Nonlinear Dynamics of the High-Speed Rotating Plate
High speed rotating blades are crucial components of modern large aircraft engines. The rotating blades under working condition frequently suffer from the aerodynamic, elastic and inertia loads, which may lead to large amplitude nonlinear oscillations. This paper investigates nonlinear dynamic responses of the blade with varying rotating speed in supersonic airflow. The blade is simplified as a pre-twist and presetting cantilever composite plate. Warping effect of the rectangular cross-section of the plate is considered. Based on the first-order shear deformation theory and von-Karman nonlinear geometric relationship, nonlinear partial differential dynamic equations of motion for the plate are derived by using Hamilton’s principle. Galerkin approach is applied to discretize the partial differential governing equations of motion to ordinary differential equations. Asymptotic perturbation method is exploited to derive four-degree-of-freedom averaged equation for the case of 1 : 3 internal resonance-1/2 sub-harmonic resonance. Based on the averaged equation, numerical simulation is used to analyze the influence of the perturbation rotating speed on nonlinear dynamic responses of the blade. Bifurcation diagram, phase portraits, waveforms and power spectrum prove that periodic motion and chaotic motion exist in nonlinear vibration of the rotating cantilever composite plate.
High-speed rotating blades are essential components of modern large aircraft engines. Blades are designed with large aspect ratios and thin-wall structure in order to raise operational efficiency. In the actual working condition, the rotating blades are frequently subjected to the aerodynamic, elastic, and inertia loads. Various types of excitation lead to large amplitude nonlinear parametric vibrations of the blades, which can result in the resonance phenomena and undesirable disasters, especially when the rotating blades operate with high speed and huge centrifugal force. According to the investigation, the vibration failure of the aircraft engine is more than 60% of the total failure, while the vibration failure of the blade accounts for more than 70% of the total vibration failure. Resonance and flutter produced by the forced vibration and self-excited vibration are the main reason leading to the blade failure. So, it is very important that a reasonable model is established to accurately predict nonlinear vibration characteristics and other complex dynamics of the blade.
In recent years, considerable attention has been given to the studies on the vibration characteristics of rotating blades. Lin et al.  deduced governing differential equations and general elastic boundary conditions of a nonuniform pretwist Timoshenko beam by using Hamilton’s principle. Yao et al.  investigated the nonlinear dynamic responses of the thin-walled rotating cantilever beam. Sarkar and Ganguli  assumed the modal function as polynomials satisfying all four boundary conditions and discussed free vibration of the nonhomogeneous Timoshenko beam. Georgiades et al.  derived equations of motion of a rotating composite Timoshenko beam by utilizing Hamilton’s principle. Xie et al. [5, 6] numerically investigated the effect of symmetric and asymmetric shroud gaps, rotational speeds, and the aerodynamic force amplitude on dynamic characteristics of the rotating Euler-Bernoulli beam with a mass point at the free end. Huang and Kuang  investigated the effect of a near root local blade crack on the stability of a bladed disk, in which an individual blade is modeled simply as a cantilever Euler-Bernoulli beam.
Currently, plenty of theoretical analyses are focused on the beam model of the blade, such as the simple Euler-Bernoulli beam and the pretwist Timoshenko beam. But, theoretical studies on the plate model of the blade are still few. Young and Liou  established equations of a rotating cantilever plate with a time-varying speed and numerically investigated the effect of the Coriolis force on boundaries of the unstable regions. Yoo and Kim  derived linear equations of motion and analyzed free vibration of rotating cantilever plates. Sinha and Turner  derived governing partial differential equations of motion for a rotating pretwist plate to investigate the static and dynamic frequencies of the blade. Li and Zhang  presented a dynamic model of a functionally graded rectangular plate undergoing large overall motions. Kouchakzadeh et al.  analyzed the nonlinear aeroelasticity of a laminated composite plate in supersonic airflow by using the classical plate theory. Malgaca et al.  tried to control the vibration of the rotating blade at different speeds by utilizing root-embedded piezoelectric materials. Kam et al.  proposed a procedure to analyze the structural failure of a composite wind blade subjected to quasi-static loads. Tang and Chen  presented the solvability conditions of nonlinear partial differential equations for in-plane moving plates of two cases, in which the one case is without internal resonance and another case is under internal resonances, respectively. Sun et al.  used a quadratic layerwise theory and a new dynamic model to study dynamic behaviors for a multilayer pretwist rotating blade. Wang et al.  established a time-dependent nonlinear model of a flexible blade-rotor-bearing system by using the Lagrange method. Shariyat et al.  performed the nonlinear dynamic analysis of rectangular composite plates. Zhang and Li  adopted the Lagrangian method to acquire dynamic equations of motion for the pretwist and predeformed rotating cantilever plate subjected to the harmonic aerodynamic force. Mendonça et al.  considered internal damping in the shaft to study dynamic behaviors of the rotors mounted on composite shafts. Banichuk et al. [21, 22] investigated the stability and bifurcation of the rotating blade under different conditions.
Based on the shallow shell theory and the Ritz method, Leissa et al.  explored frequencies and mode shapes of turbomachinery blades with the coupling of bending and twist. Kee and Kim  assumed blades as the moderately thick open cylindrical shell models. Sun et al.  applied the general shell theory to investigate the influence of parameters on natural frequency and damping characteristics of the shell model blade. Sinha and Zylka  simplified the rotating pretwisted turbomachinery cantilevered airfoil as an anisotropic shell and derived the free vibration equations of motion for the transverse deflection of the shell including the warping effect. Volker and Joachim  analyzed the aeroelastic phenomena of twenty compressor blades simplified as the spring-damper models mounted on the hub. Ekici et al.  proposed a nonlinear harmonic balance method to compute the unsteady self-excited aerodynamic of asymmetry turbomachinery blades. Farhadi and Hosseini-Hashemi  studied aeroelastic behaviors of a rotating thick plate in the supersonic airflow. Lachenal et al.  presented the design, analysis, and realization of a zero stiffness twist morphing wind turbine blade subjected to the gust loads.
Dynamic modeling of the rotating blade requires an accurate expression of the aerodynamic force. But, the physical mechanism of aerodynamic interaction has been mainly investigated by experimental methods. Models have been proposed to deal with aerodynamic interaction problems, which is done with by the piston theory widely. Ashley and Zartarian  firstly proposed the quasi-steady piston theory to deal with aerodynamic interaction problems. Navazi and Haddadpour  studied the thermal stability of the functionally graded plate subjected to the aerodynamic load obtained from the first-order piston theory. According to large deformation geometric relationship, the piston theory and the quasi-static thermal stress theory, Yuan and Qiu  established the aerodynamic model of a composite stiffened panel and used Hamilton’s principle to derive the equations of motion for the system. Yang et al.  applied a modified local piston theory to analyze aeroelastic behaviors of curved panels.
Although extensive studies have been carried out on rotating cantilever beams, studies on nonlinear dynamic responses of the pretwist, presetting rotating cantilever plates are still few. In this paper, nonlinear dynamic behaviors of the blade with varying rotating speeds under the supersonic airflow are investigated. Considering the shear deformation and the warping effect, equations of motion for the cantilever plate are derived by using Hamilton’s principle. The Galerkin approach is applied to discretize the partial differential governing equations of motion to ordinary differential equations. The asymptotic perturbation method is exploited to obtain averaged equations of the system in the case of 1 : 3 internal resonance-1/2 subharmonic resonance. Based on the averaged equations, numerical simulation is applied to investigate the bifurcation and chaotic dynamics of the rotating cantilever composite plate. In order to analyze the internal resonance, we choose the perturbation rotating speed as the controlling parameter to investigate nonlinear behaviors of the pretwist and presetting rotating cantilever plate. From the results of the numerical simulation, it is found that the system performs periodic and chaotic motions under specific conditions. It is observed that the perturbation rotating speed has a significant influence on the nonlinear dynamic behaviors of the rotating plate. Since we can control the responses of the system from the chaotic motions to the periodic motions by changing the perturbation rotating speed, we can control the large amplitude nonlinear vibrations of the blade.
2. Equations of Motion for the Rotating Cantilevered Blade
The schematic diagram of the rotating cantilever blade is shown in Figure 1. The blade is simplified as a pretwist and presetting rotating cantilever plate in the following dynamical analysis. Rotating speed of the blade is , where is the steady-state rotating speed and is the periodic perturbation. The shape of the blade is a rectangular plate, which is characterized by the span length , the chord length , and the thickness . The cross-section of the cantilever plate is shown in Figure 2. The plate is clamped to a rigid hub of radius . There are a presetting angle in the fixed end and a pretwist angle at the free end. is the inertial frame and the origin of it is in the center of the hub. At the edge of the hub, there is the rotating coordinate system . We define that the spanwise direction is , the chordwise direction is , and the thickness direction is . Another local triad system attached to the free end of the plate is introduced on the plate, which is called the sectional coordinate system.
The twist rate of the blade is expressed as and the twist angle of the blade is written as
Based on the Kirchhoff hypothesis, the first-order shear deformation theory including the warping effect is considered to establish the displacement field. Displacements of any point along , , and directions can be expressed by the displacement of the neutral plane of the plate as follows : where
Nonlinear strains of the von Karman plate theory are given as where is the membrane strain and is the flexural strain.
For the advanced fiber-reinforced composite material blade, the constitutive relation of the composite plate is expressed as follows: where where is the longitudinal modulus of the fiber, is the transverse modulus, and are the major Poisson’s ratio, respectively, and , , and are the shear modulus, respectively.
In-plane force resultants , , and ; the moment resultants , , and ; and the transverse forces and are written in matrices as follows: where represents the shear coefficient, is called the extensional stiffness, is the bending stiffness, and is the bending-extensional coupling stiffness, which are defined as , , and are mass moments of inertia, which are defined as
The rotating plate mounted in the hub is rotating with the variable speed. The displacement of a random point without deformation on the plate is expressed as and the deformation displacement of a random point on the rotating plate is written as where , , and denote displacement components along , , and directions, respectively.
When the blade rotates with a constant angular velocity , the transient angle can be written as
The instantaneous direction of the local unit vector for any typical point on the blade with respect to the stationary global Cartesian unit vector is given by
The speed and the acceleration of any random point on the plate are described as follows: where
The variation of the kinetic energy is given by
A large centrifugal force will be generated when the compressor blade of the aircraft engine operates with the high rotating speed. Components of centrifugal forces along the spanwise and chordwise directions are described by and , which can be expressed as
The variation of the potential energy consists the variation of the strain energy and the variation of the centrifugal potential energy , namely, , where
Aerodynamic forces of the blade derived by the first-order piston theory are written as where is the air density, is the speed of sound, is the speed of the airflow, and , .
The variation of the virtual work of aerodynamic forces is expressed as follows:
Nonlinear dynamic equations of motion for the rotating blade are established by using Hamilton’s principle.
Substituting (11), (13a), (13b), and (15) into (16), the nonlinear partial governing equations for nonlinear vibration of the rotating blade expressed in terms of displacement variables can be derived as follows:
The boundary conditions are given as follows:
The dimensionless variables and parameters are introduced as follows:
Substituting (23) into (17), (18), (19), (20), and (21), we obtain the dimensionless governing equations of motion for the blade.
Based on the practical working condition of the blade, and theoretical and numerical studies given by Dowell et al. [36, 37], it is known that vibrations of the first two order modes for the blade play an important role during vibration. The Galerkin approach is applied to obtain ordinary differential equations of motion in the dimensionless form. The Galerkin approach is derived by the Taylor expansion method, which is a mathematically convergent method. The first two order mode functions can produce convergent results. Mode functions of the plate , , ,