Nonlinear vibrations of rotating pretwisted composite blade reinforced by functionally graded graphene platelets under combined aerodynamic load and airflow in tip clearance

The primary resonance and nonlinear vibrations of the functionally graded graphene platelet (FGGP)-reinforced rotating pretwisted composite blade under combined the external and multiple parametric excitations are investigated with three different distribution patterns. The FGGP-reinforced rotating pretwisted composite blade is simplified to the rotating pretwisted composite cantilever plate reinforced by the functionally graded graphene platelet. It is novel to simplify the leakage of the airflow in the tip clearance to the non-uniform axial excitation. The rotating speed of the steady state adding a small periodic perturbation is considered. The aerodynamic load subjecting to the surface of the plate is simulated as the transverse excitation. Utilizing the first-order shear deformation theory, von Karman nonlinear geometric relationship, Lagrange equation and mode functions satisfying the boundary conditions, three-degree-of-freedom nonlinear ordinary differential equations of motion are derived for the FGGP-reinforced rotating pretwisted composite cantilever plate under combined the external and multiple parametric excitations. The primary resonance and nonlinear dynamic behaviors of the FGGP-reinforced rotating pretwisted composite cantilever plate are analyzed by Runge–Kutta method. The amplitude–frequency response curves, force–frequency response curves, bifurcation diagrams, maximum Lyapunov exponent, phase portraits, waveforms and Poincare map are obtained to investigate the nonlinear dynamic responses of the FGGP-reinforced rotating pretwisted composite cantilever plate under combined the external and multiple parametric excitations.


Introduction
As a key component of the aeroengine, the blade is crucial to the flight safety. To improve the flight efficiency and to enhance the thrust-weight ratio of the aero engine, the composite materials with the lighter and better mechanical properties are applied in the airplane and are highly significant. The graphene, as an advanced carbon nanomaterial, has been widely researched in recent years. The graphenes have the characteristics of the low mass density and excellent mechanical performance [1][2][3]. Therefore, the application in the blade for the graphenes is extremely meaningful. The service environment of the rotating blade is complex and changeable, which subjects to various excitations, such as the aerodynamic excitation, centrifugal force and thermal stress. During the airplane flight, the primary resonance and nonlinear vibrations of the blade often happen. The blade can be simplified to the rotating pretwisted composite cantilever plate. It is necessary for us to research the primary resonance and nonlinear vibrations of the rotating pretwisted composite cantilever plate reinforced the functionally graded graphene platelet (FGGP) under combined the external and multiple parametric excitations. The novelty of this paper is to simplify the leakage of the airflow in the tip clearance to the non-uniform axial excitation. It is found that the axial excitation and distribution of the graphene have significant influences on the primary resonance and nonlinear vibrations of the FGGP-reinforced rotating pretwisted composite cantilever plate.
Because of the excellent mechanical properties, the graphene-reinforced composites have attracted attentions for many researchers. The buckling and postbuckling properties of the FGGP-reinforced beams were investigated by Yang et al. [4]. Chen et al. [5] used Timoshenko beam theory to research the nonlinear vibrations and post-buckling on the porous nanocomposite beams reinforced by the graphene platelets. Based on the non-uniform rational B-spline formulation, Kiani [6] investigated the large amplitude free vibration of the FGGP-reinforced plate and took into account the influence of the temperature on the material properties. Mao and Zhang [7,8] investigated on the vibration and stability of the piezoelectric composite plate reinforced by the graphene platelets. Based on the high-order shear deformation theory, Wang et al. [9] presented an investigation on the frequency and bending behaviors of the graphene platelet-reinforced doubly curved shallow shell. Zhao et al. [10] employed the finite element method to research the nonlinear bending problems of the FGGPreinforced trapezoidal plate and discussed the influence of the bottom angle on the bending characteristics of the trapezoidal plates. Based on the substructure modal synthesis method and Galerkin method, Zhao et al. [11] studied the free vibration of a FGGP-reinforced disk-shaft system. Considering an open edge crack, Tam et al. [12] explored the nonlinear bending of the functionally graded graphene-reinforced beam.
Due to the complexity of the geometry features for the blade, many scholars simplified it as the rotating beam with the high aspect ratio, rotating plate and rotating shell with the low aspect ratio to analyze the vibration. The linear vibration characteristics of the rotating blades were studied by many researches. Gupta and Rao [13] presented an analysis of the torsional frequencies on the pretwisted cantilever plate. The frequency and critical load of the thick rotating blades with cracked were investigated by Chen et al. [14]. Sun et al. [15,16], respectively, proposed the plate model and shell model to study the free vibration of a rotating pretwisted blade. Cao et al. [17] applied the first-order deformation theory to investigate the free vibration of a sandwich blade considering the thermal barrier coating layers. Chen and Li [18] used Rayleigh-Ritz method to study the vibration behavior of the rotating pretwisted laminated blade. In [19], a thick shell theory was employed to analyze the linear vibration of the rotating pretwisted functionally grade sandwich blade. The model considering the couple effect among the stretching, bending and torsion was proposed by Oh and Yoo [20]. The shallow shell theory was utilized in [21] to study the influence of the initial geometric imperfection on the linear frequency and mode of the rotating pretwisted panel. Using Chebyshev-Ritz method, Niu et al. [22] and Zhang et al. [23], respectively, analyzed the natural vibration of the rotating pretwisted functionally graded composite cylindrical blade and tapered blade reinforced with the graphene platelets. The free vibrations of a rotating pretwisted beam under the axial loading were studied by Ondra and Titurus [24]. Chen et al. [25] established a cantilever varying cross sections plate to investigate the free vibrations of the FGGP-reinforced composite blades. Gu et al. [26] presented a shallow shell model to investigate the dynamic stability of a rotating twisted plate considering the initial imperfection. Using the couple model, Zhao et al. [27] researched the free vibration of a rotating pretwisted blade-shaft reinforced by graphene platelets. Maji and Singh [28] investigated the free vibration of the rotating cylindrical shell by using the third-order shear deformation theory. Zhao et al. [29] proposed the free vibration of a rotating grapheme nanoplatelet-reinforced pre-twisted blade-disk system by adopting Lagrange's equation. Based on the shallow shell theory, Li and Cheng [30] used the variable thickness model to study the free vibration of the rotating pretwisted blades. Xiang et al. [31] adopted the shell model to research the free vibration of the composite blade.
The vibration behaviors of the blade could not be solved completely by the linear vibration theory. The nonlinear vibration theory was introduced into the blade vibration analysis. Avramov et al. [32] investigated the flexural-flexural-torsional nonlinear vibrations of the rotating beam. Wang and Zhang [33] considered the geometric nonlinear model to study the stability and bifurcations of the rotating blade. Arvin et al. [34] applied the flapping nonlinear normal modes to analyze 2:1 internal resonances. Considering the influence of varying rotating speed, Yao et al. [35] explored the nonlinear vibrations of the blade. Bekhoucha et al. [36] utilized Galerkin and harmonic balance methods to research the nonlinear forced vibrations of the rotating beam. Based on Galerkin and multiple-scale methods, Arvin and Lacarbonara [37] investigated the nonlinear dynamic responses of the rotating beam. Yao et al. [38] employed a pretwisted, presetting and thin-walled rotating beam model to analyze 2:1 internal resonance and primary resonance of the compressor blade. Zhang and Li [39] studied the nonlinear vibrations of the rotating blade. Wang et al. [40] used a two-degree-of-freedom model to analyze 1:1 internal resonance of the turbine blade under the airflows. Roy and Meguid [41] investigated the nonlinear transient dynamic responses of a rotating blade. Bai et al. [42] developed a new method to promote the computational efficiency of the vibration characteristics and reliability analysis. Yao et al. [43] employed the cylindrical shell model to study the nonlinear dynamic responses of the rotating blade. Using the vortex lattice method, Zhang et al. [44] developed an analysis method of the nonlinear resonances for a rotating composite blade under the subsonic flow excitation. Niu et al. [45] utilized the backward differentiation formula and Runge-Kutta algorithm to analyze the nonlinear transient responses of the rotating FGM cylindrical panel. Considering the effect of varying cross section and aerodynamic force, Zhang et al. [46] researched the nonlinear vibrations and internal resonance of a rotating blade. Hao et al. [47] investigated the nonlinear transient responses of a rotating pretwisted cantilever plate. Zhang et al. [48] considered the strong gas pressure to study the superharmonic resonances of a rotating pre-deformed blade.
The literature reviews indicate that applying the graphene composites with the superior properties to the rotating blade is highly meaningful. Studying the nonlinear vibrations and resonances of the rotating blade is very important to the blade designs and failure analysis. Few analyses can be found on the primary resonance and nonlinear vibrations of the FGGPreinforced composite blade under the non-uniform axial excitation generated by leakage flow at the tip clearance. In this paper, the primary resonance and nonlinear dynamic responses of the FGGP-reinforced rotating pretwisted composite cantilever plate with three different distribution patterns are studied by considering the external and multiple parametric excitations. The load generated by the leakage flow at the tip clearance is simplified to the non-uniform axial excitation. There exists the rotating speed of the steady state adding a small periodic perturbation. The aerodynamic load subjecting to the surface of the blade is simplified to the transverse excitation. Utilizing the first-order shear deformation theory, von Karman nonlinear geometric relationship, Lagrange equation and mode functions satisfying the boundary conditions, three-degree-of-freedom nonlinear ordinary differential governing equations of motion are derived for the FGGP-reinforced rotating pretwisted composite cantilever plate. Runge-Kutta method is applied to analyze the primary resonance and nonlinear dynamics of the rotating pretwisted composite cantilever plate under the axial and transverse excitations. The amplitude-frequency and force-frequency response curves, bifurcation diagrams, maximum Lyapunov exponent, phase portraits, waveforms and Poincare map are obtained to investigate the nonlinear dynamic responses of the FGGP-reinforced rotating pretwisted composite cantilever plate under combined the external and multiple parametric excitations.

Dynamic modeling of vibration
The theoretical formulation is derived for a blade, which is simplified to a FGGP-reinforced rotating pretwisted composite cantilever plate. The aerodynamic load subjects to the surface of the rotating pretwisted composite cantilever plate. The leakage flows generated at the tip clearance are simulated as the non-uniform axial excitation. The rotating pretwisted composite cantilever plate is subjected to the varying rotating speed. Therefore, the rotating pretwisted composite cantilever plate is subjected to the transverse and multiple parametric excitations. The length, width and thickness of the FGGP-reinforced rotating pretwisted composite cantilever plate, respectively, are a, b and h.
There are four coordinate systems to describe the model of the blade, as shown in Fig. 1a Þis defined on the center of the root for the pretwisted composite cantilever plate, in which the axes X and Y, respectively, are parallel to X 1 and Y 1 , and the axis Z coincides with the axis Z 1 . The coordinate system x; y 0 ; z 0 ð Þis obtained by rotating the coordinate system X; Y; Z ð Þat angle u around the axis Z, in which the axis y 0 coincides with the median of the cantilever plate root. The unit vector in the coordinate system x; y 0 ; z 0 ð Þ is denoted as i ; j ; k ð Þ. The coordinate system x; y; z ð Þ is defined to describe the pretwisted of the cantilever plate. The y axis is Fig. 2 The sketch map of the airflow and axial load on the tip clearance of the blade is given, a airflow passes through tip clearance, b distribution for axial load in tip clearance equal to the y 0 axis on the root of the cantilever plate. The pretwisted angle is denoted as h at the top of the cantilever plate. The twist rate is assumed to be the linear distribution from the root to tip of the plate, which is denoted as j ¼ h=a.
As shown in Fig. 1a, the aerodynamic load subjects to the surface of the blade in transverse direction and is simplified to the transverse excitation F. Besides, the load on the top of the plate is induced by the leakage flows at the tip clearance. Based on reference [49], the sketch map of the airflow through the tip clearance of the blade is obtained, as shown in Fig. 2a. It is found that the aerodynamic load on the edge of the blade tip is small due to the presence of the separation bubble and reaches the maximum value P max after the gas flow into the tip clearance. Then, the pressure on the top of the blade gradually diminishes. Therefore, the aerodynamic load on the blade tip is simplified to the non-uniform axial excitation. As shown in Fig. 2b, the axial excitation in the tip clearance of the rotating pretwisted composite cantilever plate includes the uniform distribution static excitation P 03 and nonuniform distribution excitation P non . The non-uniform part of the axial excitation increases first and then decreases along the thickness direction.
The axial excitation P in of the FGGP-reinforced rotating pretwisted composite cantilever plate is expressed as P non ¼ P 01 þ P 02 Á cosðxtÞ; ð1cÞ where P 01 is the static part of the non-uniform distribution axial excitation, P 02 is the amplitude of the dynamic part for the non-uniform distribution axial excitation, and x is the frequency of the dynamic part for the non-uniform distribution axial excitation. The aerodynamic load subjects to the surface of the blade and is simulated as the transverse excitation with the harmonic form. Therefore, the transverse excitation F of the FGGP-reinforced rotating pretwisted composite cantilever plate is written as where F s and X 1 are the amplitude and frequency of the transverse excitation, respectively.  As shown in Fig. 1b, the location vector r 0 on the middle surface of the rotating pretwisted composite cantilever plate is obtained as Using the method given in Ref. [12], Lame parameters of the FGGP-reinforced rotating pretwisted composite cantilever plate are written as The unit vector a 1 ; a 2 ; a 3 ð Þ in the coordinate x; y; z ð Þis obtained as To describe the pretwisted properties of the rotating composite cantilever plate, the second quadratic form u 2 is derived as The radii R x and R y , respectively, are located in the x and y directions. The radii R x , R y and twisted rate R xy are obtained by The FGGP-reinforced rotating pretwisted composite cantilever plate is composed of N L layers with the same thickness Dh ¼ h=N L , as shown in Fig. 3. The reinforcement and matrix of the rotating pretwisted composite cantilever plate are the graphene platelets and epoxy polymer, respectively. It is seen from Fig. 3 that in each layer, the graphene platelets are uniformly dispersed in the epoxy polymer. The darker colors represent the more graphene platelets. The graphene platelets are uniformly distributed among the whole plate, calling as the U pattern, see Fig. 3a. The plate with the O pattern exhibited in Fig. 3b demonstrates that in the middle layer, the graphene platelet contents are high. From the middle layer to the upper and lower surfaces, the graphene platelets decrease gradually. The plate with the X pattern has the opposite distribution pattern with the O pattern, as shown in Fig. 3c.
Based on Ref. [22], the volume fractions V Þ of the kth layer graphene platelet are written as pattern U where V Ã G k ¼ 1; 2; . . .; N L ð Þdenote the total volume fractions of the graphene platelets and f G is the total weight fraction of the graphene platelets; q G and q M , respectively, denote the densities of the graphene platelets and epoxy polymer. Based on Halpin-Tsai model [4], the effective Young modulus E ðkÞ c of the kth layer for the FGGPreinforced rotating pretwisted composite cantilever plate is calculated as where where l G , w G and h G are the length, width and height of the graphene platelets, and E M and E G , respectively, denote Young's moduli of the epoxy polymer and graphene platelets.
The v ðkÞ c and q ðkÞ c are Poisson ratios and density of the kth layer for the graphene platelets, respectively, The blade model is assumed as the thin plate. For the thin plates, the first-order shear deformation theory is sufficient. Using the higher-order shear deformation theory will increase the amount of the calculation, but not greatly improve the accuracy. The displacement components u, v and w along the directions O 1 x, O 1 y and O 1 z of the FGGP-reinforced rotating pretwisted composite cantilever plate are obtained by using the first-order shear deformation theory where u 0 , v 0 and w 0 are located in the middle surface along the directions x; y; z ð Þ, respectively, and / x and / y are the angles along the axes y and x, respectively.
The strain displacement relations of the FGGPreinforced rotating pretwisted composite cantilever plate are Substituting Eq. (14) into Eq. (13), the strains can be written as The relation between the stress and the strain of the kth layer FGGP-reinforced rotating pretwisted composite cantilever plate is given as The strain energy for the FGGP-reinforced rotating pretwisted composite cantilever plate is expressed as r ðkÞ xx e xx þ r ðkÞ yy e yy þ s ðkÞ xy c xy þ s ðkÞ xz c xz þ s ðkÞ yz c yz dV where N xx N yy N xy where the K is shear correction factor and equal to 5/6. Considering the effect of the deformation, any point location vector of the FGGP-reinforced rotating pretwisted composite cantilever plate is written as The combustion airflow drives the rotation of the turbine in the aircraft engine. Because of the fluctuations of driving combustion airflow, the rotating speed of the turbine is fluctuant. The turbine drives the rotation of the compressor blades. The rotating speed of the blade is also fluctuant. Therefore, we introduced a small perturbation to simulate the fluctuations of the rotating speed for the blade. Considering a small periodic perturbation, the rotating speed vector X is derived as where X 0 is the steady-state rotating speed, and f and X p , respectively, represent the periodic perturbation amplitude and frequency of the rotating speed. The total speed v T about the rotating pretwisted composite cantilever plate consists of the rotating speed component v s and deformation speed compo- where The deformation speed component v d is given as The total speed v T can be expressed as Substituting Eqs. (24 and (25) to Eq. (26), the components V 1 ; V 2 ; V 3 ð Þof the total speed v T in the direction x; y; z ð Þare derived as where a point denotes the first-order derivative with respect to the time. The kinetic energy T of the FGGP-reinforced rotating pretwisted composite cantilever plate is derived as The components N c1 , N c2 and N c3 of the centrifugal force in the directions a 1 ; a 2 ; a 3 ð Þare obtained as where The potential energy U p of the centrifugal force is given as where displacements u i ði ¼ 1 ; 2 ; 3Þ of the centrifugal force [16,17] are given as Rayleigh dissipation function D f on the structural damping of the rotating pretwisted composite cantilever plate is written as follows where c is the damping coefficient. The potential energy U in of the axial excitation P in and energy W F of the transverse excitation F are expressed as The cantilever boundary of the FGGP-reinforced rotating pretwisted composite plate is clamped at the edge x ¼ 0 and is free at the edges x ¼ a, y ¼ À b 2 and The first-order bending, second-order bending and first-order torsional vibration modes are considered, as shown in Fig. 4. According to Ref. [50], the expansion of the middle surface displacement w 0 for the rotating pretwisted composite cantilever plate is given as Fig. 4 The first third vibration mode shapes are obtained for the FGGPreinforced rotating pretwisted composite cantilever rectangular plate, a first-order bending vibration mode, b first-order torsional vibration mode, c second-order bending vibration mode w 0 ðx; y; tÞ ¼ w 1 ðtÞX 1 ðxÞY 1 ðyÞ þ w 2 ðtÞX 1 ðxÞY 2 ðyÞ þ w 3 ðtÞX 2 ðxÞY 1 ðyÞ; where where and k i is the solution of the equation as follows The u 0 , v 0 , / x and / y are given as / y ðx; y; tÞ ¼ / y1 ðtÞ sin Based on Lagrange function, the governing equations of motion for the FGGP-reinforced rotating pretwisted composite cantilever plate are derived as d dt where q i ¼ u 1 ðtÞ ½ ; u 2 ðtÞ; u 3 ðtÞ; v 1 ðtÞ; w 1 ðtÞ; w 2 ðtÞ; w 3 ðtÞ; / x1 ðtÞ; / x2 ðtÞ; / x3 ðtÞ; / y1 t ð Þ T . Following the works given by Zhang et al. [51] and Hao et al. [52], the transverse nonlinear vibrations occupy the main aspect of the dynamic characteristics. Therefore, we mainly investigate the transverse nonlinear vibration of the FGGP-reinforced rotating pretwisted composite plate. The in-plane and rotatory inertia terms can be neglected since their influences are small compared to that of the transverse inertia term. Solving Eq. (41) on the in-plane displacement terms u 0 and v 0 , rotatory terms / x and / y , the in-plane displacements and rotatory terms are transformed into the transverse displacement w 0 . Substituting the displacements u 0 , v 0 , / x and / y into the ordinary differential equation of the transverse displacement w 0 , three-degree-of-freedom nonlinear dynamical system on the transverse vibration of the FGGP-reinforced rotating pretwisted composite cantilever plate is written as € w 1 þ l 11 _ w 1 þ n 11 f 2 cos 2 ðX p tÞ þ n 12 f cosðX p tÞ À þn 13 P 02 Á cosðxtÞÞw 1 þ m 11 þ n 14 P 01 þ 2P 03 ð Þ ð Þ w 1 þ n 15 f 2 cos 2 ðX p tÞ þ n 16 f cosðX p tÞ þ n 17 P 02 Á À € w 3 þ l 31 _ w 1 þ n 31 f 2 cos 2 ðX p tÞ þ n 32 f cosðX p tÞ À þ n 35 f 2 cos 2 ðX p tÞ þ n 36 f cosðX p tÞ þ n 37 P 02 À . . . ; 11Þ are the stiffness coefficients, and n ij ði ¼ 1 ; 2 ; 3 ; j ¼ 1 ; 2 ; . . .8Þ are the excitation coefficients.
In the following analyses, we will investigate the primary resonance and nonlinear dynamic behaviors of the FGGP-reinforced rotating pretwisted composite cantilever plate under the axial and transverse excitations in Eq. (42). The amplitude-frequency response Fig. 5 The amplitude-frequency response curves are obtained for the graphene platelet-reinforced rotating pretwisted composite cantilever plate with three different vibration modes and three different distribution types, a amplitude-frequency response curves of X pattern graphene platelet-reinforced plate with three different vibration modes, b amplitude-frequency response curves of first-order bending vibration with three different distribution types Fig. 6 The amplitude-frequency response curves of the firstorder bending vibration mode are depicted for the X pattern distribution graphene platelet-reinforced rotating pretwisted composite cantilever plate with three different axial loads P 01 and three different transverse excitations F s , a amplitudefrequency response curves with three different axial loads P 01 , b amplitude-frequency response curves with three different transverse excitations F s curves, bifurcation diagrams, maximum Lyapunov exponent, phase portraits, waveforms and Poincare map are obtained by using Runge-Kutta method.

Amplitude-frequency and force-amplitude response curves
Runge-Kutta methods are used to analyze the primary resonance of the FGGP-reinforced rotating pretwisted composite cantilever plate under combined the external and multiple parametric excitations. According to Ref. [9], the material parameters of the FGGPreinforced rotating pretwisted composite cantilever plate are given as The amplitude-frequency response curves of the firstorder bending vibration mode are given for the X distribution graphene platelet-reinforced rotating pretwisted cantilever plate with three different axial excitations P 02 and three different rotating speeds X 0 , a amplitude-frequency response curves with three different axial excitations P 02 , b amplitude-frequency response curves with three different rotating speeds X 0 Fig. 8 The force-amplitude response curves are obtained for the graphene platelet-reinforced rotating pretwisted composite cantilever plate with three different vibration modes and three different distribution types, a force-amplitude response curves of X pattern graphene platelet-reinforced plate with three different vibration modes, b force-amplitude response curves of first-order bending vibration mode with three different distribution types Unless otherwise stated, the physical dimension of the rotating pretwisted composite cantilever plate is given as follows Figure 5a depicts the amplitude-frequency response curves of the X pattern graphene plateletreinforced rotating pretwisted composite cantilever plate with three different vibration modes. Ignoring Fig. 9 The force-amplitude response curves of the first-order bending vibration mode are depicted for the X distribution graphene platelet-reinforced rotating pretwisted cantilever plate with three different axial loads P 01 and three different frequency ratios X 1 /x 1 ¼ 0.5; 1:0; 1:5 ð Þ , a force-amplitude response curves with three different axial loads P 01 , b force-amplitude response curves with three different frequency ratios X 1 /x 1 ¼ 0.5; 1:0; 1:5 ð Þ Fig. 10 The force-amplitude response curves of the first-order bending vibration mode are obtained for the X distribution graphene platelet-reinforced rotating pretwisted cantilever plate with three different axial excitations P 02 and three different rotating speeds X 0 , a force-amplitude response curves with three different axial excitations P 02 , b force-amplitude response curves with three different rotating speeds X 0 the nonlinear and damping terms, and external load in Eq. (42), the natural frequencies are determined from the eigenvalue problem of Eq. (42). The x 1 is the firstorder natural frequency of the X pattern graphene platelet-reinforced rotating pretwisted composite cantilever plate. The blue solid line indicates forward direction sweep frequency, and the red dotted line denotes backward direction sweep frequency. It is found that the amplitudes of three different vibration modes all increase when the primary resonance occurs and the hardening-spring characteristics of three different vibration modes exist for the X pattern graphene platelet-reinforced rotating pretwisted composite cantilever plate. The amplitude is the highest value of the first-order bending vibration mode and is the lowest value of the first-order torsional vibration mode when the primary resonance of the first-order bending vibration mode occurs. The amplitude-frequency response curves of the first-order bending vibration mode are demonstrated for the graphene platelet-reinforced rotating pretwisted composite cantilever plate under three different distribution types, as shown in Fig. 5b. Obviously, the hardening-spring characteristics are kept for the graphene platelet-reinforced rotating pretwisted composite cantilever plate under three different distribution types. The frequency ratio X 1 /x 1 reaching the peak value of the primary resonance amplitude with the O pattern distribution has the least value and with the X pattern distribution has the highest value. The amplitude of the primary resonance peak with the O pattern distribution is the largest value in forward direction sweep frequency. The amplitude of the primary resonance peak with the X pattern distribution is the smallest value in forward direction sweep frequency. It is because the plate reinforced with the graphene platelets in the top and bottom surfaces has better bending resistance. Therefore, comparing plate reinforced with the graphene platelets near the middle or uniform distribution, more graphene platelets near the top and bottom surfaces of the plate are better for improving the stiffness. Therefore, the stiffness of the cantilever plate with the X distribution pattern is higher than that of the plate with the U and O distribution patterns. The cantilever plate with the X distribution pattern has the larger frequency and smaller vibration amplitude than plate with the U and O distribution patterns. It is noticed that the amplitudes of the primary resonance peak for the O and U distribution patterns are close in the backward direction sweep frequency. Figure 6a describes the amplitude-frequency response curves of the first-order bending vibration mode for the X pattern distribution graphene plateletreinforced rotating pretwisted composite cantilever plate under three different axial loads P 01 . The larger the axial load P 01 is, the earlier the primary resonance occurs. The primary resonance peak is bigger under the larger axial load P 01 . Moreover, the system remains the hardening-spring characteristic under three different axial loads P 01 . Figure 6b gives the influence of three different transverse excitations F s on the amplitude-frequency curves of the first-order bending vibration mode for the X pattern distribution graphene platelet-reinforced rotating pretwisted composite cantilever plate. It is observed that the hardening-spring characteristics remain for the X pattern graphene platelet-reinforced rotating pretwisted composite cantilever plate under three different transverse excitations F s . The lager the transverse excitation is, the more obvious the hard spring characteristics is. The larger transverse excitation F s of the system has the larger primary resonance peak and primary resonance region. Figure 7a presents the influence of three different axial excitations P 02 on the amplitude-frequency response curves of the first-order bending vibration mode for the X pattern distribution graphene plateletreinforced rotating pretwisted composite cantilever plate. The frequency ratio X 1 /x 1 almost is same under three different axial excitations P 02 when the primary resonance occurs. A hardening-spring characteristic under three different axial excitations P 02 is obtained for the system. The larger the axial excitation P 02 is, the bigger the primary resonance peak is. Figure 7b describes the amplitude-frequency response curves of the first-order bending vibration mode for the X pattern distribution graphene platelet-reinforced rotating pretwisted composite cantilever plate under three different rotating speeds X 0 . The primary resonance peak is highest when X 0 ¼ 4000 rpm and is lowest when X 0 ¼ 10000 rpm. The hardening-spring characteristics are more obvious for the lower speed. The frequency ratio X 1 /x 1 reaching the peak value of the primary resonance is low when the low speed X 0 exists. Because the increase in the rotating speed leads to that, the stiffness of the FGGP-reinforced rotating pretwisted composite cantilever plate becomes larger. Therefore, the pretwisted composite cantilever plate with the high rotating speed has the larger frequency and smaller nonlinear vibration amplitude. Figure 8a exhibits the force-amplitude response curves of the X pattern graphene platelet-reinforced rotating pretwisted composite cantilever plate with three different vibration modes. It is found that the amplitudes among three different vibration modes   Fig. 14 The chaotic vibrations of the X pattern graphene platelet-reinforced rotating pretwisted composite cantilever plate are obtained when the axial excitation is P 02 ¼ 4.4 Â 10 5 N and transverse excitation is F s ¼ 7 Â 10 3 N, a, c, e phase portraits on planes w 1 ; _ w 1 ð Þ , w 2 ; _ w 2 ð Þ and w 3 ; _ w 3 ð Þ , b, d, f waveforms of first-order bending, first-order torsional and second-order bending vibration modes, g three-dimensional phase portrait in space w 1 ; w 2 ; w 3 ð Þ , h Poincare map of firstorder bending vibration mode (m) increase for the X pattern graphene platelet-reinforced rotating pretwisted composite cantilever plate with the increase of the transverse excitations F s . The increase of the transverse excitation F s obviously affects the amplitude w 1 of the first-order bending vibration mode. The effect on the amplitude w 2 of the first-order torsional vibration mode is least. Figure 8b portrays the force-amplitude response curves of the first-order bending vibration for the graphene platelet-reinforced rotating pretwisted composite cantilever plate under three different distribution types. It is concluded that the amplitudes increase with the increase of the transverse excitations F s . Obviously, when F s \1930 N, the X pattern cantilever plate has the largest amplitude and the O pattern cantilever plate has the least amplitude. When 1930 N\F s \2440 N, the U pattern cantilever plate has the largest amplitude and the O pattern cantilever plate has the least amplitude. When 2440 N\F s \2540 N, the U pattern cantilever plate has the largest amplitude and the X pattern cantilever plate has the least amplitude. When F s [ 2540 N, the O pattern cantilever plate has the largest amplitude and the X pattern cantilever plate has the least amplitude. The force-amplitude response curves of the firstorder bending vibration mode are illustrated for the X pattern graphene platelet-reinforced rotating pretwisted composite cantilever plate under three different axial loads P 01 , as shown in Fig. 9a. When F s \1600 N, the amplitudes with three different axial loads P 01 nearly equal. When F s [ 1600 N, the amplitude of the rotating pretwisted composite cantilever plate is larger with the increase of the axial loads P 01 . Figure 9b demonstrates the effect of the frequency ratio X 1 /x 1 ¼ 0.5; 1:0; 1:5 ð Þ on the forceamplitude response curves of the first-order bending vibration mode for the X pattern graphene plateletreinforced rotating pretwisted composite cantilever plate. When the frequency ratio X 1 /x 1 ¼ 1.0, the amplitude of the rotating pretwisted composite cantilever plate is the largest value. In this case, the nonlinear characteristic of the curve is most obvious. The rotating pretwisted composite cantilever plate with the frequency ratio X 1 /x 1 ¼ 0.5 has the second lager amplitude. The rotating pretwisted composite cantilever plate with the frequency ratio X 1 /x 1 ¼ 1.5 has the smallest amplitude. Figure 10a represents the influences of three different axial excitations P 02 on the force-amplitude response curves of the first-order bending vibration mode for the X pattern graphene platelet-reinforced rotating pretwisted composite cantilever plate. It is found that the amplitudes are closer under three different axial excitations P 02 when F s \1300 N. When F s [ 1300 N, the amplitude is larger with the increase of the axial excitation P 02 . Figure 10b describes the force-amplitude response curves of the first-order bending vibration mode for the X pattern graphene platelet-reinforced rotating pretwisted composite cantilever plate with three different rotating speeds X 0 . It is noticed that the nonlinear characteristic is not obvious with the X 0 ¼ 7000 rpm and X 0 ¼ 10000 rpm. Due to the dynamic stiffness of the rotating pretwisted composite cantilever plate, the amplitude is smaller with the increase of the rotating speed X 0 .

Effect of axial excitation on nonlinear vibrations
In this section, Runge-Kutta methods are used to analyze the nonlinear vibrations of the FGGP-reinforced rotating pretwisted composite cantilever plates under combined the external and multiple parametric excitations. The material parameters of the FGGPreinforced rotating pretwisted composite cantilever plate are given by Ref. [9] q G ¼ 1060 kg/m 3 ; q M ¼ 1200 kg/m 3 ; The geometric parameters of the FGGP-reinforced rotating pretwisted composite cantilever plate are Fig. 15 The chaotic vibrations of the O pattern graphene platelet-reinforced rotating pretwisted composite cantilever plate are depicted when the axial excitation is P 02 ¼ 4.4 Â 10 5 N and transverse excitation is According to Eq. (42), we research the influences of the axial excitations P 02 on the nonlinear dynamic characteristics of the FGGP-reinforced rotating pretwisted composite cantilever plate with three different distribution types. Figures 11, 12 and 13 demonstrate the bifurcation diagrams and maximum Lyapunov exponent diagram of the graphene platelet-reinforced rotating pretwisted composite cantilever plate with the X, U and O distribution patterns when the axial excitations P 02 increase from 2:4 Â 10 5 N to 4.6 Â 10 5 N, respectively. Figure (a) denotes the bifurcation diagram of the first-order bending vibration mode, namely the relation of w 1 versus P 02 . Figure (b) represents the bifurcation diagram on the relation of w 2 versus P 02 for the first-order torsion vibration mode. Figure (c) shows the bifurcation diagram on the relation of w 3 versus P 02 for the second-order bending vibration mode. Figure (d) illustrates the relation on the maximum Lyapunov exponent versus P 02 . The transverse excitation frequency X 1 and axial excitation frequency x all are 191 Hz. The transverse excitation is F s ¼ 7 Â 10 3 N. The initial conditions are chosen as w 1 ¼ À0:0014, _ w 1 ¼ À0:0026, w 2 ¼ À0:002, _ w 2 ¼ À0:0011, w 3 ¼ À0:006, _ w 3 ¼ À0:002 and t ¼ 0. It is clearly observed that the vibration laws of the FGGP-reinforced rotating pretwisted composite cantilever plates are the periodic to chaotic vibrations through twice periodic doubling bifurcations, as shown in Figs. 11, 12 and 13. For the cases of the U and O distribution patterns shown in Figs. 12 and 13, the FGGP-reinforced rotating pretwisted composite cantilever plates have a small periodic window after the chaotic vibrations. In the X distribution pattern shown in Fig. 11, the periodic window does not appear Fig. 16 The bifurcation diagram and maximum Lyapunov exponent of the X pattern graphene platelet-reinforced rotating pretwisted composite cantilever plate are obtained when the transverse excitations F s increase from 1:0 Â 10 5 N to 1:3 Â 10 5 N and axial excitation P 02 is 4 Â 10 2 N, a bifurcation diagram of the first-order bending vibration mode, b bifurcation diagram of the first-order torsional vibration mode, c bifurcation diagram of the second-order bending vibration mode, d maximum Lyapunov exponent diagram of first-order bending vibration mode in the FGGP-reinforced rotating pretwisted composite cantilever plate. Figure 11 demonstrates that in the periodic region P 02 2 2:40 Â 10 5 N $ 4:60 Â 10 5 N À Á , twice perioddoubling bifurcations occur for the X pattern graphene platelet-reinforced rotating pretwisted composite cantilever plate when the axial excitations are P 02 ¼ 3.566 Â 10 5 N and P 02 ¼ 3.876 Â 10 5 N. The short chaotic vibrations appear between the periodic vibrations when the axial excitation P 02 is around 4.0 Â 10 5 N. When the axial excitation increases to P 02 ¼ 4.20 Â 10 5 N, the chaotic vibrations happen in the X pattern graphene platelet-reinforced rotating pretwisted composite cantilever plate. Figure 12 exhibits that through a periodic region P 02 2 2:40 Â 10 5 N $ 3:76 Â 10 5 N À Á , the chaotic vibrations occur for the U pattern graphene plateletreinforced rotating pretwisted composite cantilever plate when the axial excitation is P 02 ¼ 3.76 Â 10 5 N. In addition, a short periodic window exists when P 02 2 4:34 Â 10 5 N $ 4:40 Â 10 5 N À Á in the chaotic region, which is denoted by a red circle. In the periodic region P 02 2 2:40 Â 10 5 N $ 3:76 Â 10 5 N À Á , twice perioddoubling bifurcations happen in the U pattern graphene platelet-reinforced rotating pretwisted composite cantilever plate when the P 02 ¼ 3.22 Â 10 5 N and P 02 ¼ 3.58 Â 10 5 N.
Based on the aforementioned analyses, it is found that the stiffness of the X pattern graphene plateletreinforced rotating pretwisted composite cantilever plate is greater than that of the U pattern distribution thin plate. The stiffness of the U pattern graphene platelet-reinforced rotating pretwisted composite cantilever plate is greater than that of the O pattern distribution thin plate. Therefore, it is observed from Figs. 11, 12 and 13 that the nonlinear vibrations of the O pattern graphene platelet-reinforced rotating pretwisted composite cantilever plate have the largest amplitude. The nonlinear vibrations of the U pattern distribution graphene platelet-reinforced rotating  Fig. 19 The chaotic vibrations of the U pattern graphene platelet-reinforced rotating pretwisted composite cantilever plate are obtained when the transverse excitation is F s ¼ 1:25 Â 10 5 N and axial excitation P 02 is 4 Â 10 2 N, a, c, e phase portraits on the planes w 1 ; _  Figure 14 plots the chaotic vibrations of the X pattern graphene platelet-reinforced rotating pretwisted composite cantilever plate when the axial excitation is P 02 ¼ 4.40 Â 10 5 N. Figures (a), (c) and (e), respectively, represent the phase portraits on the planes Þ. Figures (b), (d) and (f), respectively, give the waveforms of the firstorder bending, first-order torsional and second-order bending vibration modes on the planes t; w 1 ð Þ, t; w 2 ð Þ and t; w 3 ð Þ. Figure (g) demonstrates three-dimensional phase portrait in the space w 1 ; w 2 ; w 3 ð Þ . Figure (h) denotes Poincare map of the first-order bending vibration mode. Figure 15 illustrates the chaotic vibrations of the O pattern graphene platelet-reinforced rotating pretwisted composite cantilever plate when the axial excitation is P 02 ¼ 4.40 Â 10 5 N. We obtain a conclusion from Figs. 14 and 15 that the amplitudes w 1 and w 3 of the bending vibration modes are larger than the amplitude w 2 of the torsional vibration mode in the graphene platelet-reinforced rotating pretwisted composite cantilever plate.

Effect of transverse excitation on nonlinear vibrations
We investigate the effects of the transverse excitations F s on the nonlinear dynamic characteristics of the FGGP-reinforced rotating pretwisted composite cantilever plate under three different distribution types. Figures 16, 17 and 18, respectively, depict the bifurcation diagrams and maximum Lyapunov exponent of the FGGP-reinforced rotating pretwisted composite cantilever plates with the X, U and O distribution patterns when the transverse excitations F s increase from 1:00 Â 10 5 N to 1:30 Â 10 5 N. Figure (a) denotes the bifurcation diagram of the firstorder bending vibration mode, namely the relation of w 1 versus F s . Figure (b) represents the bifurcation diagram of the first-order torsion vibration mode, namely the relation of w 2 versus F s . Figure (c) shows the bifurcation diagram of the second-order bending vibration mode, namely the relation of w 3 versus F s . Figure (d) illustrates the maximum Lyapunov exponent diagram of the FGGP-reinforced rotating pretwisted composite cantilever plate. The transverse excitation frequency X 1 and axial excitation frequency x all are 318.3Hz. The axial excitation P 02 is 2 Â 10 3 N: The initial conditions are chosen as w 1 ¼ À0:0014, _ w 1 ¼ À0:0026, w 2 ¼ À0:002, _ w 2 ¼ À0:0011, w 3 ¼ À0:006, _ w 3 ¼ À0:002 and t ¼ 0. The vibration laws of the X, U and O distribution patterns for the FGGP-reinforced rotating pretwisted composite cantilever plate are demonstrated as follows: the periodic vibration ? periodic doubling bifurcation ? chaotic vibrations.
It is observed from Fig. 16 that the nonlinear vibrations of the X pattern graphene platelet-reinforced rotating pretwisted composite cantilever plate vary from the periodic to chaotic vibrations when the transverse excitation is F s ¼ 1:18 Â 10 5 N. The periodic doubling bifurcation occurs in the periodic region F s 2 1:00 Â 10 5 N $ 1:18 Â 10 5 N À Á when the transverse excitation increases to F s ¼ 1:12 Â 10 5 N. It is noticed that a quasi-periodic region F s 2 1:238 Â 10 5 N $ 1:274 Â 10 5 N À Á exists in the X pattern graphene platelet-reinforced rotating pretwisted composite cantilever plate after the chaotic vibrations happen, which is indicated by a blue circle.
When the transverse excitation increases to F s ¼ 1:21 Â 10 5 N, it is found from Fig. 17 that the nonlinear vibrations change from the periodic to chaotic vibrations in the U pattern graphene plateletreinforced rotating pretwisted composite cantilever plate. The periodic doubling bifurcation happens when F s ¼ 1:158 Â 10 5 N. Figure 18 demonstrates that through a periodic region F s 2 1:00 Â 10 5 N $ 1:23 Â 10 5 N À Á , the chaotic vibrations appear in the U pattern graphene plateletreinforced rotating pretwisted composite cantilever plate when F s ¼ 1:232 Â 10 5 N. A periodic doubling bifurcation again happens in the FGGP-reinforced rotating pretwisted composite cantilever plate when the transverse excitation is F s ¼ 1:196 Â 10 5 N.

(g) (h)
It is seen from Figs. 19 and 20 that the amplitudes of the nonlinear vibrations have little difference for the FGGP-reinforced rotating pretwisted composite cantilever plate with the U and O distribution patterns when the transverse excitations F s increase from 1:00 Â 10 5 N to 1:30 Â 10 5 N. Figure 19 illustrates the chaotic vibrations of the U pattern graphene plateletreinforced rotating pretwisted composite cantilever thin plate when the transverse excitation is F s ¼ 1:25 Â 10 5 N. Figure (a), (c) and (e), respectively, denote the phase portraits on the planes w 1 ; _ w 1 ð Þ, w 2 ; _ w 2 ð Þ and w 3 ; _ w 3 ð Þ. Figure (b), (d) and (f), respectively, give the waveforms of the first-order bending, first-order torsional and second-order bending vibration modes on the plane t; w 1 ð Þ, t; w 2 ð Þ and t; w 3 ð Þ. Figure (g) illustrates three-dimensional phase portrait in the space w 1 ; w 2 ; w 3 ð Þ . Figure (h) shows Poincare map of the first-order bending vibration mode. Figure 20 demonstrates the chaotic vibrations of the O pattern graphene platelet-reinforced rotating pretwisted composite cantilever thin plates when the transverse excitations all are F s ¼ 1:25 Â 10 5 N. It is found from Figs. 19 and 20 that the relationships among the amplitudes w 1 , w 2 and w 3 are similar to the results obtained in Figs. 14 and 15 for the FGGPreinforced rotating pretwisted composite cantilever plate under combined the external and multiple parametric excitations.

Conclusions
The primary resonance and nonlinear vibrations are investigated for the FGGP-reinforced rotating pretwisted composite cantilever plate under combined the external and multiple parametric excitations. Utilizing von Karman nonlinear geometric relationship and Lagrange equation, three-degree-of-freedom nonlinear governing equations of motion are obtained. The primary resonance and nonlinear dynamic behaviors are analyzed by Runge-Kutta method. The hardeningspring characteristics are found for the graphene platelet-reinforced rotating pretwisted composite cantilever plate. The primary resonant amplitude of the O pattern distribution plate reaches the peak firstly and is largest. The vibration amplitude of the X pattern distribution plate reaches the peak latest. The primary resonant amplitude of the X pattern distribution plate reaches the peak earlier and is larger with the increase of the axial loads P 01 . The larger the transverse excitation F s is, the more obvious the hardeningspring characteristic is.
With the increase of the axial excitations P 02 , the bigger primary resonant peak occurs. The primary resonant amplitude of the X pattern distribution plate reaches the peak later and is larger with the decrease of the rotating speed X 0 . For the force-amplitude response curves, when F s [ 2540 N, the O pattern rotating pretwisted composite cantilever plate has the largest amplitude and X pattern rotating pretwisted composite cantilever plate has the least amplitude. When F s [ 1300 N, the amplitude is larger with the increase of the axial excitations P 01 and P 02 . The amplitude of the X pattern distribution plate is larger with the decrease of the rotating speed X 0 . When the frequency ratio X 1 /x 1 ¼ 1.0 or rotating speed X 0 ¼ 4000 rpm, the nonlinear characteristic of the curve is obvious.
When the axial excitations P 02 increase from 2.4 Â 10 5 N to 4.6 Â 10 5 N, the FGGP-reinforced rotating pretwisted composite cantilever plate under the transverse excitation has the chaotic vibrations through twice periodic doubling bifurcations. When the transverse excitations F s increase from 1:00 Â 10 5 N to 1:30 Â 10 5 N, the FGGP-reinforced rotating pretwisted composite cantilever plate under the axial excitation has the chaotic vibrations through once periodic doubling bifurcation. When the transverse excitations F s increase from 1:00 Â 10 5 N to 1:30 Â 10 5 N, the amplitudes of the nonlinear vibrations for the X, U and O pattern graphene plateletreinforced rotating pretwisted composite cantilever plate have little difference.
When the axial excitations P 02 increase from 2.4 Â 10 5 N to 4.6 Â 10 5 N, the nonlinear vibrations of the O pattern graphene platelet-reinforced rotating pretwisted composite cantilever plate have the largest amplitude. The nonlinear vibrations of the U pattern graphene platelet-reinforced rotating pretwisted composite cantilever plate have the second lager amplitude. The nonlinear vibrations of the X pattern graphene platelet-reinforced rotating pretwisted composite cantilever plate have the smallest amplitude.