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Rotor systems are widely used in industrial power generation and propulsion. Once the nonlinear contact stiffness and oil film force are taken into account, the dynamics and stability of rotor systems become quite complex, often accompanied by super-harmonic and chaotic motions. Furthermore, conventional methods face limitations in real-time detection of the bifurcations and complex nonlinear motions. This research investigates the bifurcations and stability induced by nonlinear factors in a rotor-bearing system from an energy perspective. Dynamic equations of a rotor-bearing system considering cubic term stiffness are established, the steady responses are obtained by the fourth-order Runge–Kutta method. The relationship between the bifurcations and energy transfers is analyzed numerically, the proposed stability criterion is validated by comparing the Lyapunov exponents. The bistable phenomenon of period-3 m (m = 0 , 1) motion is discussed in terms of numerical results and experiments which corresponds to the asymmetric jumps of the generalized energy. It is found that bifurcations and unstable motions of the nonlinear system can be captured accurately by detecting the energy transfers, the proposed generalized energy curve exhibits more detailed information than the conventional speed-up curve. These findings provide a new perspective on studying bifurcations and stability of rotating systems, which can be further applied in the condition monitoring, stability prediction as well as the design of nonlinear energy sink, representing significant progress in converting theory into engineering applications for rotor systems. [ABSTRACT FROM AUTHOR]

In rotating machinery, aerodynamic forces and oil film forces often lead to cross-coupling stiffness. This paper is aimed at studying the stick-slip whirling vibrations induced by the piecewise smooth rotor/stator frictions in a modified flexible rotor subjected to cross-coupling stiffness. Governing equations determining the sliding region and boundaries of piecewise discontinuous friction are defined. This analytical study was conducted to discuss the complex vibrations and bifurcations. Various types of sliding motions (continuous pure rolling, continuous crossing, and grazing–sliding) were observed in this research. Further, as for discussing the impacts of the parameters on nonlinear sliding vibrations, a parametric study was conducted. The obtained results reveal that with an increase in the cross-coupling stiffness coefficient, continuous pure rolling occurs earlier, and the disk vibration time around the contact regime becomes shorter. For studying the self-excited backward whirling vibration of stick–slip nonlinear motions, analytical formulations are established. Detailed vibration amplitude and frequency studies of friction-induced backward whirling vibrations were carried out. Numerical simulations were performed to compare them with the analytical solutions and to validate the results as well. The proposed theory and results provide fresh perspectives for predicting friction-induced whirlings and creating proper designs for turbo machinery. [ABSTRACT FROM AUTHOR]

Vibration and shock control systems are of vital importance to modern vehicles when incorporating crashworthiness goals and enhancing occupant safety to protect avionics or electronics during survivable crash events. The study proposes a method denoted as the optimal generalized Bingham number (GBN) control to improve the soft-landing control accuracy of a drop-induced shock mitigation system employing a magnetorheological shock absorber (MREA). Based on the theoretical and experimental analysis of the damping force characteristics of the MREA, the dynamics of a single-degree-of-freedom shock mitigation system are established, and the GBN is defined with consideration of quadratic damping. The optimal GBN control strategy for a magnetorheological shock mitigation system is proposed to achieve a soft landing. The deceleration, velocity, and displacement formulas of the payload are derived, and the dynamic response of the magnetorheological shock mitigation system, under different GBNs, is analyzed. In terms of soft-landing control accuracy, it is shown that the optimal GBN control strategy performs better for a linear stroke MREA when the control algorithm is based on quadratic damping rather than linear damping. [ABSTRACT FROM AUTHOR]

In this paper, the nonlinear dynamic system of a brushless DC motor with voltage disturbance is studied analytically via a generalized harmonic balance method. A truncated Fourier series with time-varying coefficients is utilized to represent the analytical variations of nonlinear currents and voltages within this dynamic system. Bifurcations of periodic currents and voltages are obtained, and their stability is discussed through eigenvalue analysis. The frequency–amplitude characteristics of periodic currents and voltages exhibit complexity in the frequency domain. Comparative illustrations are provided to contrast the analytical solutions with numerical outcomes for periodic currents and voltages. These analytical findings can be effectively employed for controlling the brushless DC motors experiencing voltage disturbances. [ABSTRACT FROM AUTHOR]

In the paper, nonlinear vibration characteristics of a rotor system are investigated. Such a nonlinear rotor system is subjected to brush seal forces, which are obtained by integrating the bristle force along the entire ring. The nonlinear brush seal rotor system is constructed by merging a flexible rotor with nonlinear seal forces. The research is aimed at studying the nonlinear vibration characteristics and bifurcations of the motions under a variety of eccentricity circumstances. Different kinds of bifurcations are successfully obtained by mathematical discretization and mapping manipulation. Such a discrete mapping method successfully predicts the stable and unstable motions accurately. The period-doubling bifurcations and saddle node bifurcations of the rotor system are obtained. The sole unstable solutions are obtained, which are special, and a normal numerical integration method cannot solve this problem, which provides advantages in rotor design and motion control. According to the results, nonlinear resonances are found between the stable and unstable motions. The greater the eccentricity of the rotor, the greater the number of bifurcation points that occur during the rotor's nonlinear motions, as well as the larger the ranges of speeds where the motions are unstable. Saddle node bifurcations generate unstable nonlinear motions and non-smooth motions, which may bring damage to the mechanical rotors. The period-doubling bifurcations produce the route from period-1 to period-2 motions in the nonlinear rotor system. The research provides a new perspective to study the bifurcations and stability of the nonlinear rotor systems. [ABSTRACT FROM AUTHOR]

Dandelion seeds possess a complex three-dimensional structure and a self-adapted flying ability. To understand this fascinating flight mechanism, a three-dimensional umbrella-shaped model imitating dandelion seeds is proposed. The effects of folding angle and flow velocity on the dandelion drag force during their descent are studied, and it is found that the larger folding angle results in a smaller drag coefficient. Four different vortex structures are revealed depending on the folding angle. The effect of crosswind disturbances on the attitude stability of dandelions is also investigated by changing folding angles. It is found that dandelions with larger folding angles have better attitude stability. The proposed model suggests that when the folding angle is between 20° and 40°, the dandelion seeds might have a good balance between the drag force and attitude stability, which ensures a broad self-adapted flying ability. [ABSTRACT FROM AUTHOR]

Constructed motions and dynamic topology are new trends in solving nonlinear systems or system interactions. In nonlinear engineering, it is significant to achieve specific complex motions to satisfy expected dynamical behaviors (e.g., nonlinear motions, singularities, bifurcations, chaos, etc.), and complex motion application and control. To achieve such expected motions and global dynamical behaviors, mapping dynamics, constructed networks, random/discontinuous dynamic theorems, etc., are applied to quantitatively determine the complex motions. These theories adopt the symbolic dynamic abstracts and topological structures with nonlinear dynamics to investigate constructed complex motions to satisfy expected dynamical behaviors. [ABSTRACT FROM AUTHOR]

A brush seal has the advantages of adapting to different vibration conditions and increasing the stability of the nonlinear rotor system. In this research, the stability and bifurcations of complex vibrations in a brush-seal rotor system are studied. An analytical seal force model is obtained through the beam theory and mutual coupling dynamics of the bristles and the rotor. The interaction between the bristles and the rotor is clearly depicted by a geometric map. Periodic and chaotic vibrations as well as the corresponding amplitude–frequency characteristics are first predicted by a numerical bifurcation diagram and 3D waterfalls. Discrete dynamic eigenvalue analysis is adopted for a detailed investigation of the stability and bifurcations of nonlinear vibrations. Jumping, quasi-periodic, and half-frequency vibrations are warned during the speeding up and down process. Four separate nonlinear vibration evolving routes are discovered. Two period-doubling bifurcation trees evolving to chaos are illustrated for the observation of global and independent periodic vibrations. Nonlinear vibration illustrations are presented through displacement orbits as well as harmonic amplitudes and phases. Chaotic vibration and unstable semi-analytical vibration solutions are compared. The obtained results and analysis methods provide new perspectives on nonlinear vibrations in the brush-seal rotor system. [ABSTRACT FROM AUTHOR]

In this paper, analytical predictions of the firing cascades formed by stable and unstable firings in a Hindmarsh–Rose (HR) neuron system are completed via an implicit mapping method. The semi-analytical firing cascades present the route from periodic firings to chaos. For such cascades, the continuous firing flow of the nonlinear neuron system is discretized to form a special mapping structure for nonlinear firing activities. Stability and bifurcation analysis of the nonlinear firings are performed based on resultant eigenvalues of the global mapping structures. Stable and unstable firing solutions in the bifurcation tree exhibit clear period-doubling firing cascades toward chaos. Bifurcations are predicted accurately on the connections. Phase bifurcation trees are observed, which provide deep cognitions of neuronal firings. Harmonic dynamics of the period-doubling firing cascades are obtained and discussed for a better understanding of the contribution of the harmonics in frequency domains. The route into chaos is illustrated by the firing chains from period-1 to period-16 firings and verified by numerical solutions. The applied methods and obtained results provide new perspectives to the complex firing dynamics of the HR neuron system and present a potential strategy to regulate the firings of neurons. [ABSTRACT FROM AUTHOR]

In this paper, an independent bifurcation tree of period-2 motions to chaos coexisting with period-1 motions in a periodically driven van der Pol–Duffing oscillator is presented semi-analytically. Symmetric and asymmetric period-1 motions without period-doubling are obtained first, and a bifurcation tree of independent period-2 to period-8 motions is presented. The bifurcations and stability of the corresponding periodic motions on the bifurcation tree are determined through eigenvalue analysis. The symmetry breaks of symmetric period-1 motions is determined by the saddle-node bifurcations, and the appearance of the independent bifurcation tree of period-2 motions to chaos is also due to the saddle-node bifurcations. Period-doubling cascaded scenario of period-2 to period-8 motions are predicted analytically, and unstable periodic motions are also obtained. Numerical simulations are performed to illustrate motion complexity in such a van der Pol–Duffing oscillator. Such nonlinear systems can be applied in nonlinear circuit design and fluid-induced oscillations. [ABSTRACT FROM AUTHOR]

In this paper, a bifurcation tree of period-1 motion to chaos in a flexible nonlinear rotor system is presented through period-1 to period-8 motions. Stable and unstable periodic motions on the bifurcation tree in the flexible rotor system are achieved semi-analytically, and the corresponding stability and bifurcation of the periodic motions are analyzed by eigenvalue analysis. On the bifurcation tree, the appearance and vanishing of jumping phenomena of periodic motions are generated by saddle-node bifurcations, and quasi-periodic motions are induced by Neimark bifurcations. Period-doubling bifurcations of periodic motions are for developing cascaded bifurcation trees, however, the birth of new periodic motions are based on the saddle-node bifurcation. For a better understanding of periodic motions on the bifurcation tree, nonlinear harmonic amplitude characteristics of periodic motions are presented. Numerical simulations of periodic motions are performed for the verification of semi-analytical predictions. From such a study, nonlinear Jeffcott rotor possesses complex periodic motions. Such results can help one detect and control complex motions in rotor systems for industry. [ABSTRACT FROM AUTHOR]

In this study, we study the influence of five drilling hole arrangements on the rotordynamic and leakage performance of hole diaphragm labyrinth seal (HDLS) type seals with precise CFD analysis. Key findings reveal that hole distance significantly affects direct and cross-coupled stiffness, thereby impacting stability and leakage. Notably, the smallest hole in rectangle four holes(RF)-HDLS does not yield the minimum leakage. Vortex structure analysis, employing the Q-criterion, elucidate horizontal two holes(HT)-HDLS's superior leakage reduction owing to enhanced turbulence eddy dissipation. Seals with larger hole diameters, down single hole(DS)-HDLS and up single hole(US)-HDLS, demonstrate stable rotordynamic performance, with DS-HDLS exhibiting a notable 106.3% improvement in cross-coupled stiffness. In contrast, seals with holes positioned farther away, DS-HDLS, US-HDLS and vertical two holes(VT)-HDLS, experienced increased leakage with whirl frequency. HT-HDLS and RF-HDLS displayed a decrease, attributed to the stronger centrifugal effect. Comparing to center single hole(CS)-HDLS, VT-HDLS and US-HDLS achieved a significant 159.3% direct stiffness improvement and 432.7% reduction in cross-coupled stiffness. Despite a maximum 4.2% leakage increase, this substantial enhancement in rotor stability should not be overlooked. Moreover, all seals exhibited lower leakage rates than the traditional labyrinth seal. [ABSTRACT FROM AUTHOR]