Study of nonlinear system stability using eigenvalue analysis: Gyroscopic motion

General computational multibody system (MBS) algorithms allow for the linearization of the highly nonlinear equations of motion at different points in time in order to obtain the eigenvalue solution. This eigenvalue solution of the linearized equations is often used to shed light on the system stability at different configurations that correspond to different time points. Different MBS algorithms, however, employ different sets of orientation coordinates, such as Euler angles and Euler parameters, which lead to different forms of the dynamic equations of motion. As a consequence, the forms of the linearized equations and the eigenvalue solution obtained strongly depend on the set of orientation coordinates used. This paper addresses this fundamental issue by examining the effect of the use of different orientation parameters on the linearized equations of a gyroscope. The nonlinear equations of motion of the gyroscope are formulated using two different sets of orientation parameters: Euler angles and Euler parameters. In order to obtain a set of linearized equations that can be used to define the eigenvalue solution, the algebraic equations that describe the MBS constraints are systematically eliminated leading to a nonlinear form of the equations of motion expressed in terms of the system degrees of freedom. Because in MBS applications the generalized forces can be highly nonlinear and can depend on the velocities, a state space formulation is used to solve the eigenvalue problem. It is shown in this paper that the independent state equations formulated using Euler angles and Euler parameters lead to different eigenvalue solutions. This solution is also different from the solution obtained using a form of the Newton-Euler matrix equation expressed in terms of the angular accelerations and angular velocities. A time-domain solution of the linearized equations is also presented in order to compare between the solutions obtained using two different sets of orientation parameters and also to shed light on the important issue of using the eigenvalue analysis in the study of MBS stability. The validity of using the eigenvalue analysis based on the linearization of the nonlinear equations of motion in the study of the stability of railroad vehicle systems, which have known critical speeds, is examined. It is shown that such an eigenvalue analysis can lead to wrong conclusions regarding the stability of nonlinear systems.     

Fractional dynamics of coupled oscillators with long-range interaction

We consider a one-dimensional chain of coupled linear and nonlinear oscillators with long-range powerwise interaction. The corresponding term in dynamical equations is proportional to 1//n-m/alpha+1. It is shown that the equation of motion in the infrared limit can be transformed into the medium equation with the Riesz fractional derivative of order alpha, when 0相似文献   

Stochastic stability of a fractional viscoelastic column under bounded noise excitation

The stability of a viscoelastic column under the excitation of stochastic axial compressive load is investigated in this paper. The material of the column is modeled using a fractional Kelvin–Voigt constitutive relation, which leads to that the equation of motion is governed by a stochastic fractional equation with parametric excitation. The excitation is modeled as a bounded noise, which is a realistic model of stochastic fluctuation in engineering applications. The method of stochastic averaging is used to approximate the responses of the original dynamical system by a new set of averaged variables which are diffusive Markov vector. An eigenvalue problem is formulated from the averaged equations, from which the moment Lyapunov exponent is determined for the column system with small damping and weak excitation. The effects of various parameters on the stochastic stability and significant parametric resonance are discussed and confirmed by simulation results.     

Bethe-Salpeter Equation with Self-Energy Graphs I-Generalized Fredholm Theory,Parameter Imbedding Method

In this paper the classical Fredholm theory is generalized. The conceptions of the generalized fredholm denominator (GFD) and generalized Fredholm numerator (GFN) are defined. A set of parameter imbedding equations for GFD and GFN is deduced. In this way, the eigenvalue problem of the BS equation in ladder approximation with self-energy graphs, and the eigenvalue problem of nonlinear parameter integral equation, are carried over into an initial-value problem of a set of ordinary differential equations.     

基于Coulomb摩擦效应的一类非线性相对转动系统的混沌研究

研究一类非线性相对转动系统在负载Coulomb摩擦效应下的混沌运动行为. 根据Lagrange方程建立一类含非线性负载Coulomb摩擦阻尼的两个质量相对转动系统的动力学方程. 利用Cardano公式讨论自治系统的特征值, 在此基础上, 应用待定系数法给出系统同宿轨道的存在性, 并借助Silnikov定理研究了系统的混沌行为. 最后数值模拟了给定参数下系统的混沌运动, 并给出在Coulomb摩擦阻尼变化下系统由周期、倍周期通向混沌的途径, 验证了理论分析的正确性.     

Nonlinear nonuniform torsional vibrations of bars by the boundary element method

In this paper a boundary element method is developed for the nonuniform torsional vibration problem of bars of arbitrary doubly symmetric constant cross-section taking into account the effect of geometrical nonlinearity. The bar is subjected to arbitrarily distributed or concentrated conservative dynamic twisting and warping moments along its length, while its edges are supported by the most general torsional boundary conditions. The transverse displacement components are expressed so as to be valid for large twisting rotations (finite displacement-small strain theory), thus the arising governing differential equations and boundary conditions are in general nonlinear. The resulting coupling effect between twisting and axial displacement components is considered and torsional vibration analysis is performed in both the torsional pre- or post-buckled state. A distributed mass model system is employed, taking into account the warping, rotatory and axial inertia, leading to the formulation of a coupled nonlinear initial boundary value problem with respect to the variable along the bar angle of twist and to an “average” axial displacement of the cross-section of the bar. The numerical solution of the aforementioned initial boundary value problem is performed using the analog equation method, a BEM based method, leading to a system of nonlinear differential-algebraic equations (DAE), which is solved using an efficient time discretization scheme. Additionally, for the free vibrations case, a nonlinear generalized eigenvalue problem is formulated with respect to the fundamental mode shape at the points of reversal of motion after ignoring the axial inertia to verify the accuracy of the proposed method. The problem is solved using the direct iteration technique (DIT), with a geometrically linear fundamental mode shape as a starting vector. The validity of negligible axial inertia assumption is examined for the problem at hand.     

On the definition of the natural frequency of oscillations in nonlinear large rotation problems

Computational multibody system algorithms allow for performing eigenvalue analysis at different time points during the simulation to study the system stability. The nonlinear equations of motion are linearized at these time points, and the resulting linear equations are used to determine the eigenvalues and eigenvectors of the system. In the case of linear systems, the system eigenvalues remain the same under a constant coordinate transformation; and zero eigenvalues are always associated with rigid body modes, while nonzero eigenvalues are associated with non-rigid body motion. These results, however, cannot in general be applied to nonlinear multibody systems as demonstrated in this paper. Different sets of large rotation parameters lead to different forms of the nonlinear and linearized equations of motion, making it necessary to have a correct interpretation of the obtained eigenvalue solution. As shown in this investigation, the frequencies associated with different sets of orientation parameters can differ significantly, and rigid body motion can be associated with non-zero oscillation frequencies, depending on the coordinates used. In order to demonstrate this fact, the multibody system motion equations associated with the system degrees of freedom are presented and linearized. The resulting linear equations are used to define an eigevalue problem using the state space representation in order to account for general damping that characterizes multibody system applications. In order to demonstrate the significant differences between the eigenvalue solutions associated with two different sets of orientation parameters, a simple rotating disk example is considered in this study. The equations of motion of this simple example are formulated using Euler angles, Euler parameters and Rodriguez parameters. The results presented in this study demonstrate that the frequencies obtained using computational multibody system algorithms should not in general be interpreted as the system natural frequencies, but as the frequencies of the oscillations of the coordinates used to describe the motion of the system.     

ON THE DISCRETIZATION OF AN ELASTIC ROD WITH DISTRIBUTED SLIDING FRICTION

A one-dimensional elastic system with distributed contact under fixed boundary conditions is investigated in order to study dynamic behavior under sliding friction. A partial differential equation of motion is established and its exact solution is presented. Due to the friction the eigenvalue problem is non-self-adjoint. Mathematical methods for handling the non-self-adjoint system, such as the non-self-adjoint eigenvalue problem and the eigenvalue problem with a proper inner product, are reviewed and applied. The exact solution showed that the undamped elastic system under fixed boundary conditions is neutrally stable when the coefficient of friction is a constant. The assumed mode approximation and the lumped-parameter discretization method are evaluated and their solutions are compared with the exact solution. As a cautionary example the assumed modes approximation leads to false conclusions about stability. The lumped-parameter discretization algorithm generates reliable results.     

Nonlinear dynamics of bi-layered graphene sheet,double-walled carbon nanotube and nanotube bundle

Due to strong van der Waals (vdW) interactions, the graphene sheets and nanotubes stick to each other and form clusters of these corresponding nanostructures, viz. bi-layered graphene sheet (BLGS), double-walled carbon nanotube (DWCNT) and nanotube bundle (NB) or ropes. This research work is concerned with the study of nonlinear dynamics of BLGS, DWCNT and NB due to nonlinear interlayer vdW forces using multiscale atomistic finite element method. The energy between two adjacent carbon atoms is represented by the multibody interatomic Tersoff–Brenner potential, whereas the nonlinear interlayer vdW forces are represented by Lennard-Jones 6–12 potential function. The equivalent nonlinear material model of carbon–carbon bond is used to model it based on its force–deflection relation. Newmark’s algorithm is used to solve the nonlinear matrix equation governing the motion of the BLGS, DWCNT and NB. An impulse and harmonic excitations are used to excite these nanostructures under cantilevered, bridged and clamped boundary conditions. The frequency responses of these nanostructures are computed, and the dominant resonant frequencies are identified. Along with the forced vibration of these structures, the eigenvalue extraction problem of armchair and zigzag NB is also considered. The natural frequencies and corresponding mode shapes are extracted for the different length and boundary conditions of the nanotube bundle.     

Vibration analysis of drillstrings with self-excited stick-slip oscillations

Drillstring vibration is one of the major causes for a deteriorated drilling performance. Field experience revealed that it is crucial to understand the complex vibrational mechanisms experienced by a drilling system in order to better control its functional operation and improve its performance. Sick-slip oscillations due to contact between the drilling bit and formation is known to excite severe torsional and axial vibrations in the drillstring. A dynamic model of the drillstring including the drillpipes and drillcollars is formulated. The equation of motion of the rotating drillstring is derived using Lagrangian approach in conjunction with the finite element method. The model accounts for the torsional-bending inertia coupling and the axial-bending geometric nonlinear coupling. In addition, the model accounts for the gyroscopic effect, the effect of the gravitational force field, and the stick-slip interaction forces. Explicit expressions of the finite element coefficient matrices are derived using a consistent mass formulation. The generalized eigenvalue problem is solved to determine modal transformations, which are invoked to obtain the reduced-order modal form of the dynamic equations. The developed model is integrated into a computational scheme to calculate time-response of the drillstring system in the presence of stick-slip excitations.     

Exact dynamic stiffness matrix of a three-dimensional shear beam with doubly asymmetric cross-section

The dynamic member stiffness matrix of a three-dimensional shear beam with doubly asymmetric cross-section is derived exactly from the governing, sixth-order differential equation of motion. Such a formulation accounts for the uniform distribution of mass in the member and necessitates the solution of a transcendental eigenvalue problem. This is achieved using the Wittrick–Williams algorithm, where the necessary parameters are developed using a generalised procedure. An example is given to clarify the theory, together with a small parametric study that indicates when lateral–torsional coupling may safely be ignored. The work also holds considerable potential in its application to the approximate analysis of asymmetric, multi-storey, three-dimensional frame structures.     

The Motion of Repulsive Brownian Particles in Quenched Disorder

Brownian motion of the particles with repulsive interaction is investigated. When the potential condition is satisfied, the eigenvalue problem of interaction Fokker-Planck equation under certain conditions can be transformed to that of a many-particle Schrödinger equation. Using the Green's function method, we obtain the effective single-variable Fokker-Planck equation in the low density limit. We find that the diffusion of coupled Brownian particles in quenched disorder media is also anomalous in 2D. The Mittag-Leffler relaxation of pancake vortices is investigated by fractional Fokker-Planck equation.     

A new formulation for the existence and calculation of nonlinear normal modes

A new formulation is presented here for the existence and calculation of nonlinear normal modes in undamped nonlinear autonomous mechanical systems. As in the linear case an expression is developed for the mode in terms of the amplitude, mode shape and frequency, with the distinctive feature that the last two quantities are amplitude and total phase dependent. The dynamic of the periodic response is defined by a one-dimensional nonlinear differential equation governing the total phase motion. The period of the oscillations, depending only on the amplitude, is easily deduced. It is established that the frequency and the mode shape provide the solution to a 2π-periodic nonlinear eigenvalue problem, from which a numerical Galerkin procedure is developed for approximating the nonlinear modes. The procedure is applied to various mechanical systems with two degrees of freedom.     

1D Solitons in Saturable Nonlinear Media with Space Fractional Derivatives

Two decades ago, standard quantum mechanics entered into a new territory called space-fractional quantum mechanics, in which wave dynamics and effects are described by the fractional Schrödinger equation. Such territory is now a key and hot topic in diverse branches of physics, particularly in optics driven by the recent theoretical proposal for emulating the fractional Schrödinger equation. However, the light-wave propagation in saturable nonlinear media with space fractional derivatives is yet to be clearly disclosed. Here, such nonlinear optics phenomenon is theoretically investigated based on the nonlinear fractional Schrödinger equation with nonlinear lattices—periodic distributions of either focusing cubic (Kerr) or quintic saturable nonlinearities—and the existence and evolution of localized wave structures allowed by the model are addressed. The model upholds two kinds of one-dimensional soliton families, including fundamental solitons (single peak) and higher-order solitonic structures consisting of two-hump solitons (in-phase) and dipole ones (anti-phase). Notably, the dipole solitons can be robust stable physical objects localized merely within a single well of the nonlinear lattices—previously thought impossible. Linear-stability analysis and direct simulations are executed for both soliton families, and their stability regions are acquired. The predicted solutions can be readily observed in optical experiments and beyond.     

Estimating the nonlinear resonant frequency of a single pile in nonlinear soil

Analytical expressions are determined for the nonlinear resonant frequency (or natural frequency) of the fundamental lateral mode of a pile. A pile with a floating toe, with and without pile cap is considered in this paper. The influence of a nonlinear soil spring model that varies with depth and a nonlinear damping model that is strain amplitude dependent is considered. A non-dimensional equation of motion for the system dynamics is derived from an energy based formulation. This equation is a Duffing's type nonlinear differential system that has nonlinear damping. Harmonic balance with numerical continuation is employed to determine the nonlinear resonance curves of the system. Comparison with some experimental results is made.     

Numerical simulation for two-dimensional Riesz space fractional diffusion equations with a nonlinear reaction term

Fractional differential equations have attracted considerable interest because of their ability to model anomalous transport phenomena. Space fractional diffusion equations with a nonlinear reaction term have been presented and used to model many problems of practical interest. In this paper, a two-dimensional Riesz space fractional diffusion equation with a nonlinear reaction term (2D-RSFDE-NRT) is considered. A novel alternating direction implicit method for the 2D-RSFDE-NRT with homogeneous Dirichlet boundary conditions is proposed. The stability and convergence of the alternating direction implicit method are discussed. These numerical techniques are used for simulating a two-dimensional Riesz space fractional Fitzhugh-Nagumo model. Finally, a numerical example of a two-dimensional Riesz space fractional diffusion equation with an exact solution is given. The numerical results demonstrate the effectiveness of the methods. These methods and techniques can be extended in a straightforward method to three spatial dimensions, which will be the topic of our future research.     

Solitons and periodic solutions to a couple of fractional nonlinear evolution equations

This paper studies a couple of fractional nonlinear evolution equations using first integral method. These evolution equations are foam drainage equation and Klein–Gordon equation (KGE), the latter of which is considered in (2 + 1) dimensions. For the fractional evolution, the Jumarie’s modified Riemann–Liouville derivative is considered. Exact solutions to these equations are obtained.     

Propagation of thickness-twist waves in elastic plates with periodically varying thickness and phononic crystals

We study the propagation of thickness-twist (TT) waves in a crystal plate of AT-cut quartz with periodically varying, piecewise constant thickness. The scalar differential equation by Tiersten and Smythe is employed. The problem is found to be mathematically equivalent to the motion of an electron in a periodic potential field governed by Schrodinger’s equation. An analytical solution is obtained. Numerical results show that the eigenvalue (frequency) spectrum of the waves has a band structure with allowed and forbidden bands. Therefore, for TT waves, plates with periodically varying thickness can be considered as phononic crystals. The effects of various parameters on the frequency spectrum are examined.     

Non-steady-state electrical conduction of polymers in the model with fractional integro-differentiation

A simple dynamic model with a fractional time derivative is considered for conducting polymers. The elementary theory of electrical conduction is developed using the fractional equation of motion. In the framework of the model under consideration, the relaxation of the velocity of charge carriers is described by the Mittag-Leffler function. A kinetic equation with the fractional derivative is derived. The electrical conductivity is calculated to a first approximation with the use of the derived kinetic equation. It is demonstrated that the spectral characteristic of non-steady-state current fluctuations in a polymer should be proportional to 1/f.     

Modified perturbation method for eigenvalues of structure with interval parameters

In overcoming the drawbacks of traditional interval perturbation method due to the unpredictable effect of ignoring higher order terms,a modified parameter perturbation method is presented to predict the eigenvalue intervals of the uncertain structures with interval parameters.In the proposed method,interval variables are used to quantitatively describe all the uncertain parameters.Different order perturbations in both eigenvalues and eigenvectors are fully considered.By retaining higher order terms,the original dynamic eigenvalue equations are transformed into interval linear equations based on the orthogonality and regularization conditions of eigenvectors.The eigenvalue ranges and corresponding eigenvectors can be approximately predicted by the parameter combinatorial approach.Compared with the Monte Carlo method,two numerical examples are given to demonstrate the accuracy and efficiency of the proposed algorithm to solve both the real eigenvalue problem and complex eigenvalue problem.