Stochastic non-linear response of a flexible rotor with local non-linearities

The effects of uncertainties on the non-linear dynamics response remain misunderstood and most of the classical stochastic methods used in the linear case fail to deal with a non-linear problem. So we propose to take into account of uncertainties into non-linear models, by coupling the Harmonic Balance Method (HBM) and the Polynomial Chaos Expansion (PCE). The proposed method called the Stochastic Harmonic Balance Method (Stochastic-HBM) is based on a new formulation of the non-linear dynamic problem in which not only the approximated non-linear responses but also the non-linear forces and the excitation pulsation are considered as stochastic parameters. Expansions on the PCE basis are performed by passing via an Alternate Frequency Time method with Probabilistic Collocation (AFTPC) for estimating the stochastic non-linear forces in the stochastic domain and the frequency domain. In the present paper, the Stochastic Harmonic Balance Method (Stochastic-HBM) that is applied to a flexible non-linear rotor system, with random parameters modeled as random fields, is presented. The Stochastic-HBM combined with an Alternate Frequency-Time method with Probabilistic Collocation (AFTPC) allows us to solve dynamical problems with non-regular non-linearities in presence of uncertainties. In this study, the procedure is developed for the estimation of stochastic non-linear responses of the rotor system with different regular and non-regular non-linearities. The finite element rotor system is composed of a shaft with two disks and two flexible bearing supports where the non-linearities are due to a radial clearance or a cubic stiffness. A numerical analysis is performed to analyze the effect of uncertainties on the non-linear behavior of this rotor system by using the Stochastic-HBM. Furthermore, the results are compared with those obtained by applying a classical Monte-Carlo simulation to demonstrate the efficiency of the proposed methodology.     

Study on the stability of drum brake non-linear low frequency vibration model

A 5-DOF non-linear model is presented to simulate the vibration of a drum brake at low frequency in the course of applying the brake. Analysis and calculation are carried out to illuminate that even when the friction coefficient is constant, the vibration and instability can occur with the combination of some specific parameters. And the stable-unstable area on the parameter plane on the condition of the combination of some specific parameters is presented to illuminate the effect of the structure parameter on the system stability.     

On the coupling of non-linear normal modes

The phenomenon of internal resonance is known as the exchange of energy between the modes and the existence of coupled-mode response under a single-mode excitation. This phenomenon is observed whenever a non-linear normal mode loses its stability, called the modal coupling. The details of modal coupling are formulated in the free vibrations of two-degree-of-freedom systems, and compared with internal resonance. The theory is based on the structural change in Poincaré map due to the stability change of normal modes. It is shown that every change in stability of normal modes gives rise to a pitchfork or a period-doubling bifurcation. The functional form is derived to compute the coupled modes by the method of harmonic balance. Examples are given to describe the procedure of stability analysis of non-linear normal modes, to compute the coupled modes, and then to demonstrate that results of internal resonances can be derived by model coupling. Other examples are given to demonstrate that the results of some modal couplings cannot be obtained by internal resonances.     

A stability in the large investigation for a non-linear second order two-degree-of-freedom system with periodic excitation

An extension to an algorithm due to Simpson has been developed for the analysis of a non-linear second order two-degree-of-freedom system with external periodic excitation. The form of equations considered arises from the study of mechanical systems with a single concentrated weak non-linearity and the method assumes a solution made up of harmonic terms whose amplitudes vary slowly in time. The system considered is such that in the absence of external excitation, it possesses a stable equilibrium point and an unstable limit cycle arising from a sub-critical Hopf bifurcation. When forcing is applied, the stable equilibrium point may then be replaced by a stable periodic attractor, and the limit cycle by an unstable multi-periodic attractor. The method has been applied to the problem of locating these attractors, and if they exist, of finding the stable attractor's basin of attraction in terms of initial conditions. The method reduces the problem from a search in four-dimensional phase space to a search for a boundary in a plane defined by amplitudes a₁ and a₂ in the assumed form of the solution.The method was applied to three non-linear systems in which the non-linearity was due to either a linear spring with a small amount of cubic hardening or a linear spring with freeplay. Agreement was shown to be good in those cases where the non-linearity was weak. However, the method would not be expected to give such accurate results if the non-linear effect was more significant. This was illustrated for a case involving the freeplay non-linearity.     

Stochastic stability of non-linear SDOF systems

A semi-analytical procedure for obtaining stability conditions for strongly non-linear single degree of freedom system (SDOF) subjected to random excitations is presented using stochastic averaging technique. The method is useful for finding stability conditions for systems having highly irregular non-linear functions which cannot be integrated in closed form to yield analytical expressions for averaged drift and diffusion coefficients. In spite of numerical methods available for finding stability of SDOF system by determining Lyapunov exponent, the proposed technique may have to be adopted (i) when the excitation is non-white; and (ii) when numerical integration fails due to convergence problem. The method is developed in such a way that it lends itself to a numerical computational scheme using FFT for obtaining numerical values of drift and diffusion coefficients of Its differential equation and the corresponding FPK equation for the system. These values of averaged drift and diffusion coefficients are then fit into polynomial form using curve fitting technique so that polynomials can be used for stability analysis. Two example problems are solved as illustrations. The first one is the Van der Pol oscillator having non-linearities which can be treated purely analytically. The example is considered for the validation of the proposed method. The second one involves non-linearities in the form of signum function for which purely analytical solution is not possible. The results of the study show that the proposed method is useful and efficient for performing stability analysis of dynamic systems having any type of non-linearities.     

Detecting the position of non-linear component in periodic structures from the system responses to dual sinusoidal excitations

Based on the non-linear output frequency response functions (NOFRFs), a novel method is developed to detect the position of non-linear components in periodic structures. The detection procedure requires exciting the non-linear systems twice using two sinusoidal inputs separately. The frequencies of the two inputs are different; one frequency is twice as high as the other one. The validity of this method is demonstrated by numerical studies. Since the position of a non-linear component often corresponds to the location of defect in periodic structures, this new method is of great practical significance in fault diagnosis for mechanical and structural systems.     

Computational and quasi-analytical models for non-linear vibrations of resonant MEMS and NEMS sensors

Large-amplitude non-linear vibrations of micro- and nano-electromechanical resonant sensors around their primary resonance are investigated. A comprehensive multiphysics model based on the Galerkin decomposition method coupled with the averaging method is developed in the case of electrostatically actuated clamped-clamped resonators. The model is purely analytical and includes the main sources of non-linearities as well as fringing field effects. The influence of the higher modes and the validation of the model is demonstrated with respect to the shooting method as well as the harmonic balance coupled with the asymptotic numerical method. This model allows designers to investigate the sensitivity variation of resonant sensors in the non-linear regime with respect to the electrostatic forcing.     

Bifurcation and chaotic response of a cracked rotor system with viscoelastic supports

Cracks appearing in the shaft of a rotary system are one of the main causes of accidents for large rotary machine systems. This research focuses on investigating the bifurcation and chaotic behavior of a rotating system with considerations of various crack depth and rotating speed of the system’s shaft. An equivalent linear-spring model is utilized to describe the cracks on the shaft. The breathing of the cracks due to the rotation of the shaft is represented with a series truncated time-varying cosine series. The geometric nonlinearity of the shaft, the masses of the shaft and a disc mounted on the shaft, and the viscoelasticity of the supports are taken into account in modeling the nonlinear dynamic rotor system. Numerical simulations are performed to study the bifurcation and chaos of the system. Effects of the shaft’s rotational speed, various crack depths and viscosity coefficients on the nonlinear dynamic properties of the system are investigated in detail. The system shows the existence of rich bifurcation and chaos characteristics with various system parameters. The results of this research may provide guidance for rotary machine design, machining on rotary machines, and monitoring or diagnosing of rotor system cracks.     

Consistency,stability and convergence of the finite-difference equations for flow about a rotating sphere in an axial stream

The finite-difference equations which have previously been developed to solve the problem of laminar boundary layer flow about a rotating sphere in an axial stream are analysed according to the available numerical stability theories. This analysis is necessary to determine the restrictions on velocities and mesh sizes required to obtain a convergent numerical solution. Convergence can be achieved if both consistency and stability of the finite-difference equations are fulfilled. The analysis reported in the present paper shows that the developed finite-difference equations are consistent with their original partial differential equations. Also, the analysis proves that the developed finite-difference procedure is numerically stable for all mesh sizes as long as the downstream meridional velocity is non-negative, i.e.as long as no flow reversals occur within the domain of solution.     

Stability and bifurcation of unbalance rotor/labyrinth seal system

The influence of labyrinth seal on the stability of unbalanced rotor system was presented . Under the periodic excitation of rotor unbalance , the whirling vibration of rotor is synchronous if the rotation speed is below stability threshold, whereas the vibration becomes severe and asynchronous which is defined as unstable if the rotation speed exceeds threshold . The. Muszynska model of seal force and shooting method were used to investigate synchronous solution of the dynamic equation of rotor system. Then , based on Floquet theory the stability of synchronous solution and unstable dynamic characteristic of system were analyzed.     

Parametric vibrations and stability of an axially accelerating string guided by a non-linear elastic foundation

Parametric vibrations and stability of an axially accelerating string guided by a non-linear elastic foundation are studied analytically. The axial speed, as the source of parametric vibrations, is assumed to involve a mean speed, along with small harmonic variations. The method of multiple scales is applied to the governing non-linear equation of motion and then the natural frequencies and mode shape equations of the system are derived using the equation of order one, and satisfying the compatibility conditions. Using the equation of order epsilon, the solvability conditions are obtained for three distinct cases of axial acceleration frequency. For all cases, the stability areas of system are constructed analytically. Finally, some numerical simulations are presented to highlight the effects of system parameters on vibration, natural frequencies, frequency-response curves, stability, and bifurcation points of the system.     

Non-linear vibrations of sandwich viscoelastic shells

This paper deals with the non-linear vibration of sandwich viscoelastic shell structures. Coupling a harmonic balance method with the Galerkin's procedure, one obtains an amplitude equation depending on two complex coefficients. The latter are determined by solving a classical eigenvalue problem and two linear ones. This permits to get the non-linear frequency and the non-linear loss factor as functions of the displacement amplitude. To validate our approach, these relationships are illustrated in the case of a circular sandwich ring.     

Control of primary and subharmonic resonances of a Duffing oscillator via non-linear energy sink

The use of non-linear energy sink to passively control vibrations of a non-linear main structure under the effect of bi-frequency harmonic excitation is addressed here. The excitation is assumed to induce both 1:1 and 1:3 resonance, and the response of the system is studied after using the Multiple Scale/Harmonic Balance Method, applied to obtain amplitude modulation equations in the slow time scale. The efficiency of the non-linear energy sink to reduce or suppress vibrations of the main structure is finally discussed.     

On the local stability analysis of the approximate harmonic balance solutions

Periodic response of nonlinear oscillators is usually determined by approximate methods. In the "steady state" type methods, first an approximate solution for the steady state periodic response is determined, and then the local stability of this solution is determined by analyzing the equation of motion linearized about this predicted "solution". An exact stability analysis of this linear variational equation can provide erroneous stability type information about the approximate solutions. It is shown that a consistent stability type information about these solutions can be obtained only when the linearized variational equation is analyzed by approximate methods, and the level of accuracy of this analysis is consistent with that of the approximate solutions. It is demonstrated that these consistent stability results do not imply that the approximate solution is qualitatively correct. It is also shown that the difference between an approximate and the next higher order stability analysis can be used to "guess" the role of higher harmonics in the periodic response. This trial and error procedure can be used to ensure the qualitatively correct and numerically accurate nature of the approximate solutions and the corresponding stability analysis.     

Principal parametric resonances of non-linear mechanical system with two-frequency and self-excitations

The principal parametric resonance of a single-degree-of-freedom system with non-linear two-frequency parametric and self-excitations is investigated. In particular, the case in which the parametric excitation terms with close frequencies is examined. The method of multiple scales is used to determine the equations that describe to first-order the modulation of the amplitude and phase. Qualitative analysis and asymptotic expansion techniques are employed to predict the existence of steady state responses. Stability is investigated. The effect of damping, magnitudes of non-linear excitation and self-excitation are analyzed.     

Nonlinear resonances and self-excited oscillations of a rotor caused by radial clearance and collision

This paper investigates oscillations in a flexible rotor system with radial clearance between an outer ring of the bearing and a casing by experiments and numerical simulations. The mathematical model considers the collisions of the bearing with the casing. The following phenomena are found: (1) Nonlinear resonances of subharmonic, super-subharmonic and combination oscillation occur. (2) Self-excited oscillation of a forward whirling mode occurs in a wide range above the major critical speed. (3) Entrainment phenomena from self-excited oscillation to nonlinear forced oscillation occur at these nonlinear resonance ranges. Moreover, this study analyzes periodic solutions of the mathematical model by the Harmonic Balance Method (HBM). As the results, the nonlinear resonances of subharmonic oscillation and its entrainment phenomenon can be explained theoretically by investigating the stability of the periodic solutions. The influence of the static force and the bearing damping on these oscillation are also clarified.     

转子系统的平稳／非平稳随机地震响应分析

应用虚拟激励法结合精细时程积分计算了转子系统受平稳／非平稳随机地震激励的动力响应。采用虚拟激励分析将平稳随机激励转化为稳态简谐激励，将非平稳随机激励转化为瞬态确定性激励，即使对于非对称的油膜刚度阵和阻尼阵，算法仍然简单高效，并得到精确的结果。     

A new method for the non-linear deflection analysis of an infinite beam resting on a non-linear elastic foundation

The aim of this paper is to develop a new method of analyzing the non-linear deflection behavior of an infinite beam on a non-linear elastic foundation. Non-linear beam problems have traditionally been dealt with by semi-analytical approaches that involve small perturbations or by numerical methods, such as the non-linear finite element method. In this paper, in contrast, a transformed non-linear integral equation that governs non-linear beam deflection behavior is formulated to develop a new method for non-linear solutions. The proposed method requires an iteration to solve non-linear problems, but is fairly simple and straightforward to apply. It also converges quickly, whereas traditional non-linear solution procedures are generally quite complex in application. Mathematical analysis of the proposed method is performed. In addition, illustrative examples are presented to demonstrate the validity of the method developed in the present study.