# Global Dynamics of a Compressor Blade with Resonances

Authors: Xiaoxia Bian; Fangqi Chen; Fengxian An
Journal: Mathematical Problems in Engineering
 (2016)

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## Abstract

The global bifurcations and chaotic dynamics of a thin-walled compressor blade for the resonant case of 2 : 1 internal resonance and primary resonance are investigated. With the aid of the normal theory, the desired form associated with a double zero and a pair of pure imaginary eigenvalues for the global perturbation method is obtained. Based on the simpler form, the method developed by Kovacic and Wiggins is used to find the existence of a Shilnikov-type homoclinic orbit. The results obtained here indicate that the orbit homoclinic to certain invariant sets for the resonance case which may lead to chaos in the sense of Smale horseshoes for the system. The chaotic motions of the rotating compressor blade are also found by using numerical simulation.


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## 1. Introduction

Compressor blades are widely used in many fields of aerospace, aeronautic engineering, and mechanical industry due to their excellent mechanical properties. The problem of nonlinear dynamics of the rotating blades had attracted lots of research interest during the past decade. Various strategies and approaches have been proposed for nonlinear dynamics of rotating blades (see, e.g., [1–15]). However, theoretical analysis of global dynamics of the rotating blades has not been concerned in the current available literature. Several researchers have examined the global behaviors of plates, beams, and belt (see, e.g., [16–22]), but the results cannot be directly extended to the case of rotating blades.Yang and Tsao investigated the vibration and stability of a pretwisted blade under nonconstant rotating speed in [1], and they also predicted the time-dependent rotating speed leads to a system with six parametric instability regions in primary and combination resonances. Surace et al. [2] dealt with the coupled bending-bending-torsion vibration of rotating pretwisted blades. Şakar and Sabuncu [3] presented the static stability and the dynamic stability of an aerofoil cross section rotating blade subjected to an axial periodic force and took into account the effects of coupling due to the center of flexure distance from the centroid, rotational speed, disk radius, and stagger angle. Al-Bedoor and Al-Qaisia [4] used a reduced-order nonlinear dynamic model to research the steady-state response of the rotating blade under the main shaft torsional vibration. Tang and Dowell [5] analyzed the nonlinear response of a nonrotating flexible rotor blade subjected to periodic gust excitations theoretically and experimentally. They reported that there exists a periodic or possibly chaotic behavior in the blade. Choi and Chou [6] studied the dynamic response of turbomachinery blades with general end restraints by applying the modified differential quadrature method. A Monte Carlo approach was employed to explore a supercritical Hopf bifurcation and random bifurcation of a two-dimensional nonlinear airfoil in turbulent flow by Poirel and Price [7]. Lacarbonara et al. [8, 9] established the governing equations of the blades under the centrifugal forces and discussed linear modal properties and the nonlinear modes of vibration away from internal resonances, respectively. Yao et al. [10] performed a nonlinear dynamic analysis of the rotating blade with varying rotating speed under high-temperature supersonic gas flow; furthermore, they [11] explored the contributions of nonlinearity, damping, and rotating speed to the steady-state nonlinear responses of the rotating blade, and they also investigated the effects of the rotating speed on nonlinear oscillations of the blade. Wang and Zhang discussed the stability of a spinning blade having periodically time varying coefficients for both linear model and geometric nonlinear model and obtained the stability boundary of linear model and stability of steady-state solutions of nonlinear model in [12].In many cases, blades are usually modeled as a pretwisted, presetting, thin-walled rotating cantilever beam because the shape of the blade is very complex. Many researchers carried out studies on the dynamic behavior of the beam of this kind and obtained a lot of valuable results (see, e.g., [13–15]). Several methods have been developed to research the global bifurcation behaviors and chaotic dynamics in nonlinear systems that possess homoclinic or heteroclinic orbits. There are three methods: Melnikov method, global perturbation method, and energy-phase method. Melnikov gave the condition under which a homoclinic orbit in the unperturbation system would break under perturbation and at last lead to chaos in the system. Based on Melnikov method, Wiggins studied the global behaviors of the three basic systems [23]. Then, Kovacic and Wiggins [24] developed the global perturbation method to present Shilnikov-type homoclinic orbit for resonant system. The energy-phase method proposed by Haller and Wiggins [25, 26] detected the existence of single-pulse and multipulse homoclinic orbits in a class of near Hamilton systems. Applying the latter two methods, there were many applications to investigate the global behaviors (see, e.g., [16–22]).In this paper, we obtain a sufficient condition for the existence of Shilnikov-type homoclinic orbit of a compressor blade with 2 : 1 internal resonance and primary resonance using normal form theory and global perturbation method. Firstly, the formulas of the simpler normal form associated with a double zero and a pair of pure imaginary eigenvalues are derived by normal form theory in Section2. Then, the dynamics of unperturbed system and perturbed system are analyzed in Sections 3 and 4 in detail, respectively. The analysis indicate that Shilnikov-type homoclinic orbit exists in these cases. Finally, numerical simulations are given to confirm the result in Section 5 and the work ends in Section 6 with a short summary.


## 2. Formulation of the Problem

A thin-walled compressor blade of gas turbine engines with varying speed under high-temperature supersonic gas flow is considered in [11]. It is modeled as a pretwisted, presetting, thin-walled rotating cantilever beam, considering the geometric nonlinearity, centrifugal force, the aerodynamic load, and the perturbed angular speed.The pretwisted flexible cantilever blade, with lengthL mounted on a rigid hub with radius R0, is considered [11]. It rotates at a varying rotating speed Ω(t) around its polar axis where Ωt=Ω0+fcos⁡Ω1t, where Ω0 is the rotating speed at the steady-state and fcos⁡Ω1t is a periodic perturbation. It is also allowed to vibrate flexurally in the plane making an angle γ, as shown in Figure 1(a). The rotating blade is treated as a pretwisted, presetting, thin-walled rotating cantilever beam. The length and width of the cross section of the beam in the x and y directions are a and b, respectively, and the thickness of the thin-walled beam is h. For the purpose of describing the motion of the rotating blade, different coordinate systems are needed. The origin of the rotating coordinate system (x,y,z) is located at the blade root, xp and yp are the principal axes of an arbitrary beam cross section in the local coordinates (xp,yp,zp) (Figure 1(b)), and the transformations between two coordinate systems are shown as x=xpcos⁡(γ+β(z))-ypsin⁡(γ+β(z)) and y=xpsin⁡(γ+β(z))+ypcos⁡(γ+β(z)), z=zp. β0 is denoted as the pretwist at the beam tip; then, β(z)=β0z/L is the pretwist angle of a current beam cross section. The local coordinate system (s,t,n) is defined on the cross section of the beam to describe the geometric configuration and the cross section, where s and n are the circumferential and thickness coordinate variables in Figure 1(c); the notion (X,Y,Z) represents the points off the middle surface; it is different from the notion (x,y,z); the relationship is X=x+n(dy/ds) and Y=y-n(dx/ds). Assume that (u,v,w) and (u0,v0,w0) represent the displacements of an arbitrary point and a point in the middle surface of the rotating blades on the x, y, and z directions, respectively. θx and θy represent the rotations about the x- and y-axis, respectively.Figure 1
The rotating cantilever beam mode: (a) general view with the global coordinate system, (b) cross section of the rotating beam, and (c) local coordinate systems of the cross section.
(a) (b) (c)Based on the isotropic constitutive law, the nonlinear partial differential governing equations of motion for the pretwist, presetting, thin-walled rotating cantilever beam were derived by using Hamilton’s principle in [11]. Then, Galerkin procedure was applied to obtain the dimensionless governing differential equations of nonlinear vibration for the rotating blade by Yao et al. as follows:(1)p¨t+β12p˙t+β13q˙t+ω12pt+β11qt-2β14ptΩ0fcos⁡Ω1t-β14ptf2cos2⁡Ω1t+β5ptq2t+β5p3t=β16Ω1fsin⁡Ω1t,q¨t+β22p˙t+β23q˙t+ω22qt+β21pt-2β24qtΩ0fcos⁡Ω1t-β24qtf2cos2⁡Ω1t+β5p2tqt+β5q3t=0,where p(t) and q(t) are the amplitudes of normal modes, ω1 and ω2 are normal frequencies, and β12, β13, β22, and β23 are damping parameters. β5 plays the role of the nonlinearity, f is the amplitude of excitation, and all the expressions of the coefficients can be found in [11].We study the case of 2 : 1 internal resonance and primary resonance; the resonant relations are represented asω12=Ω12+εσ1 and ω22=(1/4)Ω22+εσ1, Ω1=1, where 0<ε≪1 and σ1 and σ2 are two detuning parameters.Using the method of multiple scales, the averaged equations were obtained as follows [11]:(2)x1˙=-μ1x1+-σ12Ω1+f28Ω1β14x2-3β58Ω1x2x12+x22+β54Ω1x2x32+x42-12β16f,x2˙=σ12Ω1-3f28Ω1β14x1-μ1x2+3β58Ω1x1x12+x22+β54Ω1x1x32+x42,x3˙=-μ2x3+-σ2Ω1-Ω0fΩ1β24+f22Ω1β24x4-β52Ω1x4x12+x22+3β54Ω1x4x32+x42,x4˙=-μ2x4+σ2Ω1-Ω0fΩ1β24-f22Ω1β24x3+β52Ω1x3x12+x22+3β54Ω1x3x32+x42.Equations (2) have a zero solution (x1,x2,x3,x4)=(0,0,0,0). Without the perturbation parameter β16, the Jacobian matrix at the origin is(3)-μ1-σ12Ω1+f28Ω1β1400σ12Ω1-3f28Ω1β14-μ10000-μ2-σ2Ω1-Ω0fΩ1β24+f22Ω1β2400σ2Ω1-Ω0fΩ1β24-f22Ω1β24-μ2.The characteristic equation corresponding to the zero solution is(4)Pλ=λ2+2μ1λ+μ12--σ12Ω1+f28Ω1β14σ12Ω1-3f28Ω1β14,λ2+2μ2λ+μ22--σ2Ω1-fΩ0Ω1β24+f22Ω1β24σ2Ω1-fΩ0Ω1β24-f22Ω1β24=λ2+2μ1λ+μ12-σ~1-38f2r1-σ~1+18f2r1λ2+2μ2λ+μ22+r22f2-σ~22.For convenience of the following analysis, letσ~1=σ1/2Ω1, σ~2=σ2/Ω1-(f2/2Ω1)β24, r1=β14/Ω1, r2=(Ω0/Ω1)β24, r3=β5/Ω1, and r4=β16. When μ1=0, μ2=0, σ~2=r2f, and σ~1>(3/8)r1f2 are simultaneously satisfied, the eigenvalues of system (2) without parameter r4 have a nonsemisimple double zero and a pair of pure imaginary eigenvalues λ1,2=±iω and λ3,4=0, where ω2=-σ~12+(1/2)r1f2σ~1-(3/64)r12f4. Assume f=1, r1=0, r2=-1/2, and σ2′=σ~2-r2f, considering σ2′, μ1, μ2, and r4 as the perturbation parameters; then, (2) without the perturbation parameters becomes(5)x1˙=-σ~1x2-38r3x2x12+x22-14r3x2x32+x42,x2˙=σ~1x1+38r3x1x12+x22+14r3x1x32+x42,x3˙=x4-12r3x4x12+x22+34r3x4x32+x42,x4˙=12r3x3x12+x22+34r3x3x32+x42.In this case, we have(6)0-σ~100σ~100000010000.Using the method in [27], a third-order normal form of (5) is obtained as(7)y1˙=-σ~1y2-38r3y2y12+y22-14r3y2y32,y2˙=σ~1y1+38r3y1y12+y22+14r3y1y32,y3˙=y4,y4˙=12r3y3y12+y22+34r3y33.Normal form with perturbation parameters of system (2) is(8)y1˙=-μ1y1-σ~1y2-38r3y2y12+y22-14r3y2y32-12r4,y2˙=σ~1y1-μ1y2+38r3y1y12+y22+14r3y1y32,y3˙=-μ2y3+1-σ2′y4,y4˙=-μ2y4+σ2′y3+12r3y3y12+y22+34r3y33.We need to transform system (8) to a desired form in order to apply the global perturbation method. Let μi→εμi(i=1,2) and r4→εr4, and use the transformations(9)y1=Icos⁡ϕ,y2=Isin⁡ϕ,y3=1-σ2′u,y4=μ2u+v.And substituting (9) into the normal form (8) yields(10)u˙=v=∂H0∂v+εgu,v˙=u12r3I+σ2′1-σ2′-μ22+34r3u3-2εμ2v=-∂H0∂u+εgv,I˙=-2εμ1I-εr4Icos⁡ϕ=∂H0∂ϕ+εgI,ϕ˙=σ~1+38r3I+14r3u2+εr4sin⁡ϕ2I=-∂H0∂I+εgϕ,(11)gu=∂H1∂v,gv=-∂H1∂u-2μ2v,gI=∂H1∂ϕ-2μ1I,gϕ=-∂H1∂I,where the Hamiltonian functions H0 and H1 are of the following form:(12)H0=12v2-14u2r3I+u22μ¯2-316r4u4-σ~1I-316r3I2,H1=-Ir4sin⁡ϕ,where μ¯2=μ22-σ2′(1-σ2′).


## 3. Dynamics of the Unperturbed System

Settingε=0 in system (10), we obtain the unperturbed system. Obviously, the variable I is a constant since I˙=0, and the first three equations are completely independent of ϕ. Thus, we obtain two uncoupled single-degree-of-freedom nonlinear systems:(13)u˙=v,v˙=u12r3I-μ¯2+34r3u3.All possible fixed points in (u,v) phase space can be classified as(14)P1:u=v=0;P2±:u=±4μ¯2-2r3I3r3,v=0,where (4μ¯2-2r3I)/3r3>0; that is, I<2μ¯2/r3 as r3<0 or r3>0. When I>2μ¯2/r3, the only solution of system (13) is P1, and from the Jacobian matrix evaluated at the trivial solution, P1 is a saddle point. At I=2μ¯2/r3, the trivial solution may bifurcate into three solutions through a pitchfork bifurcation. From the Jacobian matrices evaluated at P1 and P2±, it is known that P1 is a center and P2± are two saddle points. The phase portrait is illustrated in Figure 2.Figure 2
Phase portrait of the unperturbed system in theI-2μ¯2/r3 space.From transformation (9), the variables I and r may actually represent the amplitude and phase of nonlinear oscillations. Therefore, assume that variable I≥0 and put I1=0 and I2=2μ¯2/r3, such that, for all I∈[I1,I2], system (13) has two saddle points P2 and one center P1, which is connected by heteroclinic orbits (uh(T1,I),vh(T1,I)). In four-dimensional space (u,v,I,ϕ), the set defined by(15)M=u,v,I,ϕ∣u=±4μ¯2-2r3I3r3,v=0,0<I<2μ¯2r3,0≤ϕ≤2πis a two-dimensional invariant manifold and it is normally hyperbolic [24].M has a three-dimensional stable manifold Ws(M) and an unstable manifold Wu(M). Then, from [24], the existence of the heteroclinic orbits implies that Ws(M) and Wu(M) intersect nontransversally along a three-dimensional heteroclinic manifold denoted by Γ, which can be written as(16)Γ=WsM∩WuM=u,v,I,ϕ∣u=uhT1,I,v=vhT1,I,0<I<2μ¯2r3,ϕ=∫0T1DIH0uhs,I,vhs,I,Ids+ϕ0,where ϕ0 is a constant determined by the initial conditions. The geometric structures of the stable and unstable manifolds of M and Γ are shown in Figure 3. It is seen that (13) is a Hamilton system with Hamiltonian(17)Hu,v=12v2-η2u2+316r3u4,where η=μ¯2-(1/2)r3I. Then, we get the expressions of the pair of heteroclinic orbits as follows:(18)u=±2η3r3tanh⁡2η2T1,vT1=±23r3ηsech2⁡2η2T1.The dynamics restricted to the invariant manifold M are described by the following equations:(19a)I˙=0,(19b)ϕ˙=σ~1+38r3I+14r3u2,where I1≤I≤I2. From (19b), we have periodic orbits which are circles for each I when σ~1+(3/8)r3I+(1/4)r3u2≠0, and the corresponding circle is a circle of fixed points when σ~1+(3/8)r3I+(1/4)r3u2=0; that is, I=Ir=(-8μ¯2-24σ~1)/5r3. As homoclinic orbit in (I,ϕ) plane is a heteroclinic connection in the four-dimensional (u,v,I,ϕ) space which is shown in Figure 4, Ir is called resonance due to the vanishing frequency of rotation along the ϕ direction, and when I≠Ir, Δϕ is not defined.Figure 3
The geometric structures ofM0 and Γ.Figure 4
The geometry of trajectories homoclinic to the periodic orbits onM0 and orbits heteroclinic to fixed points on the resonances.Now, we consider the phase shift:(20)Δϕ=ϕ+∞,Ir-ϕ-∞,Ir.Substituting (18) into (19b) yields(21)ϕ˙=σ~1+38r3Ir+η3tanh2⁡2h2T1.Integrating (21) yields(22)ϕT1=σ~1+38r3Ir+η3T1-2η3tanh⁡2η2T1+ϕ0.Thus, the phase shift is expressed as(23)Δϕ=-22η3.Δϕ is a function of η. It is illustrated in Figure 5.Figure 5
The phase shiftΔϕ defined in (23).


## 4. Dynamics of the Perturbed System

As the manifoldM along with its stable manifold Ws(M) and unstable manifold Wu(M) is invariant under sufficiently small perturbations [24], under perturbation (when ε≠0), M becomes a locally invariant two-dimensional manifold Mε described as follows:(24)Mε=uε,vε,I,ϕ∣uεI,ϕ=±2η3r3+εu1I,ϕ+oε2,vεI,ϕ=0+εv1I,ϕ+oε2,0<I<2μ¯23r3,0≤ϕ≤2π.The flow on Mε is obtained by substituting (uε,vε) into (10):(25)I˙=-2εμ1I-εr4Icos⁡ϕ+oε2,ϕ˙=σ~1+38r3I+14r3u2+εr4sin⁡ϕ2I+oε2.Introduce the scale transformations(26)I=Ir+εh,τ=εT1.Substituting transformations (26) into (25) yields(27)h′=-2μ1Ir-Irr4cos⁡ϕ-εh2μ1+r42Icos⁡ϕ+oε,ϕ′=124r3h+r4sin⁡ϕ2Iε+oε.When ε=0, (27) is reduced to(28)h′=-2μ1Ir-Irr4cos⁡ϕ,ϕ′=124r3h,which is a Hamilton system with Hamiltonian(29)Hh,ϕ=-2μ1Irϕ-Irr4sin⁡ϕ+112r3h2.The fixed points of Hamilton system (28) are given by(30)p0,ϕs=0,π-arccos⁡2μ1Irr4,q0,ϕc=0,π+arccos⁡2μ1Irr4.The Jacobian matrix of (28) evaluated at these fixed points is(31)J=0Irr4sin⁡ϕs,c124r30.It is easy to find that p is a saddle point and q is a center. Therefore, there exists a homoclinic orbit connecting p to itself. The phase portrait of system (28) is shown in Figure 6. By the analysis of Kovacic and Wiggins [24], we can obtain that, for sufficiently small ε, p remains a saddle point and q becomes a hyperbolic sink qε.Figure 6
Dynamics of unperturbed system (28) on Mε.The phase portrait of perturbed system (27) is given in Figure 7; Hamilton function remains constant on the homoclinic orbit; that is, H¯(0,ϕn)=H¯(0,ϕs); then, we have(32)-2μ1Irϕn-Irr4sin⁡ϕn=-2μ1Irπ-arccos⁡2μ1Irr4-Irr4sin⁡π-arccos⁡2μ1Irr4.Figure 7
Dynamics of perturbed system (27) on Mε.In order to consider the dynamics onMε in the neighborhood of I=Ir, an annulus Aε is defined as(33)Aε=u,v,h,ϕ∣u=uIr+εh,ϕ,v=vIr+εh,ϕ,h<h0,0≤ϕ≤2π,where h0 is a constant, which is chosen sufficient large so that the unperturbed homoclinic orbits are enclosed within the annulus. Denote Ws(Aε) and Wu(Aε) as the three-dimensional stable and unstable manifolds of Aε, which are subsets of Ws(Aε) and Wu(Aε), respectively. According to the analysis of [24], the existence of an orbit homoclinic to a saddle-focus point qε can lead to chaos. This type of homoclinic orbit is called Shilnikov-type homoclinic orbit. The point qε on Aε has an orbit that comes out of Aε in the four-dimensional space and may return to the annulus; it may approach qε asymptotically as t→∞ and eventually complete a Shilnikov-type homoclinic orbit as shown in Figure 8.Figure 8
Shilnikov-type homoclinic orbit topε.We need to confirm the existence of a Shilnikov type homoclinic orbit in two steps. First, we showWu(Aε)⊂Ws(Aε) by using Melnikov theory when Melnikov function has a simple zero. Second, we determine whether or not the trajectory in Wu(Aε) comes back in the domain of attraction of qε. Based on [24], the higher dimensional Melnikov function is given as(34)MIr=∫-∞+∞∂H0∂ugu+∂H0∂vgv+∂H0∂IgIdT1.Using the division of integral method and the abovementioned analysis, (34) can be expressed as(35)MIr=∫-∞+∞-dH1dT1-2μ2u˙v+2μ1Irϕ˙dT1=M1+M2+M3.With the aforementioned analysis, the first term can be evaluated as(36)M1=-∫-∞+∞dH1dT1dT1=Ir4sin⁡ϕ+∞-sin⁡ϕ-∞=Ir4cos⁡ϕ-∞sin⁡Δϕ-sin⁡ϕ-∞1-cos⁡Δϕ=Ir4-2μ1Ir4sin⁡Δϕ+1-2μ12Irr42cos⁡Δϕ-1.The second term can be simplified as(37)M2=2μ2∫-∞+∞23r3η2sech4⁡2η2T1dT1=-162μ2η3/29r3I=Ir.The third term is changed into(38)M3=∫-∞+∞2μ1Irϕ˙dT1=2μ1IrΔϕ.By (36), (37), and (38), the Melnikov function may be expressed as(39)MIr=r4I-2μ1Ir4sin⁡Δϕ+cos⁡Δϕ-11-4μ12Irr42-162μ2η3/29r3+2μ1IrΔϕ.Now, we can require that the Melnikov function has a simple zero. That is, we require(40)r4I-2μ1Ir4sin⁡Δϕ+cos⁡Δϕ-11-4μ12Irr4-162μ29r3η3/2+2μ1IrΔϕ=0.Next, we examine whether the orbit on Wu(qε) returns to the domain of attraction of qε. The condition is given by(41)ϕs<ϕc+Δϕ+2mπ<ϕn,where m is an integer, ϕs, ϕc, and ϕn are given by (30) and (31), and Δϕ is the change of angle. According to [24], when conditions (40) and (41) are satisfied simultaneously, there exists the Shilnikov-type chaos in the sense of Smale horseshoes in system (2).


## 5. Numerical Simulation of Chaotic Motions

Now fixed parameters are used in the abovementioned theory to simplify the calculation. Letting(42)μ1=μ2=μ,β=2r4μ,Ir=1,condition (41) becomes(43)β=1-cos⁡Δϕcos⁡Δϕ-12+-36.282/9η3/2-Δϕ+sin⁡Δϕ2.From (23), Δϕ is a function of η; then, β is a function of η. The figure of β shows that β exists when η∈(0,1), so Melnikov function MIr(β,η) has a simple zero (Figure 9). ϕs, ϕc, and ϕn are presented in Figure 10; we can see ϕs<ϕc+Δϕ<ϕn; that is, condition (41) is satisfied. Then, qε has a Shilnikov homoclinic orbit for sufficiently small ε. We choose (1) and (2) to do numerical simulations. We use numerical approach to explore the existence of chaotic motions of the rotating thin-walled blade. In Figure 10, we show the existence of the chaotic responses of the thin-walled blade to the forcing excitation. β16=8.8, and other parameters and initial conditions were chosen as μ=0.001, σ1=12, β14=0, β5=-17.64, μ2=0.001, σ2=11/40, Ω0=5, β24=-4, x10=-0.052, x20=0.061, x30=0.042, and x40=-0.051. Figure 11 shows the phase portraits on the planes (x1,x2),(x3,x4),(x1,x2,x3),(x2,x3,x4) and the wave forms on plane (t,x1),(t,x3) based on (2). With the same parameters, we get the portraits on the planes (x1,x2),(x3,x4),(x1,x2,x3),(x2,x3,x4) and the wave forms on plane (t,x1),(t,x3) based on (1). They are shown in Figure 12; the chaotic motion demonstrated in Figures 11 and 12 is Shilnikov-type multipulse chaotic motion. Therefore, the numerical results agree with the theoretical predictions qualitatively.Figure 9
The zeros of Melnikov’s function.Figure 10
Graphs ofϕs, ϕc+Δϕ, and ϕn.Figure 11
The chaotic motions of the compressor blade based on (2): (a) the phase portrait on plane (x1,x2), (b) the waveform on plane (t,x1), (c) the phase portrait on plane (x3,x4), (d) the waveform on plane (t,x3), (e) the phase portrait in three-dimensional space (x1,x2,x3), and (f) the phase portrait in three-dimensional space (x2,x3,x4).
(a) (b) (c) (d) (e) (f)Figure 12
The chaotic motions of the compressor blade based on (1): (a) the phase portrait on plane (x1,x2), (b) the waveform on plane (t,x1), (c) the phase portrait on plane (x3,x4), (d) the waveform on plane (t,x3), (e) the phase portrait in three-dimensional space (x1,x2,x3), and (f) the phase portrait in three-dimensional space (x2,x3,x4).
(a) (b) (c) (d) (e) (f)


## 6. Conclusions

The global bifurcations and chaotic dynamics of the thin-walled compressor blade with varying speed are investigated for the first time by using the analytical and numerical approaches simultaneously when the averaged equations have one nonsemisimple double zero and a pair of pure imaginary eigenvalues. The study is focused on coexistence of 2 : 1 internal resonance and primary resonance. Normal theory is utilized to find the explicit expressions of the simpler normal form of the averaged equations with a double zero and a pair of pure imaginary eigenvalues. Based on the Melnikov method and its extensions to resonance cases developed by Kovacic and Wiggins, the thin-walled compressor blade can undergo homoclinic bifurcation and the Shilnikov-type homoclinic orbit; that is, there exists chaotic motion in full four-dimensional averaged system. Finally, the Dynamics software is used to perform numerical simulation. The numerical results show the existence of chaotic motions in the averaged equations, which illustrate the predictions obtained by the theoretical analysis. The chaotic motions in averaged equations can lead to the amplitude modulated chaotic oscillations in the original system under certain conditions. Therefore, there are Shilnikov-type single-pulse chaotic motions for the thin-walled rotating compressor blade. This is the extension of the results obtained by Yao et al. [11]. We believe that our results give a direct explanation for the jumping behaviors observed in this class of the compressor blade under in-plane and moment excitations.



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