Vibration analysis of harmonically excited non-linear system using the method of multiple scales

An analytical method is presented for evaluation of the steady state periodic behavior of non-linear systems. This method is based on the substructure synthesis formulation and a multiple scales procedure, which is applied to the analysis of non-linear responses. A complex non-linear system is divided into substructures, of which equations are approximately transformed to modal co-ordinates including non-linear term under the reasonable procedure. Then, the equations are synthesized into the overall system and the solution of the non-linear system can be obtained. Based on the method of multiple scales, the proposed procedure reduces the size of large-degree-of-freedom problem in solving the non-linear equations. Feasibility and advantages of the proposed method are illustrated by the application of the analytic procedure to the non-linear rotating machine system as a large mechanical structure system. Results obtained are reported to be an efficient approach with respect to non-linear response prediction when compared with other conventional methods.     

Non-linear wave and Schrödinger equations

We investigate stability of periodic and quasiperiodic solutions of linear wave and Schrödinger equations under non-linear perturbations. We show in the case of the wave equations that such solutions are unstable for generic perturbations. For the Schrödinger equations periodic solutions are stable while the quasiperiodic ones are not. We extend these results to periodic solutions of non-linear equations.Partially supported by NSERC under Grant NA7901     

Vibration of large turbo-rotors in fluid-film bearings on an elastic foundation

In the first sections of this paper the flexible rotating shaft of a turbo-rotor is treated by finite element analysis. Internal and external damping, gyroscopic forces, fluid-film forces, aerodynamic cross-coupling from steam flow and magnetic pull are taken into account. Although some hundred degrees of freedom have to be introduced to describe a realistic turbo-rotor, computational effort can be enormously reduced by making use of the banded structure of the system matrices.In the second part foundation dynamics are introduced into the rotor equations via a receptance formulation. The receptance matrices of the foundation or supporting systems can be obtained either from shaker tests or from the mode analysis of the foundation without shaft. Numerical examples are given.     

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.     

New approach to periodic solutions of integrable equations and nonlinear theory of modulational instability

A new method of finding the periodic solutions for the equations integrable within the framework of the AKNS scheme is reviewed. The approach is a modification of the known finite-band integration method, based on the re-parametrization of the solution with the use of algebraic resolvent of the polynomial defining the solution in the finite-band integration method. This approach permits one to obtain periodic solutions in an effective form necessary for applications. The periodic solutions are found for such systems as the nonlinear Schrödinger equation, the derivative nonlinear Schrödinger equation, the Heisenberg model, the uniaxial ferromagnet, the AB system, and self-induced transparency and stimulated Raman scattering equations. The modulation Whitham theory describing the slow modulation of periodic waves is expressed in a form convenient for applications. The Whitham equations are obtained for all abovementioned cases. The technique developed is applied to the nonlinear theory of modulational instability describing the transformation of a local disturbance expanding into a nonuniform region presented as a modulated periodic wave whose evolution is governed by the Whitham equations. This theory explains the formation of solitons on the sharp front of a long pulse.     

Dynamic analysis of asymmetric bladed-rotors supported by anisotropic stator

For the mathematical convenience of conventional rotor dynamic analysis, the components of the entire rotor system are often classified into two parts: the stationary parts and the rotating parts, depending upon whether or not the corresponding components rotate with respect to their axes of rotation. Even for bladed-rotors, the rotor blades have been treated along the same lines as the ordinary rotating components such as the rotor disk and the shaft. The distinct dynamic nature of the blades, therefore, has not been thoroughly taken into account in the conventional rotor dynamic analysis. In this paper, the rotating parts of a bladed-rotor system are further subdivided into the rotor blade-group and the other rotating components. The equation-of-motion for the bladed-rotor system is then developed, by modifying the conventional general rotor system equation to adopt the blade-group dynamics without loss of generality. Complex modal solutions to the bladed-rotor system are investigated based on a new modulated coordinate transformation approach, yielding newly defined directional frequency response functions that characterize the nature of asymmetry present in the rotating blade-array. Finally, the effects of the blade-group asymmetry on the rotor system dynamics are demonstrated with a pertinent numerical example.     

Abrikosov Lattices in Finite Domains

In 1957 Abrikosov published his work on periodic solutions to the linearized Ginzburg-Landau equations. Abrikosov's analysis assumes periodic boundary conditions, which are very different from the natural boundary conditions the minimizer of the Ginzburg-Landau energy functional should satisfy. In the present work we prove that the global minimizer of the fully non-linear functional can be approximated, in every rectangular subset of the domain, by one of the periodic solution to the linearized Ginzburg-Landau equations in the plane. Furthermore, we prove that the energy of this solution is close to the minimum of the energy over all Abrikosov's solutions in that rectangle.     

Variational Approach to Uniform Rigid Rotation of Ginzburg—Landau Vortices

Time periodic solutions for the hyperbolic gauged Ginzburg–Landau system, with spatial domain the unit disc, are shown to exist. Time periodic solutions representing bound states of vortices rotating about one another have been previously obtained in the near self-dual limit, using perturbative techniques. In contrast, we here take a variational approach, the solutions being obtained as critical points of an indefinite functional. We consider a special class of solutions which map out, uniformly in time, an orbit of the rotation group SO(2). It is shown that in the limit of large coupling constant the solutions have nontrivial time dependence or, as is shown to be equivalent, are not radially symmetric in any gauge.     

PARAMETRIC STABILIZATION OF A GYROSCOPIC SYSTEM

This paper studies the stabilization of a gyroscopic system using parametric stabilization near a combination resonance. The gyroscopic system is near its primary instability, i.e., the bifurcation parameter is such that the system possesses a double zero eigenvalue. The stability of the system is studied for the linear Hamiltonian system, the damped linear system, the forced linear Hamiltonian system, and finally the damped and forced linear system. The addition of the periodic excitation near the critical combination resonance provides the system with an extended stability region when the excitation frequency is slightly above the combination resonance. A non-linear numerical example shows that these results may persist for the non-linear problem. The results of this work, are then discussed in relation to an example gyroscopic problem, a rotating shaft with periodically perturbed rotation rate.     

一类相对转动非线性动力学系统周期解的唯一性与精确周期解

研究了一类具有线性刚度、非线性阻尼力和强迫周期力项的相对转动非线性动力学系统周期解的唯一性和精确周期解.讨论了一类自治系统极限环的唯一性与稳定性.应用定性分析方法,给出了一类相对转动非线性动力学系统具有唯一周期解的必要条件,并在一定条件下得到了系统的一类精确周期解.     

Periodic solution finder for an impact oscillator with a drift

In this paper, an efficient semi-analytical method is developed to compute periodic solutions for a new model of an impact oscillator with a drift, which explains the progression mechanism in vibro-impact systems and can be used to optimize their performance. The method constructs a periodic response assuming that each period is comprised of a sequence of distinct phases for which analytical solutions are known. For example, a period may consist of the following sequential phases: (I) contact with progression, (II) contact without progression, (III) no contact and (IV) contact without progression. Using this information, a system of four piecewise linear first order differential equations is transformed to a system of non-linear algebraic equations. The method allows one to accurately predict a range of control parameters for which the best progression rates are obtained.     

Analytical and numerical solutions of some integral equations in radiation transport theory

Summary We discuss some mathematical and computational problems relevant to the solutions of both linear and non-linear integral equations arising in radiation transport. By means of a functional-operator approach, analytic solutions for the FEL (Free Electron Laser) integral equations are found in terms of Bessel-Clifford functions. As for applications of this method to non-linear equations, the same technique allows us to obtain an efficient yet simple algorithm for the numerical solution of the Ambartsumian-ChandrasekharH-equation corresponding to a Neumann-series expansion. Paper presented at the I International Conference on Scaling Concepts and Complex Fluids, Copanello, Italy, July 4–8, 1994.     

A new approach with piecewise-constant arguments to approximate and numerical solutions of oscillatory problems

This paper is devoted to the development of a novel approximate and numerical method for the solutions of linear and non-linear oscillatory systems, which are common in engineering dynamics. The original physical information included in the governing equations of motion is mostly transferred into the approximate and numerical solutions. Therefore, the approximate and numerical solutions generated by the present method reflect more accurately the characteristics of the motion of the systems. Furthermore, the solutions derived are continuous everywhere with good accuracy and convergence in comparing with Runge-Kutta method. An approximate solution is developed for a linear oscillatory problem and compared with its corresponding exact solution. A non-linear oscillatory problem is also solved numerically and compared with the solutions of Runge-Kutta method. Both the graphical and numerical comparisons are provided in the paper. The accuracy of the approximate and numerical solutions can be controlled as desired by the number of terms in the Taylor series and the value of a single parameter used in the present work. Formulae for numerical computation in solving various linear and non-linear oscillatory problems by the new approach are provided in the paper.     

On the mathematical conditions for the existence of periodic fluctuations in non-uniform media

In many areas of mathematical physics where one is interested in the propagation of waves through non-uniform media, it is often assumed that periodic excitations result in periodic responses. This assumption is examined by rigorously investigating the existence of periodic solutions of linear hyperbolic differential equations whose coefficients vary with position and whose solution must satisfy periodic boundary or source data. It is shown that the nature of the coefficients of the undifferentiated terms of the differential system is crucial in determining whether or not the solution is periodic. In physical applications, these coefficients usually depend on the gradients of media properties as well as on the media properties themselves. In particular, it is shown that for a general hyperbolic system of two equations in one space dimension, the solution is not periodic. Moreover, this can remain true even if the media gradients are assumed small. However, if the media gradients vanish, or if they vanish except for a bounded region of space, the solution is shown to be periodic for a large enough time. Furthermore, if these gradients vanish asymptotically at large distances, then the disturbances will be asymptotically periodic for increasing time. Special attention is given to the propagation of infinitesimal pressure disturbances through non-uniform steady flows of a lossless fluid.     

A COUPLED FLUID-STRUCTURE ANALYSIS FOR 3-D INVISCID FLUTTER OF IV STANDARD CONFIGURATION

A three-dimensional non-linear time-marching method and numerical analysis for aeroelastic behaviour of an oscillating blade row is presented. The approach is based on the solution of the coupled fluid-structure problem in which the aerodynamic and structural equations are integrated simultaneously in time. In this formulation of a coupled problem, the interblade phase angle at which a stability (or instability) would occur is a part of the solution. The ideal gas flow through multiple interblade passage (with periodicity on the whole annulus) is described by the unsteady Euler equations in the form of conservative laws, which are integrated by use of the explicit monotonic second order accurate Godunov-Kolgan volume scheme and a moving hybrid H-H (or H-O) grid. The structure analysis uses the modal approach and 3-D finite element model of the blade. The blade motion is assumed to be a linear combination of modes shapes with the modal coefficients depending on time. The influence of the natural frequencies on the aerodynamic coefficient and aeroelastic coupled oscillations for the Fourth Standard Configuration is shown. The stability (instability) areas for the modes are obtained. It has been shown that interaction between modes plays an important role in the aeroelastic blade response. This interaction has an essentially non-linear character and leads to blade limit cycle oscillations.     

(2+1)-维色散长波方程的折叠孤立波解

本文利用一种该进的映射法和线性变量分离法,得到（2+1）-维色散长波方程大量的,带有两个任意函数的精确解。并在得到的一个周期波精确解的基础上,通过选择恰当的函数,可以观察到（2+1）-维色散长波方程的折叠孤立波的演化行为。     

Vibration of cracked shafts in bending

Cracked rotating shafts exhibit a certain particular dynamic response due to the local flexibility of the cracked section. In this response, most of the features of the response of a shaft with dissimilar moments of inertia can be identified. Moreover, the non-linear behavior of the closing crack introduces the characteristics of non-linear systems. For many practical applications, the system can be considered bi-linear and analytical methods can be applied. A de Laval rotor with an open crack is investigated by way of application of the theory of shafts with dissimilar moments of inertia. Furthermore, analytical solutions are obtained for the closing crack under the assumption of large static deflections, a situation common in turbomachinery. Finally, a solution is developed for the case in which the local flexibility function is found experimentally.     

An Approximate Approach for Systems of Singular Volterra Integral Equations Based on Taylor Expansion

In this article, an extended Taylor expansion method is proposed to estimate the solution of linear singular Volterra integral equations systems. The method is based on combining the m-th order Taylor polynomial of unknown functions at an arbitrary point and integration method, such that the given system of singular integral equations is converted into a system of linear equations with respect to unknown functions and their derivatives. The required solutions are obtained by solving the resulting linear system. The proposed method gives a very satisfactory solution, which can be performed by any symbolic mathematical packages such as Maple, Mathematica, etc. Our proposed approach provides a significant advantage that the m-th order approximate solutions are equal to exact solutions if the exact solutions are polynomial functions of degree less than or equal to m. We present an error analysis for the proposed method to emphasize its reliability. Six numerical examples are provided to show the accuracy and the efficiency of the suggested scheme for which the exact solutions are known in advance.     

Traveling periodic waves of chemical activity

It is shown that within the manifold of exact solutions a system of reaction-diffusion equations admits only travelling waves with planar symmetry. A derivation of the generic form of approximate (asymptotic) cylindrical and spiral travelling periodic wave solutions is given. If an exact solution homogeneous in space and periodic in time is admitted by the system of reaction-diffusion equations, then travelling periodic spiral waves are admissble as approximate solutions. This is the theoretical explanation for the travelling periodic waves of chemical activity observed in recent experiments.     

An Approximate Approach for Systems of Singular Volterra Integral Equations Based on Taylor Expansion

In this article, an extended Taylor expansion method is proposed to estimate the solution of linear singular Volterra integral equations systems. The method is based on combining the m-th order Taylor polynomial of unknown functions at an arbitrary point and integration method, such that the given system of singular integral equations is converted into a system of linear equations with respect to unknown functions and their derivatives. The required solutions are obtained by solving the resulting linear system. The proposed method gives a very satisfactory solution,which can be performed by any symbolic mathematical packages such as Maple, Mathematica, etc. Our proposed approach provides a significant advantage that the m-th order approximate solutions are equal to exact solutions if the exact solutions are polynomial functions of degree less than or equal to m. We present an error analysis for the proposed method to emphasize its reliability. Six numerical examples are provided to show the accuracy and the efficiency of the suggested scheme for which the exact solutions are known in advance.