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Nonlinear dynamic responses of rotating pretwisted cylindrical shells

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Abstract

A rotating pretwisted cylindrical shell model with a presetting angle is established to investigate nonlinear dynamic responses of the aero-engine compressor blade. The centrifugal force and the Coriolis force are considered in the model. The aerodynamic pressure is obtained by the first-order piston theory. The strain–displacement relationship is derived by the Green strain tensor. Based on the first-order shear deformation theory and the isotropic constitutive law, nonlinear partial differential governing equations are derived by using the Hamilton principle. Discarding the Coriolis force effect, Galerkin approach is utilized to reduce nonlinear partial differential governing equations into a two-degree-of-freedom nonlinear system. According to nonlinear ordinary differential equations, numerical simulations are performed to explore nonlinear transient dynamic responses of the system under the effect of the single point excitation and nonlinear steady-state dynamic responses of the system under the effect of the uniform distribution excitation. The effects of the excitation parameter, damping coefficient, rotating speed, presetting angle and pretwist angle on nonlinear dynamic responses of the system are fully discussed.

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Acknowledgements

The authors gratefully acknowledge the support of the National Natural Science Foundation of China (NNSFC) through Grant No. 11372015.

Author information

Authors and Affiliations

  1. College of Mechanical Engineering, Beijing University of Technology, Beijing Key Laboratory of Nonlinear Vibrations and Strength of Mechanical Structures, Beijing, 100124, People’s Republic of China

    Minghui Yao & Yan Niu

  2. College of Mechanical Engineering, Beijing Information Science and Technology University, Beijing, 100192, People’s Republic of China

    Yuxin Hao

Authors
  1. Minghui Yao
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  2. Yan Niu
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Correspondence to Minghui Yao.

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Appendices

Appendix A

Coefficients presented in Eq. (5) are given as follows

$$\begin{aligned} a_{11}= & {} \frac{1}{B}k\cos \theta , \quad a_{12} = \frac{1}{B}k\sin \theta , \nonumber \\ a_{21}= & {} -\frac{1}{B}ke\cos \theta +\frac{k^{3}e^{3}\sin ^{2}\theta \cos \theta }{B^{3}},\nonumber \\ a_{22}= & {} \frac{1}{B}\cos \theta -\frac{k^{2}e^{2}\sin ^{2}\theta \cos \theta }{B^{3}}, \nonumber \\ a_{23}= & {} \frac{1}{B}\sin \theta +\frac{k^{2}e^{2}\sin \theta \cos ^{2}\theta }{B^{3}}. \end{aligned}$$
(A1)

Appendix B

Components of the Green strain tensor in Eq. (10) are listed as follows

$$\begin{aligned} {2}f_{{11}}= & {} {2}\left[ 1+R^{2}k^{2}\sin ^{2}\theta +e_1 ^{2}k^{2}+z(-a_{11} e_1 k\right. \nonumber \\&\left. +\,a_{12} Rk\sin \theta )\right] \frac{\partial u}{\partial x}+2[e_2 Rk-z(a_{11} R\cos \theta \nonumber \\&+\,a_{12} R\sin \theta )]\frac{\partial v}{\partial x}+2za_{12} k^{2}eu\nonumber \\&+\,2[R^{2}k^{2}\sin \theta \cos \theta +e_1 Rk^{2}\sin \theta \nonumber \\&+\,z(-a_{11} Rk\sin \theta +a_{12} Rk\cos \theta )]v\nonumber \\&+\,2\left[ a_{12} Rk\sin \theta -a_{11} e_1 k+z\left( a_{11} ^{2}+a_{12} ^{2}\right) \right] w\nonumber \\&+\,G_{11} ^{2}+G_{12} ^{2}+G_{13} ^{2},\nonumber \\ {2}f_{22}= & {} {2}[Rke_2 +z(a_{21} +a_{22} e_1 k-a_{23} Rk\sin \theta )]\frac{\partial u}{\partial \theta }\nonumber \\&+\,2[R^{2}+z(a_{22} R\cos \theta +a_{23} R\sin \theta )]\frac{\partial v}{\partial \theta }\nonumber \\&+\,2z(a_{22} Rk\sin \theta -a_{23} Rk\cos \theta )u\nonumber \\&+\,2\left[ a_{22} R\cos \theta +a_{23} R\sin \theta \right. \nonumber \\&\left. +\,z\left( a_{21} ^{2}+a_{22} ^{2}+a_{23} ^{2}\right) \right] w+2z(-a_{22} R\sin \theta \nonumber \\&+\,a_{23} R\cos \theta )v+G_{21} ^{2}+G_{22} ^{2}+G_{23} ^{2},\nonumber \\ 2f_{33}= & {} 2\frac{\partial w}{\partial z}+G_{31} ^{2}+G_{32} ^{2}+G_{33}^{2},\nonumber \end{aligned}$$
$$\begin{aligned} {2}f_{12}= & {} [Rke_2 +z(a_{21} +a_{22} e_1 k-a_{23} Rk\sin \theta )]\frac{\partial u}{\partial x}\nonumber \\&+\,[R^{2}+z(a_{22} R\cos \theta +a_{23} R\sin \theta )]\frac{\partial v}{\partial x}\nonumber \\&+\,[B^{2}+e_2 ^{2}k^{2}+z(a_{12} Rk\sin \theta -a_{11} e_1 k)]\frac{\partial u}{\partial \theta }\nonumber \\&+\,[Rke_2 -z(a_{11} R\cos \theta +a_{12} R\sin \theta )]\frac{\partial v}{\partial \theta }\nonumber \\&+\,z(-a_{11} Rk\sin \theta +a_{12} Rk\cos \theta \nonumber \\&\qquad -\,a_{22} Rk^{2}\sin \theta \nonumber \\&-\,a_{23} e_1 k^{2})u+[-Rek\sin \theta +z(a_{22} Rk\sin \theta \nonumber \\&-\,a_{23} Rk\cos \theta +a_{11} R\sin \theta -a_{12} R\cos \theta )]v\nonumber \\&+\,\left[ -\frac{2kR}{B}-2z(a_{11} a_{22} +a_{12} a_{23} )\right] w\nonumber \\&+\,G_{11} G_{21} +G_{12} G_{22} +G_{13} G_{23} ,\nonumber \\ {2}f_{13}= & {} \frac{\partial w}{\partial x}+ \left[ B^{2}+k^{2}e_2 ^{2}+z(a_{12} Rk\sin \theta \right. \nonumber \\&\left. -\,a_{11} e_1 k)\right] \frac{\partial u}{\partial z}+[Rke_2 -z(a_{11} R\cos \theta \nonumber \\&+\,a_{12} R\sin \theta )]\frac{\partial v}{\partial z}+\frac{e_2 k^{2}}{B}u+\frac{Rk}{B}v\nonumber \\&+\,G_{11} G_{31} +G_{12} G_{32} +G_{13} G_{33} ,\nonumber \\ 2f_{23}= & {} \frac{\partial w}{\partial \theta }+ [Rke_2 +z(a_{21} +a_{22} e_1 k\nonumber \\&-\,a_{23} Rk\sin \theta )]\frac{\partial u}{\partial z}+[R^{2}+z(a_{22} R\cos \theta \nonumber \\&+\,a_{23} R\sin \theta )]\frac{\partial v}{\partial z}+\,\frac{Rk}{B}u-\frac{R}{B}v\nonumber \\&+\,G_{21} G_{31} +G_{22} G_{32} +G_{23} G_{33} , \end{aligned}$$
(B1)
$$\begin{aligned} e_1= & {} e-R\cos \theta , \quad e_2 =e\cos \theta -R. \end{aligned}$$
(B2)

Appendix C

Values of \(\frac{\partial \alpha ^{k}}{\partial \beta ^{i}}\) in Eq. (13) are written as follows

$$\begin{aligned} \frac{\partial \alpha ^{1}}{\partial \beta ^{{1}}}= & {} \frac{{1}}{M}\left( \frac{{1}}{B}z+R\right) , \quad \frac{\partial \alpha ^{{1}}}{\partial \beta ^{{2}}}=\frac{{1}}{M}\frac{ke\cos \theta }{B^{2}}z, \nonumber \\ \frac{\partial \alpha ^{{1}}}{\partial \beta ^{{3}}}= & {} 0, \quad \frac{\partial \alpha ^{{2}}}{\partial \beta ^{{1}}}=\frac{k}{M}\left( \frac{1}{B}z-e_2 \right) ,\nonumber \\ \frac{\partial \alpha ^{{2}}}{\partial \beta ^{{2}}}= & {} \frac{B}{M}, \quad \frac{\partial \alpha ^{{2}}}{\partial \beta ^{{3}}}=0, \quad \frac{\partial \alpha ^{{3}}}{\partial \beta ^{{1}}}=0, \nonumber \\ \frac{\partial \alpha ^{{3}}}{\partial \beta ^{{2}}}= & {} 0, \quad \frac{\partial \alpha ^{{3}}}{\partial \beta ^{{3}}}=1, \end{aligned}$$
(C1)
$$\begin{aligned} M= & {} BR\left( 1+z\frac{ee_1 k^{2}+1}{B^{3}R}\right) . \end{aligned}$$
(C2)

Appendix D

Coefficients in Eq. (35) are given as follows

$$\begin{aligned} \Theta= & {} \phi +kx, \quad e_3 =\frac{ee_1 k^{2}+1}{B^{3}R}, \end{aligned}$$
(D1)
$$\begin{aligned} c_1= & {} I_2 e_3 BR[R\sin (\theta -\phi )+e\sin \phi ][\sin (kL+\phi )\nonumber \\&-\sin (kx+\phi )]+I_0 BR\{R_0 (L-x)\nonumber \\&+\frac{1}{2}(L^{2}-x^{2}) +R^{2}\sin (\theta -\phi )[\sin (kL+\phi +\theta )\nonumber \\&-\sin (kx+\phi +\theta )]-eR\sin \theta [\sin (kL+2\phi )\nonumber \\&-\sin (kx+2\phi )]-e\sin \phi [\sin (kL+\phi )\nonumber \\&-\sin (kx+\phi )](e-2R\cos \theta )\},\nonumber \\ c_{2}= & {} I_2 e_3 BR^{{2}}\cos (kx+\phi )\nonumber \\&\quad \times \left[ \cos \left( \frac{\Phi }{2}-\phi \right) -\cos (\theta -\phi )\right] \nonumber \\&+\,I_0 BR^{2}\left[ -\frac{1}{2}R\cos \theta \cos (kx+2\phi -\theta )\right. \nonumber \\&+\,\frac{1}{2}R\theta \sin (kx+\phi -\theta )\nonumber \\&\quad +\,e\cos (\theta -\phi )\cos (kx+\phi )\nonumber \\&+\,\frac{1}{2}R\cos \left( \frac{1}{2}\Phi \right) \cos \left( kx+2\phi -\frac{1}{2}\Phi \right) \nonumber \\&\left. -\,\frac{1}{4}R\Phi \sin (kx)-e\cos \left( \frac{1}{2}\Phi -\phi \right) \cos (kx+\phi )\right] ,\nonumber \\ c_{3}= & {} I_2 e_3 R\cos (kx+\phi )\cos (\theta -\phi )\nonumber \\&+\,I_0 R[R\cos (\theta -\phi )\cos (kx+\phi -\theta )\nonumber \\&-\,e\cos (kx+\phi )\cos (\theta -\phi )-(R_0 +x)ek\sin \theta ],\nonumber \\ c_\mathrm{4}= & {} -\,I_2 e_3 BR\varphi _x [R\sin (\theta -\phi )\nonumber \\&+\,e\sin \phi ][\sin (kL+\phi )-\sin (kx+\phi )]\nonumber \\&-\,I_0 BR\varphi _x \{R_0 (L-x)+\frac{1}{2}(L^{2}-x^{2})\nonumber \\&+\,R^{2}\sin (\theta -\phi )[\sin (kL+\phi +\theta )-\sin (kx\nonumber \\&+\,\phi +\theta )]-eR\sin \theta [\sin (kL+2\phi )\nonumber \\&-\,\sin (kx+2\phi )]-e\sin \phi [\sin (kL+\phi )\nonumber \\&-\,\sin (kx+\phi )](e-2R\cos \theta )\},\nonumber \\ c_{5}= & {} -\,I_2 e_3 BR^{{2}}\varphi _\theta \cos (kx+\phi )\left[ \cos \left( \frac{\Phi }{2}-\phi \right) \right. \nonumber \\&\left. -\,\cos (\theta -\phi )\right] -I_0 BR^{2}\varphi _\theta \nonumber \\&\left[ -\frac{1}{2}R\cos \theta \cos (kx+2\phi -\theta )\right. \nonumber \\&+\,\frac{1}{2}R\theta \sin (kx+\phi -\theta )\nonumber \\&+\,e\cos (\theta -\phi )\cos (kx+\phi )+\frac{1}{2}R\cos \left( \frac{1}{2}\Phi \right) \nonumber \\&\cos \left( kx+2\phi -\frac{1}{2}\Phi \right) -\frac{1}{4}R\Phi \sin (kx)\nonumber \\&\left. -\,e\cos \left( \frac{1}{2}\Phi -\phi \right) \cos (kx+\phi )\right] , \end{aligned}$$
(D2)
$$\begin{aligned} b_{11}= & {} -BR\left( B^{2}+k^{2}e_2^2 \right) , \quad b_{12} =-BR^{2}ke_2 , \\ b_{13}= & {} -\,e_3 BR\left( B^{2}+k^{2}e_2^2 \right) , \quad b_{14} =-\,e_3 BR^{2}ke_2 ,\\ b_{15}= & {} 2\Omega BR^{{2}}\cos (\Theta -\theta ),\\ b_{16}= & {} 2\Omega R[k^{2}e^{2}\sin \theta \cos \Theta \\&-\,k^{2}eR\sin \theta \sin (\Theta -\theta )-\sin (\Theta -\theta )],\\ b_{17}= & {} 2\Omega BR^{{2}}e_3 \cos (\Theta -\theta ),\\ b_{18}= & {} \Omega ^{2}BR[k^{2}R^{2}\cos ^{2}(\Theta -\theta )\\&-\,2ek^{2}R\cos \Theta \cos (\Theta +\theta )+e^{2}k^{2}\cos ^{2}\Theta +1]\\&-\,2\dot{\Omega }ekBR\cos \Theta ,\\ b_{19}= & {} \Omega ^{2}BR^{2}k(e\cos \Theta -R)\cos (\Theta -\theta )\\&+\,\dot{\Omega }BR^{2}\cos (\Theta -\theta ),\\ b_{110}= & {} \Omega ^{2}kR[R\sin (\Theta -\theta )\cos (\Theta -\theta )\\&-\,e\sin (\Theta -\theta )\cos \Theta -e\sin \theta ]\\&+\,\dot{\Omega }R[-ek^{2}R\sin \theta \cos (\Theta -\theta )\\&+\,e^{2}k^{2}\sin \theta \cos \Theta -\sin (\Theta -\theta )],\\ b_{111}= & {} \Omega ^{2}e_3 BR[k^{2}(R\cos (\Theta -\theta )\\&-\,e\cos \Theta )^{2}+1]-2\dot{\Omega }ee_3 kBR\cos \Theta ,\\ b_{112}= & {} \Omega ^{2}e_3 kBR^{2}(e\cos \Theta -R)\cos (\Theta -\theta )\\&+\,\dot{\Omega }e_3 BR^{2}\cos (\Theta -\theta ),\\ b_{113}= & {} \Omega ^{2}\{I_2 [e_3 kR^{2}\sin (\Theta -\theta )\cos (\Theta -\theta )\\&-\,ee_3 kR\sin (\Theta -\theta )\cos \Theta -ee_3 kR\sin \theta ]\\&+\,I_0 [BR(R_0 +x)+\\&+\,kBR^{3}\sin (\Theta -\theta )\cos (\Theta -\theta )\\&+\,e^{2}kBR\sin \Theta \cos \Theta ]\}\\&+\,\dot{\Omega }\{I_2 [-ee_3 k^{2}R^{2}\sin \theta \cos (\Theta -\theta )\\&+\,e^{2}e_3 k^{2}R\sin \theta \cos \Theta -e_3 R\sin (\Theta -\theta )]\\&+\,I_0 [kBR^{2}(R_0 +x)\cos (\Theta -\theta )\\&-\,ekBR(R_0 +x)\cos \Theta \\&-BR^{2}\sin (\Theta -\theta )+eBR\sin \Theta ]\},\\ b_{21}= & {} -\,ke_2 BR^{2}, \quad b_{22} =-BR^{3}, \\ b_{23}= & {} -\,ke_2 e_3 BR^{2}, \quad b_{24} =-\,e_3 BR^{3},\\ b_{25}= & {} -\,2\Omega BR^{{2}}\cos (\Theta -\theta ),\\ b_{26}= & {} 2\Omega ekR^{2}\sin \theta \cos (\Theta -\theta ),\\ b_{27}= & {} -2\Omega BR^{{2}}e_3 \cos (\Theta -\theta ),\\ b_{28}= & {} \Omega ^{2}kBR^{3}[e\cos \Theta -R\cos (\Theta -\theta )]\cos (\Theta -\theta )\\&-\dot{\Omega }BR^{2}\cos (\Theta -\theta ),\\ b_{29}= & {} \Omega ^{2}BR^{3}\cos ^{2}(\Theta -\theta ),\\ b_{210}= & {} -\Omega ^{2}R^{2}\sin (\Theta -\theta )\cos (\Theta -\theta )\\&+\,\dot{\Omega }ekR^{2}\sin \theta \cos (\Theta -\theta ),\\ b_{211}= & {} \Omega ^{2}e_3 kBR^{2}(e\cos \Theta -R\cos (\Theta -\theta ))\cos (\Theta -\theta )\\&-\,\dot{\Omega }e_3 BR^{2}\cos (\Theta -\theta ),\\ b_{212}= & {} \Omega ^{2}e_3 BR^{3}\cos ^{2}(\Theta -\theta ),\\ b_{213}= & {} \Omega ^{2}\{-I_2 e_3 R^{2}\sin (\Theta -\theta )\\&+\,I_0 BR^{2}[e\sin \Theta -R\sin (\Theta -\theta )]\cos (\Theta -\theta )\} \\&+\,\dot{\Omega }[I_2 ee_3 kR^{2}\sin \theta \cos (\Theta -\theta )\\&-\,I_0 BR^{2}(R_0 +x)\cos (\Theta -\theta )],\\ b_{31}= & {} -BR,\\ b_{32}= & {} 2\Omega R[ek^{2}R\sin \theta \cos (\Theta -\theta )\\&-\,e^{2}k^{2}\sin \theta \cos \Theta +\sin (\Theta -\theta )],\\ b_{33}= & {} -2\Omega ekR^{2}\sin \theta \cos (\Theta -\theta ),\\ b_{34}= & {} 2\Omega Re_3 [ek^{2}R\sin \theta \cos (\Theta -\theta )\\&-\,e^{2}k^{2}\sin \theta \cos \Theta +\sin (\Theta -\theta )],\\ b_{35}= & {} -\,2\Omega ee_3 kR^{2}\sin \theta \cos (\Theta -\theta ),\\ b_{{36}}= & {} \Omega ^{2}kR[R\sin (\Theta -\theta )\cos (\Theta -\theta )\\&-\,e\sin (\Theta -\theta )\cos \Theta -e\sin \theta ]\\&+\,\dot{\Omega }R[ek^{2}R\sin \theta \cos (\Theta -\theta )\\&+\,e^{2}k^{2}\sin \theta \cos \Theta +\sin (\Theta -\theta )],\\ b_{37}= & {} -\Omega ^{2}R^{2}\sin (\Theta -\theta )\cos (\Theta -\theta )\\&-\,\dot{\Omega }ekR^{2}\sin \theta \cos (\Theta -\theta )\\ b_{38}= & {} \frac{\Omega ^{2}R}{B}[e^{2}k^{2}\sin ^{2}\theta +\sin ^{2}(\Theta -\theta )],\\ b_{39}= & {} \Omega ^{2}e_3 kR[R\sin (\Theta -\theta )\cos (\Theta -\theta ) \end{aligned}$$
$$\begin{aligned}&-\,e\sin (\Theta -\theta )\cos \Theta -e\sin \theta ]\nonumber \\&+\,\dot{\Omega }e_3 R[ek^{2}R\sin \theta \cos (\Theta -\theta ),\nonumber \\&+\,e^{2}k^{2}\sin \theta \cos \Theta +\sin (\Theta -\theta )],\nonumber \\ b_{310}= & {} -\Omega ^{2}e_3 R^{2}\sin (\Theta -\theta )\cos (\Theta -\theta )\nonumber \\&-\,\dot{\Omega }ee_3 kR^{2}\sin \theta \cos (\Theta -\theta ),\nonumber \\ b_{311}= & {} \Omega ^{2}R\left\{ \frac{I_2 e_3 }{B}[e^{2}k^{2}\sin ^{2}\theta +\sin ^{2}(\Theta -\theta )]\right. \nonumber \\&+\,I_0 [R\sin ^{2}(\Theta -\theta )+e\sin \Theta \sin (\Theta -\theta )\nonumber \\&\left. -\,ek(R_0 +x)\sin \theta ]\right\} \nonumber \\&+\,\dot{\Omega }I_0 R[ekR\sin \theta \sin (\Theta -\theta )\nonumber \\&-\,e^{2}k\sin \theta \sin \Theta +(R_0 +x)\sin (\Theta -\theta )],\nonumber \\ b_{114}= & {} 2\Omega e_3 R[-ek^{2}R\sin \theta \cos (\Theta -\theta )\nonumber \\&+\,k^{2}e^{2}\sin \theta \cos \Theta -\sin (\Theta -\theta )],\nonumber \\ b_{115}= & {} \Omega ^{2}e_3 kR[R\sin (\Theta -\theta )\cos (\Theta -\theta )\nonumber \\&-\,e\sin (\Theta -\theta )\cos \Theta -e\sin \theta ]\nonumber \\&+\,\dot{\Omega }e_3 R[-ek^{2}R\sin \theta \cos (\Theta -\theta ) \nonumber \\&+\,e^{2}k^{2}\sin \theta \cos \Theta -\sin (\Theta -\theta )],\nonumber \\ b_{116}= & {} \Omega ^{2}I_2 R[e_3 kBR^{2}\sin (\Theta -\theta )\cos (\Theta -\theta )\nonumber \\&-\,ee_3 kBR\sin (2\Theta -\theta )-e^{2}e_3 kB\sin \Theta \cos \Theta \nonumber \\&+\,kR\sin (\Theta -\theta )\cos (\Theta -\theta )\nonumber \\&-\,ek\sin (\Theta -\theta )\cos \Theta +e_3 B(R_0 +x)\nonumber \\&-\,ek\sin \theta ]+\dot{\Omega }I_2 R\nonumber \\&[+e_3 kBR(R_0 +x)\cos (\Theta -\theta )\nonumber \\&-\,ek^{2}R\sin \theta \cos (\Theta -\theta )\nonumber \\&-\,e_3 ekB(R_0 +x)\cos \Theta +e^{2}k^{2}\sin \theta \cos \Theta \nonumber \\&-\,e_3 BR\sin (\Theta -\theta )\nonumber \\&+\,e_3 eB\sin \Theta -\sin (\Theta -\theta )],\nonumber \\ b_{214}= & {} 2\Omega e_3 ekR^{2}\sin \theta \cos (\Theta -\theta ),\nonumber \\ b_{215}= & {} -\Omega ^{2}e_3 R^{2}\sin (\Theta -\theta )\cos (\Theta -\theta )\nonumber \\&+\,\dot{\Omega }e_3 ekR^{2}\sin \theta \cos (\Theta -\theta ),\nonumber \\ b_{216}= & {} \Omega ^{2}I_2 R^{2}\cos (\Theta -\theta )[-e_3 BR\sin (\Theta -\theta )\nonumber \\&+\,e_3 eB\sin \Theta -\sin (\Theta -\theta )]\nonumber \\&+\,\dot{\Omega }I_2 R^{2}\cos (\Theta -\theta )\nonumber \\&[-\,e_3 B(R_0 +x)+ek\sin \theta ]. \end{aligned}$$
(D3)

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Yao, M., Niu, Y. & Hao, Y. Nonlinear dynamic responses of rotating pretwisted cylindrical shells. Nonlinear Dyn 95, 151–174 (2019). https://doi.org/10.1007/s11071-018-4557-7

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