# Nonlinear Dynamics of the High-Speed Rotating Plate

Authors: Minghui Yao; Li Ma; Wei Zhang
Journal: International Journal of Aerospace Engineering
 (2018)

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## Abstract

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.


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## Body

## 1. Introduction

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. [1] deduced governing differential equations and general elastic boundary conditions of a nonuniform pretwist Timoshenko beam by using Hamilton’s principle. Yao et al. [2] investigated the nonlinear dynamic responses of the thin-walled rotating cantilever beam. Sarkar and Ganguli [3] assumed the modal function as polynomials satisfying all four boundary conditions and discussed free vibration of the nonhomogeneous Timoshenko beam. Georgiades et al. [4] 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 [7] 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 [8] 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 [9] derived linear equations of motion and analyzed free vibration of rotating cantilever plates. Sinha and Turner [10] 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 [11] presented a dynamic model of a functionally graded rectangular plate undergoing large overall motions. Kouchakzadeh et al. [12] analyzed the nonlinear aeroelasticity of a laminated composite plate in supersonic airflow by using the classical plate theory. Malgaca et al. [13] tried to control the vibration of the rotating blade at different speeds by utilizing root-embedded piezoelectric materials. Kam et al. [14] proposed a procedure to analyze the structural failure of a composite wind blade subjected to quasi-static loads. Tang and Chen [15] 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. [16] used a quadratic layerwise theory and a new dynamic model to study dynamic behaviors for a multilayer pretwist rotating blade. Wang et al. [17] established a time-dependent nonlinear model of a flexible blade-rotor-bearing system by using the Lagrange method. Shariyat et al. [18] performed the nonlinear dynamic analysis of rectangular composite plates. Zhang and Li [19] 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. [20] 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. [23] explored frequencies and mode shapes of turbomachinery blades with the coupling of bending and twist. Kee and Kim [24] assumed blades as the moderately thick open cylindrical shell models. Sun et al. [25] 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 [26] 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 [27] analyzed the aeroelastic phenomena of twenty compressor blades simplified as the spring-damper models mounted on the hub. Ekici et al. [28] proposed a nonlinear harmonic balance method to compute the unsteady self-excited aerodynamic of asymmetry turbomachinery blades. Farhadi and Hosseini-Hashemi [29] studied aeroelastic behaviors of a rotating thick plate in the supersonic airflow. Lachenal et al. [30] 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 [31] firstly proposed the quasi-steady piston theory to deal with aerodynamic interaction problems. Navazi and Haddadpour [32] 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 [33] 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. [34] 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 Figure1. The blade is simplified as a pretwist and presetting rotating cantilever plate in the following dynamical analysis. Rotating speed of the blade is Ωt=Ω0+Ω1cosω~t, where Ω0 is the steady-state rotating speed and Ω1cosω~t is the periodic perturbation. The shape of the blade is a rectangular plate, which is characterized by the span length L, the chord length C, and the thickness h. The cross-section of the cantilever plate is shown in Figure 2. The plate is clamped to a rigid hub of radius r. There are a presetting angle θr in the fixed end and a pretwist angle θR at the free end. i⇀,j⇀,k⇀ 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 e⇀x0,e⇀y0,e⇀z0. We define that the spanwise direction is e⇀x0, the chordwise direction is e⇀y0, and the thickness direction is e⇀z0. Another local triad system e^x,e^y,e^z attached to the free end of the plate is introduced on the plate, which is called the sectional coordinate system.Figure 1
The model of the pretwist rotating cantilever rectangular plate.Figure 2
The section of the rotating cantilever plate.The twist rateθ′ of the blade is expressed as
(1a)θ′=θR−θrR−r,and the twist angle θx of the blade is written as
(1b)θx=θr+θ′x.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 alongx, y, and z directions can be expressed by the displacement of the neutral plane of the plate as follows [35]:
(2a)ux,y,z,t=u0x,y,t+zφxx,y,t−zθ′2xω0,(2b)vx,y,z,t=v0x,y,t+zφyx,y,t,(2c)wx,y,z,t=w0x,y,t,where
(2d)ϕx=γxz−∂w0∂x,ϕy=γyz−∂w0∂y.Nonlinear strains of the von Karman plate theory are given as(3)εxxεyyγyzγxzγxy=ε0+zε1=∂u0∂x+12∂w0∂x2∂v0∂y+12∂w0∂y2∂w0∂y+φy∂w0∂x+φx∂u0∂y+∂v0∂x+∂w0∂x∂w0∂y+z∂φx∂x−θ′2w0−θ′2xw0,x∂φy∂y00∂φx∂y+∂φy∂x,where ε0 is the membrane strain and ε1 is the flexural strain.For the advanced fiber-reinforced composite material blade, the constitutive relation of the composite plate is expressed as follows:(4a)σxxσyyσxy=Q¯11Q¯12Q¯16Q¯12Q¯22Q¯26Q¯16Q¯26Q¯66εxxεyyγxy,(4b)σyzσxz=kQ¯44Q¯45Q¯45Q¯55γyzγxz,where
(4c)Q11=E11−γ12γ21,Q12=γ12E21−γ12γ21,Q22=E21−γ12γ21,Q66=G12,Q44=G23,Q55=G13,where E1 is the longitudinal modulus of the fiber, E2 is the transverse modulus, γ12 and γ21 are the major Poisson’s ratio, respectively, and G12, G23, and G13 are the shear modulus, respectively.In-plane force resultantsNxx, Nyy, and Nxy; the moment resultants Mxx, Myy, and Mxy; and the transverse forces Qx and Qy are written in matrices as follows:
(5a)NM=ABBDε0ε1,(5b)QyQx=kA44A45A45A55∂w0∂y+ϕy∂w0∂x+ϕx,where k represents the shear coefficient, Aij is called the extensional stiffness, Dij is the bending stiffness, and Bij is the bending-extensional coupling stiffness, which are defined as
(6a)Aij,Bij,Dij=∫−h/2h/2Qij1,z,z2dz,I0, I1, and I2 are mass moments of inertia, which are defined as
(6b)I0,I1,I2=∫−h/2h/2ρ1,z,z2dz.The rotating plate mounted in the hub is rotating with the variable speed. The displacement of a random pointP without deformation on the plate is expressed as
(7a)r→x,y,z,t=R0+xe⇀x+ye⇀y+ze⇀z,and the deformation displacement of a random point P on the rotating plate is written as
(7b)R→x,y,z,t=x+R0+ue⇀x+y+ve⇀y+z+we⇀z.where u, v, and w denote displacement components along x, y, and z directions, respectively.When the blade rotates with a constant angular velocityω, the transient angle Θ can be written as
(8)Θ=∫0tΩ0+Ω1cosω~tdt.The instantaneous direction of the local unit vectore⇀x,e⇀y,e⇀z for any typical point on the blade with respect to the stationary global Cartesian unit vector i⇀,j⇀,k⇀ is given by
(9a)e→x0e⇀y0e⇀z0=cosΘsinΘ0−sinΘcosΘ0000i⇀j⇀k⇀,(9b)e→xe⇀ye⇀z=1000cosβx+θrsinβx+θr0−sinβx+θrcosβx+θre→x0e⇀y0e⇀z0.The speed and the acceleration of any random pointP on the plate are described as follows:
(10a)Ṙ=u̇cosΘ−x+R0+usinΘ⋅Ω−v̇cosθsinΘ−y+vcosθcosΘ⋅Ω+ẇsinθsinΘ+z+wsinθcosΘ⋅Ωi⇀+u̇sinΘ+x+R0+ucosΘ⋅Ω+v̇cosθcosΘ−y+vcosθsinΘ⋅Ω−ẇsinθcosΘ+z+wsinθsinΘ⋅Ωj⇀+v̇sinθ+ẇcosθk⇀,(10b)R¨=u¨cosΘ−2u̇sinΘ⋅Ω−x+R0+ucosΘ⋅Ω2−x+R0+usinΘ⋅Ω̇−v¨cosθsinΘ−2v̇cosθcosΘ⋅Ω+y+vcosθsinΘ⋅Ω2−y+vcosθcosΘ⋅Ω̇+w¨sinθsinΘ+2ẇsinθcosΘ⋅Ω−z+wsinθsinΘ⋅Ω2+z+wsinθcosΘ⋅Ω̇i⇀+u¨sinΘ+2u̇cosΘ⋅Ω−x+R0+usinΘ⋅Ω2+x+R0+ucosΘ⋅Ω̇+v¨cosθcosΘ−2v̇cosθsinΘ⋅Ω−y+vcosθcosΘ⋅Ω2−y+vcosθsinΘ⋅Ω̇−w¨sinθcosΘ+2ẇsinθsinΘ⋅Ω+z+wsinθcosΘ⋅Ω2+z+wsinθsinΘ⋅Ω̇j⇀+v¨sinθ+w¨cosθk⇀,where
(10c)θ=θx=θr+θ′x.The variation of the kinetic energy is given by(11)δK=−∫VρR¨δRdV=∫sI0u¨0cosΘ−v¨0cosθsinΘ+w¨0sinθsinΘ−2u̇0sinΘ+v̇0cosθcosΘ−ẇ0sinθcosΘ⋅Ω−x+R0+u0cosΘ−y+v0cosθsinΘ+w0sinθsinΘ⋅Ω2−x+R0+u0sinΘ−y+v0cosθcosΘ−w0sinθcosΘ⋅Ω̇⋅δu0+I0u¨0sinΘ−v¨0cosθcosΘ−w¨0sinθcosΘ+2u̇0cosΘ+v̇0cosθsinΘ+ẇ0sinθsinΘ⋅Ω−x+R0+u0sinΘ+y+v0cosθcosΘ−w0sinθcosΘ⋅Ω2+x+R0+u0cosΘ−y+v0cosθsinΘ−w0sinθsinΘ⋅Ω̇⋅δv0+I0v¨0sinθ+w¨0cosθ−I2θ′2xϕ¨x−θ′2xw¨0cosΘ−ϕ¨ycosθsinΘ−2ϕ̇x−θ′2xẇ0+ϕ̇ycosθcosΘ⋅Ω−ϕx−θ′2xw0cosΘ−ϕycosθsinΘ+sinθsinΘ⋅Ω2−ϕx−θ′2xw0sinΘ+ϕycosθcosΘ−sinθcosΘ⋅Ω̇δw0+I2ϕ¨x−θ′2xw¨0cosΘ−ϕ¨ycosθsinΘ−2ϕ̇x−θ′2xẇ0+ϕ̇ycosθcosΘ⋅Ω−ϕx−θ′2xw0cosΘ−ϕycosθsinΘ+sinθsinΘ⋅Ω2−ϕx−θ′2xw0sinΘ+ϕycosθcosΘ−sinθcosΘ⋅Ω̇δϕx+I2ϕ¨x−θ′2xw¨0sinΘ+ϕ¨ycosθcosΘ+2ϕ̇x−θ′2xẇ0cosΘ−ϕ̇ycosθsinΘ⋅Ω−ϕx−θ′2xw0sinΘ+ϕycosθcosΘ−sinθcosΘ⋅Ω2+ϕx−θ′2xw0cosΘ−ϕycosθsinΘ+sinθsinΘ⋅Ω̇δϕydxdy.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 byFs and Fc, which can be expressed as
(12a)Fs=∫xLρΩ2r+xdx,(12b)Fc=∫y±C/2ρΩ2ysin2θdy.The variation of the potential energyδU consists the variation of the strain energy δU1 and the variation of the centrifugal potential energy δU2, namely, δU=δU1+δU2, where
(13a)δU1=∫Ω0∫−h/2h/2σxxδεxx0+zδεxx1+σyyδεyy0+zδεyy1+σxyδγxy0+zδγxy1+σxzδγxz0+σyzδγyz0dzdxdy,(13b)δU2=∫−h/2h/2Fs12δ∂w∂x2+∂v∂x2+Fc12δ∂w∂y2+∂u∂y2dzdxdy.Aerodynamic forces of the blade derived by the first-order piston theory are written as(14a)Δpy=C∞ρ∞∂v∂t+Uyt∂v∂x=C∞ρ∞v̇+Psinθ∂v∂x,(14b)Δpz=C∞ρ∞∂w∂t+Uzt∂w∂x=C∞ρ∞ẇ+Pcosθ∂w∂x,where ρ∞ is the air density, C∞ is the speed of sound, P is the speed of the airflow, and Uyt=Psinθ, Uzt=Pcosθ.The variation of the virtual work of aerodynamic forces is expressed as follows:(15)δW=∫SΔpzδwdxdy+∫S−Δpyδvdxdz=∫SΔpzδw+∂z∂yΔpyδvdxdy.Nonlinear dynamic equations of motion for the rotating blade are established by using Hamilton’s principle.(16)∫0tδK−δU+δWdt=0.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:
(17)δu0:A11∂2u0∂x2+2A16∂2u0∂x∂y+A66∂2u0∂y2+A16∂2v0∂x2+A26∂2v0∂y2+A12+A66∂2v0∂x∂y−2B11θ′2∂w0∂x−B11θ′2x∂2w0∂x2+A11∂w0∂x∂2w0∂x2+A66∂w0∂x∂2w0∂y2+2A16∂w0∂x∂2w0∂x∂y+A16∂w0∂y∂2w0∂x2+A26∂w0∂y∂2w0∂y2+A12+A66∂w0∂y∂2w0∂x∂y−B16θ′2x∂2w0∂x∂y+B66∂2ϕx∂y2+2B16∂2ϕx∂x∂y+B11∂2ϕx∂x2+B16∂2ϕy∂x2+B26∂2ϕy∂y2+B12+B66∂2ϕy∂x∂y−B16θ′2∂2w0∂y=Iou¨0cosΘ−v¨0cosθsinΘ+w¨0sinθsinΘ−2u̇0sinΘ+v̇0cosθcosΘ−w0sinθcosΘ⋅Ω−x+R+0u0cosΘ−y+v0cosθsinΘ+w0sinθsinΘ⋅Ω2−x+R+u0sinΘ+y+v0cosθsinΘ+w0sinθcosΘ⋅Ω̇+2I0ysin2θ∂w0∂y⋅Ω2+2I0ysin2θ∂u0∂y⋅Ω2−I0sin2θC24−y2∂2w0∂y2⋅Ω2−I0sin2θC24−y2∂2u0∂y2⋅Ω2,(18)δv0:A16∂2u0∂x2+A26∂2u0∂y2+A12+A66∂2u0∂x∂y+A66∂2v0∂x2+A22∂2v0∂y2+2A26∂2v0∂x∂y−2B16θ′2∂w0∂x−B16θ′2x∂2w0∂x2+A16∂w0∂x∂2w0∂x2+A26∂w0∂x∂2w0∂y2+A12+A66∂w0∂x∂2w0∂x∂y+A22∂w0∂y∂2w0∂y2+A66∂w0∂y∂2w0∂x2+2A26∂w0∂y∂2w0∂x∂y+B16∂2ϕx∂x2+B26∂2ϕx∂y2+B12+B66∂2ϕx∂x∂y+B66∂2ϕy∂x2+B22∂2ϕy∂y2+2B26∂2ϕy∂x∂y−B12θ′2x∂2w0∂x∂y−B12θ′2∂w0∂y+C∞ρ∞v̇0+zϕ̇y+psinθ∂v0∂x+z∂ϕy∂x∂z∂y=I0u¨0sinΘ+v¨0cosθcosΘ−w¨0sinθcosΘ+2u̇0cosΘ−v̇0cosθsinΘ+ẇ0sinθsinΘ⋅Ω−x+R0+u0sinΘ+y+v0cosθcosΘ−w0sinθcosΘ⋅Ω2+x+R0+u0cosΘ−y+v0cosθsinΘ+w0sinθsinΘ⋅Ω̇−I0r+x∂w0∂x⋅Ω2−I0r+x∂v0∂x⋅Ω2+I0rL−x+12L2−x2∂2w0∂x2⋅Ω2−I0rL−x+12L2−x2∂2v0∂x2⋅Ω2,(19)δw0:32A11∂w0∂x2∂2w0∂x2+A66+12A12∂w0∂x2∂2w0∂y2+3A16∂w0∂x2∂2w0∂x∂y+3A16∂w0∂x∂w0∂y∂2w0∂x2+3A26∂w0∂x∂w0∂y∂2w0∂y2+2A12+3A66∂w0∂x∂w0∂y∂2w0∂x∂y−3B11θ′2x∂w0∂x∂2w0∂x2−4B16θ′2x∂w0∂x∂2w0∂x∂y−3B16θ′2x∂w0∂x∂w0∂y+32A22∂w0∂y2∂2w0∂y2+A66+12A12∂w0∂y2∂2w0∂x2+3A26∂w0∂y2∂2w0∂x∂y−2B12θ′x∂2w0∂y∂x∂w0∂y−2B16θ′2x∂2w0∂x2∂w0∂y+A11∂2u0∂x2∂w0∂x+2A16∂2u0∂x∂y∂w0∂x+A66∂2u0∂y2∂w0∂x+A16∂2v0∂x2∂w0∂x+A26∂2v0∂y2∂w0∂x+A12+A66∂2v0∂x∂y∂w0∂x−2B11∂w0∂x2+B66∂2ϕx∂y2∂w0∂x+2B16∂2ϕx∂x∂y∂w0∂x+B11∂2ϕx∂x2∂w0∂x+B16∂2ϕy∂x2∂w0∂x+B26∂2ϕy∂y2∂w0∂x+B12+B66∂2ϕy∂x∂y∂w0∂x+A16∂2u0∂x2∂w0∂y+A26∂2u0∂y2∂w0∂y+A12+A66∂2u0∂x∂y∂w0∂y+A66∂2v0∂x2∂w0∂y+A22∂2v0∂y2∂w0∂y+2A26∂2v0∂x∂y∂w0∂y+B16∂2ϕx∂x2∂w0∂y+B26∂2ϕx∂y2∂w0∂y+B12+B16∂2ϕx∂x∂y∂w0∂y+B66∂2ϕy∂x2∂w0∂y+B22∂2ϕy∂y2∂w0∂y+2B26∂2ϕy∂x∂y∂w0∂y+B12θ′2∂w0∂y2+2A16∂u0∂x∂2w0∂x∂y+2A26∂v0∂y∂2w0∂x∂y+2A66∂u0∂y∂w20∂x∂y+2A66∂2v0∂x∂2w0∂x∂y+2B16∂ϕx∂x∂2w0∂x∂y−2B16θ′2w0∂2w0∂x∂y+2B26∂ϕy∂y∂2w0∂x∂y+2B66∂ϕx∂y∂2w0∂x∂y+2B66∂ϕy∂x∂2w0∂x∂y+A11∂u0∂x∂2w0∂x2+A12∂v0∂y∂2w0∂x2+A16∂u0∂y∂2w0∂x2+A16∂2v0∂x∂2w0∂x2+B11∂ϕx∂x∂2w0∂x2−B11θ′2w0∂2w0∂x2+B12∂2ϕy∂y∂2w0∂x2+B16∂2ϕx∂y∂2w0∂x2+B16∂ϕy∂x∂2w0∂x2+A12∂u0∂x∂2w0∂y2+A22∂v0∂y∂2w0∂y2+A26∂u0∂y∂2w0∂y2+A26∂v0∂x∂2w0∂y2+B12∂ϕx∂x∂2w0∂y2−B12θ′2w0∂2w0∂y2−B12θ′2x∂w0∂x∂2w0∂y2+B22∂ϕy∂y∂2w0∂y2+B26∂ϕx∂y∂2w0∂y2+B26∂ϕy∂x∂2w0∂y2−B11θ′2x∂2u0∂x2−B12θ′2x∂2v0∂x∂y−B16θ′2x∂2u0∂x∂y−B16θ′2x∂2v0∂x2−D11θ′2x∂2ϕx∂x2−2D11θ′4x∂2w0∂x−D11θ′4x∂2u0∂x2−D12θ′2x∂2ϕy∂x∂y−D16θ′2x∂2ϕx∂x∂y−D16θ′2x∂2ϕy∂x2+kA45∂2w0∂x∂y+kA45∂ϕy∂x+kA55∂2w0∂x2+kA55∂ϕx∂x+kA44∂2w0∂y2+kA44∂ϕy∂y+kA45∂2w0∂x∂y+kA45∂2ϕx∂y+C∞ρ∞ẇ0+C∞ρ∞Pcosθ∂w0∂x=I0v¨0sinθ+w¨0cosθ−I2θ′2xφ¨x−θ′2xw¨0cosΘ−φ¨ycosθsinΘ−2φ̇x−θ′2xẇ0+φ̇ycosθcosΘ⋅Ω−φx−θ′2xw0cosΘ−φycosθsinΘ+sinθsinΘ⋅Ω2−φx−θ′2xW0sinΘ+φycosθcosΘ−sinθcosΘ⋅Ω̇−2I0ysin2θ∂w0∂y⋅Ω2−2I0ysin2θ∂u0∂y⋅Ω2+I0sin2θC24−y2∂2w0∂y2⋅Ω2+I0sin2θC24−y2∂2u0∂y2⋅Ω2−I0r+x∂w0∂x⋅Ω2−I0r+x∂v0∂x⋅Ω2+I0rL−x+12L2−x2∂2w0∂x2⋅Ω2−I0rL−x+12L2−x2∂2v0∂x2⋅Ω2,(20)δϕx:B11∂2u0∂x2+2B16∂2u0∂x∂y+B66∂2u0∂y2+B16∂2v0∂x2+B12+B66∂2v0∂x∂y+B26∂2v0∂y2+B11∂w0∂x∂2w0∂x2+B11∂w0∂x∂2w0∂x2+B16∂w0∂y∂2w0∂x2+B12∂w0∂y∂2w0∂x∂y+2B16∂w0∂x∂2w0∂x∂y+B66∂w0∂y∂2w0∂x∂y+B26∂w0∂y∂2w0∂y2+B66∂w0∂x∂2w0∂y2−2D11θ′2∂w0∂x−D11θ′2x∂2w0∂x2−D16θ′2x∂2w0∂x∂y+D11∂2ϕx∂x2+2D16∂2ϕx∂x∂y+D66∂2ϕx∂y2+D16∂2ϕy∂x2+D12+D66∂2ϕy∂x∂y+D26∂2ϕy∂y2−kA45ϕy−kA55∂w0∂x−kA45∂w0∂y−kA55ϕx−kA45ϕy=I2ϕ¨x−θ′2w¨0xcosΘ−ϕ¨ycosθsinΘ−2ϕ̇x−θ′2xẇ0+ϕ̇ycosθcosΘ⋅Ω−ϕx−θ′2xw0cosΘ−ϕycosθsinΘ+sinθsinΘ⋅Ω2−ϕx−θ′2xw0sinΘ+ϕycosθcosΘ−sinθsinΘ⋅Ω̇,(21)δϕy:B16∂2u0∂x2+B12+B66∂2u0∂x∂y+B26∂2u0∂y2+B66∂2v0∂x2+B22∂2v0∂y2+2B26∂2v0∂x∂y+B16∂w0∂x∂2w0∂x2+2B26∂w0∂y∂2w0∂x∂y+B66∂w0∂y∂2w0∂x2+B66+B12∂w0∂x∂2w0∂x∂y+B22∂w0∂y∂2w0∂y2+B26∂w0∂x∂2w0∂y2−2D16θ′2∂w0∂x−D16θ′2x∂2w0∂x2−D12θ′2x∂2w0∂x∂y+D16∂2ϕx∂x2+D12+D66∂2ϕx∂x∂y+D26∂2ϕx∂y2+D66∂2ϕy∂x2+D22∂2ϕy∂y2+2D26∂2ϕy∂x∂y−kA44∂w0∂y−kA45∂w0∂x−kA45ϕx−kA44ϕy−D12θ′2x∂2w0∂x∂y+C∞ρ∞ẇ0+Pcosθ⋅∂w0∂x⋅z∂z∂y=I2ϕ¨x−θ′2xw¨0sinΘ+ϕ¨ycosθcosΘ+2ϕ̇x−θ′2xẇ0cosΘ−ϕ̇ycosθsinΘ⋅Ω−ϕy−θ′2xw¨0sinΘ+ϕycosθcosΘ−sinθcosΘ⋅Ω2+ϕx−θ′2xw0cosΘ−ϕycosθsinΘ+sinθsinΘ⋅Ω̇.The boundary conditions are given as follows:(22)x=0:u=v=w=ϕx=ϕy=0,x=L:Nxy=Nxx=Mxx=Mxy=Qx=0,y=0:Nxy=Nyy=Myy=Mxy=Qy=0,y=C:Nxy=Nyy=Myy=Mxy=Qy=0.The dimensionless variables and parameters are introduced as follows:(23)u¯0=u0L,v¯0=v0C,w¯0=w0h,x¯=xL,y¯=yC,ϕ¯x=ϕx,ϕ¯y=ϕy,θ¯=θ,θ′¯=Lθ′,r¯=rL,Aij¯=LC1/2Eh2,Bij¯=LC1/2Eh3Bij,Dij¯=LC1/2Eh4Dij,Ii¯=1ρLC1+i/2,Ω¯=ΩρCLE,t¯=tEρCL,C¯∞=C∞ρE,p¯=pρE,ρ¯∞=ρ∞L2ρA.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 ux,y,t, vx,y,t, wx,y,t, ϕxx,y,t, and ϕyx,y,t are given as follows based on a combination method of beam functions
(24a)ux,y,t=u1tsinπx2LcosπyC+u2tsin3πx2Lcos2πyC,vx,y,t=v1tsinπx2LsinπyC+v2tsin3πx2Lsin2πyC,wx,y,t=w1tX1xY1y+w2tX2xY2y,ϕxx,y,t=ϕx1tsinπx2LcosπyC+ϕx2tsinπxLcos2πyC,ϕyx,y,t=ϕy1tsinπyC1−cosπx2L+ϕy2tsin2πyC1−cosπxL,where
(24b)Xj=coshkjxa−coskjxa−βjsinhkjxa−sinkjxa,(24c)Yi=coshkiyb+coskiyb−αisinhkiyb+sinkiyb,where ki are coefficients related to frequencies, ki4=kj4=ω2ρA/EJ, κi and λj are coefficients of beam functions as follows:
(24d)κi=coshkib−coskibsinhkib−sinkibi=3,4,5,…,(24e)λj=coshkja−coskjasinhkja−sinkjaj=1,2,3,….Substitute (24a), (24b), (24c), (24d), and (24e) into the dimensionless governing equations of motion and express u, v, φx, and φy with w. Ordinary differential equations of transverse vibration for the first mode and the second mode of the rotating blade in the dimensionless form are derived through the Galerkin method as follows:
(25a)w¨1+α1+f11Ω1sinω~t+f12Ω1cosω~t+f13Ω12cos2ω~tw1+α2+f14Ω1cosω~tẇ1+α3w12+α4w1w2+α5w1w22+α6w12w2+α7w13+α8w22+α9w23+α10+f15Ω1cosω~t+f16Ω1sinω~t+f17Ω12cos2ω~tw2+α11+f18Ω1cosω~tẇ2+α12w¨2=α13+f19Ω1cosω~t+f20Ω1sinω~t+f21Ω12cos2ω~t,(25b)w¨2+β1+f31sinω~t+f32Ω1cosω~t+f33Ω12cos2ω~tw2+β14+f35Ω1cosω~tẇ2+β3w22+β4w1w2+β5w12w2+β6w22w1+β7w23+β8w12+β9w13+β10+f35Ω1cosω~t+f36Ω1sinω~t+f37Ω12cos2ω~tw1+β11+f38Ω1cosω~tẇ1+β12w¨1=β13+f39Ω1cosω~t+f40Ω1sinω~t+f41Ω12cos2ω~t,where expressions of coefficients αi, βi, and fij are described in the appendix.


## 3. Perturbation Analysis

There are both square nonlinear terms and cubic terms in (25a) and (25b), so the asymptotic perturbation method is exploited to derive average equations. The small parameter ε is introduced. The scale transformation is performed as follows:
(26)α1→ε2α1,f11⟶ε2f11,f12⟶ε2f12,f13⟶ε2f13,α2⟶ε2α2,f14⟶ε2f14,α10⟶ε2α10,f15⟶ε2f15,f16⟶ε2f16,f17⟶ε2f17,α11⟶ε2α11,f18⟶ε2f18,α12⟶ε2α12,α13⟶ε2α13,f19⟶ε2f19,f20⟶ε2f20,f21⟶ε2f21,β1⟶ε2β1,f31⟶ε2f31,f32⟶ε2f32,f33⟶ε2f33,β2⟶ε2β2,f34⟶ε2f34,β10⟶ε2β10,f37⟶ε2f37,β11⟶ε2β11,f38⟶ε2f38,β12⟶ε2β12,β13⟶ε2β13,f39⟶ε2f39,f40⟶ε2f40,f41⟶ε2f41.The case of 1 : 3 internal resonance-1/2 sub-harmonic resonance for the rotating plate is considered. Relationship betweenω1 and ω2 is expressed as
(27)3ω1=ω2,ω1=12ω~+ε2σ1,ω2=32ω~+ε2σ2,ω~=Ω,where σ1 and σ2 are two detuning parameters.Substituting (26) and (27) into (25a) and (25b), we obtain the following equation
(28a)w¨1+Ω24+Ωσ1ε2+ε4σ12w1+ε2α2+ε2f14Ω1cosω~tẇ1+α3w12+α4w1w2+α5w1w22+α6w12w2+α7w13+α8w22+α9w23+ε2α10+ε2f15Ω1cosω~t+ε2f16Ω1sinω~t+ε2f17Ω12cos2ω~tw2+ε2α11+ε2f18Ω1cosω~tẇ2+ε2α12w¨2=ε2α13+ε2f19Ω1cosω~t+ε2f20Ω1sinω~t+ε2f21Ω12cos2ω~t,(28b)w¨2+94Ω2+3Ωσ2ε2+ε4σ22w2+ε2β14+ε2f35Ω1cosω~tẇ2+β3w22+β4w1w2+β5w12w2+β6w22w1+β7w23+β8w12+β9w13+ε2β10+ε2f35Ω1cosω~t+ε2f36Ω1sinω~t+ε2f37Ω12cos2ω~tw1+ε2β11+ε2f38Ω1cosω~tẇ1+ε2β12w¨1=ε2β13+ε2f39Ω1cosω~t+ε2f40Ω1sinω~t+ε2f41Ω12cos2ω~t.The time scale transformation is introduced as(29)τ=εqt,where q is the positive rational number, which will be determined in the following deduction.Functionsxt and yt are the solutions of (28a) and (28b), which can be expressed as a power series of small parameter ε.
(30a)xt=∑n=−∞∞εrnψnτ,εe−inΩ/2t,(30b)yt=∑n=−∞∞εrnϕnτ,εe−in3Ω/2t.In (30a) and (30b), if n≠0, then rn=n, otherwise r0=r. In the following derivation, we will fix rn.The real part ofxt and yt are written as follows:
(31a)ψnτ,ε=ψ−n∗τ,ε,(31b)ϕnτ,ε=ϕ−n∗τ,ε,where the item with the asterisk on the right-hand side is the complex conjugate of the item on the left-hand side.Solutions of (28a) and (28b) can be rewritten more explicitly as follows:
(32a)xt=εrψ0τ,ε+εψ1τ,εe−iΩ/2t+ε2ψ2τ,εe−iΩt+ε3ψ3τ,εe−i3Ω/2t+ε4ψ4τ,εe−i4Ω/2t+cc+oε5,(32b)yt=εrϕ0τ,ε+εϕ1τ,εe−i3Ω/2t+ε2ϕ2τ,εe−3iΩt+ε3ϕ3τ,εe−i9Ω/2t+ε4ϕ4τ,εe−i6Ωt+cc+oε5,where cc denotes terms of the complex conjugate of functions on the right-hand side.Solutions of (28a) and (28b) can be expressed as harmonic functions, whose coefficients are related to small parameter ε, as shown in (32a) and (32b). Functions ψnτ,ε and φnτ,ε are given by
(33a)ψnτ,ε=∑i=0+∞εiψniτ,(33b)ϕnτ,ε=∑i=0+∞εiϕniτ.We suppose whenε→0, the limits of ψnτ,ε and ϕnτ,ε exist. Meanwhile, we define ψn0=ψn and ϕn0=ϕn for n≠1 and ψ10=ψ and ϕ10=ϕ for n=1. When n=2, we obtain the derivative results as follows:
(34a)ddtψne−inΩ/2t=−inΩ2ψn+εqdψndτe−inΩ/2t,(34b)ddtϕne−in3Ω/2t=−in32Ωϕn+εqdϕndτe−in3Ω/2t.In order to solve the coefficients ofψnτ,ε and ϕnτ,ε, substituting (30a) and (30b) into (28a) and (28b), we obtain equations for each harmonic with order n and a fixed order of approximation on the perturbation parameter ε.Whenn=0, we get the following expressions:
(35a)14Ω2εrψ0+2α8ϕ1ϕ1∗ε2+2α3ψ1ψ1∗ε2−f20Ω1sinω~tε2−f19Ω1cosω~tε2−f21Ω12cos2ω~tε2−α13ε2=0,(35b)94Ω2εrϕ0+2β3ϕ1ϕ1∗ε2+2β8ψ1ψ1∗ε2−β13ε2−f40Ω1sinω~tε2−f39Ω1cosω~tε2−f41Ω12cos2ω~tε2=0.The solvability of (35a) and (35b) requires r=2. Thus, we obtain the following expressions:
(36a)ψ0=−1Ω28α3ψ1ψ1∗+8α8ϕ1ϕ1∗−4f20Ω1sinω~t−4f19Ω1cosω~t−4f21Ω12cos2ω~t−4α13,(36b)ϕ0=−49Ω28β3ϕ1ϕ1∗+8β8ψ1ψ1∗−4β13−4f40Ω1sinω~t−4f39Ω1cosω~t−4f41Ω12cos2ω~t.Whenn=2, considering derivative forms of (34a) and (34b) yields
(37a)α3ψ12+α4ψ1∗ϕ1−34Ω2ψ2=0,(37b)β3ϕ12−274Ω2ϕ2=0.Then, the following equations are obtained(38a)ψ2=43α3ψ12+α4ψ1∗ϕ1Ω2,(38b)ϕ2=427β3ϕ12Ω2.Whenn=1, for the balance of the nonlinear and linear terms, we must make q=2. Equations can be derived as follows:
(39a)−IΩD1ψ1+2α3ψ0ψ1+α4ϕ0ψ1+α4ϕ1ψ2∗+3α7ψ12ψ1∗+2α3ψ2ψ1∗+α6ϕ1ψ1∗+Ωσ1ψ1−12IΩcosω~tf14Ω1ψ1+2α5ψ1ϕ1ϕ1∗−12IΩα2ψ1=0,(39b)−3IΩD1ψ1+2β3ϕ0ϕ1+β4ϕ1ψ0+β9ψ13+3Ωσ2ϕ1+2β3ϕ2ϕ1∗+3β7ϕ12ϕ1∗+2β8ψ1ψ2−32IΩcosω~tf34Ω1ϕ1+2β5ψ1ϕ1ψ1∗−32IΩβ2ϕ1=0.Substitute (36a) and (36b) and (38a) and (38b) into (39a) and (39b), differential equations about ψ1 and ϕ1 are derived as
(40a)D1ψ1=g1+Ω1g2−Iσ1+Ih3+IΩ1h1+IΩ12h2ψ1+Ih4ψ1ϕ1ϕ1∗+Ih5ψ1∗2ϕ1+Ih6ψ1∗ψ12,(40b)D1φ1=g3+Ω1g4−Iσ2+Ik3+IΩ1k1+IΩ12k2φ1+Ik4φ1ψ1ψ1∗2+Ik5φ1∗φ12+Ik6ψ13.Makeψ1 and ϕ1 as follows:
(41a)ψ1=x1+ix2,(41b)ϕ1=x3+ix4.Substitute (41a) and (41b) into (40a) and (40b), averaged equation in the Cartesian form is obtained as
(42a)ẋ1=g1+Ω1g2x1+σ1−h3−Ω1h1−Ω12h2x2−h6x2x12+x22−h4x2x32+x42−h5x4x12−x22+2h5x1x2x3,(42b)ẋ2=−σ1−h3−Ω1h1−Ω12h2x1+g1+Ω1g2x2+h6x1x12+x22+h4x1x32+x42+h5x3x12−x22+2h5x1x2x4,(42c)ẋ3=g3+Ω1g4x1+σ2−k3−Ω1k1−Ω12k2x2−k4x4x12+x22−k5x4x32+x42−k6x23x12−x22,(42d)ẋ4=−σ2−k3−Ω1k1−Ω12k2x1+g3+Ω1g4x2+k4x3x12+x22+k5x3x32+x42+k6x1x12−3x22,where expressions of coefficients σi, gi, hi, and ki are described in the appendix.


## 4. Numerical Simulation of Nonlinear Vibrations

### 4.1. Comparison about Frequencies

In this section, based on the kinetic energy and the potential energy of the blade deduced in the previous section, the Chebyshev polynomial and Ritz method are used to calculate the frequency of the model, which are compared with the frequency obtained by finite element method.To validate the model proposed, we compare frequencies theoretically calculated with frequencies obtained by the finite element method. Geometrical parameters and physical properties of the compressor blade are selected as the spanwise lengthL=0.0534m, the chordwise length C=0.03773m, the mounting angle θ=34.49o, the hub radius r=0.0845m, the density ρ=7800kg/m3, the Poisson’s ratio γ=0.3, the elastic modulus E=1.96GPa, the inlet velocity vin=39.0496m/s, the inlet mass flow Min=2.557kg/s, the real flow velocity Vin=78.472m/s. Table 1 shows frequencies of different orders obtained by two methods above. When the rotating speed increases from 0 rpm to 15000 rpm, frequencies calculated by those two methods are in good agreement. So, the model is reasonable.Table 1
Frequency comparison.
1st
2nd
3rd
FEM
C-R
FEM
C-R
FEM
C-R
0 rpm
1130.4
1076.6
4038.4
3117.2
6288.6
6253.9
30000 rpm
1133.4
1080.6
4039.1
3118.7
6291.8
6257.7
6000 rpm
1142.2
1092.8
4041
3123.2
6301.2
6269.2
9000 rpm
1156.6
1112.7
4044.2
3130.6
6316.9
6288.3
12000 rpm
1176.6
1140
4048.7
3141
6338.8
6314.9
15000 rpm
1201.7
1156.2
4054.5
3169.9
6366.9
6341


### 4.2. Numerical Simulation

Based on (42a), (42b), (42c), and (42d), we consider the influences of dimensionless parameters on nonlinear vibration behaviors of the rotating cantilever blade. Numerical simulation is utilized to investigate nonlinear dynamics of the rotating cantilever composite plate subjected to the aerodynamic force and the centrifugal force. Nonlinear oscillation of the system is investigated by choosing the perturbation rotating speed Ω1 as the controlling parameter. When other parameters and the initial condition do not vary, we only change Ω1 to detect the influence of the periodic perturbation rotating speed on vibration of the rotating cantilever composite blade. The Runge-Kutta algorithm and the Poincare map theory are used to construct numerical results of the bifurcation diagram, which describes the vibration law of the displacement x1, when Ω1 changes in a certain region.When the parameterΩ1 is chosen in the interval Ω1=0~1.5, and other parameters and the initial condition are, respectively, chosen as σ1=3.68, σ2=1.88, g1=−0.0015, g2=0.0012, g3=−0.0015, g4=0.0012, h1=5.95, h2=0.29, h3=1.2, h4=−3.57, h5=6.4, h6=6.39, k1=8.25, k2=0.2925, k3=0.18, k4=19.9, k5=−1.4, k6=1.25, x10=0.2, x20=0.101, x30=−0.26, and x40=1.2, we obtain Figure 3. Figure 3 describes the bifurcation diagram on the plane x1,Ω1 when Ω1 varies in the interval 0,1.5. It can be observed that periodic motion and chaotic motion appear in the vibration of the blade. With the increase of the perturbation rotating speed Ω1, Figure 3 presents the following law: periodic motion → chaotic motion → periodic motion → chaotic motion → periodic motion.Figure 3
Bifurcation diagram on the planex1,Ω1 when the perturbation rotating speed Ω1 varies in the interval Ω1=0~1.5. The initial condition is x10=0.2, x20=0.101, x30=−0.26, and x40=1.2.The chaotic and periodic responses can be identified by several conventional criteria. Thus, based on the bifurcation diagram, the waveform, phase portraits, and the power spectrum are utilized to further verify the existence of the chaotic and periodic motion of the blade. Figures4 and 5 illustrate that the system occurs multiperiodic motion when the perturbation rotating speed Ω1 is chosen as Ω1=0.076 and Ω1=0.412, respectively. Figures 4(a) and 4(c) represent the phase portraits on the planes x1,x2 and x3,x4, respectively. Figures 4(b) and 4(d) give the waveform on the plane t,x1 and t,x3, respectively. Figure 4(e) represents the three-dimensional phase portrait in the space x1,x2,x3. Figure 4(f) describes the power spectrum on the plane f,x1. It can be shown from Figure 4 that the amplitude of the first-order mode is larger than that of the second-order mode. Figure 6 indicates that the chaotic motion of the system appears when Ω1=0.8. Figures 7 and 8 illustrate that multiperiodic motion of the rotating composite plate occur, when the perturbation rotating speed Ω1 continues to increase and is selected as Ω1=0.93 and Ω1=1.1, respectively. Figure 9 shows that the chaotic motion of the rotating cantilever composite plate exists when Ω1=1.201. Figure 10 illustrates that the system exhibits multiperiodic motion when Ω1 increases to Ω1=1.33.Figure 4
The multiperiodic motion of the blade with varying rotating speeds whenΩ1=0.076. (a) The phase portrait on the plane x1,x2; (b) the waveform on the plane t,x1; (c) the phase portrait on the plane x3,x4; (d) the waveform on the plane t,x3; (e) the phase portrait in the three-dimensional space x1,x2,x3; and (f) the power spectrum on the plane f,x1.
(a)
(b)
(c)
(d)
(e)
(f)Figure 5
The multiperiodic motion of the blade with varying rotating speeds whenΩ1=0.412. (a) The phase portrait on the plane x1,x2; (b) the waveform on the plane t,x1; (c) the phase portrait on the plane x3,x4; (d) the waveform on the plane t,x3; (e) the phase portrait in the three-dimensional space x1,x2,x3; and (f) the power spectrum on the plane f,x1.
(a)
(b)
(c)
(d)
(e)
(f)Figure 6
The chaotic motion of the blade with varying rotating speeds whenΩ1=0.8. (a) The phase portrait on the plane x1,x2; (b) the waveform on the plane t,x1; (c) the phase portrait on the plane x3,x4; (d) the waveform on the plane t,x3; (e) the phase portrait in the three-dimensional space x1,x2,x3; and (f) the power spectrum on the plane f,x1.
(a)
(b)
(c)
(d)
(e)
(f)Figure 7
The multiperiodic motion of the blade with varying rotating speeds whenΩ1=0.93. (a) The phase portrait on the plane x1,x2; (b) the waveform on the plane t,x1; (c) the phase portrait on the plane x3,x4; (d) the waveform on the plane t,x3; (e) the phase portrait in the three-dimensional space x1,x2,x3; (f) the power spectrum on the plane f,x1.
(a)
(b)
(c)
(d)
(e)
(f)Figure 8
The multiperiodic motion of the blade with varying rotating speeds whenΩ1=1.1. (a) The phase portrait on the plane x1,x2; (b) the waveform on the plane t,x1; (c) the phase portrait on the plane x3,x4; (d) the waveform on the plane t,x3; (e) the phase portrait in the three-dimensional space x1,x2,x3; and (f) the power spectrum on the plane f,x1.
(a)
(b)
(c)
(d)
(e)
(f)Figure 9
The chaotic motion of the blade with varying rotating speeds whenΩ1=1.201. (a) the phase portrait on the plane x1,x2; (b) the waveform on the plane t,x1; (c) the phase portrait on the plane x3,x4; (d) the waveform on the plane t,x3; (e) the phase portrait in the three-dimensional space x1,x2,x3; and (f) the power spectrum on the plane f,x1.
(a)
(b)
(c)
(d)
(e)
(f)Figure 10
The multiperiodic motion of the blade with varying rotating speeds whenΩ1=1.33. (a) The phase portrait on the plane x1,x2; (b) the waveform on the plane t,x1; (c) the phase portrait on the plane x3,x4; (d) the waveform on the plane t,x3; (e) the phase portrait in the three-dimensional space x1,x2,x3; and (f) the power spectrum on the plane f,x1.
(a)
(b)
(c)
(d)
(e)
(f)Numerical simulation indicates that the system performs complex dynamic behaviors, such as multiperiodic motion and chaotic motion, whenΩ1 changes in a certain region. The perturbation rotating speed Ω1 affects the dynamic behaviors of the system significantly.


## 4.1. Comparison about Frequencies

In this section, based on the kinetic energy and the potential energy of the blade deduced in the previous section, the Chebyshev polynomial and Ritz method are used to calculate the frequency of the model, which are compared with the frequency obtained by finite element method.To validate the model proposed, we compare frequencies theoretically calculated with frequencies obtained by the finite element method. Geometrical parameters and physical properties of the compressor blade are selected as the spanwise lengthL=0.0534m, the chordwise length C=0.03773m, the mounting angle θ=34.49o, the hub radius r=0.0845m, the density ρ=7800kg/m3, the Poisson’s ratio γ=0.3, the elastic modulus E=1.96GPa, the inlet velocity vin=39.0496m/s, the inlet mass flow Min=2.557kg/s, the real flow velocity Vin=78.472m/s. Table 1 shows frequencies of different orders obtained by two methods above. When the rotating speed increases from 0 rpm to 15000 rpm, frequencies calculated by those two methods are in good agreement. So, the model is reasonable.Table 1
Frequency comparison.
1st
2nd
3rd
FEM
C-R
FEM
C-R
FEM
C-R
0 rpm
1130.4
1076.6
4038.4
3117.2
6288.6
6253.9
30000 rpm
1133.4
1080.6
4039.1
3118.7
6291.8
6257.7
6000 rpm
1142.2
1092.8
4041
3123.2
6301.2
6269.2
9000 rpm
1156.6
1112.7
4044.2
3130.6
6316.9
6288.3
12000 rpm
1176.6
1140
4048.7
3141
6338.8
6314.9
15000 rpm
1201.7
1156.2
4054.5
3169.9
6366.9
6341


## 4.2. Numerical Simulation

Based on (42a), (42b), (42c), and (42d), we consider the influences of dimensionless parameters on nonlinear vibration behaviors of the rotating cantilever blade. Numerical simulation is utilized to investigate nonlinear dynamics of the rotating cantilever composite plate subjected to the aerodynamic force and the centrifugal force. Nonlinear oscillation of the system is investigated by choosing the perturbation rotating speed Ω1 as the controlling parameter. When other parameters and the initial condition do not vary, we only change Ω1 to detect the influence of the periodic perturbation rotating speed on vibration of the rotating cantilever composite blade. The Runge-Kutta algorithm and the Poincare map theory are used to construct numerical results of the bifurcation diagram, which describes the vibration law of the displacement x1, when Ω1 changes in a certain region.When the parameterΩ1 is chosen in the interval Ω1=0~1.5, and other parameters and the initial condition are, respectively, chosen as σ1=3.68, σ2=1.88, g1=−0.0015, g2=0.0012, g3=−0.0015, g4=0.0012, h1=5.95, h2=0.29, h3=1.2, h4=−3.57, h5=6.4, h6=6.39, k1=8.25, k2=0.2925, k3=0.18, k4=19.9, k5=−1.4, k6=1.25, x10=0.2, x20=0.101, x30=−0.26, and x40=1.2, we obtain Figure 3. Figure 3 describes the bifurcation diagram on the plane x1,Ω1 when Ω1 varies in the interval 0,1.5. It can be observed that periodic motion and chaotic motion appear in the vibration of the blade. With the increase of the perturbation rotating speed Ω1, Figure 3 presents the following law: periodic motion → chaotic motion → periodic motion → chaotic motion → periodic motion.Figure 3
Bifurcation diagram on the planex1,Ω1 when the perturbation rotating speed Ω1 varies in the interval Ω1=0~1.5. The initial condition is x10=0.2, x20=0.101, x30=−0.26, and x40=1.2.The chaotic and periodic responses can be identified by several conventional criteria. Thus, based on the bifurcation diagram, the waveform, phase portraits, and the power spectrum are utilized to further verify the existence of the chaotic and periodic motion of the blade. Figures4 and 5 illustrate that the system occurs multiperiodic motion when the perturbation rotating speed Ω1 is chosen as Ω1=0.076 and Ω1=0.412, respectively. Figures 4(a) and 4(c) represent the phase portraits on the planes x1,x2 and x3,x4, respectively. Figures 4(b) and 4(d) give the waveform on the plane t,x1 and t,x3, respectively. Figure 4(e) represents the three-dimensional phase portrait in the space x1,x2,x3. Figure 4(f) describes the power spectrum on the plane f,x1. It can be shown from Figure 4 that the amplitude of the first-order mode is larger than that of the second-order mode. Figure 6 indicates that the chaotic motion of the system appears when Ω1=0.8. Figures 7 and 8 illustrate that multiperiodic motion of the rotating composite plate occur, when the perturbation rotating speed Ω1 continues to increase and is selected as Ω1=0.93 and Ω1=1.1, respectively. Figure 9 shows that the chaotic motion of the rotating cantilever composite plate exists when Ω1=1.201. Figure 10 illustrates that the system exhibits multiperiodic motion when Ω1 increases to Ω1=1.33.Figure 4
The multiperiodic motion of the blade with varying rotating speeds whenΩ1=0.076. (a) The phase portrait on the plane x1,x2; (b) the waveform on the plane t,x1; (c) the phase portrait on the plane x3,x4; (d) the waveform on the plane t,x3; (e) the phase portrait in the three-dimensional space x1,x2,x3; and (f) the power spectrum on the plane f,x1.
(a)
(b)
(c)
(d)
(e)
(f)Figure 5
The multiperiodic motion of the blade with varying rotating speeds whenΩ1=0.412. (a) The phase portrait on the plane x1,x2; (b) the waveform on the plane t,x1; (c) the phase portrait on the plane x3,x4; (d) the waveform on the plane t,x3; (e) the phase portrait in the three-dimensional space x1,x2,x3; and (f) the power spectrum on the plane f,x1.
(a)
(b)
(c)
(d)
(e)
(f)Figure 6
The chaotic motion of the blade with varying rotating speeds whenΩ1=0.8. (a) The phase portrait on the plane x1,x2; (b) the waveform on the plane t,x1; (c) the phase portrait on the plane x3,x4; (d) the waveform on the plane t,x3; (e) the phase portrait in the three-dimensional space x1,x2,x3; and (f) the power spectrum on the plane f,x1.
(a)
(b)
(c)
(d)
(e)
(f)Figure 7
The multiperiodic motion of the blade with varying rotating speeds whenΩ1=0.93. (a) The phase portrait on the plane x1,x2; (b) the waveform on the plane t,x1; (c) the phase portrait on the plane x3,x4; (d) the waveform on the plane t,x3; (e) the phase portrait in the three-dimensional space x1,x2,x3; (f) the power spectrum on the plane f,x1.
(a)
(b)
(c)
(d)
(e)
(f)Figure 8
The multiperiodic motion of the blade with varying rotating speeds whenΩ1=1.1. (a) The phase portrait on the plane x1,x2; (b) the waveform on the plane t,x1; (c) the phase portrait on the plane x3,x4; (d) the waveform on the plane t,x3; (e) the phase portrait in the three-dimensional space x1,x2,x3; and (f) the power spectrum on the plane f,x1.
(a)
(b)
(c)
(d)
(e)
(f)Figure 9
The chaotic motion of the blade with varying rotating speeds whenΩ1=1.201. (a) the phase portrait on the plane x1,x2; (b) the waveform on the plane t,x1; (c) the phase portrait on the plane x3,x4; (d) the waveform on the plane t,x3; (e) the phase portrait in the three-dimensional space x1,x2,x3; and (f) the power spectrum on the plane f,x1.
(a)
(b)
(c)
(d)
(e)
(f)Figure 10
The multiperiodic motion of the blade with varying rotating speeds whenΩ1=1.33. (a) The phase portrait on the plane x1,x2; (b) the waveform on the plane t,x1; (c) the phase portrait on the plane x3,x4; (d) the waveform on the plane t,x3; (e) the phase portrait in the three-dimensional space x1,x2,x3; and (f) the power spectrum on the plane f,x1.
(a)
(b)
(c)
(d)
(e)
(f)Numerical simulation indicates that the system performs complex dynamic behaviors, such as multiperiodic motion and chaotic motion, whenΩ1 changes in a certain region. The perturbation rotating speed Ω1 affects the dynamic behaviors of the system significantly.


## 5. Conclusions

In this paper, nonlinear vibration behaviors of the high rotating cantilever plate subjected to the centrifugal force and the aerodynamic force are investigated. Nonlinear partial differential governing equations of the rotating blade are derived by utilizing Hamilton’s principle. The Galerkin method is applied to discretize the nonlinear partial differential governing equations. The method of asymptotic perturbation is adopted to obtain the four-dimension nonlinear averaged equation of motion for the blade in the case of 1 : 3 internal resonance-1/2 subharmonic resonance. Numerical simulations are used to study nonlinear dynamic responses of the blade.The bifurcation diagram, phase portrait, and power spectrum are conducted to demonstrate that periodic motions and chaotic motions occur in nonlinear vibrations of the rotating blade under certain conditions. In this paper, the perturbation rotating speed is selected as the controlling parameter. Numerical simulation shows that nonlinear dynamic motions of the blade are sensitive to the perturbation rotating speed. The periodic motion and chaotic motion of the blade appear alternately. Periodic motions and chaotic motions exist in the averaged equations. It is well known that periodic motions and chaotic motions in the averaged equations can lead to amplitude-modulated periodic and chaotic vibrations in the original system under certain conditions. Therefore, amplitude-modulated periodic and chaotic motions occur in the rotating blade. Occurrence of the chaotic motion means that the rotating cantilever blade may perform large amplitude nonlinear vibration, which leads to the damage of the system. We can control the responses of the system from the chaotic motions to the periodic motions by changing the perturbation rotating speed. Finally, the large-amplitude vibrations of the blade can be controlled. Thus, study on nonlinear behavior of the blade is of great significance.Analytical results of this paper can be widely used in the practical aircraft engines. Large amplitude nonlinear vibration produced by the forced vibration and self-excited vibration occasionally leads to the undesirable disaster of the blade. Since chaotic motion is the large amplitude nonlinear vibration, we can adjust the perturbation rotating speed to control the responses of the system in order to avoid the appearance of the chaotic motion. Moreover, the possibility of aircraft engine failure caused by nonlinear vibration is reduced to ensure the safety of the aeroplane.



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*Source: 5610915-2018-02-27.xml*