The large-scale structure systems in engineering are complex, high dimensional, and variety of physical mechanism couplings; it will be difficult to analyze the dynamic behaviors of complex systems quickly and optimize system parameters. Model order reduction (MOR) is an efficient way to address those problems and widely applied in the engineering areas. This paper focuses on the model order reduction of high-dimensional complex systems and reviews basic theories, well-posedness, and limitations of common methods of the model order reduction using the following methods: center manifold, Lyapunov–Schmidt (L-S), Galerkin, modal synthesis, and proper orthogonal decomposition (POD) methods. The POD is a powerful and effective model order reduction method, which aims at obtaining the most important components of a high-dimensional complex system by using a few proper orthogonal modes, and it is widely studied and applied by a large number of researchers in the past few decades. In this paper, the POD method is introduced in detail and the main characteristics and the existing problems of this method are also discussed. POD is classified into two categories in terms of the sampling and the parameter robustness, and the research progresses in the recent years are presented to the domestic researchers for the study and application. Finally, the outlooks of model order reduction of high-dimensional complex systems are provided for future work.

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A Review of Model Order Reduction Methods for Large-Scale Structure Systems

Nonlinear Analysis and Characteristic Variation of Self-Excited Vibration in the Vertical Rotor System Due to the Flexible Support of the Journal Bearing

We developed a turbo compressor that has water-lubricated bearings driven at 30,000 rpm in a saturation condition, where the ambient pressure is at the saturation point of the discharged lubricant water. The bearings are supported with nonlinear elastomeric O-rings. At rotational speed exceeding 15,000 rpm, the rotor showed many subharmonic vibrations that are nonlinear phenomena unpredictable from a linear equation of motion. Instead, a stability analysis with a bifurcation diagram is an effective method to tackle these problems. In this paper, we investigated these rotor vibrations by bifurcation diagrams of the vibrations measured in experiments of saturated water journal bearings. The angular velocity was used as a bifurcation parameter. The bifurcations among synchronous, subharmonic, and chaotic vibrations were shown. Next, the nonlinear dynamics of the rotating rigid shaft were analyzed numerically with the nonlinear stiffness obtained by a commercial code that utilizes the two-dimensional (2D) Reynolds equation. The dynamic properties of the supporting structure were modeled with a complex stiffness coefficient. As a result, a Hopf bifurcation was found and a subharmonic limit cycle appeared spontaneously as observed in the experiments. The parametric studies revealed the influences of the dynamic properties of the structural components, especially the sensitive effect of the damping of the bearing support on the onset frequency and the amplitude of these vibrations. Furthermore, linear eigenvalue analysis of the motion equations clarified the mechanism of the sensitive effects.

Nonlinear Vibration of Saturated Water Journal Bearing and Bifurcation Analysis

Investigating dynamics and stability of rotors supported on journal bearings is a crucial step in the design of an efficient and reliable rotating machine. In the current work, a model for flexible rotor supported on two symmetric journal bearings is investigated. The nonlinear bearing forces are evaluated by either using direct solution of Reynolds equation or analyzing Reynolds equation to obtain linear and nonlinear bearing stiffness and damping coefficients using time dependent second order perturbation method. These coefficients are obtained for different operating conditions and bearing parameters such as length to diameter ratio, groove angle or applied groove pressure. The present results are validated with the previous literature and a perturbation analysis is used to investigate the validity range of the bearing linear and nonlinear coefficients. A novel technique based on polynomial fitting is used to present the bearing coefficient as a function of the bearing parameters. This enables the investigation of the dynamics of flexible rotor model using numerical continuation technique. Also, the effect of the bearing design parameters such as groove angle, length to diameter ratio and static pressure on the system stability is investigated. • Nonlinear journal bearing coefficients are investigated. • Time dependent perturbation method is used to obtain bearing coefficients. • Parametric analysis to the journal bearing coefficients is introduced. • Continuation analysis to the dynamics of rotor supported on Journal bearing is introduced. 

Nonlinear dynamics and bifurcation analysis of journal bearings based on second order stiffness and damping coefficients

Hydrodynamic journal bearings are used in many applications which involve high speeds and loads. However, they are susceptible to oil whirl instability, which may cause bearing failure. In this work, a flexible Jeffcott rotor supported by two identical journal bearings is used to investigate the stability and bifurcations of rotor bearing system. Since a closed form for the finite bearing forces is not exist, nonlinear bearing stiffness and damping coefficients are used to represent the bearing forces. The bearing forces are approximated to the third order using Taylor expansion, and infinitesimal perturbation method is used to evaluate the nonlinear bearing coefficients. The mesh sensitivity on the bearing coefficients is investigated. Then, the equations of motion based on bearing coefficients are used to investigate the dynamics and stability of the rotor-bearing system. The effect of rotor stiffness ratio and applied load on the Hopf bifurcation stability and limit cycle continuation of the system are investigated. The results of this work show that evaluating the bearing forces using Taylor’s expansion up to the third-order bearing coefficients can be used to profoundly investigate the rich dynamics of rotor-bearing systems.

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Bifurcation analysis of rotor/bearing system using third-order journal bearing stiffness and damping coefficients

Nonlinear Analysis of Bifurcation Phenomenon for a Simple Flexible Rotor System Supported by a Full-Circular Journal Bearing

Journal bearing has been widely used in the wide range of rotating machineries to support large loads and to add significant damping to the system. Conventionally, its fluid film force is represented by the linear model of spring and damper around its equilibrium position in the vibration analysis of the rotor system. However, the fluid film force of the journal bearing is essentially nonlinear, and it is necessary to consider its nonlinearity to expect the characteristics of the limit cycle at around the instability point. This paper investigated the order reduction and bifurcation analysis of a flexible rotor system supported by a full circular journal bearing. The fluid film force is derived under the conditions of both infinitesimal length approximation and half-Zommerfeld boundary condition, and its polynomial function approximation expression is used. Order reduction in the FEM rotor model retaining the nonlinearity of the journal bearing was performed by utilizing the substructure synthesis method. Then, its bifurcation phenomena at around the instability point are investigated by applying the center manifold theory and using the normal form theory. The influences of various parameters, such as kinematic viscosity, bearing length, and the disk position, on the bifurcation phenomena at around the instability point were investigated and explained. Furthermore, the validity of the derived analytical observation was confirmed by numerical simulation and experiment. By invoking these analytical techniques and obtained results, the bifurcation characteristics can be expected theoretically at the design stage of the journal bearing and rotor system.

Order reduction and bifurcation analysis of a flexible rotor system supported by a full circular journal bearing

This study investigates the asynchronous vibration of the flexible rotor supported by the full-circular journal bearing. The numerical method called shooting method is used, and both the stable and the unstable solutions of the asynchronous vibration are obtained. The dynamical characteristics of asynchronous vibration are clarified and explained. Particularly, the rotational speed range in which the rotor state may enter to the large amplitude asynchronous vibration by impact even if the rotational speed is lower than the limit of stability of the linearized system is shown and explained. Moreover, it is shown that the unstable solution acts as the threshold whether the state enters to the large amplitude asynchronous vibration or stays in the static equilibrium condition with zero amplitude. Also, the continuous transition of the frequency of the asynchronous vibration from the oil whirl to the oil whip after the saddle-node bifurcation from the unstable solution blanch is explained

Nonlinear Analysis of an Asynchronous Vibration in the Flexible Rotor Supported by the Full-Circular Journal Bearing under the Short Bearing Model and Half Sommerfeld Boundary Condition

Hiroshi Kano

Papers

Nonlinear Analysis of Bifurcation Phenomenon for a Simple Flexible Rotor System Supported by a Full-Circular Journal Bearing

Order reduction and bifurcation analysis of a flexible rotor system supported by a full circular journal bearing

Nonlinear Analysis of an Asynchronous Vibration in the Flexible Rotor Supported by the Full-Circular Journal Bearing under the Short Bearing Model and Half Sommerfeld Boundary Condition