Bifurcation analysis for nonlinear multi-degree-of-freedom rotor system with liquid-film lubricated bearings

The oil-film oscillation in a large rotating machinery is a complex high-dimensional nonlinear problem. In this paper, a high pressure rotor of an aero engine with a pair of liquid-film lubricated bearings is modeled as a twenty-two-degree-of-freedom nonlinear system by the Lagrange method. This high-dimensional nonlinear system can be reduced to a two-degree-of-freedom system preserving the oil-film oscillation property by introducing the modified proper orthogonal decomposition (POD) method. The efficiency of the method is shown by numerical simulations for both the original and reduced systems. The Chen-Longford (C-L) method is introduced to get the dynamical behaviors of the reduced system that reflect the natural property of the oil-film oscillation.     

Bifurcation and stability analysis of an anaerobic digestion model

This paper presents the dynamic behaviour of the anaerobic digestion process, based on a simplified model. The hydraulic, biological and physicochemical processes such as those which underpin anaerobic digestion have more than one stable stationary solution and they compete with each other. Further, the attractive domains of the stable solutions vary with the key parameters. Thus, some initial transient process moving toward one stable solution could suddenly move towards another solution, at which a so-call catastrophe takes places (e.g. washout). The paper systematically analyses the stationary solutions with their associated stability, which provides insight and guidance for anaerobic digestion reactor design, operation and control.     

Reduced-order finite element method based on POD for fractional Tricomi-type equation

The reduced-order finite element method (FEM) based on a proper orthogonal decomposition (POD) theory is applied to the time fractional Tricomi-type equation. The present method is an improvement on the general FEM. It can significantly save memory space and effectively relieve the computing load due to its reconstruction of POD basis functions. Furthermore, the reduced-order finite element (FE) scheme is shown to be unconditionally stable, and error estimation is derived in detail. Two numerical examples are presented to show the feasibility and effectiveness of the method for time fractional differential equations (FDEs).     

Comparison of POD reduced order strategies for the nonlinear 2D shallow water equations

This paper introduces tensorial calculus techniques in the framework of POD to reduce the computational complexity of the reduced nonlinear terms. The resulting method, named tensorial POD, can be applied to polynomial nonlinearities of any degree p. Such nonlinear terms have an online complexity of , where k is the dimension of POD basis and therefore is independent of full space dimension. However, it is efficient only for quadratic nonlinear terms because for higher nonlinearities, POD model proves to be less time consuming once the POD basis dimension k is increased. Numerical experiments are carried out with a two‐dimensional SWE test problem to compare the performance of tensorial POD, POD, and POD/discrete empirical interpolation method (DEIM). Numerical results show that tensorial POD decreases by 76× the computational cost of the online stage of POD model for configurations using more than 300,000 model variables. The tensorial POD SWE model was only 2 to 8× slower than the POD/DEIM SWE model but the implementation effort is considerably increased. Tensorial calculus was again employed to construct a new algorithm allowing POD/DEIM SWE model to compute its offline stage faster than POD and tensorial POD approaches. Copyright © 2014 John Wiley & Sons, Ltd.     

An efficient implementation of aeroelastic tailoring based on efficient computational fluid dynamics-based reduced order model

Current and future trends in the aerospace industry leverage on the potential benefits provided by lightweight materials that can be tailored to realize desired mechanical characteristics when loaded. For aircraft design, the deployment of aeroelastic tailoring is hindered by the need to re-compute, for any possible modification of the structure, the dependence of the aerodynamic field on the underlying structural properties. To make progress in this direction, the work presents a rapid computational fluid dynamics based aeroelastic tool which is built around a reduced order model for the aerodynamics that is updated for any modification of the structure by using the structural dynamics reanalysis method. The aeroelastic tailoring tool is demonstrated in transonic flow for the AGARD 445.6 wing, suitably modified with composite materials. It was found that the proposed method provides accurate engineering predictions for the aeroelastic response and stability when the structure is modified from the baseline model.     

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.     

Bifurcation and buckling analysis of a unilaterally confined self-rotating cantilever beam

A nonlinear dynamic model of a simple non-holonomic system comprising a self-rotating cantilever beam subjected to a unilateral locked or unlocked constraint is established by employing the general Hamilton's Variational Principle. The critical values, at which the trivial equilibrium loses its stability or the unilateral constraint is activated or a saddle-node bifurcation occurs, and the equilibria are investigated by approximately analytical and numerical methods. The results indicate that both the buckled equilibria and the bifurcation mode of the beam are different depending on whether the distance of the clearance of unilateral constraint equals zero or not and whether the unilateral constraint is locked or not. The unidirectional snap-through phenomenon (i.e. catastrophe phenomenon) is destined to occur in the system no matter whether the constraint is lockable or not. The saddle-node bifurcation can occur only on the condition that the unilateral constraint is lockable and its clearance is non-zero. The results obtained by two methods are consistent. The project supported by the National Natural Science Foundation of China (10272002) and the Doctoral Program from the Ministry of Education of China (20020001032) The English text was polished by Yunming Chen.     

Nonlinear response and bifurcation analysis of a Duffing type rotor model under sine maneuver load

This paper focuses on the nonlinear dynamic and bifurcation characteristics of an aircraft rotor system affected by the maneuvering flight of the aircraft. The equations of motion of the system are formulated with the consideration of the nonlinear supports of Duffing type and the sine maneuver load of a proposed maneuvering flight model. By utilizing the multiple scales method to solve the motion equations analytically, the bifurcation equations are obtained. Accordingly, the response and the bifurcation characteristics of the system are analyzed respectively. Basically, the increase of the maneuver load may increase the formant frequency as well as the primary resonance frequencies. Through numerical simulations, four different types of response characteristics of the system during the maneuvering flight are found, which are compared with the theoretical results, and it shows good qualitative agreements between them. Furthermore, the maneuver load can make an apparent effect on the bifurcation. The results in this paper will provide a better understanding for the effect of aircraft maneuvering flight on the dynamics and bifurcations of the rotor system.     

An indirect shooting method based on the POD/DEIM technique for distributed optimal control of the wave equation

This paper presents a fast numerical method, based on the indirect shooting method and Proper Orthogonal Decomposition (POD) technique, for solving distributed optimal control of the wave equation. To solve this problem, we consider the first‐order optimality conditions and then by using finite element spatial discretization and shooting strategy, the solution of the optimality conditions is reduced to the solution of a series of initial value problems (IVPs). Generally, these IVPs are high‐order and thus their solution is time‐consuming. To overcome this drawback, we present a POD indirect shooting method, which uses the POD technique to approximate IVPs with smaller ones and faster run times. Moreover, in the presence of the nonlinear term, to reduce the order of the nonlinear calculations, a discrete empirical interpolation method (DEIM) is applied and a POD/DEIM indirect shooting method is developed. We investigate the performance and accuracy of the proposed methods by means of 4 numerical experiments. We show that the presented POD and POD/DEIM indirect shooting methods dramatically reduce the CPU time compared to the full indirect shooting method, whereas there is no significant difference between the accuracy of the reduced and full indirect shooting methods.     

Reduced‐order modeling based on POD of a parabolized Navier–Stokes equation model I: forward model

A proper orthogonal decomposition (POD)‐based reduced‐order model of the parabolized Navier–Stokes (PNS) equations is derived in this article. A space‐marching finite difference method with time relaxation is used to obtain the solution of this problem, from which snapshots are obtained to generate the POD basis functions used to construct the reduced‐order model. In order to improve the accuracy and the stability of the reduced‐order model in the presence of a high Reynolds number, we applied a Sobolev H₁ norm calibration to the POD construction process. Finally, some numerical tests with a high‐fidelity model as well as the POD reduced‐order model were carried out to demonstrate the efficiency and the accuracy of the reduced‐order model for solving the PNS equations compared with the full PNS model. Different inflow conditions and different selections of snapshots were experimented to test the POD reduction technique. The efficiency of the H₁ norm POD calibration is illustrated for the PNS model with increasingly higher Reynolds numbers, along with the optimal dissipation coefficient derivation, yielding the best root mean square error and correlation coefficient between the full and reduced‐order PNS models. Copyright © 2011 John Wiley & Sons, Ltd.     

Transition sets analysis based parametrical design of nonlinear metal rubber isolator

This research proposes the parametrical design of Metal Rubber (MR) isolation platform based on the investigation of nonlinear vibration properties under different types of excitation. Based on the mechanical model established by experiments, the restoring force of the isolation platform is proposed as a nonlinear function in consideration of the stiffness nonlinearity and Coulomb friction of metal wires. Then, the perturbation method is utilized to solve the steady states whose local stability is studied by singularity theory. The main results obtained by singularity theory show that there are five different types of vibration property, and the critical conditions for the transformation of different vibration properties are defined by transition sets. For impact excitation, the optimum structural parameters are obtained based on the vibration dissipation time via nonlinear backbone analysis; for periodic excitation, the optimum structural parameters are determined according to multiple standards including the bandwidth for effective isolation, bandwidth for multi-steady states, resonance peak and displacement transmissibility in high frequency band etc. The vibration performances for optimum structural parameters are verified by dynamical experiments. In conclusion, this paper carries out a novel sight of choosing optimum parameters, and therefore provides the guidance for the utilization of MR isolation platform for different types of excitation in engineering practices.     

The nonlinear vibration analysis of a clamped-clamped beam by a reduced multibody method

The nonlinear response characteristics for a dynamic system with a geometric nonlinearity is examined using a multibody dynamics method. The planar system is an initially straight clamped-clamped beam subject to high frequency excitation in the vicinity of its third natural mode. The model includes a pre-applied static axial load, linear bending stiffness and a cubic in-plane stretching force. Constrained flexibility is applied to a multibody method that lumps the beam into N elements for three substructures subjected to the nonlinear partial differential equation of motion and N-1 linear modal constraints. This procedure is verified by d'Alembert's principle and leads to a discrete form of Galerkin's method. A finite difference scheme models the elastic forces. The beam is tuned by the axial force to obtain fourth order internal resonance that demonstrates bimodal and trimodal responses in agreement with low and moderate excitation test results. The continuous Galerkin method is shown to generate results conflicting with the test and multibody method. A new checking function based on Gauss' principle of least constraint is applied to the beam to minimize modal constraint error.     

Bifurcation indicator for geometrically nonlinear elasticity using the Method of Fundamental Solutions

In the present work, we propose a numerical analysis of instability and bifurcations for geometrically nonlinear elasticity problems. These latter are solved by using the Asymptotic Numerical Method (ANM) associated with the Method of Fundamental Solutions (MFS). To compute bifurcation points and to determine the critical loads, we propose three techniques. The first one is based on a geometrical indicator obtained by analyzing the Taylor series. The second one exploits the properties of the Padé approximants, and the last technique uses an analytical bifurcation indicator. Numerical examples are studied to show the efficiency and the reliability of the proposed algorithms.     

磁悬浮挠性转子系统模型复合辨识方法

针对磁悬浮挠性转子的纯有限元分析模型精度低的问题,提出一种磁悬浮挠性转子系统模型复合辨识方法。该方法首先采用有限元方法把磁悬浮挠性转子划分为多个Timoshenko梁单元,建立磁悬浮挠性转子系统模型,并对其挠性临界频率、阻尼、刚度和振型等关键参数进行分析,进而得到磁悬浮挠性转子的等效降阶数学模型;然后采用鲁棒自适应方法分析磁悬浮挠性转子系统的动态特性;最后,采用变LEVY方法从动态特性分析数据中对磁悬浮挠性转子进行系统辨识,校正有限元分析得到的降阶数学模型。实验结果表明,本方法可以得到磁悬浮挠性转子较为准确的系统模型。     

Bifurcation and nonlinear dynamic analysis of a rigid rotor supported by two-lobe noncircular gas-lubricated journal bearing system

This paper presents the study of the dynamic analysis of a rigid rotor supported by a two-lobe non-circular gas-lubricated journal bearing. A finite element method has been employed to solve the Reynolds equation in static and dynamical states and the dynamical equations have been solved using Runge–Kutta method. To analyze the behavior of the rotor center in horizontal and vertical directions under the different operating conditions, the dynamic trajectory, the power spectra, the Poincare maps, and the bifurcation diagrams are used. Results of this study indicates that by considering bearing number and rotor mass as the parameters of the system, complex dynamic behavior comprising periodic, KT-periodic, and quasi-periodic responses of the rotor center has occurred.     

基于state-space-split法的滞回非线性系统随机动力响应分析

采用SSS(state-space-split)法,建立了引入Bouc-Wen滞回模型的杜芬非线性系统在高斯白噪声激励下的概率密度函数(PDF)的近似求解方法,分析了其随机动力响应变化规律.首先,将Bouc-Wen滞回模型引入杜芬非线性系统,分别考虑非线性系统中的几何非线性和材料非线性对动力响应的影响.随后,对该模型进...     

Time series analysis of homoclinic nonlinear systems using a wavelet transform method

Homoclinic (and heteroclinic) trajectories are closed paths in phase space that connect one or more saddle points. They play an important role in the study of dynamical systems and are associated with the creation/destruction of limit cycles as a parameter is varied. Often, this creation/destruction process involves complicated sequences of bifurcations in small regions of parameter space and there is now an established theoretical framework for the study of such systems.

The eigenvalues of saddle points in the phase space determine the behaviour of the system. In this article we present a new eigenvalue estimation technique based on a wavelet transformation of a time series under study and compare it with an existing method based on phase space reconstruction. We find that the two methods give good agreement with theory using clean model data, but where noisy data are analysed the wavelet technique is both more robust and easier to implement.     

The application of reduced space harmonic balance method for the nonlinear vibration problem in rotor dynamics

The reduced space harmonic balance method is demonstrated to find the maximum vibration responses of rotor systems. Within the reduced space SQP method, transition from the high dimensional optimization space to the desired reduced space is accomplished by resorting to the null space decomposition technique, resulting in the elimination of the harmonic balance constraints. Numerical examples of rotor systems are presented to show the applicability of the proposed methodology.     

Bifurcation characteristics analysis of a class of nonlinear dynamical systems based on singularity theory

A method for seeking main bifurcation parameters of a class of nonlinear dynamical systems is proposed. The method is based on the effects of parametric variation of dynamical systems on eigenvalues of the Frechet matrix. The singularity theory is used to study the engineering unfolding(EU) and the universal unfolding(UU) of an arch structure model, respectively. Unfolding parameters of EU are combination of concerned physical parameters in actual engineering, and equivalence of unfolding parameters and physical parameters is verified. Transient sets and bifurcation behaviors of EU and UU are compared to illustrate that EU can reflect main bifurcation characteristics of nonlinear systems in engineering. The results improve the understanding and the scope of applicability of EU in actual engineering systems when UU is difficult to be obtained.