以及三角形和五边形波导的截止波数的求解结果显示等几何分析方法求解波导本征问题的高效及高精度特性.与传统方法相比,此方法以较少的自由度消耗便会达到较高的求解精度,并且数值解的收敛率较快.     

Interaction of fundamental parametric resonances with subharmonic resonances of order one-half

The interaction of fundamental parametric resonances with subharmonic resonances of order one-half in a single-degree-of-freedom system with quadratic and cubic nonlinearities is investigated. The method of multiple scales is used to derive two first-order ordinary differential equations that describe the modulation of the amplitude and the phase of the response with the non-linearity and both resonances. These equations are used to determine the steady state solutions and their stability. Conditions are derived for the quenching or enhancement of a parametric resonance by the addition of a subharmonic resonance of order one-half. The degree of quenching or enhancement depends on the relative amplitudes and phases of the excitations. The analytical results are verified by numerically integrating the original governing differential equation.     

湿蒸汽两相流动问题的耦合求解

控制湿蒸汽两相流动(汽相和液相)的方程个数多,其数值模拟通常采用将两相分开、两相间用参数传递的方法来处理。本文首次尝试将湿蒸汽两相流动汽相和液相的8个控制方程统一求解,推导了这一方程组的特征值及对应的特征向量,结合控制体积方法,采用TVD格式和时间推进法数值求解这一耦合方程组,研发了相应计算软件。对二维喷管中的自发凝结和非均质凝结算例进行的数值模拟和分析,表明了本文方法和计算软件的有效性和可靠性。     

Brownian motion and thermophoresis effects on unsteady stagnation point flow of Eyring-Powell nanofluid: a Galerkin approach

This article concerns the analysis of an unsteady stagnation point flow of Eyring-Powell nanofluid over a stretching sheet. The influence of thermophoresis and Brownian motion is also considered in transport equations. The nonlinear ODE set is obtained from the governing nonlinear equations via suitable transformations. The numerical experiments are performed using the Galerkin scheme. A tabular form comparison analysis of outcomes attained via the Galerkin approach and numerical scheme (RK-4) is available to show the credibility of the Galerkin method. The numerical exploration is carried out for various governing parameters, namely, Brownian motion, steadiness, thermophoresis, stretching ratio, velocity slip, concentration slip, thermal slip, and fluid parameters, and Hartmann, Prandtl and Schmidt numbers. The velocity of fluid enhances with an increase in fluid and magnetic parameters for the case of opposing, but the behavior is reversed for assisting cases. The Brownian motion and thermophoresis parameters cause an increase in temperature for both cases (assisting and opposing). The Brownian motion parameter provides a drop-in concentration while an increase is noticed for the thermophoresis parameter. All the outcomes and the behavior of emerging parameters are illustrated graphically. The comparison analysis and graphical plots endorse the appropriateness of the Galerkin method. It is concluded that said method could be extended to other problems of a complex nature.     

Identification of nonlinear multi-degree-of freedom structures based on Hilbert transformation

The modeling method and identified method adapted to multi-degree-of-freedom structures with strucrural nonlinearities are established. The component mode synthesis method is used to establish the nonlinear governing equations by extending the connected relationships. Based on the modeling method, the Hilbert transform method is applied to identify the nonlinear stiffness of multi-degree-of-freedom structures. Nonlinear analysis and identification of a typical folding wing configuration with three freeplay hinges are investigated. The nonlinear governing equation is established based on present methods and the computing results of different stiffness are checked by finite element programming. In order to illustrate the influence of the nonlinearities, the frequency response characteristics of the structure are analyzed and Hilbert transform is performed. The Hilbert transform identification method is utilized to identify the nonlinear stiffness of nonlinear hinges in the time domain and several parametric studies are performed. In addition, the comparison of response is made to illustrate the feasibility of the methods. The results show that the extending component mode synthesis method in the present work can be used to establish the governing equation with structural nonlinearities. Based on the modeling method, the Hilbert transform identified method can be extended to multi-degree-of-freedom structures accurately.     

Parametric excitation of two internally resonant oscillators

The response of two-degree-of-freedom systems with quadratic non-linearities to a principal parametric resonance in the presence of two-to-one internal resonances is investigated. The method of multiple scales is used to construct a first-order uniform expansion yielding four first-order non-linear ordinary differential (averaged) equations governing the modulation of the amplitudes and the phases of the two modes. These equations are used to determine steady state responses and their stability. When the higher mode is excited by a principal parametric resonance, the non-trivial steady state value of its amplitude is a constant that is independent of the excitation amplitude, whereas the amplitude of the lower mode, which is indirectly excited through the internal resonance, increases with the amplitude of the excitation. However, in addition to Poincaré-type bifurcations, this response exhibits a Hopf bifurcation leading to amplitude- and phase-modulated motions. When the lower mode is excited by a principal parametric resonance, the averaged equations exhibit both Poincaré and Hopf bifurcations. In some intervals of the parameters, the periodic solutions of the averaged equations, in the latter case, experience period-doubling bifurcations, leading to chaos.