# Analysis on Free Vibration and Stability of Rotating Composite Milling Bar with Large Aspect Ratio
## 2 Governing Equations
\tag{6}\]  
\[\left\{\begin{array}{l}A_{11}=\pi\sum\limits_{k=1}^{n}{\overline{Q}_{11}^{(k )}\Big{(}\Big{.}\Big{.}\Big{.}\Big{.}\Big{.}\Big{.}\Big{.}\Big{.}\Big{.}\Big{.} \Big{.}\Big{.}\Big{.}\Big{.}\Big{.}\Big{.}\Big{.}\Big{.}\Big{.}\Big{.  
Considering that the cutter rod rotates around the rotating axis, the speed of any point in the inertial coordinate system can be expressed as  
\[\mathbf{v}=\dot{u}\mathbf{I}+(\dot{v}-\Omega\big{(}z+w))\mathbf{J}+\big{(}\dot{w }+\Omega(y+v))\mathbf{K} \tag{9}\]  
where \(\Omega\) is the spinning speed of the bar. The kinetic energy \(T\) of the bar can be expressed as  
\[T=\frac{1}{2}\int_{V}\rho(\mathbf{v}\cdot\mathbf{v})\mathrm{d}V \tag{10}\]  
By taking the first order variation of the kinetic energy and considering the circle cross-section, the following can be obtained. \[\delta T=\int_{0}^{L}\{I_{\mathrm{m}}\dot{u}\frac{\partial\dot{w}}{\partial t} +I_{\mathrm{m}}\dot{v}\frac{\partial\dot{w}}{\partial t}+I_{\mathrm{m}}\dot{w} \frac{\partial\dot{w}}{\partial t}-I_{\mathrm{m}}\Omega^{2}(w\delta w+v\delta v )-2I_{\mathrm{m}}\Omega(\dot{v}\delta w+\dot{w}\delta v) \tag{11}\]  
where  
\[\left\{ \begin{array}{l}I_{\mathrm{m}}=\pi\sum\limits_{k=1}^{n}\rho^{(k)}({r_ {k+1}}^{2}-{r_{k}}^{2})\\ I_{\mathrm{d}}=\frac{\pi}{4}\sum\limits_{k=1}^{n}\rho^{(k)}({r_{k+1}}^{4}-{r_ {k}}^{4})\\ I_{\mathrm{p}}=\frac{\pi}{2}\sum\limits_{k=1}^{n}\rho^{(k)}({r_{k+1}}^{4}-{r_ {k}}^{4})\end{array} \right. \tag{12}\]  
By combining the energy Equation (5) and Equation (11) and adopting the [PERSON]'s principle, the following equation can be obtained. \[\int_{t_{0}}^{t_{1}}\big{[}\delta U-\delta T\big{]}dt=0 \tag{13}\]  
The equations of the motion are expressed as follows:  
\[I_{\mathrm{m}}\ddot{u}-A_{11}u^{\prime\prime}-B_{16}\varphi^{\prime\prime}=0 \tag{14}\]  
\[I_{\mathrm{m}}\ddot{w}\underline{-2\Omega I_{\mathrm{m}}\dot{w}}+D_{11}v^{ \prime\prime\prime\prime}\underline{-I_{\mathrm{m}}\Omega^{2}v}=0 \tag{15}\]  
\[I_{\mathrm{m}}\ddot{w}\underline{-2\Omega I_{\mathrm{m}}\dot{w}}+D_{11}w^{ \prime\prime\prime\prime}\underline{-I_{\mathrm{m}}\Omega^{2}w}=0 \tag{16}\]  
\[I_{\mathrm{p}}\ddot{\varphi}-B_{16}u^{\prime\prime}-D_{66}\varphi^{\prime \prime}\underline{-I_{\mathrm{p}}\Omega^{2}\varphi}=0 \tag{17}\]  
The above equations show that the bending motions are decoupled from the axial and torsional motions. The single underlined terms in Equations (15) and (16) represent gyroscopic effect, while the double underlined terms represent the inertial forces. It can be seen from Equations (14)-(17) that the vibration of the boring bar has tension-torsion coupling and bending-bending coupling. In milling chatter, bending vibration is the main vibration mode, while tension-torsional vibration and bending-bending vibration are independent and have no coupling effect. Thus, only bending vibration is analyzed below. Damping is important to restrain or enhance instability. The external viscous damping is always beneficial to the stability of the system, and the internal viscous damping may destroy the stability (depending on the rotation speed). So, it is necessary to add damping to the equation of motion for the system. Considering influence of internal and external damping and the cutting forces, the bending vibration equation can be modified as  
\[I_{\text{m}}\ddot{w}-2\Omega I_{\text{m}}\dot{w}+\frac{(c_{e}+c_{i})\dot{v}+c_{i} \Omega w}{+D_{11}v^{\prime\prime\prime\prime}}-I_{\text{m}}\Omega^{2}v=\frac{f_ {v}}{\
u} \tag{18}\]  
\[I_{\text{m}}\ddot{w}+2\Omega I_{\text{m}}\dot{v}+\frac{(c_{e}+c_{i})\dot{w}-c_{ i}\Omega w}{+D_{11}w^{\prime\prime\prime\prime}}-I_{\text{m}}\Omega^{2}w=\frac{f_ {w}}{\
u} \tag{19}\]  
in terms of underline in Equations (18) and (19), they indicate the newly added terms. \(c_{e}\) is an external viscous damping constant per unit length and \(c_{i}\) is an internal viscous damping constant per unit length. Both \(c_{e}\) and \(c_{i}\) are proportional to the rotational speed.