Non-linear dynamics of a drill-string with uncertain model of the bit–rock interaction

The drill-string dynamics is difficult to predict due to the non-linearities and uncertainties involved in the problem. In this paper a stochastic computational model is proposed to model uncertainties in the bit–rock interaction model. To do so, a new strategy that uses the non-parametric probabilistic approach is developed to take into account model uncertainties in the bit–rock non-linear interaction model. The mean model considers the main forces applied to the column such as the bit–rock interaction, the fluid–structure interaction and the impact forces. The non-linear Timoshenko beam theory is used and the non-linear dynamical equations are discretized by means of the finite element method.     

Phase Shift Adjustment for Harmonic Balance Method Applied to Vibro-impact Systems

This work addresses the phase shift adjustment between the external forcing and the responses for strongly non-linear dynamic systems calculated by Harmonic Balance Method (HBM). The HBM offers fast and robust solutions for strongly non-linear systems operating in periodic regimes, however, the phase information when applying the harmonic balance method is lost. In this paper, a practical scheme for calculating the phase difference for a piecewise oscillator mimicking a vibro-impact system is proposed.     

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.     

Non-linear dynamic of rotor–stator system with non-linear bearing clearanceDynamique non-linéaire d'un ensemble rotor–stator comportant un roulement non-linéaire avec jeu

The study deals with a rotor–stator contact inducing vibration in rotating machinery. A numerical rotor–stator system, including a non-linear bearing with Hertz contact and clearance is considered. To determine the non-linear responses of this system, non-linear dynamic equations can be integrated numerically. However, this procedure is both time consuming and costly to perform. The aim of this Note is to apply the Alternate Frequency/Time Method and the ‘path following continuation’ in order to obtain the non-linear responses to this problem. Then the orbits of rotor and stator responses at various speeds are investigated. To cite this article: J.-J. Sinou, F. Thouverez, C. R. Mecanique 332 (2004).     

Assessment of Uncertainties in Modeling of Laminar to Turbulent Transition for Transonic Flows

The effect of physical variability and uncertainty in model correlations on laminar-turbulent transition in transonic flows is computed using two different Stochastic Collocation methods. Physical variability in the boundary conditions is first investigated for a flow over a flat plate with and without pressure gradient to quantify the uncertainties on the skin friction distribution along the plate surface. Since the laboratory conditions for the flat plate test cases are well defined and the applied transition model has been tuned for these cases, good agreement with experiments is achieved and the variability in the output is low. The second investigated cases exhibit boundary layer transition on the surface of a highly loaded turbine guide vane under transonic flow conditions. Comparisons between the predicted and measured wall heat transfer are used to quantify uncertainties in the free stream turbulence and the model correlations that accounts for compressibility effects on the onset and extension of the bypass transition. The computational results show that the uncertainties have a significant impact on the transition location for the turbine guide vane simulations and, consequently, on the reliability of the predictions for compressible flows. The output uncertainty accounts to a large extent for the difference between the deterministic simulation and the experiments. The results from the Simplex Stochastic Collocation method are computationally more efficient than those of the Stochastic Collocation based on Clenshaw–Curtis quadrature.     

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.     

A non-linear study of a cracked rotor

The influence of the presence of transverse cracks in a rotating shaft is analyzed. The paper addresses the influence of crack opening and closing on dynamic response during operation. The evolution of the orbit of the cracked rotor near half and one-third of the first critical speed is investigated. The dynamic response of the rotor with a breathing crack is evaluated by expanding the changing stiffness of the crack as a truncated Fourier series and then using the Harmonic Balance Method. This method is applied to compute various parametric studies including the effects of the crack depth and location on the dynamic of a crack rotor. The evolution of the first critical speed, associated amplitudes at the critical speed and half of the critical speed, and the resulting orbits during transient operation are presented and some distinguishing features of a cracked rotor are examined.     

Non-linear dynamics of a flexible single link Cartesian manipulator

The present work deals with the non-linear vibration of a harmonically excited single link roller-supported flexible Cartesian manipulator with a payload. The governing equation of motion of this system is developed using extended Hamilton's principle, which is reduced to the second-order temporal differential equation of motion, by using generalized Galerkin's method. This equation of motion contains both cubic non-linearities of geometric and inertial type in addition to linear forced and non-linear parametric excitation terms. Method of multiple scales is used to solve this non-linear equation and study the stability and bifurcations of the system. Influence of amplitude of the base excitation and mass ratio on the steady state response of the system is investigated for both simple and subharmonic resonance conditions. Critical bifurcation points are determined from the fixed-point responses and periodic, quasi-periodic responses are also found for different system parameters. The results obtained using the perturbation analysis are compared with the previously published experimental work and are found to be in good agreement. This work will be useful for the designer of a flexible manipulator.     

精细积分时域平均法和随机扩阶系统法

讨论含随机参数结构的动力响应的计算问题,发展了精细积分时域平均法（ＴＡＰＩＭ）,它可以用来计算确定性系统受到随机激励时的动力响应;结合随机扩阶系统方法与随机有限元法,将ＴＡＰＩＭ方法应用于计算随机参数结构的动力响应,取得了较好的结果。结出了数值算例,结果表明随机扩阶系统法,随机有限元法与精细积分时域平均法的结合是计算 随机参数结构动力响应的有效方法。     

Identification of the state equation in complex non-linear systems

Building on the basic idea behind the Restoring Force Method for the non-parametric identification of non-linear systems, a general procedure is presented for the direct identification of the state equation of complex non-linear systems. No information about the system mass is required, and only the applied excitation(s) and resulting acceleration are needed to implement the procedure. Arbitrary non-linear phenomena spanning the range from polynomial non-linearities to the noisy Duffing-van der Pol oscillator (involving product-type non-linearities and multiple excitations) or hysteretic behavior such as the Bouc-Wen model can be handled without difficulty. In the case of polynomial-type non-linearities, the approach yields virtually exact results for sufficiently rich excitations. For other types of non-linearities, the approach yields the optimum (in least-squares sense) representation in non-parametric form of the dominant interaction forces induced by the motion of the system. Several examples involving synthetic data corresponding to a variety of highly non-linear phenomena are presented to demonstrate the utility as well as the range of validity of the proposed approach.     

Response and bifurcation analysis of a MDOF rotor system with a strong nonlinearity

A new HB (Harmonic Balance)/AFT (Alternating Frequency Time) method is further developed to obtain synchronous and subsynchronous whirling response of nonlinear MDOF rotor systems. Using the HBM, the nonlinear differential equations of a rotor system can be transformed to algebraic equations with unknown harmonic coefficients. A technique is applied to reduce the algebraic equations to only those of the nonlinear coordinates. Stability analysis of the periodic solutions is performed via perturbation of the solutions. To further reduce the computational time for the stability analysis, the reduced system parameters (mass, damping, and stiffness) are calculated in terms of the already known harmonic coefficients. For illustration, a simple MDOF rotor system with a piecewise-linear bearing clearance is used to demonstrate the accuracy of the calculated steady-state solutions and their bifurcation boundaries. Employing ideas from modern dynamics theory, the example MDOF nonlinear rotor system is shown to exhibit subsynchronous, quasi-periodic and chaotic whirling motions.     

Bifurcation analysis for a modified Jeffcott rotor with bearing clearances

A HB (Harmonic Balance)/AFT (Alternating Frequency/Time) technique is developed to obtain synchronous and subsynchronous whirling motions of a horizontal Jeffcott rotor with bearing clearances. The method utilizes an explicit Jacobian form for the iterative process which guarantees convergence at all parameter values. The method is shown to constitute a robust and accurate numerical scheme for the analysis of two dimensional nonlinear rotor problems. The stability analysis of the steady-state motions is obtained using perturbed equations about the periodic motions. The Floquet multipliers of the associated Monodromy matrix are determined using a new discrete HB/AFT method. Flip bifurcation boundaries were obtained which facilitated detection of possible rotor chaotic (irregular) motion as parameters of the system are changed. Quasi-periodic motion is also shown to occur as a result of a secondary Hopf bifurcation due to increase of the destabilizing cross-coupling stiffness coefficients in the rotor model.     

基于概率配点法的岩土材料参数随机场及其响应分析

概率配点法是进行不确定性问题分析的一种有效方法。通过对输入参数场进行Karhunen-Loeve展开,将其表示为一系列独立随机变量在不同权重下的线性组合,再以与之相同的随机变量组合形成混沌多项式展开对输出随机场进行分解,通过某种算法选取随机变量的值,将其作为插值点的组合(配点),在这些配点上,概率方程演化为一个确定性问题方程。由此,可以直接利用现有软件或者确定性问题计算程序进行求解,生成混沌多项式的系数矩阵后,即可得到该随机问题的各阶统计矩,从而实现参数随机场的不确定性分析。本文将该方法引进岩土工程材料参数随机场,将体积模量视为输入随机场,位移视为输出场,分别进行了弹性及塑性变形计算。结果表明该方法极大地降低了随机问题的求解难度,与MC法(Mento Carlo)相比,减少了运算消耗,提高了计算效率,具有明显的优越性。     

A probabilistic linearization method for non-linear systems subjected to additive and multiplicative excitations

A general method to obtain approximate solutions for the random response of non-linear systems subjected to both additive and multiplicative Gaussian white noises is presented. Starting from the concept of linearization, the proposed method of “Probabilistic Linearization” (PL) is based on the replacement of the Fokker–Planck equation of the original non-linear system with an equivalent one relative to a linear system subjected to additive excitation only. By means of the general scheme of the weighted residuals, the unknown coefficients of the equivalent system are determined. Assuming a Gaussian probability density function of the response process and by choosing the weighting functions in a suitable way, the equivalence of the proposed method, called “Gaussian Probabilistic Linearization” (GPL), with the “Gaussian Stochastic Linearization” (GSL) applied to the coefficients of the Itô differential rule is evidenced. In addition, the generalization of the proposed method, called “Generalized Gaussian Probabilistic Linearization” (GGPL), is presented. Numerical applications show as, varying the choice of the weighting functions, it is possible to obtain different linearizations, with a variable degree of accuracy. For the two examples considered, different suitable combinations of the weighting functions lead to different equivalent linear systems, all characterized by the exact solution in terms of variance.     

Robust optimization of the non-linear behaviour of a vibrating system

In this paper, one proposes to optimize the vibratory behaviour of an absorber of vibration related to a system subjected to a harmonic load, in the presence of uncertainties on the design parameters. The total system is modeled by two degrees of freedom (2 dof) with a shock absorber and a generalized non-linear stiffness. The resolution is carried out in the temporal field according to a traditional diagram.Two cases of non-linearity were considered. In the first case, one is interested in the study of the system comprising a combination of the two generalized non-linearities of quadratic and cubic type of stiffness and damping. The second case relates to a non-linearity of non-whole power (in this paper 1.5), combined with the cubic case. It is a question of seeking the optimal responses envelopes of the deterministic and stochastic case and this for the non-linear displacements, phases and forces.The multi-objective optimization step consists in seeking the first Pareto front of several linear and non-linear objective functions by using a genetic algorithm of type “NSGA” (Non-dominated Sorting Genetic Algorithm).The design parameters are: mass, linear and non-linear stiffness and damping of the absorber. To obtain solutions presenting a good compromise between optimality and the robustness, one introduces uncertainties on these design parameters. The robustness is then defined by the dispersion of the parameters (definite as the ratio: mean value/standard deviation) and it is introduced as additional objective function.The use of the clusters resulting from the Self-Organizing Maps of Kohonen (SOM) is also suggested for a rational management of the design space. A study of sensitivity a posteriori can be exploited in order to eliminate the non-significant design parameters.     

Bifurcation analysis and limit cycle oscillation amplitude prediction methods applied to the aeroelastic galloping problem

A global stability and bifurcation analysis of the transverse galloping of a square section beam in a normal steady flow has been implemented. The model is an ordinary differential equation with polynomial damping nonlinearity. Six methods are used to predict bifurcation, the amplitudes and periods of the ensuing Limit Cycle Oscillations: (i) Cell Mapping, (ii) Harmonic Balance, (iii) Higher Order Harmonic Balance, (iv) Centre Manifold linearization, (v) Normal Form and (vi) numerical continuation. The resulting stability predictions are compared with each other and with results obtained from numerical integration. The advantages and disadvantages of each technique are discussed. It is shown that, despite the simplicity of the system, only two of the methods succeed in predicting its full response spectrum. These are Higher Order Harmonic Balance and numerical continuation.     

Non-periodic motions of a Jeffcott rotor with non-linear elastic restoring forces

The conditions that give rise to non-periodic motions of a Jeffcott rotor in the presence of non-linear elastic restoring forces are examined. It is well known that non-periodic behaviours that characterise the dynamics of a rotor are fundamentally a consequence of two aspects: the non-linearity of the hydrodynamic forces in the lubricated bearings of the supports and the non-linearity that affects the elastic restoring forces in the shaft of the rotor. In the present research the analysis was restricted to the influence of the non-linearity that characterises the elastic restoring forces in the shaft, adopting a system that was selected the simplest as possible. This system was represented by a Jeffcott rotor with a shaft of mass that was negligible respect to the one of the disk, and supported with ball bearings. In order to check in a straightforward manner the non-linearity of the system and to confirm the results obtained through theoretical analysis, an investigation was carried out using an experimental model consisting of a rotating disk fitted in the middle of a piano wire pulled taut at its ends but leaving the tension adjustable. The adopted length/diameter ratio was high enough to assume the wire itself was perfectly flexible while its mass was negligible compared to that of the disk. Under such hypotheses the motion of the disk centre can be expressed by means of two ordinary, non-linear and coupled differential equations. The conditions that make the above motion non-periodic or chaotic were found through numerical integration of the equations of motion. A number of numerical trials were carried out using a 4th order Runge-Kutta routine with adaptive stepsize control. This procedure made it possible to plot the trajectories of the disk centre and the phase diagrams of the component motions, taken along two orthogonal coordinate axes, with their projections of the Poincaré sections. On the basis of the theoretical results obtained, the conditions that give rise to non-periodic motions of the experimental rotor were identified.     

An intelligent parameter varying (IPV) approach for non-linear system identification of base excited structures

Health monitoring and damage detection strategies for base-excited structures typically rely on accurate models of the system dynamics. Restoring forces in these structures can exhibit highly non-linear characteristics, thus accurate non-linear system identification is critical. Parametric system identification approaches are commonly used, but require a priori knowledge of restoring force characteristics. Non-parametric approaches do not require this a priori information, but they typically lack direct associations between the model and the system dynamics, providing limited utility for health monitoring and damage detection. In this paper a novel system identification approach, the intelligent parameter varying (IPV) method, is used to identify constitutive non-linearities in structures subject to seismic excitations. IPV overcomes the limitations of traditional parametric and non-parametric approaches, while preserving the unique benefits of each. It uses embedded radial basis function networks to estimate the constitutive characteristics of inelastic and hysteretic restoring forces in a multi-degree-of-freedom structure. Simulation results are compared to those of a traditional parametric approach, the prediction error method. These results demonstrate the effectiveness of IPV in identifying highly non-linear restoring forces, without a priori information, while preserving a direct association with the structural dynamics.     

A comparison of stability computational methods for periodic solution of nonlinear problems with application to rotordynamics

In this paper a comparative study of five different stability computational methods based on the Floquet theory is presented. These methods are compared in terms of accuracy and CPU performance. Tests are performed on a set of nonlinear problems relevant to rotating machinery with rotor-to-stator contact and a variable number of degrees of freedom, whose periodic solutions are computed with the Harmonic Balance Method (HBM).     

Differential tones in a damped mechanical system with quadratic and cubic non-linearities

The combination tones of differential type are studied in a non-linear damped mechanical system of two degrees of freedom with quadratic and cubic non-linearities and excited by two external harmonic forces with different frequencies. Approximate steady state solutions and the corresponding Galerkin approximations of high order are obtained and error bounds are given. For a certain frequency the existence of three exact periodic solutions is proved by Urabe's method.