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In this paper, we propose an efficient time-space-domain optimized (OptTS) finite difference scheme to model 2D and 3D scalar wave propagation. It adopts piecewise constant interpolation coefficients for several consecutive Courant number ranges, which avoids the extra time costs caused by loading the coefficients consecutively according to different wave velocities in heterogeneous media. The new scheme is capable of suppressing numerical dispersion errors using a large time interval because we introduce a Lax-Wendroff type time discretization, which utilizes the wave-field information of O (M 1 M 2) neighboring grid points in each step, with merely O (M 1 + M 2) computation, where M 1 and M 2 are the stencil radii to generate the 2nd-order and 4th-order spatial derivatives. The improved computational efficiency and accuracy of OptTS are validated by substantial experiments using both theoretical analysis and numerical modeling. They show that in heterogeneous models, the maximal Courant numbers reach 1.0 and 0.85 in 2D and 3D cases respectively without severe temporal error, and the scheme is computationally nearly three times faster than its high-order counterpart, providing a powerful tool for large-scale modeling and high-resolution imaging. [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]

Abstract: This study focuses on the nonlinear dynamic and vibration characteristics of spur gear pair with local spalling defect to explore the spalling mechanism. The dynamic model of the gear pair with spalling defect and time-variant mesh stiffness is established to investigate the effect of spalling defect on mesh stiffness and dynamic response. The analytical solutions of the system which is deduced into four different stages of the gear with the time-variant stiffness in a mesh period are obtained. The dynamic responses with the evolvement of sapll are analyzed by using time history, phase contrail, Poincaré section and spectrum analysis. The spalling characteristics are also evaluated by employing statistical techniques, which shows that the spalling failure is suitable to be detected under low velocity and small excitation. The gearbox with spalling defect is designed and the experiments are carried out to get the dynamic characteristics of the spalling vibration signals. The results obtained herein show the good agreement qualitatively with the theoretical analysis, which provides a theoretical basis to spalling fault diagnosis of gearbox. [Copyright &y& Elsevier]

Highlights • The constant excitation induced by flight maneuver is included in the rotor model. • The HB-AFT method is formulated to solve sub-harmonic resonance problems. • The bifurcation of the system affected by the constant excitation is investigated. • Different bifurcation types are identified. Abstract This paper focuses on the mechanism of a complex bifurcation behavior caused by flight maneuvers in an aircraft rub-impact rotor system with Duffing type nonlinearity. The maneuver load during flight maneuvers may induce a rub-impact phenomenon, accompanied by complex nonlinear behaviors such as periodic, sub-harmonic and quasi-periodic motions, but the bifurcation mechanism is not so clear. In this study, the harmonic balance method combined with an alternating frequency/time domain procedure (HB-AFT method) is formulated and used to derive the approximate periodic solutions of the system. In conjunction with the arc-length continuation, the solution branches for both periodic 1-T motion (synchronous oscillation) and periodic 2-T motion (sub-harmonic oscillation) are traced. Then with the aid of the Floquet theory, the stabilities of the obtained periodic solutions are examined. The changes in the number or the stability of the solutions lead to the identification of the bifurcation points, which can be classified qualitatively into three types, i.e. Neimark-Sacker bifurcation point (NSBP), quasi-periodic Hopf bifurcation point (QPHBP) and saddle-node bifurcation point (SNBP). In addition, the constant excitation significantly affects the instability as well as the bifurcation of the rotor system. In the case of a smaller constant excitation, the rotation region with respect to the quasi-periodic motions may disappear, accompanied with a PBP (pitchfork bifurcation point) instead of the two NSBPs and one QPHBP. The bifurcation analysis in this paper provides deep insight into the mechanism of the complex nonlinear phenomenon induced by the constant excitation. The results obtained will also contribute to a better understanding of the nonlinear dynamic behaviors of aircraft rotor systems during flight maneuvers. [ABSTRACT FROM AUTHOR]

Highlights • The applications of the POD method in a variety of research areas are reviewed. • The classification of the POD method is proposed. • The review focuses on the dynamical application of the POD method. • The outlooks of the POD method in actual engineering are discussed. Abstract This paper presents a review of proper orthogonal decomposition (POD) methods for order reduction in a variety of research areas. The historical development and basic mathematical formulation of the POD method are introduced. POD for parametric dynamic systems is introduced, and a physical interpretation of the POD approach based on the proper orthogonal modes (POMs) is presented. The equivalence between POD and three other order reduction methods is discussed: the first alternative method is singular value decomposition (SVD), the second is principal component analysis (PCA), and the third is Karhunen-Loeve decomposition (KLD). A classification of POD methods is described based on the parameter adaptation and sampling. Actual applications of POD methods for order reduction in engineering systems are illustrated. Finally, outlooks on the use of POD methods in high-dimensional nonlinear dynamic systems are presented in more detail to provide direct guidance for researchers in various areas of engineering. [ABSTRACT FROM AUTHOR]

• Singularity analysis reveals all kinds of periodic responses of a NES system. • Essential dimensionless parameters are defined to determine the NES dynamics. • Fourteen different kinds of periodic responses exist in the NES system. • NES effective parameter regions are located by the singularity analysis. Nonlinear energy sink (NES) can be used to reduce the forced vibration of the primary system. Many different kinds of periodic responses of the linear system coupled with NES have been presented in the literature. In order to reveal all kinds of periodic responses of the NES system on given parameter planes, this study intends to achieve the classification of different kinds of periodic responses of linear oscillator coupled with a NES under harmonic excitation, which is modeled as a 2-degree-of-freedom system. The amplitudes of periodic responses of the nonlinear system are obtained by using the harmonic balance method. Then a general method based on the singularity theory is proposed to reveal all kinds of periodic responses under a given set of parameters. The periodic responses include 10 kinds known responses and 4 kinds of newly-revealed responses. The complete classification of the responses ensure the reliability and the efficiency of a NES. [ABSTRACT FROM AUTHOR]

This paper proposes a harmonic balance method combined with dimension reduction procedure and fast Fourier transform (FFT) technique (DRF-HB) to efficiently analyze and rapidly apprehend the periodic responses of nonlinear systems. The dimension reduction procedure is employed to reduce the dimensionality of the Jacobian matrix required during the Newton-Raphson iterative procedure, while the FFT technique is utilized to expeditiously calculate the Jacobian matrix, facilitating the swift determination of the periodic responses of nonlinear systems. Computational results from two typical nonlinear systems underscore the superior performance of DRF-HB method in attaining the periodic responses for nonlinear systems. Specifically, in the case of the two-degree-of-freedom nonlinear oscillator, the DRF-HB method demonstrates an average computational efficiency 68-fold higher than that of the earlier HB-AFT (harmonic balance-alternating frequency/time domain) method and 182-fold higher than that of the RK4 (4th order Runge-Kutta) method. Similarly, in the scenario of the high-dimensional nonlinear Bernoulli beam, the DRF-HB method achieves significant computational efficiency advantage over both the HB-AFT and RK4 methods, and the superiority becoming increasingly pronounced with the augmentation of the system's degrees-of-freedom. The superior computational performance underscores the potential of the DRF-HB method presented here for application in the dynamic characteristic analysis of high-dimensional systems with complex nonlinearities in practical engineering context. [ABSTRACT FROM AUTHOR]

As we have known that it is impossible to attenuate the vibration displacement of the floating-slab depending on a strong stiffness and to improve the low-frequency vibration isolation performance of the floating-slab track (FST) depending on a soft stiffness simultaneously in engineering. In this paper, we propose a novel method to solve the paradox by introducing a "Hardening-First-Then-Softening" (HFTS) design based upon the energy distribution of wheel-rail loads to formulate a model of nonlinear floating-slab isolator. The average procedure is employed to analyze the vibration isolation ability of the isolator with parameters taken from a metro line load excitation. A simulation model of train-FST-tunnel with HFTS isolator is constructed to study the dynamic behaviors of the system. The numerical simulations show the significant suppression of displacement response of floating-slab and Z weighted acceleration vibration level of the tunnel. The results presented herein this paper provide a scheme to solve the strict requirements of displacement suppression and the vibration isolation which is of beneficial to steady and silent urban rail transit. [ABSTRACT FROM AUTHOR]

Koopman operator, acting on an infinite-dimensional Hilbert space of the observables, provides a global systematic linear representation of nonlinear systems, which is a leading candidate for data-driven modeling. In practical applications, the observable function is required to have an invertible property to predict or control the original system. However, two approaches for constructing the invertible observable functions in current research, introducing the full-state observables and adopting the auto-encoder, have drawbacks such as overly strong assumptions and lossy reconstruction. To address these issues, we propose the Invertible Koopman Network (IKN) to approximate the Koopman operator. With special invertible blocks, IKN achieves structure-level reversibility while retaining sufficient nonlinear mapping capability. This network results in a transformation from the complex dynamic system's states to the observables with simpler dynamics and affords lossless reconstruction via its invertible process. Then, we use the IKN as a dynamical embedding model and adopt the Transformer based on an autoregressive task to learn the evolution pattern of the observables. Through the greedy decoding strategy, the trained IKN and Transformer model enable data-driven prediction where only initial value is provided. The method is tested on three typical chaotic dynamic systems (Lorenz system, Chen system, and Kuramoto-Sivashinsky equation) and is shown to capture the systems' intrinsic evolution patterns, such as the structure of the strange attractor. Comparative experiments with alternative data-driven modeling approaches illustrate that our proposed method can model a longer evolutionary process with less error, further demonstrating its effectiveness and superiority. • A novel framework (IKN) for approximating the Koopman operator is presented. • IKN gives an invertible map from the states to observables with simpler dynamics. • A better data-driven modeling method is proposed based on the IKN and Transformer. • The proposed method can capture complex dynamics, such as the strange attractor. [ABSTRACT FROM AUTHOR]

This paper presents a modified incremental harmonic balance (IHB) method to analyze the nonlinear dynamic and thermal coupling characteristics of rotor-bearing systems after thermal balance. The IHB method is modified by tensor contraction and fast Fourier transform (FFT) to efficiently calculate the residual vector and Jacobian matrix. Nonlinear dynamic and thermal full coupling is implemented through the nonlinear part of the Jacobian matrix. The presented method achieves the simultaneous and efficient calculation of the motion and heat balance equations in the frequency domain according to the improvement measures. The effectiveness of the presented method is tested using the vibration response and temperature variation analysis for the coupled thermo-mechanical model of a rotor-ball bearing system. Furthermore, the performance comparisons show the good consistency of results between the presented method and the step-by-step (S-S) method, which is a partitioned iterative solution method for dealing with fully coupled problems in the time domain. The presented method has greater computational efficiency, more accurate temperature prediction, and better recognition of nonlinear phenomena in the system rather than the S-S method. Consequently, the presented method has a broad application prospect in solving fully coupled thermo-mechanical problems of more complex and higher dimensional rotor-bearing systems. • The motion and heat balance equations can be calculated simultaneously. • Greater calculation efficiency is illustrated than the step-by-step method. • Temperature can be predicted more accurately by time-varying nonlinear forces. • Nonlinear phenomenon recognition of coupled thermo-mechanical problems is improved. [ABSTRACT FROM AUTHOR]

The proper orthogonal decomposition (POD) method is a main and efficient tool for order reduction of high-dimensional complex systems in many research fields. However, the robustness problem of this method is always unsolved, although there are some modified POD methods which were proposed to solve this problem. In this paper, a new adaptive POD method called the interpolation Grassmann manifold (IGM) method is proposed to address the weakness of local property of the interpolation tangent-space of Grassmann manifold (ITGM) method in a wider parametric region. This method is demonstrated here by a nonlinear rotor system of 33-degrees of freedom (DOFs) with a pair of liquid-film bearings and a pedestal looseness fault. The motion region of the rotor system is divided into two parts: simple motion region and complex motion region. The adaptive POD method is compared with the ITGM method for the large and small spans of parameter in the two parametric regions to present the advantage of this method and disadvantage of the ITGM method. The comparisons of the responses are applied to verify the accuracy and robustness of the adaptive POD method, as well as the computational efficiency is also analyzed. As a result, the new adaptive POD method has a strong robustness and high computational efficiency and accuracy in a wide scope of parameter. [ABSTRACT FROM AUTHOR]

Microvascular lesions of the body are one of the most serious complications that can affect patients with type 2 diabetes mellitus. The blood-brain barrier (BBB) is a highly selective permeable barrier around the microvessels of the brain. This study investigated BBB disruption in diabetic rhesus monkeys using dynamic contrast-enhanced magnetic resonance imaging (DCE-MRI). Multi-slice DCE-MRI was used to quantify BBB permeability. Five diabetic monkeys and six control monkeys underwent magnetic resonance brain imaging in 3 Tesla MRI system. Regions of the frontal cortex, the temporal cortex, the basal ganglia, the thalamus, and the hippocampus in the two groups were selected as regions of interest to calculate the value of the transport coefficient K trans using the extended Tofts model. Permeability in the diabetic monkeys was significantly increased as compared with permeability in the normal control monkeys. Histopathologically, zonula occludens protein-1 decreased, immunoglobulin G leaked out of the blood, and nuclear factor E2–related factor translocated from the cytoplasm to the nuclei. It is likely that diabetes contributed to the increased BBB permeability. [ABSTRACT FROM AUTHOR]

• A modified HB-AFT method is proposed, which can solve high-dimensional differential equations with complex nonlinearities programmatically and efficiently. • All periodic solutions of a dual-rotor-bearing-casing system are obtained, the nonlinear dynamics of the system is analyzed. • The nonlinear dynamic behaviors such as vibration jumping, bi-stable and resonance hysteresis are revealed. • The evolution of the inter-shaft bearing's nonlinear dynamic load is analyzed, jumping phenomena are also found here. This paper presents a modified harmonic balance-alternating frequency/time domain (HB-AFT) method to analyze the nonlinear dynamics of a dual-rotor-bearing-casing system. The motion equations of a dual-rotor-bearing-casing system subjected to the unbalanced excitations of the two rotors are formulated with 284 degrees of freedom, in which, the complex nonlinear factors of the inter-shaft bearing such as exponential nonlinear restoring force, radial clearance and varying stiffness excitation are taken into consideration. The HB-AFT method is modified by applying the AFT procedure to obtain the harmonic expending coefficients of the harmonic balance residuals rather than that of the nonlinear terms, hence, the Jacobian matrix is calculated programmatically and efficiently, as a result the performance of the HB-AFT method is improved greatly. The primary resonance of the system is analyzed by employing the three-dimensional diagrams of amplitude-frequency responses of each node of the system, including the inner casing and the outer casing. In addition, the amplitude-frequency response curves with separated frequencies and orbits of the two rotors are also discussed. Furthermore, the pseudo-arclength continuation procedure is introduced into the modified HB method, then all periodic solution branches of the system including the unstable solutions are obtained. The results show that there are two resonant peaks in the amplitude-frequency response curves of the system and the vibration of each part of the system, i.e., LPC, LPT, HPC, HPT, inner casing and outer casing, is synchronous. Besides, the nonlinear dynamic behaviors such as vibration jumping, bi-stable and resonance hysteresis are revealed. Accordingly, the dynamic load of the inter-shaft bearing is calculated. The results show that the evolution of the inter-shaft bearing's dynamic load is associated with the vibration responses of the two rotors. The nonlinear characteristics of the force-frequency responses of the inter-shaft bearing are consistent with that of the two rotors and casings. In addition, with the increase of the inter-shaft bearing clearance, the nonlinear characteristics of the force-frequency responses of the inter-shaft bearing are more significant. In comparison with the Newmark method, the modified HB-AFT method proposed in this paper has a great advantage in computing efficiency, and it can also grasp all solution branches of the system including the unstable solutions. Consequently, it has great potential to deal with high-dimensional systems with complex nonlinearities and multifrequency excitations in practical engineering. [ABSTRACT FROM AUTHOR]

• A dual rotor experimental rig of an aero-engine is designed and manufactured. • Nonlinear dynamic model of the test rig is established based on finite element method. • Reduced-order model of dual rotor-bearing system is obtained by CMS method. • Local blade-casing rubbing fault feature is revealed by numerical simulation and experiments. In order to reveal the nonlinear vibration characteristics of an actual dual-rotor-aero-engine with blade-casing rubbing, a dual-rotor-bearing-coupling misalignment system with blade-casing rubbing are investigated based on numerical simulation and experimental measurement in this paper. A dual-rotor experimental rig is designed and manufactured, that is a simplified structure according a dual-rotor-aero-engine. The dynamic model of the dual- rotor-bearing system of the test rig is established based on the finite element method (FEM) taking into account the coupling misalignment, blade-casing rubbing and the nonlinearities of all of the supporting rolling element bearings. The component mode synthesis (CMS) method is applied to the model order reduction of the high-degrees of freedom (DOFs) dynamic model. The numerical and experimental nonlinear vibration responses of the dual-rotor system are compared with each other at different rotational speeds at different rotational speed ratios. The numerical results are in agreement with the experiment. These results reveal the nonlinear vibration characteristics of the dual-rotor-bearing-coupling misalignment system with local blade-casing rubbing fault. [ABSTRACT FROM AUTHOR]

• The dual rotor-bearing system with misalignment is established. • POD method is the first time apply to dimension reduction of dual rotor-bearing system. • Efficiency and accuracy of POD method is verified. • Nonlinear characteristics of dual rotor-bearing system with misalignment are revealed. In this paper, the proper orthogonal decomposition (POD) method is first time apply to the dimension reduction of dual rotor-bearing experiment rig which is similar to an aero-engine rotor. The basic theory of the POD method and the application in dynamical system are introduced. The supporting bearing nonlinearity and the coupling misalignment of high and low pressure rotors are considered, and the dynamical model is established by finite element method based on dual rotor experiment rig. The frequency behaviors of dual rotor-bearing coupling misalignment response are discussed via comparing the numerical and experiment results to verify the efficiency of the model. The POD method is used for dimension reduction of rotor-bearing system, and the dynamical behaviors of the reduced-order model (ROM) are compared with the full order system (FOM) and experiment to verify the higher computational efficiency and accuracy of the POD method. These results in this paper can provide engineering guidance to actual dual rotor-bearing system with coupling misalignment fault. [ABSTRACT FROM AUTHOR]