ISSN 1068�798X, Russian Engineering Research, 2010, Vol. 30, No. 5, pp. 530–532. © Allerton Press, Inc., 2010.Original Russian Text © A.V. Lobusov, 2010, published in STIN, 2010, No. 2, pp. 36–38.

530

The machining of metal is accompanied by plasticdeformation, which may be enhanced by vibration. Itis important here to determine the eigenfrequencies ofthe microstructure in plastically deformed metal.When one eigenfrequency of the microstructure isequal to the frequency of tool vibration, resonance willpermit more effective oscillation of the metal micro�structure and reduce the cutting force.

The fundamental (lowest) frequency of the intrin�sic oscillations is of particular interest. In the presentwork, we find this frequency for the early stages ofplastic deformation. In particular, we consider colddeformation. We first turn our attention to a singlecrystal.

Numerous experiments have established that plasticdeformation in a crystal with a cubic or hexagonal lat�tice is mainly due to slip. Slip is localized in slip bands,which consist of a series of atomic planes affected bydeformation. The distance between the slip bands isabout 1 μm [1]. In the deformation of a single crystalwith an fcc lattice, the distance between the slip bandsin the easy�slip stage is a few tens of nm (for example,40–65 nm, for copper), according to [2]. With furtherdeformation, this distance increases to 1 μm.

It is significant that, in the early stages, the metalbetween the bands is not involved in plastic deforma�tion; in other words, it is elastic. Later, this metalbetween the slip bands forms a layer of thickness h,which is the same for all the layers.

At the beginning of plastic flow, the elasticallydeformed metal layers between the slip bands moverelative to one another. In the dynamic model, theinertial properties of a layer (mass mi) are representedby an elastic body (mass mi); the elastic properties arerepresented by two elastic elements, each of which isequivalent to a half�layer (thickness h/2). The elasticelements of layer i are connected to the elastic ele�ments of adjacent layers at points Oi and Oi – 1 (which,when the metal passes to the plastic state, correspondto the adjacent slip bands): the dynamic model of themicrostructure for a single crystal, represented as amultimass chain system, is shown in Fig. 1. We assumethat, in the early stages of plastic deformation, the lay�ers move linearly along the axes Xi. We also replace theelastic elements of the adjacent layers by the equiva�lent elastic coupling. Then the calculation scheme for

determining the eigenfrequency of the single crystal’smicrostructure on transition to the plastic state willtake the form in Fig. 2. This system has n degrees offreedom, where n is the number of metal layersbetween the slip bands. Taking account of the grainsize of most iron alloys, we conclude that n = 5–250.

Assuming that all the metal layers between the slipbands are of the same size, we write a formula for themass of layer i

(1)

where i = ρ is the metal density; S is the area ofthe layer.

The rigidity coefficients c0, c1, c2, …, cn of the elas�tic couplings in Fig. 2 will also be equal. The rigiditycoefficient of an elastic coupling is the sum of therigidity coefficients of adjacent half�layers connectedat point Oi (Fig. 1).

According to Hooke’s law, the tangential stress τrequired to create shear strain x/d in a layer of thick�ness d is τ = Gx/d, where G is the shear modulus. In the

mi ρSh,=

1 n, ;

Determining Resonant Frequencies for Vibrational Metal CuttingA. V. Lobusov

Krasnodar State Technical University, Krasnodar

DOI: 10.3103/S1068798X10050230

i + 1 mi + 1

mi

Oi – 1

mi – 1

Oi

i

i – 1

h

h

h

Fig. 1. Dynamic model of the microstructure of a metalsingle crystal: i – 1, i, i + 1 are the numbers of adjacentmetal layers; mi – 1, mi, mi + 1 are the masses of the corre�sponding layers; Oi – 1 and Oi are the points where the elas�tic elements are connected to the adjacent metal layers.

RUSSIAN ENGINEERING RESEARCH Vol. 30 No. 5 2010

DETERMINING RESONANT FREQUENCIES FOR VIBRATIONAL METAL CUTTING 531

present case, d = h/2 and x = xi. The force F creatingstress τ is

(2)

The factor preceding xi in Eq. (2) is the rigiditycoefficient of the half�layer (thickness h/2). The totalrigidity coefficient of two identical elements in serieswill be half as large. Therefore

(3)

Determining mi and ci in Fig. 2, we may calculatethe fundamental oscillation frequency of the mathe�matical system approximately on the basis of Dunker�ley’s method [3]. In this method, the square of thefundamental frequency is found from the formula

(4)

where pi is the partial oscillation frequency of mass mi,under the condition that all the other masses are zero.

The elastic couplings below masses mi (Fig. 2) arein series. Therefore, the total rigidity coefficient cB iscB = ci/i.

Analogously, for elastic couplings above the massesmi, the total rigidity coefficient is cA = ci(n + 1 – i).

Since a parallel coupling is formed when the upperand lower branches interact with mass mi, the total rigid�ity coefficient is cTi = cB + cA = ci(n + 1)/[i(n + 1 – i)].

Taking account of Eqs. (1) and (3), the partial fre�quency is

Then, from Eq. (4), we find the fundamental fre�quency p of the mechanical structure corresponding tothe single crystal’s microstructure in the early stages ofplastic deformation

(5)

For steel, we obtain the following results fromEq. (5). When n = 5, the fundamental frequency inplastic deformation is p = 212 MHz; when n = 40, p =30.7 MHz; when n = 250, p 4.94 MHz. (Note that theDunkerley’s formula underestimates the fundamentalfrequency [3].) These frequency are very high formechanical oscillations and cannot be obtained inpractice using any exciters. However, for polycrystal�line bodies, the number of degrees of freedom of theplastically deformed microstructure increases in pro�portion to the number of crystals (grains) in the body,since intracrystalline deformation predominates incold plastic deformation [1]. Therefore, the number ofterms in Dunkerley’s formula increases for a polycrys�

F 2Sch

�������xi.=

ci SG/h.=

1/p2 1/pi2,

i 1=

n

∑=

pi2 cTi/mi G n 1+( )/ iph2 n 1 i–+( )[ ].= =

p G n 1+( )

ph2 i n 1 i–+( )

i 1=

n

∑

����������������������������������� .=

talline body. This decreases the fundamental fre�quency determined from Dunkerley’s formula.

For the metal layers between the slip bands, theirthickness is small relative to their other two dimen�sions. Therefore, the interaction area of a single layerwith the boundary of the adjacent grain is considerablyless than the interaction area with adjacent layerswithin the grain. As a result, the forces from the adja�cent grains preventing motion of the layer along theaxes Xi are much less than shear forces from the adja�cent layers within the grain Therefore, the adjacentgrains have little influence on the eigenfrequencies ofthe layer. Then the fundamental frequency Pp of layersin a polycrystalline body is

(6)

where N is the number of grains in the polycrystallinebody.

Using Eqs. (5) and (6), we may approximately cal�culate the fundamental frequency of the microstruc�ture in a plastically deformed metal, if we know thevolume of the metal and its grain size (so as to deter�mine the number of grains N).

The problem in determining the resonant fre�quency of vibrational cutting is that it is practically

Pp p/ N,=

Xn – 1

Xn

Xi

X1

X2

cn – 1

ci – 1

cn

ci

c1

c0

mn – 1

mn

mi

m1

m2

Fig. 2. Determining the fundamental vibration frequencyof the single crystal’s microstructure: X1, X2, Xi, Xn – 1, Xn,coordinate axes; m1, m2, mi, mn – 1, mn, masses of the cor�responding metal layers; c0, c1, ci – 1, ci, cn – 1, cn, rigiditiesof the elastic couplings.

532

RUSSIAN ENGINEERING RESEARCH Vol. 30 No. 5 2010

LOBUSOV

impossible to identify precisely the boundaries of theplastically deformed metal in the chip�formationzone. This metal is in front of the tool’s front surface,in the primary�deformation zone [4]. It takes the formof a wedge with its vertex at the cutting blade. Its upperboundary is shorter than its lower boundary by a factorof 2–4. A primary�deformation zone of this shape istypical for low cutting speeds, large cut�layer thick�nesses, and small front angles of the tool.

For the cutting speeds, cut�layer thicknesses, andfront angles of the tool that predominant in practice,the upper and lower boundaries are shifted and equal�ized in length; they approach a line inclined to the cut�ting surface at the shear angle [4]. Therefore, we mayassume that the shear deformation is localized in a thinlayer, whose thickness is a hundredth or sometimes atenth of a mm. Thus, according to our assumptions,the volume of plastically deformed metal in the chip�formation zone varies widely, depending on the cut�ting parameters.

For example, in the machining of steel with a cut�layer thickness of 1 mm and a width of 4 mm, this vol�ume may vary from 2.4 × 10–8 m3 to 8 × 10–10 m3. Cal�culations by Eqs. (5) and (6) give the followingresults. The fundamental frequency for a volume of2.4 × 10–8 m3 is 15.330 kHz with a grain size of 5 μm,50.159 kHz with a grain size of 40 μm, and126.274 kHz with a grain size of 250 μm. The funda�mental frequency for a volume of 8 × 10–10 m3 is

83.961 kHz with a grain size of 5 μm, 275.200 kHzwith a grain size of 40 μm, and 689.904 kHz with agrain size of 250 μm. Hence, vibrational cutting atthe resonant frequencies of the metal microstructureis only possible in rare cases, since the highest�fre�quency magnetostrictional exciters do not operateefficiently at frequencies above 50 kHz.

To reduce the fundamental frequency of the micro�structure for metal plastically deformed in the cuttingzone, we must adopt cutting conditions that ensure thelargest possible primary�deformation zone. Reducingthe grain size of the machined metal also reduces thefundamental frequency of its microstructure in plasticdeformation.

REFERENCES

1. Gromov, N.P., Teoriya obrabotki metallov davleniem(Theory of the Pressure Treatment of Metal), Moscow:Metallurgiya, 1978.

2. Podukin, P.I., Gorelik, S.S., and Vorontsov, V.K.,Fizicheskie osnovy plasticheskoi deformatsii (PhysicalPrinciples of Plastic Deformation), Moscow: Metal�lurgiya, 1982.

3. Biderman, V.L., Teoriya mekhanicheskikh kolebanii(Theory of Mechanical Vibrations), Moscow: VysshayaShkola, 1980.

4. Bobrov, V.F., Osnovy teorii rezaniya metallov (Funda�mentals of Metal�Cutting Theory), Moscow: Mashi�nostroenie, 1975.