Three to one internal resonances of a pre-deformed rotating beam with quadratic and cubic nonlinearities

Highlights

• The evolutions of steady state solution branches with parameters are presented.

• The stability regions are obtained for different dynamic models.

• The analytical results are supported by the numerical simulation.

Abstract

The primary resonances of a pre-deformed rotating beam model including the quadratic and cubic nonlinearities are investigated in the presence of the 3:1 internal resonance. The steady state responses of the beam are analyzed in two cases of the primary resonance with the method of multiple scales. The original dynamic equation is integrated numerically in two frequency sweep directions. The theoretical results are consistent with those obtained in the numerical simulation. The contributions of quadratic nonlinearities and cubic nonlinearities to the primary resonances behavior of the rotating beam are clarified. The frequency response curves are discussed by considering different model parameters such as the thermal gradient, the rotating speed, the damping coefficient and the gas pressure. The stability regions of coupled mode solutions are compared between the models with different nonlinearities in the case of the primary resonance of the second mode. A series of interesting findings are presented.

Introduction

A rotating beam is a typical structure in modern industries such as aircraft engines, gas turbines, robotic manipulators, helicopter propellers et al.. The investigations on the dynamic behaviors of a rotating beam can be traced back to Sutherland [1]. Rafiee et al. [2] reported a comprehensive review of the research progress in the field of rotating beam vibrations and controls in the past decades.

In traditional research work of this field [3], [4], [5], [6], [7], [8], [9], [10], [11], researchers provided with various rotating blade dynamic models in the framework of linear vibration theory. However, the highly unsteady and turbulent gas flow through the turbines results in the large amplitude vibration and even high-cycle fatigue failure of the rotating blades [12]. It is reasonable to apply the nonlinear von-Karman formula to describe the relationship between the displacement and the strain. Consequently, cubic nonlinearities appear in the transverse (including flapping and chordwise) vibration equations. Zhou et al. [13] studied the dynamic characteristics of a rotating functionally graded beam based on von Karman geometric nonlinearity theory. Thomas et al. [14] showed the relationship between the hardening/softening behavior and the rotating speed of a rotating beam under the framework of the large amplitude nonlinear vibration theory. Sina and Haddadpour [15] derived the nonlinear axial-torsional equations of motion of a rotating composite beam based on the nonlinear strain–displacement relations. Yao and coauthors [16], [17] built a nonlinear dynamic model and demonstrated the chaotic motion in rotating plates and rotating cylindrical shells, respectively. Wang and Zhang [18] addressed the dynamic stability of a rotating blade with a varying speed. Fang and Zhou [19] performed the modal analysis of a rotating tapered beam based on Timoshenko Beam Theory. Baghani et al. [20] clarified the effect of the rotating speed, surface elastic constants and the small-scale parameter on the stability of a rotating nanobeam. The contributions of the centrifugal and gyroscopic forces to the dynamic characteristics of rotating beams were clarified in Yang’s works [21], [22]. Guo et al. [23] presented the overall contour of the complex modal vibrations of a rotating tapered Timoshenko beam. Georgiades et al. [24] established the dynamic model of a rotating composite beam considering the nonconstant rotating speed and arbitrary preset angle based on Hamilton principle. Lin et al. [25] investigated the effect of the elastically restrained root on the dynamic characteristics of a rotating composite beam. Li et al. [26] applied Love’s nonlinear shell theory to build the dynamic model of a rotating composite shell considering the effects of hygrothermal environment. Niu et al. [27] derived the equation of motion of a rotating blade reinforced with graphene platelets and investigated the variation of natural frequencies with material properties and geometry parameters. Turhan and Bulut [28] addressed the nonlinear vibration of a rotating beam discarding the probability of internal resonance. Arvin and Bakhtiari-Nejad [29] used method of direct multiple scales to analyze the nonlinear free vibration of a rotating composite beam based on Timoshenko assumptions.

For nonlinear rotating blade dynamic models, a series of interesting physical phenomena arise in the dynamic response when some of the natural frequencies are commensurable, namely, the existence of an internal resonance [30], [31]. Since 2010s, the internal resonance of rotating blade has become a major area of interest in the field of nonlinear vibration of rotating blades. Lacarbonara et al. [32] derived the dynamic equation of a rotating blade and numerically determined the rotating speeds at which potential 1:1, 2:1, 3:2 and 3:1 internal resonances may occur. Serval researchers [33], [34], [35] have applied the invariant manifold to build the nonlinear normal modes for rotating blades with 3:1 internal resonances. Arvin and Bakhtiari-Nejad [36] investigated the stabilities of nonlinear normal modes of a rotating beam in the case of 3:1 and 2:1 internal resonance. Sina et al. [37] investigated the 3:1 internal resonance and nonlinear normal modes of thin-walled beam. Wang et al. [38], [39] studied the bifurcation of the vortex-induced vibrations for turbine blades in the presence of 1:1 internal resonance. Yao et al. established the nonlinear dynamic governing equation of a rotating thin-walled beam subjected by gas flow, and investigated the nonlinear behavior of the rotating beam in the case of 1:1 internal resonance [40] and 2:1 internal resonance [41], respectively. Kandil and El-Gohary [42] designed a nonlinear saturation controller making use of the saturation phenomenon in the internal resonance to suppress the vibration of the rotating beam, which was adopted from Yao’s rotating blade model. Zou and coauthors derived the equation of motion of a propulsion shafting and investigated the nonlinear dynamic behavior of the rotating shaft without internal resonance [43] and with internal resonances [44], respectively.

The turbine blades often serve under a huge thermal gradient which can deform the central axis of the blades. As a result, the quadratic nonlinearities are introduced into the transverse vibration equations. A better understanding the dynamic behavior of pre-deformed rotating blade lies a foundation for vibration suppression [45], [46] of such structure. Invernizzi and Dozio [47] reported that the pre-deformation of a rotating blade can strongly affect the natural frequencies of the blade. Zhang and Li [48] established a novel nonlinear dynamic model for a rotating blade considering the pre-deformation raised from the thermal gradient and demonstrated the possibility of 2:1 internal resonance in the model. Zhang et al. [49] investigated the saturation phenomenon of the pre-deformed blade and proved that the cubic nonlinearities are ignorable when investigate the 2:1 internal resonance. Recently, Zhang, Liu and Siriguleng [50] studied the dynamic response of a rotating laminated composite blade in the case of 2:1 internal resonance and the author reported the interesting saturation phenomena in the rotating composite blade. Zhang et al. [51] presented the super-harmonic response of a blade based on the pre-deformed blade model [48]. Obviously, it is of great significance to study the dynamic response of pre-deformed rotating blade. However, in the previous research works on the dynamic behavior of rotating blades in presence of 3:1 internal resonance mentioned above, the blade axis is always assumed as a straight line. Consequently, these studies are limited by the absence of quadratic nonlinearities in the transverse vibration equations. Up to now, far too little attention has been paid to the 3:1 internal resonance of rotating blades considering the contributions of quadratic and cubic nonlinearities simultaneously. The primary aim of this study is to develop a better understanding of the contributions of quadratic nonlinearities to the 3:1 internal resonance behavior of the rotating blade. Zhang’s rotating pre-deformed blade model [48] is adopted in the present study.

An outline of the subsequent paper is as follows. Section 2 begins with the description of the dynamic model of rotating beam considered in the present paper and then presents the theoretical analysis of the steady state response of the rotating beam with the 3:1 internal resonance based on the method of multiple scales. Section 3 is divided into two distinct parts to illustrate the dynamic behavior of the rotating beam with the 3:1 internal resonance in the case of first primary resonance and the second primary resonance, respectively. Section 4 summarizes the main findings of this research work.