S. Shestakov¹, R. Rink²

¹ Moscow State University of Technology and Management, Russia,

e-mail: sdsh@mail.ru

² Oil Tech Production OY, Estonia,

e-mail:info@oiltech-nordic.eu

A Criterial Assessment of Quality Transformations a Plane Elastic Wave in the Solid-State Resonant Sonotrodes

Abstract

The criteria are described optimization of the form of solid-state resonant sonotrodes, which allow without full-scale simulation make a complex assessment of the qualitative characteristics of the transformation of the plane elastic waves:

- the resistance to the bending vibrations at the high value of the coefficient of transformation;

- disposition to the emergence of the bending a vibration modes;

- measure of sharpness of the resonance, which determined by the shape of the profile function.

It is shown that the best is in all respects is sonotrode, the function of the profile of which is described by the equation of the line of the flow path of the wave vector in the canonical stepped sonotrode.

Optimization of the design of resonant sonotrodes (acoustic waveguide) that are part of the ultrasonic vibrating systems is an important engineering problem. Is correct solution of this problem provides the highest values of energy efficiency in the transmission of elastic waves in the load.

Oscillatory systems operating in devices for the treatment of the solid (for example, plastic deformation) may be described by characteristics of idling. Optimization of waveguide transformers such systems is carried out by means of quality factor f Eisner [1]. Was proposed a universal criterion q [2]. These criteria are non-trivial criterias of similarity that allow at the choice of the profile function the sonotrode is minimize mechanical stresses in it by optimizing the cross-section and the curvature of the surface in the plane of the antinodes of oscillatory deformations at obtaining maximum coefficient of transformation. Thus, they provide a reliability and reduce the possibility of bending vibrations of the sonotrode, which can dissipate part of the energy.

More stronger load have systems transmitting oscillations in the substance which have not elasticity of shape (the liquids). If sonotrodes of such systems do not contain the stress raisers – grooves, grooves or holes – near the antinodes of strain, then criterion:

, (1)

where: K – stress concentration factor; m – the transformation ratio of the optimized sonotrode; M – transformation ratio sonotrode with stepped profile with the same areas ratio on the ends; e_max – the maximum by length of sonotrode value vibrational strain; e_max,0 – the maximum strain along the length of sonotrode with a coefficient of transformation to 1, can be written, as:

(2)

In this case graphics functions q(m) for the most common forms of the profiles of sonotrodes will have the form shown in Fig. 1.

FIG. 1. Functions of the quality criterion q from coefficient transformation m, excluding the effect of stress concentrators.

1 - equiline sonotrode [3,4], the profile of which is described by lines of the flow path of the wave vector in the stepped sonotrode at the height of the canonical conformal invariant h = 0,95;

2 - sonotrode, the profile of which is composed of line parallel to the axis x, fillets, chamfer and another line parallel to the axis of x, which conjugate (congruent) together;

3 - hollow chamfer sonotrode;

4 - cylindrical-conical sonotrode.

For correctness of comparison the node of vibrational displacement all sonotrodes is located at a distance of 0,25l from the end of the larger diameter. In FIG. 2 the transformation ratio of all sonotrodes is 4.

FIG. 2. The profiles of investigated sonotrodes.

In the new interpretation criteria q(m) its graph for stepped canonical sonotrode in FIG. 1 will look line gray.

In this case m = 1, .

In FIG. 1 clearly shows that in the range of transformation ratios, which provide condition for the preservation of a flat wave front, the worst is sonotrode with cylindrical-conical shape, and the best - with equiline profil. The rest almost do not differ from each other and from the stepped sonotrode.

For a comprehensive evaluation of the quality the sonotrodes oscillatory systems operating under high mechanical loads, it is proposed two new criterion - a criterion for passive mass s (the term "passive mass" introduced E. Kikuchi [5]) and quality factor of form d = дL/дf. The first of these is given by:

, (3)

where: v, V, V₀ – volumes of sonotrode what assessed, the stepwise sonotrode with the same coefficient transformation and a sonotrode with coefficient transformation m = 1, the sectional area of which equals the area of the smaller end of stepwise sonotrode, respectively; L – a resonant length of the studied sonotrode; s(x) – the function of the cross-sectional of studied sonotrode along its length.

Such type of criterion selected from the consideration that the sonotrode with a constant cross-section not has a passive mass, so masses of the sonotrodes from the same material or volumes of sonotrodes from different materials as in this case, must be compared without volumes which equal the length of the waveguides, multiplied by area of its smaller end.

The graphical comparison profiles of sonotrodes, which were considered by availability a passive mass shown in Fig. 3.

FIG. 3 Functions of the criterion of passive masses s from transformation ratio m.

Sonotrode of the equiline profile only one may have a volume of passive mass smaller than in the stepped transformer, where it is equal to 1 (solid gray line). The waveguide 2 with increasing m the criterion is increases, so as the ratio of the radius of chamfer and internal rounding (fillet) constantly, whereas, for example, the radius of curvature equilines flux of the field of wave vector at the canonical sonotrode in the wide portion more than in the narrow (see Fig. 2). Larger values criterion s at the waveguide cylindrical-conical profile has the strong difference from profile the type of equiline flow in a stepped transformer with the same ratio of the area of input and output ends. He has willed, conditionally, with up to 8 time’s larger coefficient of transformation.

Criterion q and the criterion s show that the advantages of the waveguide 3 before hollow chamfer transformer in given conditions occur only in the field of transformation ratios below 2.3. Passive masses are responsible for the presence of additional modes of oscillation in the sonotrode (other than main: strictly longitudinal and strictly transverse), hence and for additional uncontrolled energy dissipation at a time when the sonotrode fully immersed in the environment in which it should transfer energy fundamental modes of oscillation.

Therefore, smaller value of the criterion s (especially the smaller then 1) point to the possibility of obtaining high efficiencies of sonotrode. It is essential for the acoustic transformers on which are based work, for example, the sonochemical reactors cylindrical wave [6].

The second newly proposed criterion - the criterion "quality factor" of form d, shows amount by which the hypothetical should be changed the resonant length of sonotrode for a given change of sound velocity in the material of which it is made, or at change frequency of the oscillator.

The need to analyze the sonotrodes with this criterion arises from need to pre-assessment "quality factor" oscillating system without its full-scale simulation. Criterion d in the form in which it is proposed, allows evaluating the resonance characteristic of the projected waveguide independently of characteristics scattering of the material from which it is made. Clearly, the larger on magnitude of this criterion values indicate higher sensitivity to changes sonotrode resonance condition, i.e. the sharpness of the peak, and therefore the possibility of obtaining large amplitude values vibrational displacement at resonance.

Table 1 shows the values of the criterion "quality factor" of form the considered above sonotrodes at a constant value of m and the position node of the vibrational displacement х_node/L. The values were obtained as the average in the range of variation of frequency (0,9...1,1)f.

Table 1

Criterion	Value
d canonical	1,00
d 1	1,10
d 2	0,86
d 3	0,89
d 4	0,91

Here, as in all previous cases, the qualitative benefits remain with the equiline sonotrode.

REFERENCES

1. Physical Acoustics / eds. W. Mason, V.1, Part B, New York, London: Academic Press, 1964

2. Shestakov S. The optimization criterion of acoustic waveguide transformer // Proceedings of the X Session Russian Acoustical Society, V2.- Mоscow.: GEOS, 2000.- pp. 115-119

3. Patent RU 2183141, 2002

4. Patent RU 2311971, 2007

5. Ultrasonic transducers / eds. Y. Kikuchi, Tokyo: Corona Publishing Company, Ltd., 1969

6. Rink R. et al. The Sonochemical Reactors with Symmetric oscillatory Systems of the Acoustic Cells // International Journal of Research and Reviews in Applied Sciences, V12, 3, pp. 391-396, 2012