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International Journal of Product Design 2015

STRATEGY OF FEM MODELING OF ULTRASONIC HORN FOR WORKPIECE VIBRATION USED AT MICRO-ELECTRICAL DISCHARGE MACHINING

Daniel Ghiculescu1, George Seritan2, and Ovidiu Alupei3 1 Politehnica University of Bucharest, Romania, daniel.ghiculescu@upb.ro

2 Politehnica University of Bucharest, Romania, serigeorge1973@yahoo.com 3 Politehnica University of Bucharest, Romania, ovidiu.alupei@gmail.com

ABSTRACT: The paper deals with a strategy for obtaining resonance condition within an ultrasonic chain used for micro-electrical discharge machining, based on Finite Element Method (FEM) determination of a profiled ultrasonic horn geometry needed to vibrate a disk shape workpiece. 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 Multipysics FEM modeling of horn profile integrating workpiece at its end. Different horn constructive elements were changed gradually to assure the control of horn natural frequency and elongation amplification. The entry data for horn geometry determination were the dimensions and natural frequency of transducer subassembly, which is joined with ultrasonic horn and workpiece. Based on this applied strategy, the horn integrating the workpiece was machined by CNC, and real natural frequency obtained was very close to the result provided by FEM modeling. KEY WORDS: Finite Element Method, ultrasonic horn, micro-electrical discharge machining.

1. INTRODUCTION Ultrasonics aiding of micro-Electrical Discharge Machining (µEDM+US) by vibrating the tool or workpiece (in this case) proved spectacular improvement of main technological parameters in terms of machining rate, precision/volumetric relative wear, and surface quality [1], [2], [3], and new materials with low machinability [4], [5], [6].

But horn fabrication encounters a critical condition, the equality between own frequency of transducer subassembly (as entry data) and of horn which integrates the workpiece to be vibrated – resonance condition. The considerable time consuming for this condition achievement especially for horns complex shape, comprising several iterative stages (horn machining – own frequency measurement) is justified on relative great volumes of fabrication. From this derives disagreeable lack of flexibility of EDM+US technology. Under these conditions, the betterment of this drawback, using FEM modeling of ultrasonic horn, associated with Computer Aided Machining of the horn, based on FEM results, becomes of utmost interest nowadays when the response of companies to client demands is critical.

Achieving new shape (profiles in case of revolution surfaces) of ultrasonic horn appropriate for diverse application [7], taking into account the unique character of horn shape in the relation to the type of machining [8], is a very important research direction. In case of present paper, the complex horn integrating the workpiece positioned at its end as antinodal point.

2. PRELIMINARY DIMENSIONING OF CYLINDRICAL STEPPED HORN

A preliminary dimensioning was approached as a starting point for an usual cylindrical stepped horn, using some known relations, aiming finally at determination of complex profiled horn, integrating the workpiece to be vibrated during µEDM+US.

The amplification (K) in case of a stepped horn, after Merkulov and Kharitonov [9], is equal with:

K = (D1 / D2 )2 (1)

where: D1 is the entry diameter [m]; D2 is the output diameter of the stepped horn [m]. This is applicable when entry length (l1) and output one (l2) of the stepped horn (upper and lower steps in figure 4) meet the following conditions:

l1 = 1.5 / α [m] (2)

l2 = 1.6 / α [m] (3)

α is the wave number calculated with the formula:

α = 2 π / λ [m-1] (4)

and λ is the wave length calculated as it follows:

λ = c / f [m] (5)

where f is the oscillation ultrasonic frequency. The ultrasonics velocity (c) within a solid material is determined with the relation:

ρEc = [m/s] (6)

where E is Young’s modulus of horn material [Pa], ρ - density of horn material [kg/m3].

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The transducer assembly provided by the Institute of Solid Mechanics of the Romanian Academy (fig. 1) has the own series frequency ftr= 41760 Hz – of interest for this type of applications – [10], and the output diameter of the radiant bush was =35 mm.

Figure 1. Transducer assembly used in ultrasonic chain for

assisted micro-EDM

In case of our horn made from OLC 35 steel with characteristics E=2.1 1011 Pa, ρ = 7850 kg/m3, the entry frequency of the horn was fo1=40.0 kHz, lower than that of the transducer ftr. Thus, the values of physical parameters from the relations presented above are: λ=0.1293 m, α =48.5674 m-1. Therefore, the steps lengths resulted: l1=30.885 mm; l2=32.944 mm. The entry diameter of the horn, is equal with that of radiant bush, D1=35 mm.

These above preliminary results represented entry data for the first stage of FEM modeling.

3. STRATEGY OF FEM MODELING OF ULTRASONIC HORN WITH INTEGRATED WORKPIECE

Comsol Multiphysics with Structural Mechanics module, and Eigenfrequency submodule were used for FEM modeling and simulation of an ultrasonic horn, which included a disk shape workpiece at its end, as antinodal point under conditions of standing waves.

The used strategy for modeling introduced or changed only one constructive parameter of horn geometry in each stage, following the control of oscillation physical parameters, i.e. own frequency and amplification. Consequently, in order to achieve the needed profile of the horn, more than 20 modeling stages were covered.

The logical scheme depicting the modeling strategy is presented in fig. 2. The following parameters of horn geometry were the subject to change successively in each modeling stage: steps lengths, fillet radius between main steps, disk shape radius of wokpiece at the chain end, fillet radius and horn

neck dimensions between workpiece and horn, nodal channel radius and its vertical position, threaded hole for workpiece and horn assembling to transducer. After the most part of FEM stages was done, a first variant of horn was machined, measuring its own frequency in order to validate the model. After step lengths adjustments, the horn was CAM-ed, and resonance condition was fulfilled.

Figure 2. The strategy logical scheme for FEM modeling of

the profiled stepped horn with integrated workpiece and resonance condition obtaining

Radiant bush

Reflecting bush

PZT plates Supply copper blades

fo1, ftr, D1, D2, E, ρ

Calculation of λ, α, and lengths l1, l2 with relations 2-6 at frequency fo1

Insert of fillet radius at main steps joint; determination of frequency fo2 and amplification K2

Insert of integrated workpiece; determination of frequency fo3 and amplification K3

Increase of workpiece radius and input fillet radius to workpiece joint; determination of frequency fo4, …

and amplification K4, …

Insert of horn neck to workpiece joint; determination of frequency fo5 and amplification K5

Modification of workpiece height; determination of frequency fo6,… and amplification K6, …

Modification of steps lengths; determination of frequency fo7,… and amplification K7, …

Modification of neck height; determination of frequency fo8,… and amplification K8, …

Insert of nodal channel for clamping; determination of frequency fo9 and amplification K9

Modification of workpiece radius; determination of frequency fo10, … and amplification K10, …

Modification of nodal channel radius and its position; determination of frequency fo11, … and amplification K11, …

Modification of lower step radius; determination of frequency fo12, …and amplification K12, …

Validation of model by machining and measuring real horn fo

Modification of horn lengths for obtaining resonance condition

Input threaded hole for assembling workpiece and horn; determination of frequency fo13 and amplification K13

Horn CAM with dimensions resulted from FEM

Yes

No

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A parametric model was achieved, facilitating the rapid changes of geometry and making the strategy operational. The initial needed parameters were established in Global Parameters, and presented in fig. 3.

Figure 3. The defined parameters in the first stage

The other main elements for the generic modeling are synthesized underneath. The modeled horn geometry was created using axis symmetric dimensional space, based on defined parameters from Global definitions. The needed horn material properties were provided by Comsol Multiphysics Materials Library and were adjusted for Romanian OLC 45 steel used in this case. The physics boundary conditions for horn eigenfrequency (resonance) frequency were set on free for all geometry limits. The mesh was set on extrafine, with more than 2300 elements for final models and average quality around 0.96 on a 0-1 scale.

4. ANALYSIS OF FEM RESULTS AND ULTRASONIC HORN ACHIEVEMENT

The analysis of the most important results obtained by FEM modeling, and the geometrical parameters of horn construction are presented below.

Figure 4. Resonance frequency and displacements at

initial stage

The corresponding FEM results for the parameters defined in fig. 3, in terms of eigenfrequency and deformations are presented in fig. 4.

In the next stage, a new parameter was introduced in Global Parameters model as fillet radius between main steps, as it is presented in fig. 5.

Figure 5. Parameterized fillet radius of horn main steppes

Consequently, the corresponding FEM results are presented in fig. 6. It can be noticed that fillet radius insertion produced increase of horn natural frequency and amplification, i.e. the ration between displacements at the two ends of the horn – 1 is considered at superior (entry) surface of the horn.

Figure 6. Resonance frequency and displacements at insertion

of fillet radius between horn main steps

The new parameters needed for workpiece insertion by modification of horn shape end are presented in fig. 7.

Figure 7. Parameters introduced for workpiece and its fillet radius modeling

The results obtained in terms of resonance frequency and displacements with parameters values presented above are shown in fig. 8.

Entry step (r1, l1)

Output step (r2, l2)

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Figure 8. Resonance frequency and displacements at insertion

of workpiece and its fillet radius

A decrease of horn resonance frequency and of the amplification was perceptible in this stage in comparison with the previous one.

The parameters needed to insert the neck between the workpiece and the rest of the horn, having the goal of stiffness increase, are presented in fig. 9.

Figure 9. Parameters introduced for neck modeling

In fig. 10, the resonance frequency obtained in this stage and the corresponding displacements of the horn are presented. Therefore, a slight decrease of horn own frequency and amplification was produced in comparison with previous shape.

Figure 10. Resonance frequency and displacements at insertion

of neck between horn and workpiece

The next stage was focused on the nodal channel insertion needed to clamp the ultrasonic chain on electrical discharge machine. A nodal level was determined (minimum oscillation amplitude) as it is shown in fig. 11.

Figure 11. Nodal level determination

At this level, a radial channel was introduced, which serves at horn clamping by four pointed radial screws, which enter this channel.

The parameters needed to model this nodal channel are presented in fig. 12.

Figure 12. Parameters needed to insert nodal channel The own frequency and displacements obtained by nodal channel insertion were presented in fig. 13.

Figure 13. Resonance frequency and deformations at nodal

channel insertion

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A decrease of horn natural frequency was produced by the nodal channel insertion, amplification being slightly increased.

In order to extend the diameters range of the workpieces to be machined by µEDM+US, the radius of workpiece was increased gradually. In this stage, the value of this parameter is specified in fig. 14.

Figure 14. Current value assigned to workpiece radius

The natural frequency and displacements corresponding to this stage were presented in fig. 15. Thus, the workpiece radius increase determined resonance frequency decrease and amplification increase.

Figure 15. Resonance frequency and displacements at increase

of workpiece radius

The parameters modified in the following stages are presented in fig. 16, and horn natural frequency and displacements were presented in figure 17.

Figure 16. Values assigned to parameters

The above values of the parameters led to increase of natural frequency and of amplification. Through increase of own horn frequency, it is aimed at obtaining the resonance condition, i.e. equality between the one of horn and transducer, which is higher.

Figure 17. Resonance frequency and displacements at

adjustment of channel radius, lower step radius, neck height, and workpiece radius

After horn CAM based on FEM results, measuring the resonance frequency of ultrasonic horn and chain assembly was done, under working conditions - cavitation in the liquid in which the workpiece is immersed - as it is presented in fig. 18. This was needed to validate the model before machining the final shape of the horn. This setup is also necessary for the accordance of the ultrasonic generator on nominal frequency of around 40 kHz, its adaptive control working within ± 1kHz range.

Figure 18. Measuring stand for natural frequency of ultrasonic horn and chain used at assisted EDM and generator accordance A good agreement was achieved between FEM modeling and measured natural frequency, less than 1%. On this basis, the final horn model was created, including longitudinal threaded holes (M6 and M12)

Transducer subassembly

Support

Ultrasonic horn

Integrated workpiece

immersed in liquid

Versatester Tektronix

oscilloscope

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for workpiece assembling and junction with transducer, represented in fig. 19.

Figure 19. Final modeling of ultrasonic horn

The horn was obtained by CAM, and the workpiece was detached from the horn by WEDM, being later assembled with the horn. Using these constructive solutions, the resonance condition was achieved, existing a reserve of horn lengths that can be used for shorten them to get the target of ftr= 41760 Hz.

5. CONCLUSIONS The influence of some constructive elements of ultrasonic horn was identified based on FEM modeling strategy: fillet radii are able to increase the natural frequency and its amplification, against other elements like radii extension of disk shapes inserted in horn construction, which produced decrease of natural frequencies and its amplification. The same influence was observed in case of radial channels insertion, but their profile radius diminishing can increase the horn natural frequency.

FEM modeling strategy led to ultrasonic horn complex shape achievement integrating disk shape wokpiece at its end as antinodal point. Thus, a good agreement was obtained between real horn natural frequency and that provided by FEM, less than 1%, this facilitating the resonance condition achieving for an ultrasonic chain used at µEDM+US.

The presented modeling strategy was able to significantly decrease machining preparation time needed for ultrasonic assistance of EDM, thus contributing to increase its flexibility, considered as its main drawback for industrial applications.

6. ACKNOWLEDGEMENT

The presented results are obtained in Joint Applied Research Project, C4, 222/2014.

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