Nonlinear dynamic behaviors of a rotor-labyrinth seal system

The nonlinear model of rotor-labyrinth seal system is established using Muszynska’s nonlinear seal forces. We deal with dynamic behaviors of the unbalanced rotor-seal system with sliding bearing based on the adopted model and Newmark integration method. The influence of the labyrinth seal one the nonlinear characteristics of the rotor system is analyzed by the bifurcation diagrams and Poincare’ maps. Various phenomena in the rotor-seal system, such as periodic motion, double-periodic motion, quasi-periodic motion and Hopf bifurcation are investigated and the stability is judged by Floquet theory and bifurcation theorem. The influence of parameters on the critical instability speed of the rotor-seal system is also included.     

Non-linear dynamic analysis of the ultra-short micro gas journal bearing-rotor systems considering viscous friction effects

Due to the micro-fabrication limitations and the low thickness of the silicon wafer, the length-to-diameter ratio (L/D) of the gas journal bearings in Power MEMS is about one order lower than that of the conventional bearings, which suggests that the viscous friction force in the micro-bearing is comparable to the load capacity. The effects of viscous friction force on non-linear dynamic characteristics of the ultra-short micro-bearing-rotor system are studied in this paper. The molecular gas-film lubrication model, which valid for arbitrary Knudsen numbers, is systematically coupled with the rotor kinetic equations and solved simultaneously to investigate the non-linear dynamic behavior of the system. The center orbits, phase portraits, Poincaré maps, and FFT spectra of the system response at different L/D ratio, rotor mass, and bearing number, and the corresponding bifurcation diagrams for cases of ignoring and considering viscous friction force are inspected and compared. The results indicate that, if the viscous friction force is not taken into account in the case of low L/D ratio, the low-frequency large-amplitude self-excited whirl motion will be predicted as the increase of the rotor mass and the bearing number. However, when the viscous friction force is included in the non-linear dynamic model, the rotor motion becomes more stable under the same conditions, as the synchronous motion with smaller amplitude prevails.     

Non-linear dynamic analysis of a HSFD mounted gear-bearing system

An investigation is carried out on the systematic analysis of the dynamic behavior of the hybrid squeeze-film damper (HSFD) mounted a gear-bearing system with strongly non-linear oil-film force and gear meshing force in the present study. The dynamic orbits of the system are observed using bifurcation diagrams plotted using the dimensionless unbalance coefficient, damping coefficient and the dimensionless rotating speed ratio as control parameters. The non-dimensional equations of the gear-bearing system are solved using the fourth order Runge-Kutta method. The onset of chaotic motion is identified from the phase diagrams, power spectra, Poincaré maps, bifurcation diagrams, maximum Lyapunov exponents and fractal dimension of the gear-bearing system. The results presented in this study provide some useful insights into the design and development of a gear-bearing system for rotating machinery that operates in highly rotating speed and highly non-linear regimes.     

Periodic and quasiperiodic motion for complex non-linear systems

A damped complex non-linear system corresponding to two coupled non-linear oscillators with a periodic damping force is investigated by an asymptotic perturbation method based on Fourier expansion and time rescaling. Four coupled equations for the amplitude and the phase of solutions are derived. Phase-locked solutions with period equal to the damping force period are possible only if the oscillators amplitudes are equal. On the contrary, if the oscillators amplitudes are different, periodic solutions exist only with a period different from the damping force period. These solutions are stable only for perturbations that conserve the phase difference and the square amplitude sum of the oscillators. Energy considerations are used in order to study existence and characteristics of quasiperiodic motion. We demonstrate that modulated motion can be also obtained for appropriate values of the detuning parameter and in this case an approximate analytic solution is easily constructed. If the detuning parameter decreases the modulation period increases and then diverges, an infinite-period bifurcation occurs and the resulting motion becomes unbounded. Analytic approximate solutions are checked by numerical integration.     

Linear and non-linear vibrations of fluid-filled hollow microcantilevers interacting with small particles

Linear and non-linear vibrations of a U-shaped hollow microcantilever beam filled with fluid and interacting with a small particle are investigated. The microfluidic device is assumed to be subjected to internal flowing fluid carrying a buoyant mass. The equations of motion are derived via extended Hamilton's principle and by using Euler-Bernoulli beam theory retaining geometric and inertial non-linearities. A reduced-order model is obtained applying Galerkin's method and solved by using a pseudo arc-length continuation and collocation scheme to perform bifurcation analysis and obtain frequency response curves. Direct time integration of the equations of motion has also been performed by using Adams-Moulton method to obtain time histories and analyze transient cantilever-particle interactions in depth. It is shown that exploiting near resonant non-linear behavior of the microcantilever could potentially yield enhanced sensor metrics. This is found to be due to the transitions that occur as a matter of particle movement near the saddle-node bifurcation points of the coupled system that lead to jumps between coexisting stable attractors.     

轴向运动导电板磁弹性非线性动力学及分岔特性

研究磁场环境中轴向运动导电薄板磁弹性动力学及分岔特性。考虑几何非线性因素,在给出薄板运动的动能、应变能及外力虚功的基础上,应用哈密顿变分原理,得到磁场中轴向运动薄板的非线性磁弹性振动方程,并给出洛伦兹电磁力的确定形式。针对横向磁场环境中条形板共振特性进行分析,应用多尺度法和奇异性理论,得到稳态运动下的分岔响应方程以及普适开折对应的转迁集。通过算例,分别得到以磁感应强度、轴向运动速度和激励力为分岔控制参数的分岔图、最大李雅普诺夫指数图和庞加莱映射图等计算结果,讨论不同分岔参数对系统呈现的倍周期和混沌运动的影响。结果表明,通过相应参数的改变可实现对系统复杂动力学行为的控制。     

迷宫密封转子系统非线性动力稳定性的研究

研究迷宫密封对转子系统动力稳定性的影响，迷宫密封的气动力采用Muszynska非线性力模型，计算了单盘Jeffcott转子非线性动力学特性。对Jacobi矩阵的分析表明，在密封力的影响下，转子达到一定转速后开始失稳，发生Hopf分岔，进入周期涡动状态，涡动幅度随转速的提高而增大，提高到一定程度，密封和转子发生碰摩，采用Runge-Kutta法数值模拟了转子的轴心轨迹。最后分析了迷宫密封的物理和结构参数对系统运动特性的影响。     

Dynamic modeling and extended bifurcation analysis of flexible-link manipulator

Abstract

In this article, the nonlinear dynamic analysis of a flexible-link manipulator is presented. Especially, the possibility of chaos occurrence in the system dynamic model is investigated. Upon the occurrence of chaos, the system dynamical behavior becomes unpredictable which in turn brings about uncertainty and irregularity in the system motion. The importance of this investigation is pronounced in similar systems such as double pendulum and single-link flexible manipulator. What makes this study distinct from previous ones is the increase in the number of links as well as the changing the bifurcation parameters from system mechanical parameters to force and torque inputs. To this aim, the motion equations of the N-link robot, which are derived with the aid of the recursive Gibbs-Appell formulation and the assumed modes method, are used. In the end, the equations of motion are developed for a two-link flexible manipulator, and its nonlinear dynamical behavior is analyzed via numerical integration of discrete equations. The results are presented in the form of bifurcation diagrams (for variation of torque amplitude), time histories, phase-plane portraits, Poincaré sections, and fast Fourier transforms. The outcomes indicate that when there is no offset, the decrease in damping results in chaotic generalized modal coordinates. In addition, as the excitation frequency decreases from 2π to π, a limiting amplitude is created at 0.35 before which the behavior of generalized rigid and modal coordinates is different, while this behavior has more similarity after this point. An experimental setup is also used to check the torques as the system input.     

Effects of supported angle on stability and dynamical bifurcations of cantilevered pipe conveying fluid

The effects of the supported angle on the stability and dynamical bifurcations of an inclined cantilevered pipe conveying fluid are investigated. First, a theoretical model of the pipe is developed through the force balance and stress-strain relationship. Second, the response surfaces, stability, and critical lines of the typical hanging system (H-S) and standing system (S-S) are discussed based on the modal analysis. Last, the bifurcation diagrams of the pipe are presented for different supported angles. It is shown that pipes will undergo a series of bifurcation processes and show rich dynamic phenomena such as buckling, Hopf bifurcation, period-doubling bifurcation, chaotic motion, and divergence motion.     

Coupled, forced response of an axially moving strip with internal resonance

In this paper, the forced response of a non-linear axially moving strip with coupled transverse and longitudinal motions is studied. In particular, the response of the system is examined in the neighborhood of a 3 : 1 internal resonance between the first two transverse modes. The equations of motion are derived using the Hamilton's Principle and discretized by the Galerkin's method. First, with the longitudinal motion neglected, the forced transverse response is investigated by applying the method of multiple scales to assess the effects of speed and the internal resonance. In general, the speed is shown to affect each mode differently. The internal resonance results in the constant solutions having transition to instability of both a saddle-node type and a Hopf bifurcation. In the region where the Hopf bifurcation occurs, steady-state periodic motion does not exist. Instead the stable motion is amplitude- and phase-modulated. When the coupled system with longitudinal motion is examined with internal resonance, results reveal that the modulated motions disappear. Thus, the presence of the longitudinal motion has a stabilizing effect on the transverse modes in the Hopf bifurcation region. The second longitudinal mode is shown to drift due primarily to a direct excitation of the first transverse mode. Effects of the longitudinal motion on the transverse response are shown to be significant for speeds both away from and close to the critical speed.     

Non-linear vibration of a single link viscoelastic Cartesian manipulator

In this present work, the non-linear behavior of a single-link flexible visco-elastic Cartesian manipulator is studied. The temporal equation of motion with complex coefficients of the system is obtained by using D’Alembert's principle and generalized Galarkin method. The temporal equation of motion contains non-linear geometric and inertia terms with forced and non-linear parametric excitations. It may also be found that linear and non-linear damping terms originated from the geometry of the large deformation of the system exist in this equation of motion. Method of multiple scales is used to determine the approximate solution of the complex temporal equation of motion and to study the stability and bifurcation of the system. The response obtained using method of multiple scales are compared with those obtained by numerically solving the temporal equation of motion and are found to be in good agreement. The response curves obtained using viscoelastic beams are compared with those obtained from a linear Kelvin-Voigt model and also with an equivalent elastic beam. The effect of the material loss factor, amplitude of base excitation, and mass ratio on the steady state responses for both simple and subharmonic resonance conditions are investigated.     

Non-linear dynamic instability of a double-sided nano-bridge considering centrifugal force and rarefied gas flow

Double-sided electromechanical nano-bridges can potentially be used as angular speed sensors and accelerometers in rotary systems such as turbine blades and vacuum pumps. In such applications, the influences of the centrifugal force and rarefied flow should be considered in the analysis. In the present study, the non-linear dynamic pull-in instability of a double-sided nano-bridge is investigated incorporating the effects of angular velocity and rarefied gas damping. The non-linear governing equation of the nanostructure is derived using Euler-beam model and Hamilton׳s principle including the dispersion forces. The strain gradient elasticity theory is used for modeling the size-dependent behavior of the system. The reduced order method is also implemented to discretize and solve the partial differential equation of motion. The influences of damping, centrifugal force, length scale parameters, van der Waals force and Casimir attraction on the dynamic pull-in voltage are studied. It is found that the dispersion and centrifugal forces decrease the pull-in voltage of a nano-bridge. Dynamic response of the nano-bridge is investigated by plotting time history and phase portrait of the system. The validity of the proposed method is confirmed by comparing the results from the present study with the experimental and numerical results reported in the literature.     

Nonlinear lateral-torsional coupled motion of a rotor contacting a viscoelastically suspended stator

Interaction of a rotor with a stationary part is a kind of serious malfunction that could result in a catastrophic failure if remained undetected. Past analytical and numerical simulation work on rotor?Cstator interactions mainly focus on the vibrations along the lateral directions. The torsional degree of freedom (dof) is usually ignored. The present work is aimed to study the influence of a rotor to stator contact on the lateral-torsional coupled vibrations. A mathematical model consisting of interacting vibratory systems of rotor and stator is presented. The contact is modeled using contact stiffness, damping and Coulomb friction. Equations derived for kinetic, potential and dissipation energies and non-conservative external forces are used in the Langrange??s equations for deriving the motion equations for the rotor?Cstator system. Equations revealed that the lateral-torsional motion coupling exists twofold for the rotor. The unbalance couples lateral-torsional motion of rotor through inertia and damping matrices. Coupling due to the rotor?Cstator friction occurs through a force vector. The nonlinear equations are solved using a Runge?CKutta fourth-order numerical integration scheme using relatively small time step. Results obtained through the proposed model are compared with the identical rotor?Cstator system without torsional dof and differences are identified. Effect of several parameters such as speed, relative inertia, coefficient of friction and contact damping on the bifurcation behavior of the rotor?Cstator motion has been investigated. Vibration motions presented in the forms of spectrum cascade of the coast-up response, and orbit and Poincaré plots of the steady-state response are exhibiting rich dynamic behavior of the system.     

考虑密封的悬臂转子系统的裂纹故障分析

以双盘悬臂立式转子-轴承系统为研究对象，建立了系统运动微分方程，并用数值方法分析了在非线性密封力和非线性油膜力作用下的裂纹转子的动力学特性。分析表明，在一定深度裂纹下，转子系统响应随不同角频率比表现出复杂的非线性现象，出现了周期k运动、拟周期运动和混沌运动等多种运动形式。在一定角速度时，工作在远离临界角速度区的转子系统对裂纹非常敏感，而工作在近临界角速度区的转子系统对裂纹不是特别敏感，但是裂纹对它的运动状态影响较大。该研究结果为该类转子-轴承系统的安全运行与故障诊断提供了一定的理论参考。     

Chaotic attitude motion of gyrostat statellites in a central force field

The nonlinear attitude motion of gyrostat satellites in a central force field is investigated, with particular emphasis on their long-time dynamic behavior for a wide range of parameters. The numbers of equilibrium solutions, as well as their stability, vary with the rotor speed, and bifurcation diagrams have been obtained. Various dynamic behaviors of gyrostat satcllites, e.g. periodic, quasiperiodic, and chaotic, are studied via the Poincaré map technique. It is shown that the rotor speed has a significant effect on the dynamic behavior of gyrostat satellites.     

Resonance and Bifurcation in a Nonlinear Duffing System with Cubic Coupled Terms

The dynamic behaviors of two-degree-of-freedom Duffing system with cubic coupled terms are studied. First, the steady-state responses in principal resonance and internal resonance of the system are analyzed by the multiple scales method. Then, the bifurcation structure is investigated as a function of the strength of the driving force F. In addition to the familiar routes to chaos already encountered in unidimensional Duffing oscillators, this model exhibits symmetry-breaking, period-doubling of both types and a great deal of highly periodic motion and Hopf bifurcation, many of which occur more than once. We explore the chaotic behaviors of our model using three indicators, namely the top Lyapunov exponent, Poincaré cross-section and phase portrait, which are plotted to show the manifestation of coexisting periodic and chaotic attractors.     

Linear and non-linear interactions between static and dynamic bifurcations of damped planar beams

The critical and post-critical behavior of a non-conservative non-linear structure, undergoing statical and dynamical bifurcations, is analyzed. The system consists of a purely flexible planar beam, equipped with a lumped visco-elastic device, loaded by a follower force. A unique integro-differential equation of motion in the transversal displacement, with relevant boundary conditions, is derived. Then, the linear stability diagram of the trivial rectilinear configuration is built-up in the parameter space. Particular emphasis is given to the role of the damping on the critical scenario. The occurrence of different mechanisms of instability is highlighted, namely, of divergence, Hopf, double zero, resonant and non-resonant double Hopf, and divergence-Hopf bifurcation. Attention is then focused on the two latter (codimension-two) bifurcations. A multiple scale analysis is carried-out directly on the continuous model, and the relevant non-linear bifurcation equations in the amplitudes of the two interactive modes are derived. The fixed-points of these equations are numerically evaluated as functions of two bifurcation parameters and some equilibrium paths illustrated. Finally, the bifurcation diagrams, illustrating the system behavior around the critical points of the parameter space, are obtained.     

含三次耦合项的两自由度Duffing系统的共振及混沌行为

研究了一类含三次耦合项的两自由度Duffing系统的动力学行为。首先应用多尺度方法近似求解系统的一阶稳态响应。通过讨论系统的主共振和1∶1内共振,分析了三次耦合项对系统响应的影响。随后研究系统随外加周期力强度变化的分岔过程,发现除了常见的倍周期分岔通向混沌外,还存在一种直接由周期运动进入混沌的突发路径。结合对系统的最大Lyapunov指数,相轨图及Poincar啨映射的分析验证了上述结论。     

Two-to-one resonant multi-modal dynamics of horizontal/inclined cables. Part II: Internal resonance activation,reduced-order models and nonlinear normal modes

Resonant multi-modal dynamics due to planar 2:1 internal resonances in the non-linear, finite-amplitude, free vibrations of horizontal/inclined cables are parametrically investigated based on the second-order multiple scales solution in Part I [1] (in press). The already validated kinematically non-condensed cable model accounts for the effects of both non-linear dynamic extensibility and system asymmetry due to inclined sagged configurations. Actual activation of 2:1 resonances is discussed, enlightening on a remarkable qualitative difference of horizontal/inclined cables as regards non-linear orthogonality properties of normal modes. Based on the analysis of modal contribution and solution convergence of various resonant cables, hints are obtained on proper reduced-order model selections from the asymptotic solution accounting for higher-order effects of quadratic nonlinearities. The dependence of resonant dynamics on coupled vibration amplitudes, and the significant effects of cable sag, inclination and extensibility on system non-linear behavior are highlighted, along with meaningful contributions of longitudinal dynamics. The spatio-temporal variation of non-linear dynamic configurations and dynamic tensions associated with 2:1 resonant non-linear normal modes is illustrated. Overall, the analytical predictions are validated by finite difference-based numerical investigations of the original partial-differential equations of motion.