Rotor Blade Finite Element

Abstract

Previous chapters have focused on rotating beams with only the out-of-plane motion. In this chapter, we study rotating blades which have in-plane bending, torsion and axial degrees of freedom. Rotor blades are essential and critical components of helicopters, turbines, compressors and other rotatory machinery (Xiong and Yu, J Sound Vib 302(4–5):821–840, 2007 [1], Thakkar and Ganguli, J Sound Vib 270(4–5):729–753, 2004 [2], Hwang and Kim, J Sound Vib 270(1–2):1–14, 2004 [3], Yoo et al. J Sound Vib 302(4–5):789–805, 2007 [4], Das et al. J Sound Vib 301(1–2):165–188, 2007 [5], Choi et al. J Sound Vib 300(1–2):176–196, 2007 [6]). Blade failure is a severe accident causing the entire machine to shut down.

Appendix 1

Mass matrix is derived as,

$$\begin{aligned} \mathbf{[M]} = \left[ \begin{matrix} M_{uu} &{} 0 &{} 0 &{} 0\\ 0 &{} M_{vv} &{} 0 &{} M_{v\phi }\\ 0 &{} 0 &{} M_{ww} &{} M_{w\phi }\\ 0 &{} M_{\phi v} &{} M_{\phi w} &{} M_{\phi \phi }\\ \end{matrix}\right] \end{aligned}$$

(7.83)

where,

$$\begin{aligned} M_{uu}= & {} \int _{0}^{l}m\mathbf{H}_u^T\mathbf{H}_udx\end{aligned}$$

(7.84)

$$\begin{aligned} M_{vv}= & {} \int _{0}^{l}m\mathbf{H}_v^T\mathbf{H}_vdx\end{aligned}$$

(7.85)

$$\begin{aligned} M_{v\phi }= & {} -\int _{0}^{l}me_g\sin \theta _0\mathbf{H}_v^T\mathbf{H}_{\phi }dx\end{aligned}$$

(7.86)

$$\begin{aligned} M_{ww}= & {} \int _{0}^{l}m\mathbf{H}_w^T\mathbf{H}_wdx\end{aligned}$$

(7.87)

$$\begin{aligned} M_{w\phi }= & {} \int _{0}^{l}me_g\cos \theta _0\mathbf{H}_w^T\mathbf{H}_{\phi }dx \end{aligned}$$

(7.88)

$$\begin{aligned} M_{\phi v}= & {} -\int _{0}^{l}me_g\sin \theta _0\mathbf{H}_{\phi }^T\mathbf{H}_vdx\end{aligned}$$

(7.89)

$$\begin{aligned} M_{\phi w}= & {} \int _{0}^{l}me_g\cos \theta _0\mathbf{H}_{\phi }^T\mathbf{H}_wdx\end{aligned}$$

(7.90)

$$\begin{aligned} M_{\phi \phi }= & {} \int _{0}^{l}mK_m^2\mathbf{H}_{\phi }^T\mathbf{H}_{\phi }dx \end{aligned}$$

(7.91)

Stiffness matrix is derived as,

$$\begin{aligned} \mathbf{[K]} = \left[ \begin{matrix} K_{uu} &{} K_{uv} &{} K_{uw} &{} K_{u\phi }\\ K_{vu} &{} K_{vv} &{} K_{vw} &{} K_{v\phi }\\ K_{wu} &{} K_{wv} &{} K_{ww} &{} K_{w\phi }\\ K_{\phi u} &{} K_{\phi v} &{} K_{\phi w} &{} K_{\phi \phi }\nonumber \end{matrix}\right] \end{aligned}$$

where,

$$\begin{aligned} K_{uu}= & {} \int _{0}^{l}EA\mathbf{B}_u^T\mathbf{B}_udx -\int _{0}^{l}m\mathbf{H}_u^T\mathbf{H}_udx\end{aligned}$$

(7.92)

$$\begin{aligned} K_{uv}= & {} -\int _{0}^{l}EAe_A\cos \theta _0\mathbf{B}_u^T\mathbf{N}_vdx\end{aligned}$$

(7.93)

$$\begin{aligned} K_{uw}= & {} -\int _{0}^{l}EAe_A\sin \theta _0\mathbf{B}_u^T\mathbf{N}_wdx\end{aligned}$$

(7.94)

$$\begin{aligned} K_{u\phi }= & {} \int _{0}^{l}EAK_A^2\theta _0'{} \mathbf{B}_u^T\mathbf{B}_{\phi }dx\end{aligned}$$

(7.95)

$$\begin{aligned} K_{vu}= & {} -\int _{0}^{l}EAe_A\cos \theta _0\mathbf{N}_v^T\mathbf{B}_udx \end{aligned}$$

(7.96)

$$\begin{aligned} K_{vv}&=\int _{0}^{l}\left( EI_z\cos ^2\theta _0+EI_y\sin ^2\theta _0\right) \mathbf{N}_v^T\mathbf{N}_vdx -\int _{0}^{l}m\mathbf{H}_v^T\mathbf{H}_vdx +\int _{0}^{l}\left[ \int _{x}^{l}mxd\xi \right] \mathbf{B}_v^T\mathbf{B}_vdx\qquad \qquad \end{aligned}$$

(7.97)

$$\begin{aligned} K_{vw}&=\int _{0}^{l}\left( EI_z-EI_y\right) \sin \theta _0\cos \theta _0\mathbf{N}_v^T\mathbf{N}_wdx \end{aligned}$$

(7.98)

$$\begin{aligned} K_{v\phi }&=-\int _{0}^{l}EB_2\theta _0'\cos \theta _0\mathbf{N}_v^T\mathbf{B}_{\phi }dx +\int _{0}^{l}me_g\sin \theta _0\mathbf{H}_v^T\mathbf{H}_{\phi }dx -\int _{0}^{l}mxe_g\sin \theta _0\mathbf{B}_v^T\mathbf{\mathbf{H}}_{\phi }dx\qquad \qquad \end{aligned}$$

(7.99)

$$\begin{aligned} K_{wu}=-\int _{0}^{l}EAe_A\sin \theta _0\mathbf{N}_w^T\mathbf{B}_udx\qquad \qquad \end{aligned}$$

(7.100)

$$\begin{aligned} K_{wv}=\int _{0}^{l}\left( EI_z-EI_y\right) \sin \theta _0\cos \theta _0\mathbf{N}_w^T\mathbf{N}_vdx\qquad \qquad \end{aligned}$$

(7.101)

$$\begin{aligned} K_{ww}=\int _{0}^{l}\left( EI_y\cos ^2\theta _0+EI_z\sin \theta _0\right) \mathbf{N}_w^T\mathbf{N}_wdx +\int _{0}^{l}\left[ \int _{x}^{l}mxd\xi \right] \mathbf{B}_w^T\mathbf{B}_wdx\qquad \qquad \end{aligned}$$

(7.102)

$$\begin{aligned} K_{w\phi }=-\int _{0}^{l}EB_2\theta _0'\sin \theta _0\mathbf{N}_w^T\mathbf{B}_{\phi }dx +\int _{0}^{l}mxe_g\cos \theta _0\mathbf{B}_w^T\mathbf{H}_{\phi }dx\qquad \qquad \end{aligned}$$

(7.103)

$$\begin{aligned} K_{\phi u}=\int _{0}^{l}EAK_A^2\theta _0'{} \mathbf{B}_{\phi }^T\mathbf{B}_udx\qquad \qquad \end{aligned}$$

(7.104)

$$\begin{aligned} K_{v\phi }=-\int _{0}^{l}EB_2\theta _0'\cos \theta _0\mathbf{N}_v^T\mathbf{B}_{\phi }dx +\int _{0}^{l}me_g\sin \theta _0\mathbf{H}_v^T\mathbf{H}_{\phi }dx\qquad \qquad \nonumber \\ -\int _{0}^{l}mxe_g\sin \theta _0\mathbf{B}_v^T\mathbf{\mathbf{H}}_{\phi }dx\qquad \qquad \end{aligned}$$

(7.105)

$$\begin{aligned} K_{\phi w}=-\int _{0}^{l}EB_2\theta _0'\sin \theta _0\mathbf{B}_{\phi }^T\mathbf{N}_wdx +\int _{0}^{l}mxe_g\cos \theta _0\mathbf{H}_{\phi }^T\mathbf{B}_wdx\qquad \qquad \end{aligned}$$

(7.106)

$$\begin{aligned} K_{\phi \phi }=\int _{0}^{l}\left( GJ+EB_1{\theta _0'}^2\right) \mathbf{B}_{\phi }^T\mathbf{B}_{\phi }dx -\int _{0}^{l}m\left( K_{m_1}^2-K_{m_2}^2\right) cos2\theta _0\mathbf{H}_{\phi }^T\mathbf{H}_{\phi }dx\qquad \qquad \end{aligned}$$

(7.107)

Damping matrix is derived as,

$$\begin{aligned} \mathbf{[C]} = \left[ \begin{matrix} 0 &{} C_{uv} &{} 0 &{} 0\\ C_{vu} &{} C_{vv} &{} C_{vw} &{} 0\\ 0 &{} C_{wv} &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 0\\ \end{matrix}\right] \end{aligned}$$

(7.108)

where,

$$\begin{aligned} C_{uv}= & {} -\int _{0}^{l}2m\mathbf{H}_u^T\mathbf{H}_vdx\end{aligned}$$

(7.109)

$$\begin{aligned} C_{vu}= & {} \int _{0}^{l}2m\mathbf{H}_v^T\mathbf{H}_udx\end{aligned}$$

(7.110)

$$\begin{aligned} C_{vv}= & {} -\int _{0}^{l}2me_g\cos \theta _0\mathbf{H}_v^T\mathbf{B}_vdx -\int _{0}^{l}2me_g\cos \theta _0\mathbf{B}_v^T\mathbf{H}_vdx\end{aligned}$$

(7.111)

$$\begin{aligned} C_{vw}= & {} -\int _{0}^{l}2m\beta _p\mathbf{H}_v^T\mathbf{H}_wdx -\int _{0}^{l}2me_g\sin \theta _0\mathbf{H}_v^T\mathbf{B}_wdx\end{aligned}$$

(7.112)

$$\begin{aligned} C_{wv}= & {} -\int _{0}^{l}2m\beta _p\mathbf{H}_w^T\mathbf{H}_vdx -\int _{0}^{l}2me_g\sin \theta _0\mathbf{B}_w^T\mathbf{H}_vdx \end{aligned}$$

(7.113)

Load vector is derived as,

$$\begin{aligned} \mathbf{\{R\}} = \left\{ \begin{matrix} -\int _{0}^{l}mx\mathbf{H}_u^Tdx\\ -\int _{0}^{l}\left( me_g\cos \theta _0\mathbf{H}_v^T -me_g\sin \theta _0\ddot{\theta _0}{} \mathbf{H}_v^T+mxe_g\cos \theta _0\mathbf{B}_v^T\right) dx\\ \int _{0}^{l}\left( mx\beta _p\mathbf{H}_w^T +me_g\cos \theta _0\ddot{\theta _0}{} \mathbf{H}_w^T+mxe_g\sin \theta _0\mathbf{B}_w^T\right) dx\\ \int _{0}^{l}\left( \left( K_{m_2}^2-K_{m_1}^2\right) \frac{sin2\theta _0}{2}m\mathbf{H}_{\phi }^T+mx\beta _pe_g\cos \theta _0\mathbf{H}_{\phi }^T\right) dx\\ \end{matrix}\right\} \nonumber \\ \end{aligned}$$

(7.114)