Ultrasonic vibration has been widely used as a tool in many different applications, such as ultrasonic-assisted machining [1] , ultrasonic welding [2] , ultrasonic wire-drawing [3] , ultrasonic motors [4] , medical insertion for treatments [5] , etc. The action of ultrasonic vibration can induce intermittent contact, reduce friction, increase stiffness, etc. [6] The traditional ultrasonic vibrator only generates its vibration along one specific direction, which is defined by the polarization direction of Lead Zirconate Titanate (PZT) ceramic stack. Most ultrasonic vibrators generate longitudinal vibration due to the polarization direction of PZT. To extend the advantages of ultrasonic vibration on various directions, researchers are now looking into the development of 2-dimensional or 3dimensional ultrasonic hybrid vibration, such as hybrid longitudinal-radial (L&R) vibration, hybrid longitudinalbending (L&B) vibration, hybrid longitudinal-torsional (L&T) vibration, or hybrid bending-torsional (B&T) vibration [7] .
There are two different ways to generate hybrid L&T vibration. The first method is orthogonally placing two pairs of piezo-ceramic stacks along with the two correspondent polarization directions [7] [8] [9] [10] . The two pairs of piezo-ceramic stacks are then excited by two coupled power systems. This method can precisely control the vibration locus by controlling the amplitudes and phases of the two independent signals. However, the power systems and the driving circuits are relatively complex to design and also expensive to implement. The tangentially polarized piezo-ceramic discs are difficult to be manufactured. The second method is to geometrically change and modify the external structures of a longitudinalexcited vibrator, such as engraving diagonal slits or helical
Ultrasonic vibration has been widely used as a tool in many different applications, such as ultrasonic-assisted machining [1] , ultrasonic welding [2] , ultrasonic wire-drawing [3] , ultrasonic motors [4] , medical insertion for treatments [5] , etc. The action of ultrasonic vibration can induce intermittent contact, reduce friction, increase stiffness, etc. [6] The traditional ultrasonic vibrator only generates its vibration along one specific direction, which is defined by the polarization direction of Lead Zirconate Titanate (PZT) ceramic stack. Most ultrasonic vibrators generate longitudinal vibration due to the polarization direction of PZT. To extend the advantages of ultrasonic vibration on various directions, researchers are now looking into the development of 2-dimensional or 3dimensional ultrasonic hybrid vibration, such as hybrid longitudinal-radial (L&R) vibration, hybrid longitudinalbending (L&B) vibration, hybrid longitudinal-torsional (L&T) vibration, or hybrid bending-torsional (B&T) vibration [7] .
There are two different ways to generate hybrid L&T vibration. The first method is orthogonally placing two pairs of piezo-ceramic stacks along with the two correspondent polarization directions [7] [8] [9] [10] . The two pairs of piezo-ceramic stacks are then excited by two coupled power systems. This method can precisely control the vibration locus by controlling the amplitudes and phases of the two independent signals. However, the power systems and the driving circuits are relatively complex to design and also expensive to implement. The tangentially polarized piezo-ceramic discs are difficult to be manufactured. The second method is to geometrically change and modify the external structures of a longitudinalexcited vibrator, such as engraving diagonal slits or helical
Ultrasonic vibration has been widely used as a tool in many different applications, such as ultrasonic-assisted machining [1] , ultrasonic welding [2] , ultrasonic wire-drawing [3] , ultrasonic motors [4] , medical insertion for treatments [5] , etc. The action of ultrasonic vibration can induce intermittent contact, reduce friction, increase stiffness, etc. [6] The traditional ultrasonic vibrator only generates its vibration along one specific direction, which is defined by the polarization direction of Lead Zirconate Titanate (PZT) ceramic stack. Most ultrasonic vibrators generate longitudinal vibration due to the polarization direction of PZT. To extend the advantages of ultrasonic vibration on various directions, researchers are now looking into the development of 2-dimensional or 3dimensional ultrasonic hybrid vibration, such as hybrid longitudinal-radial (L&R) vibration, hybrid longitudinalbending (L&B) vibration, hybrid longitudinal-torsional (L&T) vibration, or hybrid bending-torsional (B&T) vibration [7] .
There are two different ways to generate hybrid L&T vibration. The first method is orthogonally placing two pairs of piezo-ceramic stacks along with the two correspondent polarization directions [7] [8] [9] [10] . The two pairs of piezo-ceramic stacks are then excited by two coupled power systems. This method can precisely control the vibration locus by controlling the amplitudes and phases of the two independent signals. However, the power systems and the driving circuits are relatively complex to design and also expensive to implement. The tangentially polarized piezo-ceramic discs are difficult to be manufactured. The second method is to geometrically change and modify the external structures of a longitudinalexcited vibrator, such as engraving diagonal slits or helical
Ultrasonic vibration has been widely used as a tool in many different applications, such as ultrasonic-assisted machining [1] , ultrasonic welding [2] , ultrasonic wire-drawing [3] , ultrasonic motors [4] , medical insertion for treatments [5] , etc. The action of ultrasonic vibration can induce intermittent contact, reduce friction, increase stiffness, etc. [6] The traditional ultrasonic vibrator only generates its vibration along one specific direction, which is defined by the polarization direction of Lead Zirconate Titanate (PZT) ceramic stack. Most ultrasonic vibrators generate longitudinal vibration due to the polarization direction of PZT. To extend the advantages of ultrasonic vibration on various directions, researchers are now looking into the development of 2-dimensional or 3dimensional ultrasonic hybrid vibration, such as hybrid longitudinal-radial (L&R) vibration, hybrid longitudinalbending (L&B) vibration, hybrid longitudinal-torsional (L&T) vibration, or hybrid bending-torsional (B&T) vibration [7] .
There are two different ways to generate hybrid L&T vibration. The first method is orthogonally placing two pairs of piezo-ceramic stacks along with the two correspondent polarization directions [7] [8] [9] [10] . The two pairs of piezo-ceramic stacks are then excited by two coupled power systems. This method can precisely control the vibration locus by controlling the amplitudes and phases of the two independent signals. However, the power systems and the driving circuits are relatively complex to design and also expensive to implement. The tangentially polarized piezo-ceramic discs are difficult to be manufactured. The second method is to geometrically change and modify the external structures of a longitudinalexcited vibrator, such as engraving diagonal slits or helical 48th SME North American Manufacturing Research Conference, NAMRC 48 (Cancelled due to grooves onto the vibrator structure [2, 4] . The logic behind this method is to use the reflection and combination of the propagated wave to achieve hybrid vibration [11, 12] . Although multiple theoretical calculations and simulation analyses are available to guide the designs, it is computationally costly to find an appropriate node plane by solving both the longitudinal and the torsional vibration modes of a vibrator simultaneously.
To achieve high efficiency of hybrid vibration conversion, this paper presents a new design of the L&T vibrator. Inspired by the ideas of modifying the vibrator structure, this paper proposed to use the waveguide principle to design and to optimize the new geometric shape of the L&T vibrator performance. The waveguide can constrain the acoustic wave propagation along the defined path. Compared to the traditional methods that only modify the external structure, the presented new waveguide structure can achieve better L&T conversion efficiency and reduce the counteraction of the reflected wave during vibration propagation.
The new L&T vibrator design comes with a new challenge. Due to the complex geometric shape of the new L&T vibrator design, the traditional machining processes become incompetent in realizing the new L&T design. After searching for various processes, it is identified that 3D additive manufacture (AM) is an appropriate approach to fabricate the complex internal and external features of the L&T vibrators. Selective laser melting (SLM) is one of the AM techniques for the fabrication of near net-shaped parts directly through layerby-layer build-up processing. The SLM uses high powerdensity laser selectively to melt and fuse metallic powders, such as iron, titanium, nickel, aluminum, copper, etc. [13] . This technique has been proven to produce the components with high density and good mechanical properties which are comparable to that of wrought materials [14] [15] [16] . In the paper, we chose the AlSi10Mg powder as the material and use SLM to fabricate the designed L&T vibrators. The microstructure of the as-built AlSi10Mg SLM-AM components is reported to be extremely fine cellular dendritic microstructure. And it also can obtain excellent mechanical behavior and fatigue performance before or after heat treatment, which is meet the requirement of the ultrasonic vibrator [14] [15] [16] .
In this paper, a new design of L&T vibrator is presented. A torsional-rod array structure functioning as waveguide to transform ultrasonic waves into hybrid L&T vibration at the end of the vibrator. SLM-AM process is used to fabricate the complex geometric design of L&T vibrator of AlSi10Mg metal powder. To optimize the design parameters, a finite element (FE) analysis was carried out in studying the vibration modes. Laboratory experiments were conducted to validate the electromechanical and vibration performance of the new design of L&T ultrasonic vibrator. Details are discussed in the following sections.
There are two conventional designs to deliver longitudinal vibration to L&T vibration by modifying external structures. One is adding a series of diagonal slits around the circumference of the vibrator, as shown in Figure 1(a) . In this case, the axially propagating wave can be reflected in the slits and produce the desired L&T hybrid vibration at the end of the vibrator [3, 4, 17] . The composited vibration is affected by the size, depth, width, and length of the slits. Another method is to use helical grooves on the vibrators [11, 12] , as shown in Figure  1 (b). Compared to the first method of using straight slits arrays, the helical grooves are more difficult to be machined due to the complex geometric shapes, even though the torsional conversion efficiency sometimes can be higher than that of the diagonal slits. However, both the two methods require the resonant frequency of the torsional vibration mode to be the same as that of their longitudinal mode. This is challenging particularly when they are sharing only one node plane. helical grooves type [11, 12] .
Inspired by the idea of modifying the structure of the longitudinal-excited vibrator, this paper proposed to design a waveguide structure with internal and external geometric features. An illustrative example of L&T vibrator design is shown in Figure 2 . An acoustic waveguide structure can guide a wave propagation direction with minimal loss of energy, hence high vibrations conversion efficiency. Therefore, a waveguide can constrain the wave propagation along the defined path to further achieve a 2-dimensional hybrid vibration. Our objective is that the overall composite vibration wave can form a hybrid L&T vibration on the same plane, as shown in Figure 2 . As shown in Figure 2 , the new waveguide-based vibrator was designed by using a CAD software SolidWorks 2019. The basic dimension of the vibrator is Φ28 mm ×37 mm, as shown in Figure 3 . The diameter of the flange is Φ28 mm. Based on the size of the flange, the wave propagation path needs to be analyzed and defined. To achieve a hybrid L&T vibration, a circular array of helical paths is used in which each helical path can guide the wave to a certain angle (see Figures 2 and 3) . The defined wave path is a circular helix trajectory, which can be represented as follows:
Where D is the diameter of the circular helix, and the L is the length of the rods. Based on the wave path, six patterned helical rods were created as the final waveguide structure (see Figure 2 ). The diameter of the rods was designed as Φ4.5 mm, which is determined by the maximum sustainable stress based on our FE analysis, as discussed in the next section. The design and configuration of the assembled new L&T vibrator are shown in Figure 4 . To obtain high electromechanical conversion efficiency and reasonably stable performance, a bolt-clamped type Langevin transducer is used to generate the longitudinal vibration. The Lead Zirconate Titanate (PZT-8) piezoelectric ceramic rings are polarized along the thickness direction. The PZT-8 used in the design are commonly used in power ultrasonic applications. As shown in Figure 4 , every two adjacent piezo-ceramics rings are oriented in the opposing direction, and among them, the copper electrodes are alternatively sandwiched. For safety concerns, the copper electrodes attached to the mass bolt are connected to the ground as negative terminals, while another pair of the copper electrodes are connected to the ultrasonic driver (see Figure 4 ). When sinusoidal voltages are applied to the PZT rings, all the PZT ceramic rings will synchronously generate longitudinal vibrations. 
To analyze and optimize the design parameters of the waveguide-based L&T vibrator, a finite element model is constructed for analyzing the resonance frequencies, mode shapes, and steady-state behaviors of the geometric design. The modeling software used was ANSYS 19.1 Workbench at our lab. The material used for the vibrator is aluminum alloy AlSi10Mg. The connecting bolt and back mass is made of 1045 steel. The heavier back end is used to efficiently transfer the ultrasonic vibration energy forward. The Lead Zirconate Titanate (PZT-8) is used as the material for the piezoelectric ceramics, which possesses a high mechanical quality factor, elastic property, and stability of piezoelectricity.
For the FE simulations, Table 1 lists the parameters of the material properties. The relevant parameters of PZT-8 are provided by the piezoelectric ceramics manufacturer. Detailed vibration-related parameters of the piezoelectric ceramics PZT-8 are shown in Equations (2)-(4). These functional parameters include the compliance matrix, the piezoelectric coupling matrix, and the relative permittivity matrix, respectively. The compliance matrix of piezoelectric ceramic PZT-8 is shown as follows: In the FE analysis, a fixed constraint was added on the flange of the vibrator (see Figure 5 ).
To study the vibration characteristics of the L&T vibrator design, FEA modeling and analysis was conducted. The objective is to examine the design parameters for the vibration dynamic analysis and the harmonic response analysis. During the FEA simulations, the range of frequency was set to be from 20 kHz to 30 kHz, which is suitable for most power ultrasonic vibration devices. Table 2 shows the vibration response frequencies from the 1st modal frequency to the 5th modal frequency. Figure 5 shows the corresponding mode shapes of the vibrator from 1st mode to the 5th mode. It can be seen that the mode shapes of the 1st and the 2nd modes are two pairs of flexing modes with perpendicular vibration directions, as shown in Figure 5 (a) and (b). The same occurs to the 3rd and the 4th modes (see Figure 5 (c) and (d)). More importantly, the frequencies in each pair of two flexing modes are very close, which results in a preferred hybrid L&T vibration. This observed result is difficult for the traditional vibrators to achieve.
By further examining the FEA simulation data, one can find that the 5th mode shape is already a hybrid L&T vibration (see Figure 5 (e)), and its frequency is very close to the 3rd and 4th modal frequency. Hence, a hybrid L&T vibration resonant frequency can be located at around the frequency of 26 kHz based on the results of Figure 5 (c), (d) and (e). This observation is later validated by the harmonic response analysis and the actual laboratory experiments, as presented in the next section. 
To analyze the harmonic response of waveguide-based vibrator design, an ANSYS 19.1 Workbench software was used for harmonic response analysis and simulation. Based on the analysis data, the harmonic response was identified in the vicinity of the 3 rd to the 5 th modal frequencies, which is from 25 kHz to 27 kHz. In the vibration simulation, it is assumed one end (the heavy mass end) is fixed and a fixed constraint was added to the flange ring (see Figure 5 ). An exciting voltage of 45 Vrms (Root-Mean-Squared voltage), of about 64 Vp-p Peak-to-Peak voltage, was applied to the piezoelectric ceramic rings to simulate the actual power level in vibrations.
As shown in Figure 6 , the waveguide-based vibrator with the design parameters (geometric dimensions and waveguide paths) is able to deliver harmonic responses on both the longitudinal direction and the torsional direction, as shown in Figure 6 . Figure 6 shows the vibrator has a resonant frequency at 25,514 Hz. Figure 7 shows a steady-state hybrid L&T vibration at the corresponding resonant frequency. The vibration displacement vectors represent the exciting energy propagating alone the six waveguides to the front end with directional changes. At the front end of the vibrator, a steadystate hybrid L&T vibration can be observed (see Figure 7 ). 
To validate the mechanical property of the vibrator, this paper also analyzes the maximum stress of the vibrator at the resonant frequency under different exciting voltages. The study would be important for dynamic applications, especially on fatigue behavior. In the FE analysis, the maximum principal stress method was used to analyze the stress distribution, as shown in Figure 8 . The maximum stress increases as the applied driving voltage are increased, as shown in Figure 9 . The maximum voltage is 60 Vrms based on the capacity of PZT stacks and the power. According to the reported stress-life (S-N) behavior of the SLM-AM AlSi10Mg parts, the part material can withstand 10 8 and more cycles under the stress 60MPa [16] . After heat treatment, the SLM-AM AlSi10Mg can sustain longer repetitive cycles under higher stress [15] . In our study, the predicted stress is far less than that in fatigue testing, and consequently, the theoretic fatigue life should be much longer. For the ultrasonic-assisted vibrator in our case size, its load will not be much high. For example, in our earlier works presented in [18] [19] [20] vibration-assisted medical tools, like needles or surgical cutting tools, the load force is only a few newtons (Ns). 
In this paper, our proposed new design waveguide-based vibrator was fabricated by using SimpNeed SLM 280 SLM-AM machine with AlSi10Mg powder. The particle size of AlSi10Mg powder is in the range of 24-45 µm. During the process, the following parameters were applied:
Laser beam power of 80 W Laser beam diameter of 80µm Layer thickness of 0.06 mm Laser scanning speed of 900 mm/s The 3D printed L&T vibrator is shown in Figure 10 . Due to the relatively low achieved tolerance and reducing to use supportive structures, the vibrator was printed without the connecting hole (see Figures 10(a), (b) and (c) ). And a 0.5 mm allowance was left on the flange region for finishing. After the AM, a turning process was applied to finishing the back end and the profile of the flange for the further assembly (see Figure  10 (d)). A drilling process was performed followed by a threading operation with M5x0.5 tap for the connecting bolt. Since the SLM-AM introduces porosities in the material during the melting process, which would affect the density and mechanical performance of the component. In this paper, we did not apply any heat treatment to eliminate the defects during or after the AM. Therefore, the porosity and density of the vibrator were primarily verified by means of testing on samples. Based on the measurement, the relative density of the as-built part condition is around 99.85% meaning 0.15 % porosity. This porosity of 0.15% needs to be considered in the materials strength and mechanical property analysis.
After the waveguide-based L&T vibrator and the other vibrator components were assembled, an impedance analysis was first conducted to confirm the actual resonance frequency. Impedance analysis was performed using a Wayne Kerr 6500B precision impedance analyzer as shown in Figure 11 . According to the FE analysis, the variations of the impedance and the phase angle with frequency were scanned from 24 kHz to 27 kHz during the testing. The measurement results of the impedance analysis are shown in Figure 12 . In the scanning period, there is only one minimum impedance occurred at 25,350 Hz which indicates the resonant frequency of the vibrator. The corresponding phase angle is 23.7 o , which means the entire system is inductive. Latter, an extra capacitance was used to compensate it.
Comparing the experiment result and the simulation result, one can find the discrepancy between the simulation result (25,514 Hz) and the actual measured result (25,350 Hz) is only about 164 Hz off. The small discrepancy is considered acceptable. There might be many reasons causing such discrepancy, like the pressing force (tight or loose) on the PZT stacks, or the inaccurate definition of boundary conditions in FEA, etc. Some further fine-tune procedures could reduce the discrepancy. Author name / Procedia Manufacturing 00 (2019) 000-000 7
To measure the vibration amplitudes on both longitudinal and torsional direction, a SIOS Meβtechnik GmbH laser interferometer was employed. The experimental setup is shown in Figure 13 . A table vise was used to hold the flange of the vibrator during the experiment. Due to the limited capability of our laser interferometer, it is difficult to directly measure the torsional amplitude (torsional arc length). Therefore, a small rectangular wood block was attached to the front end of the vibrator to measure the torsional displacement, as shown in Figure 13 (a). The distance between the measuring laser spot and the center of the vibrator is 4 mm measured by caliper. Since the ultrasonic vibration amplitude is relatively small, the measured linear displacement can be approximated to the torsional displacement. Figure 14 shows the results of vibration amplitudes at both the longitudinal and the torsional directions under 64V peakto-peak drive voltage from 24 kHz to 26.5 kHz. The resonant frequency of longitudinal vibration is about the same as the impedance analysis and it also has a good agreement with the simulation predictive results. As for the torsional vibration, the frequency discrepancy between the actual measurement and the simulation predicted result is about 300 Hz. The most possible reason results from the inaccurate measurement method by attaching the wood block, which might affect the natural frequency of the overall system. And the deflection of the reflected laser also might cause the frequency shift. Other possible causes could be responsible for it, such as clamping position on the flange of the vibrator, the measurement errors, the difference of boundary conditions between the actual test and FE simulation, etc. The real resonant frequency of the torsional vibration should be close to longitudinal vibration, according to the FE analysis and impedance analysis. Figures 15 (a) and (b) show the co-relationship between the measured amplitudes and the different levels of drive voltages (16V, 32V, 48V, and 64V). Figure 16 shows that either the longitudinal amplitude or the torsional amplitude sustains a good linear change upon different driving voltages. As shown in Figure 16 , the L&T amplitude ratio displays good consistency and high efficiency, which is about 2:1. This observation also confirms that our proposed waveguide-based vibrator is capable of delivering an excellent L&T conversion efficiency. It can be applied in power ultrasonic or ultrasonic motor. 
As shown in Table 3 , the performances of the developed waveguide-based vibrator are compared with two typical longitudinal-excited L&T vibrators. Although the two existing vibrators represent a higher torsional amplitude, the actual torsional angle is related to the working end diameter (measured diameter). To compare the actual L&T conversion efficiency, a ratio of longitudinal amplitude to torsional angle (L/TA) was calculated based on the data. It can be observed that the developed vibrator has a similar conversion efficiency to the vibrator with diagonal slits, but it has much higher conversion efficiency than the vibrator with helical grooves. It can also be noticed that the developed L&T vibrator is able to achieve higher energy efficiency when the driving voltages or corresponding longitudinal amplitudes remain the same.
Based on the analysis data and experiment results, it is apparent that the developed waveguide-based L&T vibrator can successfully achieve hybrid vibration with higher conversion efficiency and energy efficiency. 
In this paper, a waveguide-based design of a longitudinalexcited L&T ultrasonic vibrator was presented. Both the FE analysis and the laboratory experiments show that the waveguide-based L&T vibrator design is able to achieve high L&T conversion efficiency. This is due to each pair of orthogonal vibration modes having highly similar frequencies.
The SLM-AM was used to fabricate the designed L&T vibrator using AlSi10Mg metal powders. Impedance analysis and displacement measurements were conducted to validate the electromechanical performance and acoustics characteristics of the developed L&T vibrator. The laboratory experiments demonstrated that the developed waveguide-based L&T vibrator can successfully achieve hybrid vibration with higher conversion efficiency and energy efficiency. The presented new L&T vibrators can be used for ultrasonic-assisted manufacturing and ultrasonic surgical tools for medical applications.