# Research on a Composite Power-Superimposed Ultrasonic Vibrator for Wire Drawing
## 2 Structure and Mathematical Model
### Amplification Ratio  
According to [PERSON] (2016), any variable cross-section amplitude horn can be equivalent to a four-terminal network, as shown in Figure 4. The free vibration equations can be transformed to transport equations in the form of a transport matrix. As for the variable cross-section, such as a cylinder, cone, exponent surface, _etc._, the transport matrix was presented in the reference. For the left part of the node section, the transport equation is expressed by Equation (11). \[\left[ \begin{array}{c}u_{2}\\ F_{2}\end{array} \right]=\left[ \begin{array}{c}a_{11}a_{12}\\ a_{21}a_{22}\end{array} \right]\left[ \begin{array}{c}u_{1}\\ F_{1}\end{array} \right] \tag{11}\]  
The transport matrix is shown as follows:  
\[\left[ \begin{array}{c}u_{2}\\ F_{2}\end{array} \right]=\left[ \begin{array}{cc}a_{11}{}^{1}a_{12}{}^{1}\\ a_{21}{}^{1}a_{22}{}^{1}\end{array} \right]\left[ \begin{array}{cc}a_{11}{}^{2 }a_{12}{}^{2}\\ a_{21}{}^{2}a_{22}{}^{2}\end{array} \right]\left[ \begin{array}{cc}a_{11}{}^{3} a_{12}{}^{3}\\ a_{21}{}^{3}a_{22}{}^{3}\end{array} \right]\left[ \begin{array}{c}u_{1}\\ F_{1}\end{array} \right] \tag{12}\]  
where  
\[\left\{ \begin{array}{c}a_{11}{}^{1}=a_{22}{}^{1}=\cos\!\sigma L_{1},\,a_{12}{ }^{1}=\frac{-j{\rm sin}\sigma L_{1}}{\rho cS_{1}},\,a_{21}{}^{1}=-j\rho cS_{1} {\rm sin}\sigma L_{1}\\ a_{11}{}^{2}=\frac{-(\alpha/\sigma){\rm sin}\sigma L_{2}}{1-2L_{2}},\,a_{12}{ }^{2}=\frac{-j{\rm sin}\sigma L_{2}}{\rho cS_{1}(1-2L_{2})}\\ a_{21}{}^{2}=\frac{\rho cS_{1}}{j\sigma}\left\{\left[\sigma(1-2L_{2})+\frac {\alpha^{2}}{\sigma}\right]{\rm sin}\sigma L_{2}-\alpha^{2}L_{2}{\rm cos} \alpha L_{2}\right\}\\ a_{22}{}^{2}=(1-2L_{2}){\rm cos}\sigma L_{2}+\frac{\alpha}{\sigma}\sigma L_ {2}\\ a_{11}{}^{3}=a_{22}{}^{3}=\cos\!\sigma L_{3},\,a_{12}{}^{3}=\frac{-j{\rm sin} \sigma L_{3}}{\rho cS_{3}},\,a_{21}{}^{3}=-j\rho cS_{3}{\rm sin}\sigma L_{3} \end{array} \right. \tag{13}\]  
where, \(\alpha=S_{2}/S_{1}\), \(\sigma=w/c\). Figure 4: Equivalent four-terminal network of amplitude horn. Figure 3: Variation of \(R_{\rm res}\) with \(r_{0}\). The speed ratio is calculated as follows:  
\[\frac{u_{2}}{u_{1}}=\cos\!\sigma L_{1}\times\frac{(\alpha/\sigma)\text{sin} \sigma L_{2}}{\sigma L_{2}-1}\times\cos\!\sigma L_{3} \tag{14}\]  
We set the cylindrical coordinate system and simplified the energy concentrating structure to be composed of a rod and a disc, as is shown in Figure 5.  
For the overlapping part:  
\[\left\{ \begin{array}{l}u_{r}=ak_{p}r,u_{z}=bkz\\ u_{rr}=\frac{\partial u_{r}}{\partial r}=ak,u_{\theta\Theta}=\frac{1}{r}\frac{ \partial u_{\Theta}}{\partial\phi}+\frac{u_{r}}{r},u_{zz}=\frac{\partial u_{ z}}{\partial z}=bk\\ \sigma_{rr}=\frac{E(1-v)}{(1+v)(1-2v)}[ak_{p}/(1-v)+bkv/(1-v)]\\ \sigma_{zz}=\frac{E(1-v)}{(1+v)(1-2v)}[2 vak_{p}/(1-v)+bk]\end{array} \right. \tag{15}\]  
For the other parts:  
\[\left\{ \begin{array}{l}u_{z}=\pm D\text{cos}(H_{0}\mp z)\\ \sum_{zz}=E\times kD\text{sin}k(H_{0}\mp z)\\ u_{r}=A_{\text{I}}(k_{p}r)+BN_{\text{I}}(k_{p}r)\\ \sum_{zz}=\frac{E}{1-v^{2}}[\frac{\partial u_{r}}{\partial r}+v\frac{u_{r}}{r }]=\frac{Ek_{p}}{1-v^{2}}[A_{y}(k_{p}r)+B_{n}(k_{p}v)]\end{array} \right. \tag{16}\]  
where  
\[\left\{ \begin{array}{l}y(k_{p}r)=J_{0}(K_{p}r)-(1-v)\frac{J_{1}(K_{p}r)}{K_ {p}r}\\ n(k_{p}r)=N_{0}(K_{p}r)-(1-v)\frac{N_{0}(K_{p}r)}{K_{p}r}\end{array} \right. \tag{17}\]  
\(k_{\text{p}}=w/C_{\text{p}}\), \(C_{\text{p}}=[E/\rho(1\text{-}v_{2})]^{1/2}\) is the longitudinal wave velocity in the disc, \(J_{\text{i}}\) and \(N_{\text{i}}\) are Bessel functions of the first kind and second kind respectively. The boundary conditions are as follows:  
\[\left. \begin{array}{l}z=\pm h,\ r=r_{0}:\sum_{ii}=\sigma_{ii},\ U_{i}=u_{i}( i=z,r)\\ r=R:\sum_{rr}=0\end{array} \right\} \tag{18}\]  
Based on above equations, the amplification ratio is derived as follows:  
\[\frac{y}{x}=\frac{\frac{h}{r_{0}}\times\frac{2v}{(1+v)(1-2v)}}{[\frac{(1-v)( H_{0}-h)}{(1+v)(1-2v)}\text{cos}k-kh\text{sin}k(H_{0}-h)]} \tag{19}\]  
Figure 5: Simplified model of energy concentrating structure.