Tipos de movimientos y sus bases neurológicas

Dr. Merchant was the first scientist who proposed a model for the cutting of metals. Merchant’s model was based on the principle of minimum energy. Figure 5.50 shows a simple scheme of the metal cutting process in which a wedge-shaped tool removes material from the workpiece. The

removed material appears in the form of chips. According to Merchant, following the natural law of minimum energy, the shear plane angle, f, assumes a value, depending on the rake angle a of the tool, friction angle b at the chip-tool interface, and a constant C that expresses the machinability of the workpiece material. The relationship developed by Dr. Merchant is expressed as

(2f + b – a) = C

Fig. 5.50 Merchants circle diagram.

Derivation of shear angle relation (2f + b – a) = C: The chip is treated as a free body held in equilibrium due to equal and opposite forces acting on the chip (see Fig. 5.50).

The first force is the force R exerted by the tool on the chip. This force is the resultant of the chip-tool interface friction, F and the normal force N at the chip-tool interface.

The second force is the force R¢ exerted by the workpiece on the chip, and the shear plane normal force, F_n. Merchant represented the vector diagram of forces on circle diagram as shown. Now, from

the principle of minimum energy, the energy required in cutting depends only on the cutting force F_c (cutting force is not affected much by the cutting speed), for which the minimum is to be sought by setting (dF_c/df)) = 0.

From the geometry of the circle diagram.

F_c = R cos (b – a) (5.9) F_s = R cos (f + b – a) (5.10) Therefore,

FsAcos(

F ssCB ¹ CB)(5.11)c cos(GC B) cos(G C B)

Now, the shear stress S on the shear plane depends on the normal stress s, according to the relation S = (S₀ + Ks)

Also, since F_s = F_n cot(f + b – a) we have S = s cot(f + b – a) (from Merchant’s circle). Therefore, S₀ = S – K × S tan(f + b – a) = S[1 – K tan(f + b – a)] (5.12)

Substituting the value of S from Eq. (5.12) in Eq. (5.11), we have

SA

F 0 s cos(CB) (5.13)c _{[1 Ktan(})] cos( _{)GC B G C B} Further, substituting A_S = (A/sin f), we get

SA _cos(CB) F 0s

c _{[sin cos( )][1 tan(GGCB GCB)]}

NumeratorFc _{[sin cos( )][1 tan(GGCB GCB)]} Differentiating F_c w.r.t. f, we have

cot (2f + b – a) = K Setting K = cot C, 2f + b – a = C (5.14)

The interpretation of C is the slope of S-s curve. The value of C can be estimated by material testing. Its value usually lies between 70 and 80° for metals and 90° for plastics. Table gives the C values of some important metals.

(i) Utility of Merchant’s analysis. The relationship

sA cos(CB)Fc sin cos(GGCB) is of great importance. It can be used for estimating the cutting forces in metal cutting. Rearranging, we have

scos(CB) _cFA_{sin cos(GGCB)}Ap[]

(5.15)

where p is called specific cutting pressure. Its value depends on yield shear stress s of the workpiece material and thickness of cut. Practically, above a thickness of cut of 0.15 mm, p is constant. To be able to estimate the cutting force accurately, the Merchant’s constant C should also be known (Table 5.3). Substituting (b – a) from Merchant’s relation, we get

2f + b – a = C

or (b – a) = (C – 2f) Thus,

F G

cos(C 2 ) (5.16)c = As _{sin cos(CGG2 )}

where A denotes uncut area. The yield shear stress of the workpiece material can be estimated from indentation test and calculated as follows: Brinell Hardness Number (BHN) » (6s) where s is the yield stress of the material in shear.

For good cutting conditions yielding continuous chips, polished tool surface and efficient cutting fluid, shear plane angle f is of the order of magnitude » 20°.

Table 5.3 Values of Merchant’s constant C Material C (degrees)

AISI 1010 69.8 AISI 1020 69.6 AISI 1045 78.0 AISI 2340 76.2 AISI 3140 70.6 AISI 4340 74.5

Stainless 303 92.0 Stainless 304 82.0

For cylindrical turning, A = (f × d), where f is the feed rate in mm/revolution and d is the depth of cut. In drilling operation, A = (f/2) (D – C)/2, where D is the drill diameter and C the chisel edge length. The thrust component F_t in metal cutting can be estimated from:

F_t = F_c tan (b – a)

F_t = F_c tan (C – 2f)

Figure 5.40 indicates force system of drilling and turning showing F_i and F_t and other force components in the machine reference system.

(ii) Drilling torque and thrust. In drilling, the torque can be calculated from mechanics of cutting. As can be seen from Fig. 5.51,

Torque = F_c × (D + C)/2 = p × f/2[(D – C)/2] [(D + C)/2 (5.17) = pf(D2 – C2)/8 Thrust due to cutting = 2F_t sin p (5.18)

Fig. 5.51 Mechanics of drilling process.

where p is the semi-point angle of the drill. In the drilling process, the chisel edge of a drill acts as an indentor rather than as a cutting tool. It functions as a wedge causing semi-hot extrusion. It has been shown by Bhattacharya [4] that the chisel edge contributes 50% to the total drill thrust. Hence, total drill thrust (due to cutting action of the drill tips + thrust due to indenting action of the chisel edge) is 2[2F_t sin p] = 4F_t sin p (5.19) (iii) Analysis of turning operation. The various forces in turning are illustrated in Fig. 5.52.

Fig. 5.52 Mechanics of turning.

Tangential cutting force = F_c = p × A = p.f.d. (5.20) Thrust due to cutting = F_t = (p.f.d.) tan (b – a) (5.21)

The thrust force can be resolved parallel to the axis of workpiece, i.e. in the direction of feeding. The feed force F_f, is given by the relation

F_f = F_t sin (90 – C) = F_t cos C (5.22) where C is the side cutting edge angle of the tool. The radial force F_r can be expressed as F_r = F_t sin C (5.23)

Cutting tools with special geometry having a high value of C are used for bar peeling operation, in which a hard skin on the surface of bar stock is to be removed at a high feed rate. On the other hand, slender workpieces are machined with cutting tools with low values of side cutting edge angle C, so that small radial forces are created. This helps maintain accuracy of the workpiece due to reduced deflection of the workpiece as a beam subjected to radial loading, owing to reduced value of F_r.

(iv) System approach to metal removal processes. Some authors like Alting, Leo [6] prefer to employ an approach different from shear plane analysis, to estimate power and forces in cutting. The following elementary relationships should be noted:

1. Energy/min consumed in cutting = F_c V N m/min

2. Power consumed in watts = F_cV/60 Nm/s

3. Metal removal rate, W = (fdV × 103)/60 mm3/s

The power required to remove a unit volume of metal per minute is of high utility to machine tool engineers.

p = specific cutting pressure = F_c/d_f N/mm2

U_c = power/W = (F_cV/60) (60/fdV × 103) = F_c/(d_f × 103) = p/103 An illustrative example on system approach to metal cutting is now taken up to make the notion clearer.

Example. A lathe running idle consumes 325 W. When cutting an alloy steel at 24.5 m/min, the power input rises to 2580 W. Find the cutting force and torque at the spindle when running at 124 rpm. If the depth of cut is 3.8 mm and feed = 0.2 mm, find the power criterion.

Solution. The power consumed in cutting = 2580 – 325 = 2255 W

W = 0.2 × 3.8 × 24.5 × 103/60 = 310 mm3/s Power criterion = 2255/310 = 7.27 W/mm3/s

Torque at spindle = 2255 × 60/(2p × 124) = 174 Nm Cutting force, F_c = 2255 × 60/24.5 = 5522 N

The merit of the system approach is that it can be modified to multitooth operations of forming as well as generating, such as milling, gear generation, e.g. hobbing and shaping. The reader is referred to authentic works on the subject of metal cutting by, among others Alting, Leo [6], Bhattacharya [4], and Ghosh and Mallik [5] have expressed power criterion as dependent on undeformed chip thickness. That is, c FUUf₀ 0.4 ​​ÍÝ Ff 0.4 U c 0 _1000​​df FcU0 _1000​​df0.6

Rearranging, we obtain F_c = (1000 × d × f 0.6 U₀) newtons The values of U₀ are as given in Table