# Analysis on Free Vibration and Stability of Rotating Composite Milling Bar with Large Aspect Ratio
## 2 Governing Equations  
The bar is modeled as a rotating cantilever shaft as shown in Figure 1. It rotates at a constant angular velocity around its axis. \(\Omega\), \(b\), \(t\), \(r\) and \(L\) represent the outer diameter, wall thickness, the radius of midline of cross-section profile and the length of the milling bar, respectively. (\(X\), \(Y\), \(Z\)) denote the inertial coordinate system, and the corresponding unit vectors are (**I**, **J**, **K**); (\(x\), \(y\), \(z\)) denote the coordinate systems rotating with shaft axis and the corresponding unit vectors are (**i**, **j**, **k**). (\(\xi\), \(s\), \(x\)) is a local coordinate system, \(s\) refers to the tangential direction of midline of cross-section profile, \(\xi\) designates the normal line, and \(\eta\) is the ply angle. Based on Euler Bernoulli beam theory which does not consider transvers shear, the linear strain-displacement relationship can be expressed as  
\[\left\{ \begin{array}{l}\varepsilon_{xx}=u^{\prime}-zw^{\prime\prime}-yv^{ \prime\prime}\\ \varepsilon_{xy}=-zq^{\prime}/2\\ \varepsilon_{xz}=yq^{\prime}/2\\ \varepsilon_{yy}=\varepsilon_{zz}=\varepsilon_{yz}=0\end{array} \right. \tag{1}\]  
where \(u,v,w\) denote the \(X\), \(Y\), and \(Z\) components of the extension and bending displacements of any point on the bar's axis, as shown in Figure 2, and \(\varphi\) is the rotating angle of the cross-section around \(x\)-axis. Since the geometry of the bar is a hollow cylinder, it is convenient to use the cylindrical coordinate to express the stress-strain relationship of the bar. The strain components in cylindrical coordinate can be expressed as  
\[\left\{ \begin{array}{l}\varepsilon_{xx}=u^{\prime}-r\sin\theta w^{\prime \prime}-r\cos\theta v^{\prime\prime}\\ \varepsilon_{x0}=\frac{1}{2}rq^{\prime}\\ \varepsilon_{xr}=\varepsilon_{rr}=\varepsilon_{\theta\theta}=\varepsilon_{ r\theta}=0\end{array} \right. \tag{2}\]  
Figure 1: Coordinate systems and geometry of rotating bar. Based on the stress-strain relationship of the composite, the stresses in the \(k\)th layer of the bar can be expressed as  
\[\left\{ \begin{array}{l}\sigma_{xx}^{(k)}=\overline{Q}_{11}^{(k)}\left(u^{\prime }-r\sin\theta w^{\prime\prime}-r\cos\theta v^{\prime\prime}\right)+\frac{1}{2} \overline{Q}_{16}^{(k)}r\varphi^{\prime}\\ \tau_{x\theta}^{(k)}=\overline{Q}_{16}^{(k)}\left(u^{\prime}-r\sin\theta w^{ \prime\prime}-r\cos\theta v^{\prime\prime}\right)+\frac{1}{2}\overline{Q}_{66}^ {(k)}r\varphi^{\prime}\\ \tau_{x\tau}^{(k)}=0\end{array} \right. \tag{3}\]  
where \(\overline{Q}_{ij}^{(k)}\) is listed in reference [29]. The strain energy can be expressed as  
\[U_{s}=\frac{1}{2}{\int_{V}{[\sigma]}^{T}{[\varepsilon]}dV}=\frac{1}{2}{\int_{ V}{(\sigma_{xx}\varepsilon_{xx}+\sigma_{rr}\varepsilon_{rr}+\sigma_{\theta \theta}\varepsilon_{\theta\theta}+2\tau_{x\varepsilon}\varepsilon_{x\tau}+2 \tau_{x\theta}\varepsilon_{x\theta}+2\tau_{r\theta}\varepsilon_{r\theta})}} \mathrm{d}V \tag{4}\]  
Taking variation of the above strain energy expression and considering Equations (2) and (3), the following equation can be obtained  
\[\delta U_{s}=\frac{1}{2}{\int_{L}{(N_{x}\frac{\partial\delta u}{\partial x}-M_ {y}\frac{\partial^{2}\delta w}{\partial x^{2}}-M_{z}\frac{\partial^{2}\delta v }{\partial x^{2}}+M_{x0}\frac{\partial\delta\varphi}{\partial x})}}\mathrm{d}L \tag{5}\]  
where  
\[\left\{ \begin{array}{l}N_{x}=A_{11}u^{\prime}+B_{16}\varphi^{\prime}\\ M_{y}=M_{z}=-D_{11}w^{\prime\prime}\\ M_{x0}=B_{16}u^{\prime}+D_{66}\varphi^{\prime}\end{array} \right.