*2.1. Design of the Piezoelectric Transducer and the Ladder Horn*

When a force is applied to the piezoelectric crystal in a proper direction, its internal polarization state will change. An internal electric field will then be generated, and a bound charge proportional to the external force will appear on two surfaces of the piezoelectric crystal. On the contrary, when an external electric field is applied to the piezoelectric crystal, the electric energy can be converted into mechanical energy through the piezoelectric crystal due to the inverse piezoelectric effect. The piezoelectric transducer is made of this inverse piezoelectric effect. Piezoelectric transducer has no eddy loss, hysteresis loss and resistance loss, which brings it fairly simple structure and high stability. Moreover, piezoelectric transducer usually has high sensitivity as well as outstanding electromechanical coupling characteristics. All these superior properties make piezoelectric transducer the most popular transducer in ultrasonic machining industry. In this investigation, a piezoelectric transducer with Lead Zirconate Titanate (PZT) is designed to convert electrical signals into ultrasonic vibration. Parameters of PZT are presented Table 1.

**Table 1.** Parameters of piezoelectric ceramics.


The longitudinal sound velocity in PZT slices *cPZT* can be calculated in Equation (1).

$$c\_{\rm PZT} = \sqrt{\frac{E}{\rho} \frac{1}{1 - \sigma^2}}\tag{1}$$

where, *E* is young's modulus, ρ is density and σ is Poisson's ratio of PZT-8.

The wavelength in PZT slices λ*PZT* could then calculated by λ*PZT* = *cPZT*/ *f*. According to the design requirements of the longitudinal vibration transducer, the equivalent diameter of the PZT slices *DPZT* should less than λ*PZT*/4, considering the inner holes, set the *DPZT* as 38 mm.

The output power of the PZT transducer is determined by the volume of PZT ceramic slice and its power capacity. In order to amplify the output power to satisfy the turning process, several pieces of PZT slices are piled. The number *mPZT* could be acquired by Equation (2).

$$m\_{\rm PZT} = \frac{P}{P\_d f \pi (D^2 - d^2) \frac{\epsilon}{4}} \tag{2}$$

where *P<sup>d</sup>* is power capacity in W/cm<sup>3</sup> , *f* is vibration frequency and *D*, *d* and *t* represent outside diameter, inside diameter and thickness of ceramic slice, respectively.

The power capacity of PZT ceramic slice *P<sup>d</sup>* is 2–3 W/cm<sup>3</sup> , the required output power *P* is set as 800 W, considering the slices need to be piled in pairs, set the number of slices as 4. Four pieces of PZT slices are piled with copper electrodes separating between them, the polarization of two adjacent slices are opposite so that the pile could output an overlapped amplitude. The whole transducer used in this investigation is shown in Figure 1.

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**Figure 1.** Structure diagram of the PZT transducer. 1. Back block; 2. electrodes; 3. PZT ceramics; 4. connect bolt.

The output vibration amplitude the piezoelectric transducer is much less than required. To amplify the amplitude and to install the turning tool, a two-stage ladder horn is designed. The back end of the horn is contact with the PZT slices, so the diameter of the back end *D*<sup>1</sup> is 38 mm. The front end is designed much smaller than the back end and the energy of ultrasound is concentrated on this comparative smaller area. Due to the energy gathering effect, the front end of the horn outputs the amplified amplitude. The diameter of the front end *D*<sup>2</sup> determines the amplification *M*. The less *D*<sup>2</sup> is, the higher *M* will be. However, the strength may decrease when *D*<sup>2</sup> is reduced because the horn can be seen as cantilever structure during turning process and the components of cutting force can lead to enormous stress concentration on abrupt change in section. To ensure the enough strength and satisfy the amplification, set *D*<sup>2</sup> as 10 mm. Thus, the area index is *N* = *D*1/*D*<sup>2</sup> = 3.8.

Longitudinal sound wave will bounce back and forth between the back and front end, the standing-wave effect happens when the incident wave and the reflected wave overlap. At specific points, the amplitude can be decreased to zero. These points can be called as displacement node, which can be used to fix the horn to the machine tool. Likewise, the amplitude reaches its peak at particular points, where the turning tool should be installed. The displacement node and the peak point can be acquired though wave equation which is shown in Equation (3).

$$\frac{\partial^2 \xi}{\partial \mathbf{x}^2} + \frac{1}{S} \times \frac{\partial \mathcal{S}}{\partial \mathbf{x}} \times \frac{\partial \xi}{\partial \mathbf{x}} + k^2 \xi = \mathbf{0} \tag{3}$$

where, ξ = ξ(*x*) is particle displacement function, *S* = *S*(*x*) is cross-section area function, *k* = ω/ p *E*/ρ is circle wavenumber, ω is angular frequency, *E* is elastic modulus, ρ is the density of the horn.

In this investigation, the material of horn is 1045 steel, the parameters are listed in Table 2. Therefore, resonance length δ can be acquired as 258 mm, the length of the horn *l* is half of δ so that the front end located on the peak of the standing wave, outputting the largest amplitude.



The displacement node located at the middle of the horn, where a flange plate is designed to install the horn to the machine tool. The ladder horn is shown in Figure 2.

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**Figure 2.** Two-stage ladder horn.
