# The Development of Piezo-Driven Tools for Cellular Piercing
## Materials and Methods
### Design of the Piezo-Drill
The initial condition of the cantilever is  
\[w(x,0)=w_{0}(x)=0,\ v(x,0)=v_{0}(x)=0. \tag{4}\]  
According to the initial condition, we can conclude that  
\[q_{r}(0)=\frac{1}{M_{r}}\int_{0}^{l}\rho Av_{0}(x)W_{r}(x)dx=0,\ \text{and that} \tag{5a}\] \[\dot{q}_{r}(0)=\frac{1}{M_{r}}\int_{0}^{l}\rho Av_{0}(x)W_{r}(x)dx=0. \tag{5b}\]  
The generated external periodic point force at tip of the pulled section, the coordinate of which is defined as \(x=x_{l}\), is derived as  
\[f(x_{l},t)=F_{y}=G(t)\cdot\sin\theta, \tag{6a}\]  
Figure 2: **(A)** Structure of the conventional injector. **(B)** Partial enlarged detail of the junction where the front of the piezoelectric actuator is connected to the holders in the conventional injector and a slight deviation \(\theta\) is generated from installation errors. where \(G(t)=\frac{k\cdot d_{\text{max}}}{V_{\text{max}}}\cdot g(t)\). \[M(x_{l},t)=f(x_{l},t)\cdot a, \tag{6b}\]  
where \(l\) is the length of the pulled section of the injection pipette, and \(a\) represents the distance between the front of the piezoelectric actuator and the tip of the pulled section of the injection pipette. The obtained modal force is  
\[F_{y}=F\cdot\sin\theta=\frac{k\cdot d_{\max}}{V_{\max}}\cdot g(t)\cdot\sin\theta, \tag{1}\]0  
Thus, the lateral vibration of the pulled section of the injection pipette is  
\[F_{y}=F\cdot\sin\theta=\frac{k\cdot d_{\max}}{V_{\max}}\cdot g(t)\cdot\sin\theta, \tag{1}\]1  
When \(r\geq 3\), \(\frac{W_{r}(x_{l})}{W_{r}(x_{l})}=\frac{1+sb(rx_{l})\cdot\text{sin}(rx_{l})}{ cht(rx_{l})\cdot\text{sin}(rx_{l})}\) and \(rx_{l}\approx(r-\frac{1}{2})\tau\); therefore, \(W_{r}(x_{l})\) and \(W_{r}^{\prime}(x_{l})\) have the same sign. Equation (7) can be changed to the following equation:  
\[F_{y}=F\cdot\sin\theta=\frac{k\cdot d_{\max}}{V_{\max}}\cdot g(t)\cdot\sin\theta, \tag{1}\]2  
where \(H_{r}(a,x_{l})=W_{r}(x_{l})\cdot(W_{r}(x_{l})+a\cdot W_{r}^{\prime}(x_{l}))\) and \(H_{r}(a,x_{l})\) are always positive when \(r\geq 3\). When the value of distance \(a\) increases, \(H_{r}(a,x_{l})\) will increase accordingly. If other parameters remain constant, the deflection of the injection pipette tip \(w\) will increase. In addition, when the deviation angle \(\theta\) generated from the installation error increases, and other parameters remain constant, the deflection of the injection pipette tip \(w\) will increase. Kinetic energy (\(T\)) of pipette's pulled section is  
\[F_{y}=F\cdot\sin\theta=\frac{k\cdot d_{\max}}{V_{\max}}\cdot g(t)\cdot\sin\theta, \tag{1}\]3  
Equation (7) can be changed to  
\[F_{y}=F\cdot\sin\theta=\frac{k\cdot d_{\max}}{V_{\max}}\cdot g(t)\cdot\sin\theta, \tag{1}\]4  
As a single injection pipette, actuator, and voltage are used in different injectors, all parameters are identical in Equation (10) except distance \(a\) and deviation angle \(\theta\). Because the pulled pipette is recognized as a cantilever, \(W_{r}(x_{l})\) and \(W_{r}(x)\) have the same sign, and \(H_{r}(a,x)\) is also positive when \(r\geq 3\). When \(a\) becomes smaller, \(H_{r}(a,x)\) will become smaller, which will lead to a smaller kinetic energy \(T\). In addition, when the deviation angle \(\theta\) generated from the installation error decreases, kinetic energy \(T\) will decrease. If the pipette's kinetic energy from the lateral vibration increases, more energy is required, further complicating the driving circuit. According to the analysis mentioned above, decreasing the distance \(a\) and deviation angle \(\theta\) is beneficial not only to the cellular piercing, but also for the simplification of the driving circuit. Here, we introduce two modified injectors where we change the distance \(a\) and deviation angle \(\theta\), respectively. The analysis above reveals that, in order to reduce the lateral vibration of the pulled section of the injection pipette and to avoid wasting energy, distance \(a\) needs to be smaller. Hence, we placed the piezoelectric actuator closer to the injection pipette, while other parameters remained the same as the conventional injectors.