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Purpose: Liquid sloshing problem in liquid-filled container under excitation is important in engineering field. When the resonant conditions are met, resonances of different types occur, which will lead to unexpected sloshing behaviors. To explore the fundamentals of the resonances in sloshing dynamics, the study is carried out. Methods: In this work, analytical analysis is approached. The nonlinear governing equations of the liquid sloshing modals in cylindrical boundary basin under multi-frequency excitations are adopted and then established. By means of multi-scale method, the possible combinations of frequencies which will cause resonances are found. Two general cases of single combination resonance and hybrid combination resonance are investigated analytically. Results: The bifurcation equations of the sloshing problem are got analytically and the results show that in single combination resonance mode, the frequency curve merges hard spring characteristic and with the bifurcation parameters varying, the qualitative results would not change. While in hybrid combination resonance case, the behaviors are more complex. Seven non-trial solutions exist at the same time. Conclusion: The case studies show that adequate dynamical behaviors exist in sloshing problem when resonances occur. And varied resonant conditions can be achieved when system is disturbed by different frequency components. The results enrich the exploration of the dynamical analysis of the liquid sloshing problem and provide theoretical basis for system design and optimization. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This paper is focused on the relationship between rigid body translation and rigid body precession in a squeeze film damper-rigid-rotor system with unsymmetrical stiffness supports. Two cases are considered: the precession motion non-resonance and internal resonance when translation motion occur primary resonant. In the first case, the amplitudes of translation and precession can be connected with an integration parameter about rotor parameters such as geometric size, stiffness, and mass. Some combination of system parameters will make the amplitudes of precession motion reach the same magnitude of the amplitudes of translation motion, so this integration parameter become an index to reflect the precessional motion degree, it is can be used to judge the feasibility of neglecting processional motion and simplifying asymmetry rotor as reasonable symmetry model. In the second case, the translation motion and precessional motion are strongly coupled, the vibration energy transfer between two kind of motion and the system occur internal resonant, which is possible appear in the rotor with cantilever disk. Each of case may appear nonlinear phenomenon, which is closely related with system parameters. The bifurcation analysis by using singularity methods is carried out to delimit the range of applicative operation parameters to avoid harmful phenomenon in unsymmetrical rotor system. The results of this paper provide a theoretical foundation for the convenient model simplification judgment and parameters optimization of the squeeze film damper-unsymmetrical rotor systems. [ABSTRACT FROM AUTHOR]

The singularity theory is applied to study the bifurcation behaviors of a reduced rotor model obtained by nonlinear transient POD method in this paper. A six degrees of freedom (DOFs) rotor model with cubically nonlinear stiffness supporting at both ends is established by the Newton's second law. The nonlinear transient POD method is used to reduce a six-DOFs model to a one-DOF one. The reduced model reserves the dynamical characteristics and occupies most POM energy of the original one. The singularity of the reduced system is analyzed, which replaces the original system. The bifurcation equation of the reduced model indicates that it is a high co-dimension bifurcation problem with co-dimension 6, and the universal unfolding (UN) is provided. The transient sets of six unfolding parameters, the bifurcation diagrams between the bifurcation parameter and the state variable are plotted. The results obtained in this paper present a new kind of method to study the UN theory of multi-DOFs rotor system. [ABSTRACT FROM AUTHOR]

A reduced model is proposed and analyzed for the simulation of vortex-induced vibrations (VIVs) for turbine blades. A rotating blade is modelled as a uniform cantilever beam, while a van der Pol oscillator is used to represent the time-varying characteristics of the vortex shedding, which interacts with the equations of motion for the blade to simulate the fluid-structure interaction. The action for the structural motion on the fluid is considered as a linear inertia coupling. The nonlinear characteristics for the dynamic responses are investigated with the multiple scale method, and the modulation equations are derived. The transition set consisting of the bifurcation set and the hysteresis set is constructed by the singularity theory and the effects of the system parameters, such as the van der Pol damping. The coupling parameter on the equilibrium solutions is analyzed. The frequency-response curves are obtained, and the stabilities are determined by the Routh-Hurwitz criterion. The phenomena including the saddle-node and Hopf bifurcations are found to occur under certain parameter values. A direct numerical method is used to analyze the dynamic characteristics for the original system and verify the validity of the multiple scale method. The results indicate that the new coupled model is useful in explaining the rich dynamic response characteristics such as possible bifurcation phenomena in the VIVs. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

When an aircraft is hovering or doing a dive-hike flight at a fixed speed, a constant additional inertial force will be induced to the rotor system of the aero-engine, which can be called a constant maneuver load. Take hovering as an example. A Jeffcott rotor system with a biased rotor and several nonlinear elastic supports is modeled, and the vibration characteristics of the rotor system under a constant maneuver load are analytically studied. By using the multiple-scale method, the differential equations of the system are solved, and the bifurcation equations are obtained. Then, the bifurcations of the system are analyzed by using the singularity theory for the two variables. In the EG-plane, where E refers to the eccentricity of the rotor and G represents the constant maneuver load, two hysteresis point sets and one double limit point set are obtained. The bifurcation diagrams are also plotted. It is indicated that the resonance regions of the two variables will shift to the right when the aircraft is maneuvering. Furthermore, the movement along the horizontal direction is faster than that along the vertical direction. Thus, the different overlapping modes of the two resonance regions will bring about different bifurcation modes due to the nonlinear coupling effects. This result lays a theoretical foundation for controlling the stability of the aero-engine's rotor system under a maneuver load. [ABSTRACT FROM AUTHOR]

This paper investigates the bifurcation and resonant hysteresis of varying compliance (VC) vibrations in a rigid-rotor ball bearing system with Hertzian contact and radial internal bearing clearance. With the aid of arc-length continuation, the harmonic balance and alternating frequency time (HB-AFT) technique is employed to trace the branches of periodic responses, for which the stability is also investigated using Floquet theory. It is found that the soft resonant hysteresis resulted from the Hertzian contact resonance dominates the primary parametric resonance. In addition, this paper also demonstrates that the mechanism of the period doubling in the case of small bearing clearances is arisen from one to two parametrical internal resonance in the primary resonant area, which can lead nonlinear responses such as quasi-period and chaotic motions to the system, and the evolvements of these complex behaviors are also explored. [ABSTRACT FROM AUTHOR]

The bifurcation analysis of a simple electric power system involving two synchronous generators connected by a transmission network to an infinite-bus is carried out in this paper. In this system, the infinite-bus voltage are considered to maintain two fluctuations in the amplitude and phase angle. The case of 1:3 internal resonance between the two modes in the presence of parametric principal resonance is considered and examined. The method of multiple scales is used to obtain the bifurcation equations of this system. Then, by employing the singularity method, the transition sets determining different bifurcation patterns of the system are obtained and analyzed, which reveal the effects of the infinite-bus voltage amplitude and phase fluctuations on bifurcation patterns of this system. Finally, the bifurcation patterns are all examined by bifurcation diagrams. The results obtained in this paper will contribute to a better understanding of the complex nonlinear dynamic behaviors in a two-machine infinite-bus (TMIB) power system. [ABSTRACT FROM AUTHOR]

This paper focuses on the bifurcation behaviors of a rigid rotor-squeeze film damper system considering the effect of fluid inertia. The equations of motion of the system are formulated considering π oil film model with fluid inertia. Then the averaging method is employed to obtain the bifurcation equation of the system model. By using the C-L method, three different modes of bifurcation behaviors are found from the three regions divided by the transition sets, namely bifurcation set and hysteresis set in system parameter plane. By changing the value of Reynolds number that reflects the fluid inertia of the squeeze film damper, the hysteresis set is moved obviously; it is shown that the fluid inertia plays an important role in determining the bifurcation behaviors of the system. Meanwhile, the bifurcation behaviors of system are affected significantly by the fluid inertia when the bearing coefficient locates within a certain region. Thus in this situation, the fluid inertia must be taken into account for theoretical analysis. Direct numerical simulation is also carried out by using the 4th order Runge-Kutta method to verify the theoretical results. The results obtained in this paper will provide a fundamental theory for designing an effective squeeze film damper. [ABSTRACT FROM AUTHOR]

In this paper the models of the transmission line with one DOF, two DOFs and three DOFs are constructed respectively by using Hamilton principle, where the initial location, the geometric nonlinearity caused by deformation and the aerodynamic nonlinearity caused by flow are considered. These simple models are accurate and can be used for the theoretical analysis. In actual engineering, the critical wind speed of galloping has been paid more and more attention. Therefore, in this paper the critical wind speeds are obtained by Lyapunov stability theory and the effects of structural parameters to the critical wind speed are discussed for different models. And the response curves of the system with one DOF and two DOFs (out-of-plane motion) are obtained by using harmonic balance method and multiple scale method, respectively. The bifurcation of the system moving out-of-plane is analyzed by singularity theory. For the systems whose motion is coupled with torsion, the numerical results are given by using the fourth order Runge-Kutta method. Additionally, the effects of the dynamic vibration absorber and detuning pendulum to the anti-galloping of the transmission line are studied for different cases. [ABSTRACT FROM AUTHOR]

Duffing equation is a representative nonlinear equation in practical engineering systems, where the most existing research focuses on local solutions of weakly nonlinear systems. In this paper, we study global bifurcations and chaos of the standard Duffing system by employing the Arnold tongue, and use the basin of attraction to investigate the properties of the Arnold tongue overlap. Our results show that a resonance solution and chaos could coexist, when the parameters are on the Arnold tongue overlap. The phenomenon does not exist in a system described by a weak Duffing equation. Numerical solutions for these bifurcations and chaos are also provided to demonstrate the theoretical results. [ABSTRACT FROM AUTHOR]

Abstract: The global bifurcations and chaotic motions are investigated analytically for an arch structure with parametric and forced excitation. The critical curves separating the chaotic and non-chaotic regions are drawn, which show that the system in the case of 1:1 resonance is more easily chaotically excited than the case of 1:2 resonance. There exist “uncontrollable regions” or “chaotic bands” for the system as the natural frequency varies. There also exists a “controllable frequency” for the system with linear and cubic parametric excitation. The system can be chaotically excited through infinite subharmonic bifurcations of odd/even orders. Numerical results agree with the analytical ones. [Copyright &y& Elsevier]

The method of multiple scales is applied for constructing nonlinear normal modes (NNMs) of a three-degree-of-freedom system which is discretized from a two-link flexible arm connected by a nonlinear torsional spring. The discrete system is with cubic nonlinearity and 1:3 internal resonance between the second and the third modes. The approximate solution for the NNM associated with internal resonance are presented. The NNMs determined here tend to the linear modes as the nonlinearity vanishes, which is significant for one to construct NNM. Greatly different from results of those nonlinear systems without internal resonance, it is found that the NNM involved in internal resonance include coupled and uncoupled two kinds. The bifurcation analysis of the coupled NNM of the system considered is given by means of the singularity theory. The pitchfork and hysteresis bifurcation are simultaneously found. Therefore, the number of NNM arising from the internal resonance may exceed the number of linear modes, in contrast with the case of no internal resonance, where they are equal. Curves displaying variation of the coupling extent of the coupled NNM with the internal-resonance-deturing parameter are proposed for six cases. [Copyright &y& Elsevier]

In this paper we investigate the bifurcations and the chaos of a piecewise linear discontinuous (PWLD) system based upon a rig-coupled SD oscillator, which can be smooth or discontinuous (SD) depending on the value of a system parameter, proposed in [18] , showing the equilibrium bifurcations and the transitions between single, double and triple well dynamics for smooth regions. All solutions of the perturbed PWLD system, including equilibria, periodic orbits and homoclinic-like and heteroclinic-like orbits, are obtained and also the chaotic solutions are given analytically for this system. This allows us to employ the Melnikov method to detect the chaotic criterion analytically from the breaking of the homoclinic-like and heteroclinic-like orbits in the presence of viscous damping and an external harmonic driving force. The results presented here in this paper show the complicated dynamics for PWLD system of the subharmonic solutions, chaotic solutions and the coexistence of multiple solutions for the single well system, double well system and the triple well dynamics. [ABSTRACT FROM AUTHOR]