A knife for cutting and processing food products with ultrasonic vibration should have a resonant frequency of 20 kHz, and a longitudinal vibration mode and maximum vibration amplitude should appear at the knife blade that is cut by direct contact with the food products. In addition, it should basically be suitable for the shape or size of the food products.

First, the approximate length of the cutting knife is determined using the vibration equation. Other final shapes or dimensions are determined by changing the shape of the knife and performing modal analysis in a trial & error method. Since there may be a difference between the theoretical analysis results and the vibration characteristics of the actually manufactured knife, the vibration performance must be verified through a vibration test on the manufactured knife. Proper tuning may be required in the manufacturing process until the required vibration characteristics are obtained. Fig. 5 shows the overall vibration analysis and manufacturing processes.

Fig. 5 
Design and manufacturing process of cutting knife

Since the knife blade cuts the material with vibration parallel to the food products, the longitudinal vibration mode must appear and the frequency separation should also be clear to have only one natural frequency in the resonant frequency band. Using the longitudinal vibration equation of the elastic rod, the length of the horn can be calculated as follows.⁶

When vibration is applied to a rod with a uniform cross section, longitudinal vibration occurs. The vibration equation of an isotropic uniform elastic rod with free boundary conditions can be defined as the following Eq. (1).

Where u(x,t) is the displacement at time t, c is the sound velocity propagating through the rod, E is the Young's modulus, and ρ is the density. Using the variable separation method, the solution can be obtained as follows.

The cylindrical horn transmits the vibration displacement U₀ of the booster at x = 0 and x = L, and the displacement becomes maximum at both ends of the rod, so the boundary condition can be written as the follows.

Using these boundary conditions Eq. (3), the solution of the vibration equation can be summarized by the follows.

Therefore, the characteristic equation can be obtained as follows.

Since the length of the knife is an integer multiple of the half wavelength of the sound wave, the length L of the knife at a specific frequency can be obtained by the following Eq. (9).

Where λ is the wavelength of sound wave.

As for the material of the knife, strength and hardness are the most important depending on the shape or cutting object. It should also be considered that the length of the wavelength is determined by the sound velocity of the wave in the material. This is because the length of the wavelength has a great influence on determining the length of the ultrasonic knife.

The ultrasonic knife should be a half-wavelength resonator, and the material is generally aluminum alloy or titanium alloy. Among them, titanium alloy is widely used because it has excellent acoustic and strength properties. In this study, Ti-6Al-4V alloy is used as the knife material, and its mechanical properties used in vibration analysis are shown in Table 1.

To develop the ultrasonic cutting system, modal analysis was performed by using ANSYS tool for validation via the FE modeling approach. As the transducer, the booster and the knife body are fastened by stud bolts each, boundary conditions must be given accordingly. As constraints on FE analysis, radial and tangential displacements of the cylindrical surface of all stud bolts holes were restrained and only longitudinal displacements were allowed. The number and the shape of meshes were automatically generated by ANSYS tool. The modal analysis was repeated to obtain the most efficient design of each part.

Fig. 6 shows the knife model 1 proposed in this study. Modal analysis was performed while varying dimensions such as the thickness and length of the knife. The final shape of the knife that satisfies the required resonant frequency 20 kHz and longitudinal vibration mode were derived. (The content of this study was submitted for a patent, so only the critical dimensions were noted here.)

Fig. 6 
Proposed knife model 1

Table 2 shows the modal analysis results for the shape of Fig. 6. It can be seen that the longitudinal vibration mode satisfying the resonance frequency of 20 kHz ±500 Hz is the 3rd mode and the natural frequency is 19,519 Hz. In addition, as shown in the mode shape in Fig. 7, the maximum amplitude occurs at the front end of the knife blade, and although smaller than that, longitudinal vibration occurs at the rest of the blade. This means that cutting can be the easiest at the front end of the knife blade. As the vibration amplitude decreases toward the rear of the knife, the cutting performance may be degraded. Considering that it is impossible to obtain a uniform amplitude across the knife blade due to the characteristics of ultrasonic vibration, it is judged that the proposed design model 1 can be sufficiently used for cutting food products.

Fig. 7 
Mode shape of the final model 1

The overall length of the knife of the model 1 was 115 mm. In consideration of the convenience of use, the length of the blade of the model 2 was doubled. Table 3 shows the results of modal analysis while changing the remaining dimensions. The longitudinal vibration mode with a resonant frequency of 20 kHz ±500 Hz is the 3rd mode and the natural frequency is 19,856 Hz. However, as shown in the vibration mode of Fig. 8, the maximum amplitude occurs at the both front and middle blades, and a part without vibration appears between them. This vibration mode is judged not to be desirable because a problem such as non-uniform cutting may occur depending on the position of the knife blade.

Fig. 8 
Mode shape of the model 2

The blade of a commonly used kitchen knife is located below the handgrip. In consideration of this and the user's convenience, previous knife models were modified to a new model similar to the existing knife. As a result of repeating the modal analysis, the final shape of the new knife model is obtained as shown in Fig. 9. The modal analysis results for this shape are shown in Table 4. In mode 3, longitudinal vibration with a natural frequency of 19,782 Hz appears, and the remaining vibration modes are bending vibration modes. Fig. 10 shows the mode shape. As in model 1, the vibration amplitude decreases toward the rear, so the cutting performance may deteriorate. Considering that it is impossible to obtain a uniform amplitude across the knife due to the characteristics of ultrasonic vibration, it is judged that the proposed design model can be used for cutting food products while enhancing user convenience.

Fig. 9 
Proposed knife model 3

Fig. 10 
Mode shape of the knife model 3

The transducer and booster assembly shown in Fig. 4 generates longitudinal vibration and transmits it to the knife. The assembly should have the resonant frequency of 20 kHz and longitudinal vibration mode too. The dimensions of this assembly (Right Rear Part of the Whole System) are shown in Fig. 12. The Transducer is composed of PZT and backing mass generally. For the FE analysis, the materials of the backing mass, PZT and booster were assumed to be SUS316, PZT8, and titanium alloy Ti-6Al-4V, respectively. Table 5 shows the mechanical properties of each material.

The results of the modal analysis of this assembly are shown in Table 6 and Fig. 11. From Table 6, it is found that the longitudinal vibration of the third mode satisfies the resonant frequency of 20 kHz ±500 Hz. In addition, the maximum amplitude occurs at the left end of the booster where the knife is fixed by a stud bolt. It is shown that vibration is minimized where the knife handgrip is fixed. This means the generated ultrasonic vibration is not transmitted to the user’s hand, so there are no problems in use. These results indicate that the proposed assembly model can be used effectively in an ultrasonic cutting knife system.

Fig. 11 
Mode shape of the booster and transducer assembly

The proposed cutting system consists of a knife body, a booster and a transducer as shown in Fig. 12. The knife body uses the model 3 (Fig. 9) because it is similar to a real kitchen knife and has high user convenience. The assembly of the booster and transducer described above are fixed to the knife body via a stud bolt. The overall system is 205 mm long, which is similar to the length of a knife commonly used at home.

Fig. 12 
2D drawing of the proposed food cutting system

The result of modal analysis of this system is shown in Table 7. A desirable longitudinal vibration mode appears in the third mode with a resonant frequency of 20,652 Hz. Normally, the frequency of FE analysis is designed to be slightly higher than 20 kHz, but since it is tuned to the working frequency during manufacturing, the vibration characteristics of the entire system are judged to have no problem in use.

The mode shape is shown in Fig. 13. The magnitude of the vibration amplitude increases toward the left end of the blade, and there is little vibration at the right end of the blade. In addition, the vibration of the knife body does not show a uniform longitudinal vibration that is symmetrical vertically, which is an inevitable phenomenon that appears because the entire cutting system is not vertically symmetrical. The longitudinal vibration is minimized where the knife handgrip is fixed.

Fig. 13 
Mode shape of the proposed food cutting system

Considering these results, it is judged that the cutting system proposed in this study can be sufficiently practical at home. In addition, it plans to apply for a patent that reflects these results.