Jtam-A4.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 56, 1, pp. 31-42, Warsaw 2018 DOI: 10.15632/jtam-pl.56.1.31 DYNAMIC AND RESONANCE RESPONSE ANALYSIS FOR A TURBINE BLADE WITH VARYING ROTATING SPEED Dan Wang, Zhifeng Hao School of Mathematical Sciences, University of Jinan, Jinan 250022, China e-mail: danwang2014518@hotmail.com Yushu Chen School of Astronautics, Harbin Institute of Technology, Harbin 150001, China Yongxiang Zhang School of Mathematical Sciences, University of Jinan, Jinan 250022, China A couplingmodel between turbine blades with a varying rotating speed and oncoming vor- tices is constructed,where the coupling of the structure and the fluid is simulated by the van derPol oscillation.Partial differential governing equations ofmotions for the coupled system are obtained and discretized by using theGalerkinmethod. The 1:2 subharmonic resonance and the 1:1 internal resonanceare investigatedwith themultiple scalemethodandfirst-order averaged equations are then derived.Nonlinear responses and bifurcation characteristics are studied by a numerical integration method. Stability of bifurcation curves is determined by utilizing the Routh-Hurwitz criterion. The effect of system parameters including the detuning parameter, steady-state rotating speed, amplitude of periodic perturbation for the rotating speed and freestream velocity on vibration responses are investigated. Keywords: varying rotating speed, van der Pol oscillation, multiple scale method, nonlinear response, bifurcation curve 1. Introduction The blade is an important component in the turbomachinery, such as gas axial compressors, wind turbines, aero-engine turbines, etc. Rotating blades are subjected to high centrifugal and aerodynamic loads which can lead to aeroelastic problems of the blades, like flutter and vortex- -inducedvibrations (Gostelow et al., 2006). To keep safe runningof the turbomachinery, analysis of the dynamic response characteristics is of importance for the blade design. Owing to a variety of engineering applications, dynamic analysis of rotating blades has re- ceived broad interest. Transverse and rotational motion as well as control of vibrations for a rotating uniformEuler-Bernoulli beamwere studied by Yang et al. (2004). In addition, the ear- ly research that focused on the problem of rotating nonconstant speed was done in the work by Kammer and Schlack (1987). Nonlinear vibration of a variable speed rotating beam was studied by Younesian and Esmailzadeh (2010), where the influence of various parameters was investigated. The nonlinear dynamic response of a rotating blade with varying rotating speed was investigated byYao et al. (2012), and the results showed that the dynamic responses of the rotating blade changed from periodicmotions to chaotic motions with different rotating speeds. Nonlinear oscillations and steady-state responses of a rotating compressor bladewith varying ro- tating speedwere investigated byYao et al. (2014). Staino andBasu (2013) formulated amulti- -modal flexible wind turbine model with variable rotor speed by using a Lagrangian approach, and anlysed the effect of the rotational speed on the edgewise vibration of the blades. The equations of motion of a rotating composite Timoshenko beam were derived in the study of 32 D.Wang et al. Georgiades et al. (2014), and the results showed that the variable rotating speed as well as a nonzero pitch angle have important effects on the system dynamics. Amore accurate nonlinear model of a rotating cantilever beam was proposed by Kim and Chung (2016). Geometrically nonlinear vibrations of beamswith properties periodically varying along the axis were investiga- ted by Domagalski and Jędrysiak (2016). A new model for a spinning beam under deployment was proposed and the dynamic responses and characteristics were analyzed by Zhu and Chung (2016). The study of forced nonlinear vibrations of a simply supported Euler-Bernoulli beam resting on a nonlinear elastic foundationwith quadratic and cubic nonlinearities was carried out by Shahlaei-Far et al. (2016) with the homotopy analysis method. Vibration of a rotating beam with variable speed/acceleration has been controlled by using the sandwich beam filled with an ER fluid (Wei et al., 2006). Moreover, Warmiński and Latalski (2016) applied a nonlinear saturation control strategy to suppress vibration of the rotating hub-beam structure. The vortex-induced vibration of a rotating blade with the steady-state rotating speed was investigated by Wang et al. (2016c), where the time-varying characteristic of the vortex shed- ding was represented by a van der Pol oscillator. Moreover, the van der Pol oscillator has been introduced as a reduced model in a number of articles to model the time-varying characteri- stics of the fluid (Hartlen and Currie, 1970; Barron and Sen, 2009; Hemon, 1999; Gabbai and Benaroya, 2005; Wang et al., 2016a) or the fluid-structure interaction (Barron, 2010; Lee et al., 2006; Facchinetti et al., 2004; Keber and Wiercigroch, 2008; Wang et al., 2016b) according to experimental and numerical studies. In addition, the effect of structural vibration on the mo- tion of the fluid was also investigated in the above articles. Under different air flow conditions, the dynamic behaviour of the blades becomes very complex when the rotating speed is time- -varying, which could convert to a nonlinear system with the coupling of parametric-excitation and self-excitation. The motivation of this paper is to investigate the dynamic response and bifurcation cha- racteristics of blades with varying rotating speed. The coupling model of the blade with the varying rotating speed and the time-varying flows is derived based on the results by Wang et al. (2016c). The analysis of the 1:2 subharmonic resonance and 1:1 internal resonance is carried out with the multiple scale method. Four-dimensional nonlinear averaged equations are then derived. Bifurcation curves are obtained and the effect of the system parameters on dynamic responses are discussed in detail. 2. Modeling 2.1. Modeling of the coupling for the structure and vortices The blade with length r and varying rotating speed Ω is assumed as a continuous uniform straight cantilever beam based on the Euler-Bernoulli formulation in the centrifugal force field as shown in Fig. 1. Fig. 1. A beamwith varying rotating speed Similar to the derivation process of formulas in the study by Wang et al. (2016c), the go- verning equation of transversemotion of a uniform cantilever beamwith varying rotating speed can be obtained as follows Dynamic and resonance response analysis for a turbine blade... 33 EI ∂4w(x,t) ∂x4 + m̃ ∂2w(x,t) ∂t2 + c̃ ∂w(x,t) ∂t =Ff −ρAΩ 2x ∂w(x,t) ∂x + 1 2 ρAΩ2(r2−x2) ∂2w(x,t) ∂x2 (2.1) wherew(x,t) denotes the transverse displacement of the blade,EI is the flexural rigidity of the structure, c̃ is the viscous damping coefficient, m̃=(ρ+ρf)A is the total mass of the structure and fluid, ρ and ρf are densities of the structure and air flow, respectively, A is the area of the cross-section of the cantilever beam, Ff =0.5ρfU 2DCL(x,t) is the lift force effecting the blade and induced by the vortex,U = √ V 2+(Ωx)2 is the total velocity, V is the freestream velocity, D denotes characteristic length of the cross-section of the beam. Here, the varying rotating speed is expressed as Ω = Ω0 +Ω1cosωt, representing the periodic perturbation Ω1cosωt on the steady-state rotating speedΩ0. Letting v(x,t)=w(x,t)/D, z=x/r, Eq. (2.1) can be rewritten as ω20 ∂4v(z,t) ∂z4 + ∂2v(z,t) ∂t2 +ζ ∂v(z,t) ∂t = 1 4m̃ CL0ρf[V 2+(rΩz)2]q(z,t)− ρAΩ2 m̃ z ∂v(z,t) ∂z + ρAΩ2 2m̃ (1−z2) ∂2v(z,t) ∂z2 (2.2) with the boundary conditions v(0, t) = 0, v′(0, t) = 0, v′′(1, t) = 0, v′′′(1, t) = 0, where ω0 = √ EI/(m̃r4), ζ = c̃/m̃ is the damping ratio, q(z,t) = 2CL/CL0 represents a time-varying variable of the vortical flows,CL0 is the reference lift coefficient. Similarly, the van der Pol oscillator is applied to simulate time-varying characteristics of the vortices as follows ∂2q(z,t) ∂t2 +sωf[q 2(z,t)−1] ∂q(z,t) ∂t +ω2fq(z,t)=M ∂2v(z,t) ∂t2 (2.3) where ωf is the shedding frequency of the vortex, s is the van der Pol damping coefficient, M is the linear coupling parameter representing the impact of structural vibration on the fluid motion. 2.2. The Galerkin discretization of the coupled system Discretization of partial differential equations (2.2) and (2.3) into afinite-dimensional system is done according to the study by Clough and Penzien (2003), Wang et al. (2016c), letting v(z,t) = ∞∑ i=1 vi(t)ṽi(z) (2.4) represent an arbitrary oscillation of the structure and q(z,t) = ∞∑ i=1 qi(t)q̃i(z) (2.5) denote an arbitrary oscillation of the vortical flows. Themodal functions of the structure and the fluid are expressed as those used in the study of Wang et al. (2016c), that is ṽi(z)= cosh(βiz)− cos(βiz)− coshβi+cosβi sinhβi+sinβi [sinh(βiz)− sin(βiz)] (2.6) and q̃i(z)= sin(iπz) i=1,2, . . . (2.7) 34 D.Wang et al. where βi (i = 1,2, . . .) satisfy the equation cosβcoshβ + 1 = 0 that is obtained from the boundary conditions for the cantilever beam. Repeating the discretization process again, the first modemotion of the structure and fluid can be derived as follows d2v1(t) dt2 + ζ dv1(t) dt +ω20β 4 1v1(t)+a(Ω0+Ω1cosωt) 2v1(t) = [b+ d̃(Ω0+Ω1cosωt) 2]q1(t) d2q1(t) dt2 +sωf [3 4 q21(t)−1 ]dq1(t) dt +ω2fq1(t)=M d2v1(t) dt2 (2.8) where a= ρA 2m̃ 2 ∫1 0 dṽ1(z) dz ṽ1(z)z dz− ∫1 0 d2ṽ1(z) dz2 (1−z2)ṽ1(z) dz ∫1 0 ṽ 2 1(z) dz b= CL0ρfV 2 4m̃ ∫1 0 q̃1(z)ṽ1(z) dz∫1 0 ṽ 2 1(z) dz d̃= CL0ρfr 2 4m̃ ∫1 0 q̃1(z)ṽ1(z)z 2 dz ∫1 0 ṽ 2 1(z) dz Equations (2.8) model the interactions between the vortical flows and the structure, which is also aMathieu-van der Pol type oscillation. 3. Analysis with the multiple scale method The research by Hao and Cao (2015), Hao et al. (2016) showed that nonlinear systems can present rich dynamic characteristics when the resonance occurs, like the primary resonance, superharmonic/subharmonic resonance as well as the internal resonance, etc. Themultiple scale method is often utilized to understand qualitative characteristics of the system which present resonant conditions (Nayfeh andMook, 1979). Introducing the scaling parameters ζ → εζ,CL0 → εCL0, s→ εs,Ω1 → εΩ1,M → εM into Eqs. (2.8), one can obtain d2v1(t) dt2 +εζ dv1(t) dt +ω2sv1(t)+a(ε 2Ω21 cos 2ωt+2εΩ0Ω1cosωt)v1(t) = [ εb+εd̃(Ω0+εΩ1cosωt) 2 ] q1(t) d2q1(t) dt2 +εsωf [3 4 q21(t)−1 ]dq1(t) dt +ω2fq1(t)= εM d2v1(t) dt2 (3.1) where ωs = √ ω20β 4 1 +aΩ 2 0 denotes the uncoupled natural frequency of the first-order mode of the beam. Considering the possible 1:1 internal resonance between the structure and the fluid as well as the 1:2 subharmonic resonance conditions, the relations of frequencies can be expressed as ω=2ωs+εσ, ωf =ωs+εσ1, where σ, σ1 are the detuning parameters, respectively. Assume the approximate form of the solutions as shown in the following v1(t)= v10(T0,T1)+εv11(T0,T1)+ . . . q1(t)= q10(T0,T1)+εq11(T0,T1)+ . . . (3.2) Substituting solutions (3.2) into Eqs. (3.1) and equating the coefficients of like powers of ε, one can obtain: Dynamic and resonance response analysis for a turbine blade... 35 —order ε0 D20v10(T0,T1)+ω 2 sv10(T0,T1)= 0 D20q10(T0,T1)+ω 2 fq10(T0,T1)= 0 (3.3) — order ε1 D20v11(T0,T1)+2D0D1v10(T0,T1)+ω 2 sv11(T0,T1)= bq10(T0,T1)+ d̃Ω 2 0q10(T0,T1) − ζD0v10(T0,T1)−2av10(T0,T1)Ω0Ω1cosωt D20q11(T0,T1)+2D0D1q10(T0,T1)+ω 2 fq11(T0,T1)=MD 2 0v10(T0,T1) −sωf [3 4 q210(T0,T1)−1 ] D0q10(T0,T1) (3.4) where d dt =D0+εD1+ε 2D2+ . . . d2 dt2 =D20 +2εD0D1+ . . . Dn = ∂ ∂Tn General solutions to Eqs. (3.3) can be obtained in the complex form v10 =A(T1)e iωsT0 +A(T1)e −iωsT0 q10 =B(T1)e iωfT0 +B(T1)e −iωfT0 (3.5) Substituting (3.5) into Eqs. (3.4) and considering the resonance conditions yields D20v11+ω 2 sv11 = bBe i(ωsT0+σ1T1)+ d̃Ω20Be i(ωsT0+σ1T1) − iζωsAe iωsT0 −aΩ0Ω1 [ Aei(ω+ωs)T0 +Aei(ωsT0+σT1) ] −2iωsD1Ae iωsT0 + c.c. D20q11+ω 2 fq11 =−Mω 2 sAe i(ωf−εσ1)T0 −2iωfD1Be iωfT0 − isω2f [3 4 B3e3iωfT0 + (3 4 BB−1 ) BeiωfT0 ] + c.c. (3.6) where c.c. stands for the complex conjugate of the proceeding terms. The solvability conditions ofEqs. (3.6) can be obtained by equating the coefficients of secular terms to zero, which reads bBeiσ1T1 + d̃Ω20Be iσ1T1 − iζωsA−aΩ0Ω1Ae iσT1 −2iωsD1A=0 −Mω2sAe −iσ1T1 − isω2f (3 4 BB−1 ) B−2iωfD1B=0 (3.7) The derivatives of amplitudesA andB with respect to T1 can be obtained by Eqs. (3.7), that is D1A= 1 2ωs [ −ibBeiσ1T1 − id̃Ω20Be iσ1T1 − ζωsA+iaΩ0Ω1Ae iσT1 ] D1B= 1 2ωf [ iMω2sAe −iσ1T1 −sω2f (3 4 BB−1 ) B ] (3.8) Assume that the functionsA andB are expressed in polar co-ordinates, which reads A(T1)= a1(T1) 2 eiθ1(T1) B(T1)= a2(T1) 2 eiθ2(T1) (3.9) 36 D.Wang et al. where ak, θk (k=1,2) represent the amplitudes and phase angles of the responses, respectively. Thefirst-order averaged equations can be obtained after separating the real and imaginary parts by substituting (3.9) into Eqs. (3.8), that is a′1 = 1 2ωs [(b+ d̃Ω20t)a2 sinϕ− ζωsa1−aΩ0Ω1a1 sinφ] θ′1 = 1 2a1ωs [−(b+ d̃Ω20)a2cosϕ+aΩ0Ω1a1cosφ] a′2 = 1 2ωf [ Mω2sa1 sinϕ−sω 2 f ( 3 16 a22−1 ) a2 ] θ′2 = 1 2a2ωf Mω2sa1cosϕ (3.10) where (′) denotes the derivatives with respect to T1 andϕ= θ2+σ1T1−θ1, φ=σT1−2θ1. The derivatives of ϕ and φwith respect to T1 can be derived by eliminating θ1 and θ2 from Eqs. (3.10)2,4 ϕ′ = Mω2sa1cosϕ 2a2ωf +σ1+ (b+ d̃Ω20)a2cosϕ−aΩ0Ω1a1cosφ 2a1ωs φ′ =σ+ (b+ d̃Ω20)a2cosϕ−aΩ0Ω1a1cosφ a1ωs (3.11) The equilibrium solutions of Eqs. (3.10)1,3 and (3.11) correspond to periodic motions of the coupled system. The steady-state solutions for system (2.8) can be obtained when assuming a′1 =0, a ′ 2 =0, ϕ ′ =0, φ′ =0. 4. The nonlinear response and bifurcation analysis with different system parameters The research of Facchinetti et al. (2004), Keber and Wiercigroch (2008), Wang et al. (2016c) showed that during the interaction process of the fluid and structure, the structural motion can affect formation of the fluid as well. Therefore, the effects of the system parameters including the detuning parameter σ, steady-state rotating speedΩ0 and the amplitude of periodic pertur- bationΩ1 as well as freestream velocity V on the amplitudes and phase angles of the responses under different coupling parameters M are investigated. The bifurcation curves are computed and stability is determined by examining the eigenvalues of the corresponding characteristic equation to Eqs. (3.10)1,3 and (3.11). Figures 2 and 3 show the varying trends of the amplitudes a1 and a2 and phase angles ϕ and φ (mod T) with respect to the detuning parameter σ for the coupling parameters M = 0.1,0.2,0.3, respectively. The other parameters are fixed at A = 4.2 · 10−4m2, ρ = 7800kg/m3, ρf = 1.225kg/m 3, EI = 300Nm, V = 110m/s, Ω0 = 350rad/s, Ω1 = 0.1rad/s, D = 0.1m, r = 0.3m, c̃ = 6Ns/m, CL0 = 0.01, ωf = 552.64rad/s, s = 0.03, respectively. It can be seen from Figs. 2a,b and 3a,b that as the detuning parameter σ increases, the trivial solutions of the amplitudes a1, a2 and the phase angles ϕ, φ jump to large two-mode solutions via a saddle-node bifurcation at SN1, leading to the occurrence of a stable and an unstable solution. Similarly, as the dutuning parameter σ decreases, the trivial solutions of the amplitudes a1 and a2 and the phase angles ϕ, φ become other two-mode solutions via a saddle-node bifurcation at SN2, resulting in a stable and an unstable solution, respectively. Dynamic and resonance response analysis for a turbine blade... 37 Fig. 2. Frequency-response curves of the amplitude a1 and the phase angleϕ of the structure with respect to the detuning parameter σ Fig. 3. Bifurcation curves of the amplitude a2 and the phase angle φ of the fluid with respect to the detuning parameter σ Figures 2a and 3a show that the amplitudes a1 and a2 have the same varying trends with respect to the varying detuning parameter. The phase anglesϕ and φ have the opposite varying trendswith respect to the varyingdetuningparameter,whichmeans that there is transformation between the two vibration modes. In addition, Figs. 2 and 3 show that the absolute value of σ for the critical bifurcation increases and the amplitudes as well as the period of the steady-state solutions can be increased as the coupling parameter M increases. Figures 4 and 5 show the varying trends of the responseswith respect to the steady-state ro- tating speedΩ0 for the couplingparametersM =0.1,0.2,0.3, respectively.Theotherparameters are fixed at A = 4.2 ·10−4m2, ρ = 7800kg/m3, ρf = 1.225kg/m 3, EI = 300Nm,V=110m/s, Ω1 = 0.1rad/s, D = 0.1m, r = 0.3m, c̃ = 6Ns/m, CL0 = 0.01, ωf = 552.64rad/s, s = 0.03, σ=0.01, respectively. It can be seem from Figs. 4 and 5 that the trivial solution jumps to large solutions via a saddle-node bifurcation, resulting in the occurrence of a two-mode solution consisting of a stable solution and an unstable one. Figure 4a shows that the steady-state solutions of the amplitude a1 decrease as the steady-state rotating speedΩ0 increases, which indicates that the increasing of the steady-state rotating speed can suppress the large-amplitude vibrations of the structure. Figure 5a shows that the steady-state solutions of the amplitude a2 increase as the steady-state rotating speedΩ0 increases,whichdisplays an inverse varying trendof the responses comparing with those for the amplitude a1. By comparison, the stable and unstable solutions for the phase angle ϕ as shown in Fig. 4b increase simultaneously as the steady-state rotating 38 D.Wang et al. Fig. 4. Bifurcation curves of the amplitude a1 and the phase angleϕ of the structure with respect to the steady-state rotating speedΩ0 Fig. 5. Bifurcation curves of the amplitude a2 and the phase angle φ of the fluid with respect to the steady-state rotating speedΩ0 speedΩ0 increases while the stable solutions of the phase angleφ decrease and the unstable one increases when the steady-state rotating speedΩ0 increases, as shown in Fig. 5b.Moreover, the critical steady-state rotating speed can decrease for the saddle-node bifurcation as the coupling parameter M increases, that is: Ω0 = 257.9793rad/s for M = 0.1, Ω0 = 146.8258rad/s for M =0.2,Ω0 =101.3993rad/s forM =0.3, respectively. In addition, an increase in the coupling parameterM can increase the amplitudes a1 and a2 of the responses. It can be illustrated from system (2.8) that the increasing of the coupling parameterM can excite large vibrations of the fluid, which can in turn promote the oscillations of the structure. Figures 6 and 7 show the bifurcation characteristics of the system responses with respect to the amplitude Ω1 of the periodic perturbation for different coupling parameters M. The other parameters are fixed at A = 4.2 ·10−4m2, ρ = 7800kg/m3, ρf = 1.225kg/m 3, EI = 300Nm, V =110m/s,Ω0 =350rad/s,D=0.1m, r=0.3m, c̃=6Ns/m,CL0 =0.01,ωf =552.64rad/s, s=0.03, σ=0.01, respectively. It can be seen from Figs. 6 and 7 that the trivial solution jumps to the large solution via a saddle-node bifurcation (SN), leading to the occurrence of a two-mode solution including a stable solution andanunstable oneas theparameterΩ1 increases. Figures 6a and7adisplay that the varying trends of the amplitudes a1 and a2 with respect to the parameterΩ1 are the same to each other, that is, the stable solutions of the amplitudesa1 anda2 increase asΩ1 increaseswhile the unstable solutions decrease asΩ1 increases. In comparison, the stable and unstable solutions of the phase angles ϕ and φ have the opposite varying trend. In addition, the increasing of the coupling parameterM canmake the critical bifurcation value of the parameterΩ1 smaller, that is, Ω1 = 0.0674rad/s for M = 0.1, Ω1 = 0.0383rad/s for M = 0.2 and Ω1 = 0.0286rad/s for M =0.3, respectively. Dynamic and resonance response analysis for a turbine blade... 39 Fig. 6. Bifurcation curves of the amplitude a1 and the phase angleϕ of the structure with respect to the amplitudeΩ1 of the periodic perturbation Fig. 7. Bifurcation curves of the amplitude a2 and the phase angle φ of the fluid with respect to the amplitudeΩ1 of the periodic perturbation Figures 8 and 9 display the varying trends of the responses with respect to the freestream velocity V under different values of the coupling parameter M. The other parameters are fixed at A = 4.2 · 10−4m2, ρ = 7800kg/m3, ρf = 1.225kg/m 3, EI = 300Nm, Ω0 = 350rad/s, Ω1 =0.1rad/s,D=0.1m, r=0.3m, c̃=6Ns/m,CL0 =0.01, s=0.03, σ=0.01, respectively. Fig. 8. Bifurcation curves of the amplitude a1 and the phase angleϕ of the structure with respect to the freestream velocity V It can be seen from Figs. 8 and 9 that the trivial solution jumps to a large solution via a saddle-node bifurcation as the freestream velocity V increases, resulting in the occurrence of a two-mode solution consisting of a stable solution and an unstable one. The stable and unstable solutions of the amplitudesa1 anda2 increasewhen the freestreamvelocityV increases, as shown in Figs. 8a and 9a which present the similar varying trends for the solutions of the amplitudes 40 D.Wang et al. Fig. 9. Bifurcation curves of the amplitude a2 and the phase angle φ of the fluid with respect to the freestream velocity V a1 and a2. Figures 8b and 9b indicate the opposite varying trends for the stable solutions of the phase angles ϕ and φ, namely, the solution of the phase angle ϕ increases while the solution of the phase angle φ decreases as the freestream velocity V increases. Similarly, the increasing of the coupling parameter M can decrease the critical freestream velocity V for the saddle-node bifurcation of the responses. 5. Conclusions The dynamic responses and bifurcation characteristics of turbine blades under variable rotating speed have been investigated, where the rotating blade was modeled as a cantilever beam and the effect of the oncoming vortices was represented as the van der Pol oscillation. And the acceleration coupling was considered to simulate the influence of the vibration of the beam on the van der Pol oscillation. The first-ordermode vibrations of the coupled systemwere obtained by theGalerkin discretization. The1:2 subharmonic resonance and the 1:1 internal resonance for thecoupled systemwere studiedbyusingthemultiple scalemethod.Theaveraged equationswere derived and the bifurcation curves were computed. Effects of the system parameters including the dutuning parameter σ, the steady-state rotating speedΩ0, the amplitudeΩ1 of the periodic perturbation as well as the freestream velocity V on the responses were investigated. The phenomenon of saddle-node bifurcation was found to occur under certain parameter conditions. The bifurcation analysis indicates that the increasing of the coupling parameter M can delay the saddle-node bifurcation of the responses with respect to the detuning parameter while the increasing of the coupling parameter M can make the saddle-node bifurcation occur earlier for the responses with respect to the steady-state rotating speed Ω0, the amplitude Ω1 of the periodic perturbation as well as the freestream velocity V . Moreover, the increasing of the steady-state rotating speed can suppress large vibration of the structure. The amplitudes of vibrations of the structure and fluid can be increased as the freestream velocity V increases. The opposite varying trends for the amplitudes and phase angles with respect to the system parameters indicate the energy transfer between the vibrations of the fluid and the structure. The results can help one to understand the interaction of the fluid and the structure. Acknowledgement The authors acknowledge the financial support from the National Natural Science Foundation of China (No. 11702111, 11572205, 11732014), the Natural Science Foundation of Shandong Province (No. ZR2017QA005, ZR2017BA031) and the University of Jinan (No. 160100210). Dynamic and resonance response analysis for a turbine blade... 41 References 1. Barron M.A., Sen M., 2009, Synchronization of coupled self-excited elastic beams, Journal of Sound and Vibration, 324, 209-220 2. 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