Bifurcation and dynamic behavior analysis of a rotating cantilever plate in subsonic airflow

Turbo-machineries, as key components, have a wide utilization in fields of civil, aerospace, and mechanical engineering. By calculating natural frequencies and dynamical deformations, we have explained the rationality of the series form for the aerodynamic force of the blade under the subsonic flow in our earlier studies. In this paper, the subsonic aerodynamic force obtained numerically is applied to the low pressure compressor blade with a low constant rotating speed. The blade is established as a pre-twist and presetting cantilever plate with a rectangular section under combined excitations, including the centrifugal force and the aerodynamic force. In view of the first-order shear deformation theory and von-Kármán nonlinear geometric relationship, the nonlinear partial differential dynamical equations for the warping cantilever blade are derived by Hamilton’s principle. The second-order ordinary differential equations are acquired by the Galerkin approach. With consideration of 1:3 internal resonance and 1/2 sub-harmonic resonance, the averaged equation is derived by the asymptotic perturbation methodology. Bifurcation diagrams, phase portraits, waveforms, and power spectrums are numerically obtained to analyze the effects of the first harmonic of the aerodynamic force on nonlinear dynamical responses of the structure.


Introduction
Blades, which are vital rotating structural units, are widely used in turbo-machinery of aeronautical and aerospace industries. It is well-known that blades are often exposed to harsh working conditions, especially for gas turbine blades. Vibration of blades under airflow excitation has been recognized as one of the major causes of system failure in the engineering field.
Transverse vibrations of compressor blades often exhibit dynamic behaviors, which may lead to bifurcation phenomena. Dynamic behaviors of blades are very complex due to the coupling effect of fluid-structure interaction and strong nonlinear and unsteady characteristics. To design such structures properly, we need a profound understanding of their dynamic characteristics to avoid some undesirable disasters such as resonance phenomena and flutter.
Blade, a core component of modern rotating machine, endures the coupling effects of the centrifugal force and the aerodynamic load. In order to get a profound understanding of the vibration characteristics of the system, the establishment for a reasonable dynamical model is the most pressing concern for researchers. Plenty of theoretical analyses focus on the vibration characteristics of blades, which have been simplified as Euler-Bernoulli beams [1] and Timoshenko beams [2][3] under various conditions. However, if the blade is thin, the aspect ratio is small, or higher frequencies and modes are needed, the model of the cantilever beam is highly inaccurate. Thus, the cantilever plate or the shell model is needed to describe the rotating blade. Sabuncu and Thomas [4] numerically investigated the free vibration characteristics of the shrouded pre-twist rotating aerofoil blade without shear deformation and rotary inertia effects. Li and Zhang [5] observed frequency loci veering phenomena rather than crossing in either rotating homogeneous plates or functionally graded (FG) plates. Mahi et al. [6] and Bennoun et al. [7] proposed a refined plate theory and a novel hyperbolic shear deformation theory to study the bending and free vibration of different kinds of composite plates. Xie et al. [8] put forward a novel breathing model for the blade with cracks of the rotating shaft-disk-blade system subject to bending stresses and centrifugal stresses and compared the vibration responses of this model with two different crack models. Niu et al. [9] studied free vibrations of a rotating blade, which is modeled as a pre-twisted FG cylindrical cantilever panel reinforced with graphene.
In addition to linear vibrations, some scholars have also studied the nonlinear oscillations of blades. Zhang and Zhao [10] analyzed the nonlinear oscillations and amplitude-frequency characteristics for the composite laminated rotary plate enduring lateral and horizontal excitations. Chen et al. [11] discussed the nonlinear steady-state responses of thin-walled structures in supercritical flow. Subrahmanyam et al. [12] and Yao et al. [13] studied the effects of the Coriolis terms and several other nonlinear terms on the oscillation and stability of the rotating thin-walled linearly pre-twist blade. Roy and Meguid [14] found that the centrifugal force field decaying leads to a severe change in the dynamical responses of blades. Wang et al. [15] explained Hopf bifurcation and saddle-node based on the reduced fluid-structure interaction model of the vortex-induced oscillations for turbine blades. The double Hopf bifurcations, global bifurcations, and multi-pulse chaotic dynamics of the laminated plate under certain internal resonances were investigated [16][17] . Zhang et al. [18][19] studied sub-harmonic and superharmonic resonances combined with different resonance cases of a rotary blade under strong gas pressure. Steady-state responses, saturation, jump, and hysteresis are focused on. Ding et al. [20] further illustrated the energy transfer between the saturation-like phenomena and the internal resonance modes. Liu et al. [21] gave phase portraits, time history, Poincaré maps, and bifurcation diagrams with different parameters of an FG cylindrical shell with small initial geometric imperfect defects under complex excitations. Yao et al. [22] explored the nonlinear transient dynamic responses and steady-state dynamic vibrations of the pre-twisted cylindrical shell model for the aero-engine compressor blade subject to the single point excitation and the uniform distribution excitation.
Enormous studies have been implemented about blades. However, few of them devoted to the nonlinear responses for the pre-twist and presetting rotating cantilever blade under subsonic airflow. The difficulty lies in the fact that we do not have an explicit expression for the blade in subsonic airflow. Therefore, in this paper, the explicit time dependent aerodynamic force expression of the blade in the subsonic airflow is introduced. The nonlinear dynamical behaviors of the blade subject to aerodynamic forces numerically obtained are investigated. Consider-ing the shear deformation and von-Kármán nonlinear geometric relationship, the equations of motion for the cantilever plate are derived by Hamilton's principle. The asymptotic perturbation methodology is introduced to deduce the averaged equation of the structure in case of 1:3 internal resonance and 1/2 sub-harmonic resonance. Bifurcation and chaotic dynamics of the system are analyzed numerically. Numerical simulation shows that the aerodynamic force has a significant effect on the periodic and chaotic motions of the rotating blade under specific cases. Since we can avoid chaotic motions of the system by changing the aerodynamic force, the large amplitude nonlinear vibration of the blade can be further controlled.

Equations of motion
The structure diagram of the rotary cantilever plate is shown in Fig. 1 [23] . The blade is modeled as a presetting and pre-twist rotating cantilever plate with a rectangular section, which is clamped to a rigid hub with the radius R. The blade rotates with a constant rotating speed Ω 0 . The span length and the chord length of the blade are L and C, respectively, where L = R 0 − R. The thickness of the plate is marked as h. The pre-twist angle φ R and the mounting angle θ r exist at the free end and the fixed end of the plate, respectively. Three coordinate systems, i.e., the inertial coordinate system (i, j, k), the rotating coordinate system (e 0 x , e 0 y , e 0 z ), and the sectional coordinate system (e x , e y , e z ), are introduced, respectively. The origin of the inertial frame (i, j, k) is the center of the hub. (e 0 x , e 0 y , e 0 z ) is located at the edge of the hub. e 0 x , e 0 y , and e 0 z represent the spanwise direction, the chordwise direction, and the thickness direction, respectively. Based on the research done by Yao et al. [23] , considering the first-order shear deformation and Kirchhoff hypothesis [24] , the dimensionless partial differential governing equations are expressed by the displacements of an arbitrary points on the neutral plane of the plate u 0 , v 0 , and w 0 along the x-, y-, and z-directions. The nonlinear dynamic oscillations of the rotating blade are investigated qualitatively. The dimensionless parameters are introduced as follows: The dimensionless nonlinear governing equations for the structure are described as follows: δϕ y : Here, θ is the twist angle of the plate, which is defined as θ = θ r + (θ R /L) · x; ϕ x denotes the rotation of a transverse normal about the y-axis, and ϕ y denotes the rotation about the x-axis, where ϕ x = − ∂w0 ∂x + γ xz , and ϕ y = − ∂w0 ∂y + γ yz ; Θ is the rotary angle of the plate, which can be written as Θ = Ω 0 · t. These parameters can be found in Ref. [23].
The Galerkin method, which is derived by the Taylor expansion approach, is proposed for transferring a continuous operator problem to a discrete one. It is a convergent method in mathematics. This method may provide useful qualitative discussion of nonlinear behaviors for a dynamical system. The second-order ordinary differential dimensionless equations for the system are derived by applying the Galerkin method given by Ding et al. [25] . In this study, only the first two order modes of the later vibration are considered. The lateral displacement w(x, y, t) and the angular displacement functions ϕ x (x, y, t) and ϕ y (x, y, t) can be found in Eq. (26) of Ref. [23]. The in-plane displacements are expressed as follows: where u, v, and w are the displacements of an arbitrary point on the plate along the x-, y-, and z-directions, respectively.
The transverse ordinary differential dimensionless equations of the rotating structure are deduced by using the Galerkin method as follows: where Ω 1 is the frequency of the aerodynamic force [23] .

Perturbation analysis
The perturbation method, which is called the small parameter expansion method, aims to find an asymptotic solution to a problem, especially a nonlinear problem. In Eq. (5), the square and cubic nonlinear terms are presented. Thus, it is reasonable to exploit the asymptotic perturbation methodology [26] to derive the average equations, which guarantees the accuracy of the solution to a certain extent.
Considering 1/2 sub-harmonic resonance and 1:3 internal resonance, the relationship between ω 1 and ω 2 for the structure is written as follows: where σ 1 and σ 2 are two detuning parameters, and Ω is the frequency of the external excitation. By introducing the following time scale transformation, the positive rational number q is constant in the following deduction: The power series solutions of the equations for the blade under 1/2 sub-harmonic resonance and 1:3 internal resonance are written as follows: In the above expression, if n = 0, then r n = |n|, otherwise r 0 = r. In the following derivation, we assume that the value of r n does not change.
When n = 0 and r = 2, the following expressions are obtained: where the items with asterisk are the complex conjugate of the corresponding items. When n = 2, considering the derivative form of Eq. (9) yields When n = 1 and q = 2, the equations can be established. We obtain the following equations: Substitute Eqs. (10) and (11) into Eq. (12). Then, the differential equations about ψ 1 and φ 1 are obtained as follows: Let ψ 1 and φ 1 be in the following forms: The following polar coordinate averaged equation is acquired by extraction of the real and imaginary parts after substituting Eq. (14) into Eq. (13): Whenȧ 1 =ȧ 2 =φ 1 =φ 2 = 0, the following equations of the steady-state motions of the system are obtained: Let ψ 1 and φ 1 in Eq. (13) be rewritten in the following forms: Substitute the above formula into Eq. (13). Then, the corresponding averaged equations in the Cartesian coordinates are given as follows: 4 Numerical simulation

Numerical simulation of frequency-response
The variation trends of the stiffness characteristic of the steady-state rotating cantilever blade enduring the aerodynamic force in the series form in the subsonic airflow are analyzed numerically in this section. Considering the operating condition of the compressor blade, it is obvious that the nonlinear responses of the blade can be affected by various factors, such as the linear parameters, the nonlinear parameters, and the aerodynamic force. Therefore, their effects on the frequency-responses for the plate are analyzed by numerical simulation.
Considering the weakly coupled condition, namely, let a 2 = 1 in Eq. (16a) and a 1 = 1 in Eq. (16b), the frequency-response curves of the cantilever blade structure for the first two order modes are obtained, as shown in Fig. 2, which illustrates that the plate performs hard spring properties. The effects of different factors on the frequency-responses of the cantilever plate are depicted in Fig. 3. Due to the nonlinearity, the multi-valued phenomenon is exhibited in Fig. 3. , and 3(f) show that, when the first harmonics of f 1 , h 6 , and k 3 increase, respectively, the nonlinear vibration amplitudes and the response bandwidths of the first two modes both increase. When the nonlinear parameters |h 8 | and |k 6 | increase, the dynamical response amplitudes decrease. Moreover, the larger |h 8 | and |k 6 | are, the greater the curvatures of the curves become, as shown in Figs. 3(c) and 3(d).
With the decrease in the frequencies σ 1 and σ 2 , the frequency-response curves jump down to lower amplitudes for soft spring properties while jump up to higher amplitudes for hard spring properties. The signs of h 8 and k 6 can change the spring characteristics of the system. When the

Numerical simulation of nonlinear behaviors
Besides the steady-state analysis, the nonlinear dynamical behaviors of the compressor blade subject to the aerodynamic load and the centrifugal force are also numerically investigated based on Eq. (18). Except for the first harmonic of the aerodynamic force f 1 or f 2 , keep other parameters and the initial condition unchanged so as to investigate the effects of the first harmonic of the aerodynamic force on nonlinear oscillations of the rotating plate. When f 1 or f 2 changes in a certain interval, bifurcation diagrams, in which complicated oscillation behaviors can be found, are numerically depicted based on the Poincaré map theory and Runge-Kutta algorithm. Unless otherwise stated, the parameters are set as σ 1 = 1.384 2, σ 2 = 1.   The chaotic and periodic responses can be identified by several conventional criteria. Thus, the bifurcation diagrams, waveforms, phase portraits, and power spectra are utilized to further verify the existence of the chaotic and periodic motions of the blade. Figure 6 illustrates that the blade performs period-1 motion as the first harmonic of the aerodynamic force f 1 is chosen as f 1 = 0.9. Figure 7 shows that the system exhibits period-3 motion as the first harmonic of the aerodynamic force f 1 is chosen as f 1 = 1.98. The bifurcation diagrams in Fig. 4 demonstrate that, as f 1 increases, the multi-periodic motion of the blade changes with the chaotic motion. Due to the limited space, only two diagrams are depicted here. When the first harmonic of Fig. 6 The period-1 motion of the blade when f1 = 0.9: (a) the phase portrait in the x1x2-plane; (b) the waveform for x1; (c) the phase portrait in the x3x4-plane; (d) the waveform for x3; (e) the three-dimensional phase portrait; (f) the power spectrum the aerodynamic f 1 is changed as f 1 = 3, the system performs multi-periodic motion, as shown in Fig. 8. Figure 9 exhibits the chaotic motion of the system when the parameter f 1 is selected as f 1 = 3.91. It can be seen in Fig. 5 that the multi-periodic motion of the plate changes with its chaotic motion when f 2 is larger than 2.6. When f 2 is chosen as f 2 = 2.6, the system exhibits the multi-periodic motion, as shown in Fig. 10. When f 2 increases to 15, the system performs chaotic motion, as shown in Fig. 11. According to the numerical simulation, the system performs complex dynamic behaviors when the parameters f 1 and f 2 change in a certain region. The first harmonic of the aerodynamic force significantly affects the dynamical vibrations of the blade.

Conclusions
In this paper, the traditional piston theory is not used. The nonlinear vibration of the plate rotating with a constant speed subject to the aerodynamic load in series form is studied. The nonlinear partial differential governing equations of the rotating blade are derived by using Hamilton's principle. Ordinary differential equations are obtained by discretizing the nonlinear partial differential governing equations with the Galerkin method. Asymptotic perturbation methodology is used to obtain the averaged equation for 1:3 internal resonance and 1/2 subharmonic resonance rotating blade. The nonlinear dynamics and frequency-responses for the blade are investigated numerically. Jumping of the solutions is described clearly in frequency-response curves. The effects of various factors on the frequency-response curves show that the system spring property is determined by whether the certain nonlinear parameter is larger or smaller than zero. The increase in the aerodynamic force and linear parameters results in the existence of the hard spring characteristic of the blade. Meanwhile, the increase in the aerodynamic force and linear parameters causes the increase in the vibration amplitudes and the response bandwidth of the first two modes. However, when the nonlinear parameters increase, the amplitude of the dynamic responses decreases.
Based on Eq. (18), bifurcation diagrams, phase portraits, waveforms, and power spectra are obtained to demonstrate that periodic motions and chaotic motions occur in the nonlinear vibrations of the rotating blade under certain conditions. The bifurcation diagrams, phase portraits, waveforms, and power spectra numerically depicted demonstrate that system nonlinear oscillations can be easily affected by the first harmonic of the aerodynamic force. Period-1 motion, period-3 motion, multi-periodic motion, and chaotic motion occur when f 1 and f 2 are chosen in specified intervals. From mathematical points of view, in the certain initial condition, the appearance of amplitude-modulated periodic and chaotic vibrations for original systems is explained by the bifurcation diagrams that we have obtained. The large-amplitude vibrations of the blade with a constant rotating speed that may lead to undesirable damage of the aero-engine can be controlled by adjusting the aerodynamic force. Therefore, it is of great significance to investigate the nonlinear behaviors of blades to reduce the probability of aero-engine failure.