# Research on a Composite Power-Superimposed Ultrasonic Vibrator for Wire Drawing
## 2 Structure and Mathematical Model
### Vibration Equations  
In order to obtain the value of \(R_{\text{res}}\) and \(r_{0}\), the vibration equations of the ultrasonic vibration system should be established at first. Before this some assumptions have to be made to simplify the physical model. We assume that disc 1 vibrates radially, rods 3 and 4 vibrate longitudinally, while disc 2 vibrates both radially and longitudinally. The approximate vibration equation of the first part can be described as follows:  
\[\left(1-\frac{k_{p}^{2}v_{p}^{2}T^{2}\left(r\right)}{12}\right) \frac{d^{2}\overline{u}_{r1}}{d^{2}r}+\left[\frac{1}{r}+\frac{1}{T\left(r \right)}\frac{dT\left(r\right)}{dr}-\frac{k_{p}^{2}v_{p}^{2}T^{2}\left(r\right) }{12}\left(\frac{1}{r}+\frac{3}{T\left(r\right)}-\frac{dT\left(r\right)}{dr} \right)\right]\frac{d\overline{u_{r1}}}{dr} \tag{1}\] \[+\left[K_{p}^{2}+\frac{v}{rT\left(r\right)}\frac{dT\left(r\right) }{dr}-\frac{1}{r^{2}}+\frac{k_{p}^{2}v_{p}^{2}T^{2}\left(r\right)}{12}\left( \frac{1}{r^{2}}-\frac{3}{rT\left(r\right)}\frac{dT\left(r\right)}{dr}\right) \right]\overline{u_{r1}}=0\]  
The approximate vibration equation of second part can be described as:  
\[\overline{u_{r2}}=A_{1}J_{1}\left(K_{p}r\right)+A_{2}Y_{1}\left(K_{p}r\right) \tag{2}\]  
\[\overline{u_{z2}}=A_{3}K_{1}z \tag{3}\]  
Similarly, the vibration equation of the rods would be:  
\[\overline{u_{r3}}=A_{4}cosK_{1}\left(z-H\right) \tag{4}\]  
where \(J_{1}(K_{\text{p}}r)\) and \(Y_{1}(K_{\text{p}}r)\) is the Bessel function, \(A_{i}\) (\(i\) = 1, 2, 3, 4) are constant, \(K_{1}\) = \(w/C_{1}\), \(C_{1}\) = \((E/\rho)^{1/2}\), and the boundary conditions of the composite power-superimposed ultrasonic vibration system can be expressed as follows:  
\[\left\{\begin{array}{l}r=r_{res}:\ s_{rr1}=0\\ r=r_{0}:\ s_{rr1}=0\\ r=r_{1}:\ s_{rr1}=s_{rr2},\ \overline{u_{r1}}=\overline{u_{r2}}\\ z=H:\ s_{z22}=s_{z23},\end{array}\right. \tag{5}\]  
where  
\[s_{rr1}=\frac{E}{1-v^{2}}\left[\frac{d\overline{u_{r1}}}{dr}+\frac{v\overline {u_{r1}}}{r}\right] \tag{6}\]  
\[s_{rr2}=EA_{1}\left[\frac{\left(1-v\right)}{\left(1+v\right)} \frac{K_{p}J_{0}\left(K_{p}r\right)}{\left(1-2v\right)}-\frac{J_{1}\left(K_{p} r\right)}{\left(1+v\right)r}\right] \tag{7}\] \[+EA_{2}\left[\frac{\left(1-v\right)}{\left(1+v\right)}\frac{K_{p} Y_{0}\left(K_{p}r\right)}{\left(1-2v\right)}-\frac{Y_{1}\left(K_{p}r\right)}{ \left(1+v\right)r}\right]+EA_{3}\frac{vK_{1}}{\left(1+v\right)\left(1-2v\right)}\]  
\[s_{z22}=\frac{1}{S}\iint s_{p22}dS=\frac{E}{\left(1+v\right)\left(1-2v\right)} \left\{\left(1-v\right)K_{1}A_{3}+\frac{2 vK_{p}}{r_{1}^{2}-r_{0}^{2}}\left\{ \begin{array}{l}A_{1}\left[r_{1}J_{1}\left(K_{p}r\right)-r_{0}Y_{1}\left(K_ {p}r\right)\right]\\ +A_{2}\left[r_{1}Y_{1}\left(K_{p}r_{1}\right)-r_{0}Y_{1}\left(K_{p}r_{0} \right)\right]\end{array}\right\}\right\} \tag{8}\]  
\[S=\pi\left(r_{1}^{2}-r_{0}^{2}\right) \tag{9}\]  
\[\sigma_{\text{zz3}}=-Ek_{1}A_{4}\text{sink}\left(z-H\right) \tag{10}\]  
When the resonance frequency was set as 21 kHz, and \(r_{1}\) = 31.5 mm, \(T_{0\,\text{min}}\) = 70 mm, \(H\) = 35 mm, \(H_{1}\) = 100 mm, the variation between \(R_{\text{res}}\) and \(r_{0}\) could be obtained, as shown in Figure 3.