P ROTEÓLISIS LIMITADA EN CÉLULAS PERMEABILIZADAS

Consider a work gear to be stationary. A Cartesian coordinate system XgYgZg is associated

with the work gear as shown in Figure 8.1. If a form gear cutting tool is considered a rigid body, then it can perform a variety of single parametric motions relative to the reference system XgYgZg. Generally speaking, all feasible single parametric relative motions of the

form cutting tool relative to the reference system XgYgZg are either a type of a screw motion

or a degenerated kind of screw motion, namely, either a translation or a rotation. Under such a scenario, the secondary generating surface T2. of the form tool makes line contact

with the work gear tooth flank surface G.

A Cartesian coordinate system Xc2.Yc2.Zc2. is the cutting tool reference system. After it has

been designed, the form gear cutting tool will be represented in this reference system. In the most general case, location and orientation of the coordinate system Xc2.Yc2.Zc2. with

respect to the coordinate system XgYgZg can be specified by the following parameters.

Coordinates of the origin of the coordinate system Xc2.Yc2.Zc2. are completely specified by

three vectors: agx, agy, and agz. These vectors are along the corresponding coordinate axes Xg, Yg, and Zg. Therefore, position vector ac2. of the origin of the cutting tool coordinate

system Xc2.Yc2.Zc2. can be expressed in terms of vectors agx, agy, and agz of the elementary

linear displacements

In reality, every elementary linear displacement can be performed either along a coor- dinate axis or along two or even three coordinate axes simultaneously. Therefore, the total number na of the feasible displacements is limited to the following seven linear displace-

ments: agx, agy, agz, (agx+ agy), (agy+ agz), (agx+ agz), and (agx+ agy + agz).

Orientation of the coordinate system Xc2.Yc2.Zc2. with respect to the work gear reference

system XgYgZg is completely specified by three rotation vectors, φgx, φgy, and φgz, which

are a type of angular displacements. These vectors are about the corresponding coordi- nate axes Xg, Yg, and Zg. Therefore, the resultant orientation vector φc2. of the cutting tool

coordinate system Xc2.Yc2.Zc2. can be expressed in terms of vectors φgx, φgy, and φgz of the

elementary angular displacements

ϕϕc2. =ϕϕgx+ϕϕgy+ϕϕgz (8.2.)

Each elementary angular displacement can be performed either about a coordinate axis or about two or even three coordinate axes simultaneously. Therefore, total number nφ of

the feasible displacements of this type is limited to just seven angular displacements: φgx,

φgy, φgz, (φgx+ φgy), (φgy+ φgz), (φgx+ φgz), and (φgx+ φgy + φgz).

The total number of possible displacements naφ of the coordinate system Xc2.Yc2.Zc2. with

respect to the coordinate system XgYgZg is equal to

naϕ =(na+1) (nϕ+ − =1 1 63) (8.3)

The cutting tool is allowed to travel along any of the coordinate axes, Xc2., Yc2., or Zc2..

Velocities of these translational motions are denoted by Vcx, Vcy, and Vcz, respectively. Any

two or even all three translations can be performed simultaneously. Without going into

g Z g X g Y g O gx Xc2 2 c Y 2 c Z cx V cy V cz V cz ω cy ω cx ω gy gz gz gx a gy a a FIGURE 8.1

An arbitrary configuration of the reference system Xc2.Yc2.Zc2. associated with the form gear cutting tool with

Nontraditional Methods of Gear Machining with Form Cutting Tools 165

details of the analysis, it is easy to see that in addition to the elementary translations, Vcx,

Vcy, and Vcz, three pairs of translations—(Vcx, Vcy), (Vcy, Vcz) and (Vcz, Vcx), and one triple

translation (Vcx, Vcy, Vcz) are feasible. Therefore, a total of seven combinations of elementary

translations are possible. All possible combinations of translations Vcx, Vcy, and Vcz cause

the resultant motion, which is also a kind of translational motion

VcΣ =Vcx+Vcy+Vcz (8.4)

Ultimately, the total number nv of possible translations of the cutting tool coordinate sys-

tem Xc2.Yc2.Zc2. relative to the coordinate system XgYgZg associated with the gear is limited

to nv = 7 translational motions.

The cutting tool can rotate about any of the coordinate axes, Xc2., Yc2., or Zc2.. Angular

velocities of these rotations are respectively denoted by ωcx, ωcy, and ωcz. Any two or three

rotations can be performed simultaneously. Without going into details of the analysis, it is easy to see that in addition to the elementary rotations, ωcx, ωcy, and ωcz, three pairs of rota-

tions—(ωcx, ωcy), (ωcy, ωcz) and (ωcz, ωcx)—and one triple rotation (ωcx, ωcy, ωcz) are feasible.

Therefore, seven combinations of elementary rotations in total are possible. All possible combinations of the rotations ωcx, ωcy, ωcz cause the resultant rotation ωcΣ. The rotation ωcΣ

can be expressed in terms of elementary rotations as

ωωcΣ =ωωcx+ωωcy+ωωcz (8.5)

Similarly, the total number n_ω of possible rotations of the cutting tool coordinate system

Xc2.Yc2.Zc2. relative to the coordinate system XgYgZg associated with the gear is limited to nω =

7 rotational motions.

Translations Vcx, Vcy, and Vcz and rotations ωcx, ωcy, and ωcz represent the components of

single parametric motions.

The translations and rotations can be performed simultaneously—that is, they can be combined with one another. The total number of possible motions n_vω of the coordinate system Xc2.Yc2.Zc2. with respect to the coordinate system XgYgZg is equal to

nvωω =(nv+1) (nωω+ − =1 1 63) (8.6)

Neither the linear nor angular displacements of the cutting tool reference system with respect to the work gear reference system are strictly required. They can be observed or they can be of zero value. It is the relative motion of the coordinate system Xc2.Yc2.Zc2. with

respect to the coordinate system XgYgZg that is essential. At least one of the possible rela-

tive motions must be observed. This means that the total number ncg of all possible combi-

nations of displacements and relative motions is limited to

ncg =(naϕ+1)nvωω=4032. (8.7)

Design of a cutting tool for machining of a given gear cannot be performed in compli- ance with all possible combinations [see Equation (8.7)], because many of the combinations are not physically feasible.

In the most general case, the resultant translation VcΣ and resultant rotation ωcΣ can

be performed simultaneously. In this event, the motions VcΣ and ωcΣ are timed with one

of the resultant screw motion can be expressed in terms of the parameters of the elemen- tary translations and elementary rotations.

8.2 Implementation of the Single Parametric Motions