Finite element modeling of an ultrasonic horn integrating the tool for micro-electrical discharge machining

GHICULESCU Daniel1, a, MARINESCU Niculae-Ion1,b, KLEPKA Tomasz2,c and ALUPEI-COJOCARIU Ovidiu-Dorin1,d

1Politehnica University of Bucharest, Romania,

2Lublin University of Technology, Poland. adaniel.ghiculescu@upb.ro, bniculae.marinescu@yahoo.com, ct.klepka@pollub.pl

dovidiu.alupei@gmail.com

Keywords: ultrasonic, horn, micro-electrical discharge machining, finite element modeling.

Abstract. The paper deals with a strategy to obtain a complex conical ultrasonic horn, which integrates the tool for micro-drilling through electrical discharge machining (EDM+US) based on Finite Element Method (FEM) modeling. This approach comprises more than 20 stages, starting with preliminary determination of a basic profile stepped horn through mathematical relations. The next stages are based on Comsol Multiphysics FEM modeling, changing gradually different horn constructive elements and integrating the tool and their clamping parts. The entry data for horn geometry determination were the dimensions and own frequency of transducer subassembly, joining ultrasonic horn. Finally, the CAD file of horn was transferred to CAM file and CNC machined. Horn real own frequency obtained was very close to the result provided by FEM modeling.

Introduction

Ultrasonics aiding of micro-Electrical Discharge Machining (EDM+US) by vibrating the tools (in this case) or workpiece proved advances in surface quality, precision/volumetric relative wear, and machining rate [1], [2], [3], also in case of new materials with low machinability [4], [5], [6].

Nevertheless obtaining the sine qua non conditions of resonance - equality between assemblies of transducer (fotr) and ultrasonic horn (foh) - mainly in case of complex shape ultrasonic horn, is an intricate and time consuming process, containing several iterative stages (horn dimensional adjustment own frequency measurement). This is a technologic task, justified on relative great volumes of fabrication, and consequently showing lack of flexibility of EDM+US technology. But FEM modeling of ultrasonic chains facilitates the resonance condition obtaining. In this case a conic shape horn - leading to low cost manufacturing mainly by conventional machining against other complicate curve profiles - that integrates the tool for micro-holes is approached.

Coupling FEM with Computer Aided Machining (CAM) of the horn becomes of utmost interest nowadays, new shapes of ultrasonic horn appropriate for diverse application being achieved [7], taking into account the unique character of horn shape in relation to type of machining [8].

Preliminary Dimensioning of an Usual Cylindrical Stepped Horn

A preliminary dimensioning was carried on for usual cylindrical stepped horn, standing as entry data for FEM modelling of complex profiled horn, integrating the tool to be axially vibrated during micro-drilling by EDM+US. Some known relations were used. The upper (l1) and lower (l2) steps lengths in case of a cylindrical horn (see Fig. 4), after Merkulov and Kharitonov [9], are equal with:

l1 = 1.5 / [m], (1) l2 = 1.6 / [m], (2)

where is the wave number calculated with relation:

= 2 / [m-1], (3) and is the wave length calculated as it follows:

= E

ffc 1

= [m], (4)

where f is the oscillation ultrasonic frequency [Hz]; c - ultrasound velocity within a solid material [m/s]; E - Youngs modulus of horn material [Pa], - density of horn material [kg/m3].

The previous dimensional conditions lead to amplification (K), which is equal with:

K =2

2

1

DD (5)

where: D1, D2 are the entry diameter, and the output diameter of the stepped horn [m] - see Fig. 4.

The transducer assembly provided by the Institute of Solid Mechanics of the Romanian Academy - IMSAR (Fig. 1) has the own series frequency fotr= 40805 Hz entry data for this application type [10]. Its radiant bush, which joints the concentrator, has output diameter Db=35 mm. In case of our horn made from Romanian OLC 35 steel, the needed characteristics are: E=2.1x1011 Pa, = 7850 kg/m3. Its entry frequency for calculation was fo1=40.0 kHz, lower than that of the transducer fotr. Therefore, values of physical parameters from the above relations are: =0.1293 m, =48.5674 m-1. So, resulted steps lengths are: l1=30.885 mm; l2=32.944 mm. The horn entry diameter is D1= Db=35 mm.

Strategy Based on Finite Element Method

Using FEM modeling strategy, only one constructive parameter of horn geometry was introduced or changed in each stage, aiming at control of own frequency foi and amplification Ki, and thus having a permanent feed-back. Therefore, more than 20 modeling stages of strategy were covered, whose generic logical scheme is presented in Fig. 2.

Comsol Multiphysics with Structural Mechanics module, and Eigenfrequency submodule were used for FEM modeling and simulation of a complex ultrasonic horn, which included tool and its clamping at horn end - antinodal point under conditions of standing waves.

The needed material properties were provided by Comsol Library and were adjusted for real (measured sound velocity) OLC 45 steel used for horn and real copper 99.5 for tool. The physics boundary conditions for horn eigenfrequency were set on free for all geometry limits. The modeled horn geometry was created using 2D axis symmetric space, using defined parameters like those presented in Fig. 3 for the first stage.

Fig. 1. 40 kHz Transducer - IMSAR

Fig. 2. Logical scheme of FEM modeling strategy applied to complex horn with tool

radiant bush

reflecting bush

PZT rings

supply and

cooling flange

Calculation of , , and lengths l1, l2 with relations 1-4 at frequency fo1

ftr,fo1, E, , D1, D2

Insert, modification of filet radius Insert, modification of taper angle

Insert, modification of threaded hole Insert, modification of nodal channel

Insert, modification of tool and clamping Modification of l1, l2

Determination of foi and Ki

Validation of model by foi

Horn CAM adjusting l1, l2 for resonance condition Yes

No

Other parameters introduced during modeling strategy will be presented below in correspondence with results analysis.

The mesh was set on extrafine, with more than 2000 elements for final models and average quality around 0.96 on 0-1 scale.

Results Analysis

In comparison with usual horn from initial stage, the amplification increased, i.e. the ratio between relative displacements at two ends of horn (value 1 is considered at entry (superior) surface of the horn), and the own frequency of conical shape horn also increased as it is presented in Fig. 6, 7. At taper angle increase, as it can be noticed, the horn own frequency also increased gradually.

Fig. 6. Own frequency and displacements at 1o taper angle Fig. 7. Own frequency and displacements at 5o taper angle

Fig. 3. Defined parameters for initial stage

The analysis of the results obtained in most important stages of FEM modeling of complex conical horn, and their related geometrical parameters are presented in this section. In the initial stage, the horn own frequency and its amplification (in direct relation with relative displacements) are presented in Fig. 4 in correspondence with parameters previously defined (Fig. 3).

In the next stages, the taper angle (alpha) was introduced at lower step of the horn and the needed parameters for geometry construction are presented in Fig. 5.

Fig. 5. Parameters at taper insertion at lower horn step Fig. 4. Own frequency and displacements at initial stage

D1

l 1 l 2

D2

In the subsequent stages, the tool was inserted and its clamping parts from the same material with the horn. It is considered that these parts make perfect contact with the conical hole for assembly machined within the horn. This determines no modification of corresponding own frequency of horn assembly as previous running of programmed model pointed out. The dimensions of remained cavity unfilled by clamping parts were parameterized as well as tool dimensions, and presented in Fig. 8.

Fig. 9. Own frequency and displacements at tool insertion and clamping parts

Fig. 10. Own frequency and displacements at tool insertion and clamping parts with 6o taper angle of lower

step As one can notice, insertion of tool and its clamping system determined decrease of horn

assembly own frequency together with amplification in comparison with previous model (Fig. 9). Increase of taper angle at 6 is able to compensate some reduction of foi and Ki values (Fig. 10).

In the next stages, a preliminary determination of a nodal point was carried on in order to create a nodal channel for clamping the ultrasonic chain assembly. A limited range of amplification, in terms of visualization of colored zone, led to coordinate z = 39.3 mm for nodal point (see Fig. 12).

This was followed by insertion of radial nodal channel in horizontal plane, comprising nodal point with coordinate z, mentioned above. The defined parameters for nodal channel construction are presented in Fig. 11.

As it can be noticed, from Fig. 13, the insertion of nodal channel determined a decreased of own frequency and amplification as well, in comparison with previous model.

Fig. 8. Parameters needed for tool insertion

Fig. 11. Parameters for nodal channel insertion

In the following stages the hole threaded and stud were inserted, which are used to assembly the transducer with the horn at its upper end, with the parameters for modeling presented in Fig. 14. These constructive changes produced own frequency and amplification decrease, as Fig. 15 pointed out.

In the final stages, for resonance condition reaching, the step lengths (l1 and l2) were increased; von Mises stress was decreased in the zone of pointed inner surfaces (as previous running models indicated) through their radii increase; the position of nodal channel was changed to correct the displacement of nodal point produced by previous constructive changes ( Fig. 16).

Fig. 15. Own frequency and displacements at threaded

hole and stud insertion Fig. 16. Corrections needed to reach resonance condition

and proper horn working

Fig. 12. Preliminary determination of nodal point

with z coordinate of 39.3 mm Fig. 13. Own frequency and displacements at insertion of

nodal channel

Fig. 14. Parameters for threaded stud and hole

The strategy of complex horn was finalized through machining by CAM of its final version resulted from FEM, whose real own frequency proved to be very close under entry frequency fotr= 40805 Hz. Only fine adjustments of step lengths by their reduction were executed for resonance.

Conclusions

A strategy for ultrasonic complex horn achievement that integrated the tool and its clamping parts was completed based on FEM modeling and horn CAM in order to ease the ultrasonic chain resonance obtaining. FEM results indicated that positioning of clamping elements at horn end in antinodal point led to proper working of ultrasonic chain - the tool appropriate for micro-drilling through EDM exerted very low influence on own frequency of horn assembly. Other constructive elements of complex conical shape horn, introduced step by step, like filled radius between exterior surfaces, taper angle, cavities and threaded holes increased horn own frequency. Other elements like nodal channel, tool and threaded stud insertion decreased horn frequency. Such CAE-CAM approached strategy proved its capacity to improve flexibility of EDM+US technology.

Acknowledgement

The paper was achieved in Joint Applied Research Project supported by MEN-UEFISCDI, project no. PN-II-PT-PCCA-2013-4-0236, Contract no. 222/2014.

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