Design and Magnetic Field Uniformity of Giant Magnetostrictive Ultrasonic Transducer for Progressive Sheet Forming

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1.The Ministry of Education Key Laboratory of NC Machine Tools and Integrated Manufacturing Equipment，Xi’an University of Technology，Xi’an 710048，P.R.China；

2.Key Laboratory of Manufacturing Equipment of Shaanxi Province，Xi’an University of Technology，Xi’an 710048，P.R.China

（Received 17 August 2019；revised 3 January 2020；accepted 27 May 2020）

Abstract: Design of a giant magnetostrictive ultrasonic transducer for progressive sheet forming was presented. A dynamic analysis of the theoretically designed ultrasonic vibration system was carried out using the finite element method（FEM）. In addition，simulations were performed to verify the theoretical design. Then，a magnetically conductive material was added between the giant magnetostrictive rod and the permanent magnet. Besides，magnetic field simulations of the transducer were performed. The influence of the material thickness of the magnetically conductive material on uniformity of the induced magnetic field was studied. Furthermore，the impedance analysis and amplitude measurement were performed to compare the performance of transducers with and without the magnetically conductive material. The experimental results show that the magnetic field uniformity is the highest when the magnetically conductive material has a thickness of about 1.6 mm. The output amplitude of the giant magnetostrictive transducer is improved by adding the magnetically conductive material. Moreover，the mechanical quality factor and impedance are reduced，while the transducer operates more stably.

Key words：giant magnetostrictive material（GMM）；ultrasonic transducer；magnetic field uniformity；finite element

0 Introduction

Ultrasonic vibration-assisted sheet forming is the progressive forming of ordinary sheet material with the addition of ultrasonic vibrations to improve the process. During sheet forming，the metallic ma?terial undergoes plastic deformation， however，adopting ultrasonic vibrations that periodically change direction can significantly reduce the flow stress required for plastically deforming the materi?al，thereby improving the forming limit of the mate?rial and the quality of the finished product［1］. After applying ultrasonic vibrations to the sheet，the aver?age axial force can reduce by 23.5%，and the re?bound and surface roughness can also reduce［2］. Hou et al.［3］carried out a comparative experiment of lami?nated drilling with three methods of conventional drilling，rotary ultrasonic assisted drilling and low frequency vibration assisted drilling. Compared with the conventional drilling，the introductions of rotary ultrasonic and axial low frequency vibration can re?duce drilling temperature and improve drilling quali?ty.

At present，piezoelectric ceramics are widely used in machining as materials for ultrasonic trans?ducers. However，several disadvantages such as the low power density of piezoelectric ceramics，fre?quent overheating failures，and fragility of the mate?rial have limited large-scale application of piezoelec?tric ceramics in high power applications. To this end，giant magnetostrictive material（GMM）offer several advantages including a large magnetostric?tion coefficient，high power capacity，and fast re?sponse speed，thus becoming research focus in the development of high-power and large-amplitude ul?trasonic processing systems［4-5］.

A number of studies have recently been carried out on magnetostrictive materials. Jammalamadaka et al.［6］developed and tested a 100 kHz giant magne?tostrictive transducer for detecting defects in con?crete structures using ultrasonic transmission tech?nique. Karunanidhi et al.［7］designed a magnetostric?tive actuator for a high dynamic servo valve and found that valves with magnetostrictive actuators have a faster time response compared with conven?tional servo valves. In addition，their results suggest that the valve has good static and dynamic character?istics and is suitable for high speed drive systems.

To test magnetostrictive materials，Zhao et al.［8］designed a test analysis system. They used giant magnetostrictive materials and analyzed the nonlin?ear hysteresis characteristics using theoretical meth?ods，meanwhile，they studied the effects of damp?ing，prestress and spring stiffness on the maximum amplitude. Sheykholeslami et al.［9］conducted an ex?perimental comparison of the first and second longi?tudinal modes of a giant magnetostrictive transduc?er. In their analysis，they demonstrated a higher quality factor of the second mode of the ultrasonic transducer，while sensitivity of the Young’s modu?lus was lower. Furthermore，they examined the variation in Young’s modulus（ΔEeffect）of the GMM rod using both theory and experiments［10］.

Mathematical models and numerical simula?tions have been previously used to study GMM.Zeng et al.［11］analyzed the magnetic energy conver?sion and vibration characteristics of magnetostrictive power ultrasonic transducers by establishing a math?ematical model of the transducer vibrator and per?forming computer simulations. Zhu et al.［12］also used a theoretical approach to develop an accurate giant magnetostrictive actuator with a thermal dis?placement suppression system. The thermal dis?placement control mechanism consisted of a temper?ature control module and a thermal displacement compensation module and the system significantly improved the output characteristics of the giant mag?netostrictive drive，especially the displacement accu?racy. Zheng et al.［13］established a dynamic model of the radiant panel of the magnetostrictive transducer under the action of the magnetostrictive rod through a spring and discussed the effect of its stiffness，po?sitioned between the magnetostrictive rod and the ra?diant panel，on the amplitude of the radiant panel.By selecting a spring with suitable properties，the magnetostrictive rod can be prestressed，thus，pro?viding the system with the dynamic characteristics required for vibration and allowing large displace?ment of the radiant panel.

Wang et al.［14］proposed a giant magnetostric?tive actuator based on permanent magnets. The use of permanent magnets to drive the magnetic field of magnostrictive Terfenol-d can inhibit temperature effects associated with the skin effect caused by ed?dy currents of the solenoid.

While GMM are widely used in the fields of ul?trasonic chemistry，industrial processing，and medi?cine［15］，current studies on GMM are mainly focused on low frequency systems and detection devices.For example，the high-frequency magnetostrictive transducers are widely used for ultrasonic detection；however，giant magnetostrictive power ultrasonic transducers are still rarely used in machining.

Li et al.［16］studied the geometry of the coil and the influence of magnetic circuit components on the magnetic field characteristics and output displace?ment of the transducer. They showed that installing a soft-iron core at both ends of the GMM rod can in?crease the magnetic field strength，reduce magnetic flux leakage，and increase the transduction of the output shift，while reducing the drive current. Cai et al.［17］studied the influence of different material prop?erties of the magnetizer on the vibration perfor?mance of an ultrasonic system. Highly permeable ferrite can improve the electromechanical energy conversion efficiency of low-power ultrasonic vibra?tion system；whereas high-power ultrasonic vibra?tion systems must take into account the magnetic permeability rate and saturation flux density，allow?ing the magnetizer to operate in a non-magnetic satu?ration state to improve system conversion efficiency.

Permanent magnets are typically placed at both ends of the GMM rod to generate the bias magnetic field of the giant magnetostrictive transducer. How?ever，the magnetic field distribution in the GMM rod will be non-uniform，owing to the low magnetic permeability of permanent magnets，which limits the performance of the giant magnetostrictive trans?ducer. Adding a highly permeable magnetic conduc?tive material between the permanent magnet and GMM rod of the giant magnetostrictive transducer can create a more uniform magnetic field distribu?tion in the GMM rod［18-19］，but further research of the magnetic conductive material influence on the gi?ant magnetostrictive transducer is needed.

In this paper，a theoretical design of a novel gi?ant magnetostrictive ultrasonic transducer was pre?sented. Dynamics of the ultrasonic vibration device was analyzed by using finite element analysis and the transducer was experimentally tested. Further，a magnetically conductive material（electrician pure iron）of various thicknesses was placed between the GMM rod and the permanent magnet to study the influence of thickness on the axial magnetic field uni?formity. Finally，performances of transducers with or without magnetically conductive materials were compared.

1 Structural Design of Ultrasonic Transducer

1.1 Principle analysis of ultrasonic transducer

Fig.1 shows a schematic illustration of the pro?posed giant magnetostrictive ultrasonic vibration de?vice. The ultrasonic generator produces a high-fre?quency electrical signal that is transmitted from the primary winding of the non-contact power transmis?sion device to the secondary winding and through the lead wire to the excitation coil of the GMM rod.Then，the excitation coil generates a high-frequency alternating magnetic field，and the GMM rod gener?ates high-frequency magnetostriction under an alter?nating magnetic field and the ultrasonic vibrations are amplified through the horn and transmitted to the tool head.

Fig.1 Schematic of ultrasonic vibration device

GMM rod is a brittle material with a tensile strength of about 28 MPa. Therefore，an appropri?ate prestress must be applied to the GMM rod in or?der to reduce tensile stress under actual working conditions. Moreover，adopting a certain amount of prestress can increase the magnetostriction coeffi?cient［20］. Thus，pre-tightening bolts are applied a suitable prestress to the GMM rod.

In this study，the rear cover，magnetic conduc?tive block，permanent magnet，GMM rod and horn were connected by a bolt，made of non-magnetic stainless steel. Cylindrical permanent magnets were placed at both ends of the GMM rod. The shank，a magneticallly conductive material，and horn were fixed by screws. Finally，a quarter-wavelength coni?cal transition step-type composite horn was de?signed to reduce the longitudinal length of the ultra?sonic vibration device.

1.2 Composite ultrasonic vibrator design

The front and rear cover plates of the vibrator and GMM can be regarded as continuous elastic me?dia. Thus，elastic equations were established for each and the boundary conditions were used to ob?tain various design parameters. Fig.2 shows a onedimensional segment within a variable section of the vibrator body，where the axis of symmetry is thexaxis and tensile stress acting on the segment defined by a small-volume element（x，x+ dx）isσxdx.

Fig.2 One-dimensional segment of a variable section of the vibrator body

According to Newton’s laws，the kinetic equa?tion can be written as

whereσis the stress function andσ=hereEis the Young’s modulus；Sis the function of the cross-sectional area of the rod andS = S（x）；ξis the particle displacement function andξ =ξ（x）；andρis the density of the rod material.

When the rod performs a simple harmonic mo?tion，ξ（x，t）=ξ（x）ejωt，the equation for the simple harmonic vibration state of the variable section rod can be obtained as

wherekis the number of circular waves andk = ω/c，herecis the speed of sound inside the member andc =（E/ρ）1/2.

The velocity of vibration，V =jωξ，was sub?stituted into Eq.（2）to obtain the vibration velocity equation for longitudinal vibration of the variable cross section bar

If the cross section of the bar is uniform andSis the constant，Eq.（3）can be simplified as

The general solution to the equations of vibra?tion velocity and force of a body with a uniform cross section is

whereZis the characteristic acoustic impedance of the vibrator andZ= ρcS.

The magnetic conductive material outside the coil and the coil does not participate in the vibra?tion，therefore，the ultrasonic vibrator can be simpli?fied as a rear cover，two magnetic conductive blocks，a GMM rod and a horn，as shown in Fig.3.The nodal plane is located at the joint surface be?tween the horn and the magnetic conductive block.The front surface vibration velocity of the transduc?er vibrator isVfand the elastic forceF=-ZωVf，whereZωis the input impedance of the front surface of the transducer. Ends of the vibrator are in a free state，soF1=F8=0. Furthermore，the velocity of the particle at the nodal plane is zero，thusV4（l4）=V5（0）=0.

On the left-hand-side（LHS）of the section，the boundary conditions are

Fig.3 Simplified ultrasonic vibrator

According to the boundary conditions and gen?eral solutions of the vibration velocity and force equations，the frequency equations on LHS of the nodal plane can be obtained as

On the right-hand-side（RHS）of the section，the boundary conditions are

Dimensions of the various parts of the transduc?er assembly are determined from Eq.（8） and Eq.（9）by using known parameters. The rear cover of the transducer is primarily used to achieve unob?structed unidirectional radiation，ensuring minimal energy loss through the rear surface of the transduc?er，thereby increasing the forward radiated ultrason?ic power.

In general，the rear cover of a transducer is con?structed from some types of heavy metals. If the rear cover is completely comprised of heavy metals，the total mass of the transducer and rotational mo?ment of inertia increases，which can affect the per?formance of the vibration system［21-22］. However，combining non-magnetic 316 stainless steel and an aluminum alloy can limit the radiation of energy from the rear cover，and also reduce the quality of the transducer.

Since the diameter of the magnetic conductive block is much smaller than that of the large end of the horn，the magnetic conductive block was de?signed in a stepped shape in order to reduce the im?pedance. The rear cover and horn were made of 316 non-magnetic stainless steel to avoid magnetic leak?age from the transducer. Properties of the selected materials are listed in Table 1.

Table 1 Material properties

The length of the transducer：l2=l4=4.5 mm，l3=42 mm，l5=10 mm，l6=21 mm，l7=20 mm.Diameter：D1=35 mm，D2=D3=D4=15 mm，D5=49 mm，D7=25 mm，D8=15 mm.

Substituting the above parameters into Eq.（8）and solving Eq.（9）yield equations forl5，l6，l7，andl8，which can be solved in MATLAB to obtain val?ues forl1andl8：l1=36.5 mm，l8=51.7 mm.

In theory，the length of the tool bar should be selected according to the half-wavelength theory.However，in actual production，the length of the tool bar is usually less than 1/4 wavelength and the lateral dimension is less than 1/10 wavelength，thus，enhancing rigidity of the tool. Furthermore，resonance of the whole system at the working fre?quency reduces；the length of the end of the original variable amplitude also appropriately shortens. The length is related to the equivalent quality of the tool，while it can be approximated using the follow?ing equcations［23］

whereMtis the tool equivalent mass，ρthe end horn density，Sthe end horn face area，mtthe tool bar actual mass，ktthe round wave number，ρtthe density of the tool bar，andltthe length of the tool bar.

The tool is made of steel，designed for highspeed tools. DiameterDt=7 mm and lengthlt=20 mm were obtained for the tool headl8=51.7-5=46.7（mm）

2 Finite Element Analysis of Trans?ducer

2.1 Modal analysis of ultrasonic transducer

A modal analysis of the transducer was per?formed in ANSYS using the piezoelectric-compres?sive magnetic comparison method. The pre-tighten?ing bolt was simplified and the rear cover was con?sidered a unified body. The SOLID95 element was selected for the front and rear cover plates as well as the magnetic conductive block. In the coupled field analysis，the SOLID98 element was adopted for the piezoelectric ceramic［24］. Therefore，the SOLID98 element was also selected for the GMM rod.

The frequency range was set to 16—24 kHz and the Lanczos method was used to perform the calculations. During post-processing，the required longitudinal vibration mode frequency is 19.65 kHz and the vibration mode of the transducer is shown in Fig.4. The vibration output at the top of the horn is the largest，while the vibration at the back cover plate and the flange of the horn is relatively small.The natural frequency of the transducer is very close to the design frequency（20 kHz），which basically verifies the correctness of the theoretical design.

Fig.4 Transducer vibration mode

2.2 Magnetic field analysis of transducer

A static magnetic field analysis of the transduc?er was performed in ANSYS using the two-dimen?sional（2-D）symmetrical PLANE13 element. Be?cause of the large longitudinal size of the front cover plate and the rear cover plate，the effect on the mag?netic field is small. Therefore，only a part of the model is established. Magnetic permeability of each material is listed in Table 2 and the finite element model of the transducer is shown in Fig.5.

Table 2 Relative magnetic permeability of different ma?terials

Fig.5 Finite element model of transducer

Since GMM is relatively brittle，non-uniformi?ty of the internal magnetic field generates internal stress that can reduce the service life of the materi?als. Thus，realizing a uniform magnetic field distri?bution of the GMM rod during expansion and con?traction can improve the performance of the trans?ducer，and increasing uniformity of the internal mag?netic field is important.

Simulations were carried out with a magnetical?ly conductive material of various thicknesses be?tween the permanent magnet and the GMM rod.The magnetic field intensity distribution in the axial direction of the GMM rod is shown in Fig.6. Thick?ness values of the guide can be calculated［25］and when the magnetic sheet is used，the axial magnetic field uniformity of the GMM rod is

whereHiis the magnetic field strength of theith point on the GMM rod，nthe number of points con?sidered，andHmaxthe maximum magnetic field strength in the GMM rod.

Fig.6 Axial magnetic field intensity distribution in GMM rod

Fig.7 shows the axial magnetic field uniformity of the GMM rod at various thicknesses of magneti?cally conductive material. When the magnetically conductive material is added between the permanent magnet and the GMM rod，the magnetic field uni?formity in the GMM rod improves. Moreover，the highest uniformity was obtained with a magnetic conductive material thickness of about 1.6 mm.Above 1.6 mm，the magnetic field uniformity de?creases. Therefore，to improve energy conversion efficiency，the length of the GMM rod should in?creased as much as possible，such that the thickness of the magnetically conductive material minimizes.

Fig.7 Magnetic field uniformity of GMM rod

3 Experimental Procedure

The theoretical design and simulation analysis were validated by experimentally measuring the res?onant frequency and amplitude of the ultrasonic transducer. A prototype of the giant magnetostric?tive ultrasonic vibration device was fabricated，as shown in Fig.8. The experimental measurement sys?tem consists of a high-speed bipolar power supply（BP4620）to generate voltage signals of different frequencies and amplitudes，an impedance analyzer to measure the impedance of the transducer，and a laser oscillator used to measure the amplitude of the transducer close to the resonant frequency.

Fig.8 Prototype of giant magnetostrictive ultrasonic vibra?tion device for progressive sheet forming

The effect of magnetically conductive material between the GMM rod and the permanent magnet on the performance of the giant magnetostrictive transducer was studied，while the validity of simula?tion analysis was verified. According to Fig.7，the magnetic field uniformity is the highest when the thickness of the magnetic conductive material is about 1.6 mm. Therefore，the thickness of the mag?netically conductive material was set to 1.6 mm in subsequent experiments.

Impedance analysis was performed to compare the results of the magnetostrictive transducer with or without the additional magnetic material.The me?chanical quality factor decreases， as shown in Table 3，after the magnetically conductive material is added to the giant magnetostrictive transducer.

Table 3 Results of impedance analysis of giant magneto?strictive transducer

Figs.9 and 10 show impedance circles of the gi?ant magnetostrictive transducer without and with the magnetically conductive material，respectively.It can be observed that the impedance decreases when the magnetically conductive material is added.

Fig.9 Impedance circle without magnetically conductive material

Fig.11 shows the amplitude of the top of the transducer tool head at different frequencies. The test conditions were as follows：Voltage of 60 V，square waveform，and the frequency varied from 19.0 kHz to 19.6 kHz，with a step size of 50 Hz.The output amplitude of the transducer with and without magnetically conductive material is mea?sured three times at each frequency point，and then the average value is obtained.

Fig.10 Impedance circle with magnetically conduc?tive material

Fig.11 Transducer amplitude at different frequencies

The results show that the maximum amplitude of the giant magnetostrictive transducer is 37.86 μm.However，when the magnetically conduc?tive material is placed between the GMM rod and the permanent magnet，the maximum amplitude of the giant magnetostrictive transducer increases to 41.24 μm. In conclusion，the maximum amplitude of the transducer increases when a magnetically con?ductive material is added between the GMM rod and the permanent magnet.

4 Conclusions

The theoretical design of a giant magnetostric?tive ultrasonic transducer was presented. Finite ele?ment analysis was performed to analyze the dynam?ics of the ultrasonic vibration system. Then a proto?type of the giant magnetostrictive ultrasonic vibra?tion device was fabricated and an impedance analy?sis of the transducer was performed. Amplitude measurements show that the actual resonant fre?quency of the transducer is 19.25 kHz and the maxi?mum amplitude of the transducer is 37.86 μm，thus validating the theoretical design and simulation.

Furthermore，a magnetic field analysis of the giant magnetostrictive transducer was carried out.Adding a magnetically conductive material，com?prised of pure iron between the GMM rod and the permanent magnet，resulted in a more uniform mag?netic field，improved the utilization ratio of the GMM rod，and achieved transduction. In addition，the overall working performance of the device was more stable. Moreover，the magnetic field uniformi?ty of the GMM rod was the highest when the thick?ness of the magnetic conductive material was about 1.6 mm.

Performance of the giant magnetostrictive transducer with and without the magnetically con?ductive material was compared. The results show that the addition of the magnetically conductive ma?terial reduces the mechanical quality factor and im?pedance of the giant magnetostrictive transducer and increases the maximum amplitude. This work has great significance to the optimal design of high-pow?er and large-amplitude ultrasonic transducers.