In this paper, the Nonlinear Auto-Regressive with exogenous inputs (NARX) model with parameters of interest for design (NARX-M-for-D), where the design parameter of the system is connected to the coefficients of the NARX model by a predefined polynomial function is studied. For the NARX-M-for-D of nonlinear systems, in practice, to predict the output by design parameter values are often difficult due to the uncertain relationship between the design parameter and the coefficients of the NARX model. To solve this issue and conduct the analysis and design, an improved algorithm, defined as the Weighted Extended Forward Orthogonal Regression (WEFOR), is proposed. Firstly, the initial NARX-M-for-D is obtained through the traditional Extended Forward Orthogonal Regression (EFOR) algorithm. Then a weight matrix is introduced to modify the polynomial functions with respect to the design parameter, and then an improved model, which is referred to as the final NARX-M-for-D is established. The genetic algorithm (GA) is used for deriving the weight matrix by minimizing the normalized mean square error (NMSE) over the data sets corresponding to the design parameter values used for modeling and first prediction. Finally, both the numerical and experimental studies are conducted to demonstrate the application of the WEFOR algorithm. The results indicate that the final NARX-M-for-D can accurately predict the system output of a nonlinear system. The new algorithm is expected to provide a reliable model for dynamic analysis and design of the nonlinear system.
This article presents a nonlinear vibration characteristics study of a bolted joint dual-rotor system. The motion equations are derived through the lumped mass modeling method and a two-node bolted joint element. Nonlinear time-varying bending stiffness at the joint interface is considered in the numerical integration. Qualitative analysis of the effect of preload on system responses is conducted through changing the value of the transition point of bending stiffness of the bolted joint. Moreover, the transfer path of vibration in the dual-rotor system through inner-shaft bearing was discussed in the present work. The nonlinear differential equations are solved using the Newmark integration method to predict the dynamic characteristics of the dual-rotor system. The results show that the maximum vibration displacement of LP rotor is positively correlated with that of HP rotor as preload changes. Moreover, the maximum amplitude of the time-domain responses of the HP rotor will decrease and the minimum amplitude will increase with the increase of preload. The difference between the maximum and minimum values of the time-domain response will decrease with the increase of preload. This can be explained by the “stiffness hardening” phenomenon of the bolted joint. The research results can help understanding the dynamic properties of the bolted joint dual-rotor system and the vibration transfer path of the rotor system through inner-shaft bearing.
Bolted joints are widely used in aeroengine rotor systems to connect multiple components into an integrated structure and provide sufficient stiffness. The mechanical properties of a bolted joint have a significant effect on rotor dynamics. For modern aeroengine designs, the blade-tip clearance is gradually reduced to improve efficiency, which may lead to rubbing damage and affect safe operation. The mechanical properties of a bolted joint change significantly during the blade–casing rubbing process and influence the dynamic properties of the rotor system. Based on the finite element (FE) modeling method, a 15-node bolted joint rotor system model is established in this paper, in which the bolted joint is represented by a 2-node joint element, and the blade–casing rubbing force is considered. The Newmark method is used to solve the motion equations. The dynamic model is validated by comparing the frequency response characteristics for different numbers of blades with the results provided in other published studies. Based on the established model, the effects of the rotational speed, number of blades, and rubbing stiffness on the dynamic responses, normal rubbing forces, and bending stiffness of the bolted joint are evaluated by numerical simulation. The results show that the response amplitude and bending stiffness of the bolted joint change significantly under blade–casing rubbing faults, and the mean value of the vibration response deviates significantly from 0 as the number of blades increases. Meanwhile, the amplitude of the frequency component fVC and the maximum value of the normal rubbing force also increase as the number of blades increases. The main contribution of this paper is the establishment of a new model for a bolted joint rotor system, considering the time-varying bending stiffness of the bolted joint and the blade–casing rub fault, comparing the simulation results to obtain some general results bridging the current research gap. Meanwhile, the numerical results in this paper can provide a cognitive basis for the blade–casing rubbing fault mechanism of a bolted joint rotor system under the influence of speed, number of blades, and rubbing stiffness. The nonlinear dynamic characteristics observed in the present paper can be applied to the blade–casing rubbing fault diagnosis of turbomachines.
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