Kinematics of Surface Generation 332.1.2 Elementary Relative Motions Instant relative motion of the cutting tool can be interpreted as an instant screw motion.. It is eliminated from th
Trang 1Kinematics of Surface Generation 31
unit vectors t1.P,t2.P, and nP Thus, the moving coordinate system x P y P z P is
established at every point of the sculptured surface P.
Most of the equations are significantly simplified when Darboux’s
trihe-dron is used However, computation of the unit tangent vectors of the
princi-pal directions is often a computation-consuming procedure
Second, in order to compose a right-hand Cartesian local coordinate system
x P y P z P with the origin at point K of contact of the surfaces P and T, the unit
tan-gent vectors uP,vP, and nP can be employed Generally speaking, pairs of the
vectors uP and nP, vP and nP are orthogonal to each other, while the unit tangent
vectors uP andvP are not orthogonal to each other Aiming the composing of the
right-hand Cartesian coordinate system, the third unit vector vP* =uP×nP can
be used The three unit vectors uP, vP*, and nP make up a right-hand trihedron
The orthogonal trihedron uP, vP*, and nP can be constructed at any surface
point However, in order to construct the trihedron, computation of the
vec-tor v*P is necessary at every surface point
Third, the initial parameterization of the surface P can be changed when
it becomes an orthogonal parameterization Under such a scenario, the unit
tangent vectors uP and vP are always orthogonal to each other
In order to change the initial surface P parameterization, it is necessary to
replace the parameters U P and V P with new parameters U P* =U U V P*( P, P) and
P P
P P
P P
U
V U
P P
P P
P P
U
V V
Y U
Z U X
V
Y V
Z V
P P
P P
P P P
P
P P
P P
P P
U V V
U
V V
P P
P P P
P
P P
Trang 232 Kinematic Geometry of Surface Machining
The new fundamental matrix is given by
It can be shown on the premises of Equation (2.1), Equation (2.2), and
Equation (2.6) that the unit surface normal nP is invariant under the
transfor-mation, as it could be expected
The transformation of the second fundamental matrix can similarly be
by differentiating Equation (2.1) and Equation (2.2) and using the invariance of
nP It can be shown from Equation (2.6) and Equation (2.7) that the principal
cur-vatures and the principal directions are invariant under the transformation
Equation (2.6) and Equation (2.7) yield the natural form of the surface P
representation with the new U P* and V P* parameters
It can be concluded that the unit normal vector nP and the principal
direc-tions and curvatures are independent of the parameters used and are
there-fore geometric properties of the surface They should be continuous if the
surface is to be tangent and curvature continuous
At that point it is necessary to establish a set of constraints onto the
rela-tionships U P* =U U V P*( P, P) and V P*=V U V P*( P, P) under which an orthogonal
parameterization of the sculptured surface P can be obtained In order to
obtain an orthogonal parameterization of the sculptured surface P, the
rela-tions U P* =U U V P*( P, P) and V P*=V U V P*( P, P) have to satisfy the following two
conditions, F P≡ 0 and M P≡ 0, which are the necessary and sufficient
condi-tions for the orthogonally parameterized sculptured surface P.
Once reparameterized, the surface P yields easy computation of the
orthog-onal trihedron uP,vP, nP at every surface point
The presented consideration clearly illustrates the feasibility of composing
the right-hand trihedron uP,vP, nP using one of the methods discussed above
The three unit vectors uP,vP, nP can be used as the direct vectors of the local
Cartesian coordinate system x P y P z P with the origin at point K of contact of the
sculptured surface P and the generating surface T of the cutting tool.
Trang 3Kinematics of Surface Generation 33
2.1.2 Elementary Relative Motions
Instant relative motion of the cutting tool can be interpreted as an instant screw
motion Depending on the actual configuration of the local reference system
x P y P z P, the instant screw motion of the cutting tool can be decomposed on not
more than six elementary motions — that is, on three translations along and
onto three rotations about axes of the local coordinate system x P y P z P Not all
of the six elementary relative motions are feasible The translational motion of
the cutting tool along z P is not feasible It is eliminated from the instant
kine-matics of sculptured surface generation because of two reasons.*
First, the elementary motion of the cutting tool in the +nP direction results in
interruption of the surface-generating process, which is not allowed Second,
the motion of the cutting tool in the −nP direction results in unavoidable
interference of the surfaces P and T Any interference of the surfaces P and
T is not allowed Hence, the speed of the translational motion of the cutting
tool along the common perpendicular must be equal to zero (Figure 2.1):
V z≡V n= ∂ k
z t
P
Here time is designated as t.
The speed of translational motion of the cutting tool along x P and y P axes
is designated as V x and V y, respectively Then x, y, and z designate
rota-tions about axes of the local coordinate system x P y P z P
According to the principal instant kinematics of sculptured surface
genera-tion (Figure2.1), the cutting tool instant screw motion relative to the surface P
can be decomposed on not more than five elementary instant relative motions
This set of five feasible elementary relative motions includes two translations
about axes of the local reference system x P y P z P Here the angles of rotation of
the cutting tool about axes of coordinate system x P y P z P are designated as f x,
f y, and f z, respectively
Trang 434 Kinematic Geometry of Surface Machining
2.2 Generating Motions of the Cutting Tool
After being machined on a multi-axis NC machine, the sculptured surface is
represented as a series of tool-paths Generation of a sculptured surface by
consequent tool-paths is a principal feature of sculptured surface machining
on a multi-axis NC machine The motion of the cutting tool along a tool-path
can be considered as a permanent following motion, and a side-step motion
of the cutting tool (in the direction that is orthogonal to the tool-path) can be
considered as a discrete following motion
The surface P can be generated as an enveloping surface to consecutive
posi-tions of the moving surface T when the surfaces P and T make either linear
or point contact When linear contact of the surface is observed, then a
one-degree-of-freedom relative motion of the cutting tool is sufficient for
generat-ing the surface P Such relative motion is referred to as one-parametric motion of
the cutting tool When the surfaces make a point contact, then a
two-degrees-of-freedom relative motion of the cutting tool is required for generating the
entire surface P Such relative motion is referred to as two-parametric motion
of the cutting tool The number of available degrees of freedom can exceed
2 degrees This results in multiparametric motion of the cutting tool.
Known methods of developing machining operations do not provide a
single solution to the problem of synthesis of the most efficient (i.e.,
opti-mal) machining operations Known methods return a variety of solutions to
the problem, of which the efficiency of each is not the highest possible For
the computing of parameters of relative motion, known methods are based
mostly on the equation of contact*: nP ∙ VΣ = 0 Here VΣ denotes the speed of
the resultant motion of the cutting tool relative to the work
The equation of contact nP ∙ VΣ = 0 imposes restrictions onto only one
component of the relative motion of the cutting tool and of the work This
means that projections of speed of the resultant motion onto the direction
specified by the unit normal vector nP must be equal to zero Prn VΣ = 0 ≡ 0 No
restrictions are imposed by the equation of contact onto other projections of
the vector of resultant motion of the cutting tool and of the work In
compli-ance with the equation of contact, the magnitude and direction of VΣ within
the common tangent plane can be arbitrary
Evidently, the infinite number of the vectors VΣ satisfies the equation of
contact nP ∙ VΣ = 0 All are within the tangent plane to the sculptured surface
P This is the principal reason why the implementation of known methods
returns an infinite number of solutions to the problem under consideration
No doubt, performance of a sculptured surface generation depends on
direc-tion of the vector VΣ For a certain direction of VΣ, it is better; for another
1940s/early 1950s [38].
Trang 5Kinematics of Surface Generation 35
direction of VΣ, it is poor This yields intermediate conclusions that the
opti-mal direction of VΣ exists, that this direction satisfies the equation of contact
nP ∙ VΣ = 0, and that the optimal parameters of the direction of VΣ can be
computed
For the computation of the optimal parameters of the instant relative
motion VΣ of the work and of the cutting tool, an appropriate criterion of
optimization is necessary The major purpose of the criterion of optimization
is to select the optimal direction of the vector VΣ from the infinite number of
feasible directions that satisfy the equation of contact nP ∙ VΣ = 0
In order to satisfy the equation of contact, vector VΣ of the resultant relative
motion of the cutting tool must be within the tangent plane to the surfaces
P and T at the CC-point K This is the geometrical interpretation of the
equa-tion of contact
Consider a local reference system x P y P z P with the origin at CC-point K
(Figure2.2) In the coordinates system x P y P z P, vector VΣ can be described
analytically by vector equation:
y t
z t
P
In that same local coordinate system, the equality nP = kP is observed
Substituting Equation (2.11) and the relationship nP= kP along with
Equa-tion (2.8) into the equaEqua-tion of contact nP ∙ VΣ = 0, one can obtain
Trang 636 Kinematic Geometry of Surface Machining
In order to satisfy the equation of contact, projection of the vector VΣ of the
resultant motion on the direction perpendicular to the surfaces P and T must
equal zero This is proof that the vector VΣ must be within the common
tan-gent plane to the surfaces P and T.
It is important to point out here that the condition prn VΣ < 0 can be
consid-ered as the condition of roughing Portions of the surface T that perform such
motion remove stock while machining the work Condition prn VΣ = 0 that is
equivalent to the condition of contact nP ∙ VΣ = 0 corresponds to generating
the surface being machined Finally, the condition prn VΣ > 0 relates to
por-tions of the surface T that are departing from the machined surface P These
conditions are presented in more detail in Chapter 5
The equation of contact nP ∙ VΣ = 0 does not uniquely determine the instant
kinematics of sculptured surface generation In addition to the infinite
num-ber of feasible directions for the vector VΣ, one more reason can affect the
indefiniteness of the equation of contact
Location and orientation of the common perpendicular nP are uniquely
specified by the geometry of the surface P Usually it cannot be changed
However, in special cases of machining, the orientation of the common
per-pendicular nP can be changed for manufacturing purposes For example,
when machining a thin-wall part (Figure2.3), an elastic deformation can be
applied to the work Under the applied load, unit normal vectors nm a, nm b, and
nm c to the part surface P become parallel to each other.
In the deformed stage, the work is machining on a lathe with simple motion
of the cutter relative to the work The elastic deformation results in the plane
machining instead of machining of the concave sculptured surface (for this
purpose, the magnitude of the applied load may vary according to the cutter
feed rate — a distributed load of variable magnitude can be applied as well)
After being released, the work gets it original shape, and the machined plane
a
n
c P
n
b
n
c m
n
b m
n
a m
n
Feed
Load
P Cutter
a b c
a b c
FiguRE 2.3
An example of the application of elastic deformation of the work for the purpose of machining
the surface.
Trang 7Kinematics of Surface Generation 37
transforms into a concave sculptured surface P Unit normal vectors n P a, nP b,
and nP c to the machined surface P are not parallel to each other.
If elastic deformation is used for manufacturing purposes, then the
equa-tion of contact must be satisfied in the deformed stage of the surface being
machined Elastic deformation of a work for manufacturing purposes is
observed when machining flex-spline that is an essential machine element
of a harmonic drive, and in other applications
The capability to change the orientation of the unit normal vector to the
sur-face is limited; however, such a capability exists, and it affects the generation of
the surface P The last is of principal importance.
Capabilities of variation of orientation of the unit normal vector nT to the
generating surface T of a cutting tool are significantly wider, especially when
implementing cutting tools of special design with variable shape and
param-eters of the surface T for machining a given sculptured surface [28,29,33].
When machining a sculptured surface on a multi-axis NC machine, the
cutting tool is performing a continuous follow motion along every tool-path
[30,33] Therefore, the generating motion can be considered as a continuous
follow motion of the cutting tool relative to the work This motion results in
the CC-point traveling along the tool-path
After the machining of a certain tool-path is complete, then the cutting tool
feeds across the path in a new position The machining of another
tool-path begins from the new position of the cutting tool Hence, the feed motion
of the cutting tool can be represented as a discontinuous follow motion of the
cutting tool relative to the work This motion results in the CC-point
travel-ing across the tool-path
The generating motion of the cutting tool can be described analytically
For this purpose, the elementary motions that make up the principal instant
kinematics of surface generation are used (Figure 2.1) The elementary relative
motions are properly timed (synchronized) with one another in order to
pro-duce the desired instant generation motion of the cutting tool (Figure 2.4)
The following equations can easily be composed based on the premises of
the analysis of the instant kinematics of sculptured surface generation:
| | | |V x=ωωy⋅R P.x and | | | |V y= ωx⋅R P.y (2.13)
Here R P.x and R P.y designate the normal radii of curvature of the sculptured
surface P The radii of curvature R P.x and R P.y are measured in the plane
sec-tions through the unit tangent vectors uP and vP, respectively Equation (2.13)
yields a generalization of the following kind:
| | |VΣ = ωT P- |⋅R P.Σ (2.14)
where VΣ is the vector of the resultant motion of the CC-point along the
tool-path; T −P is a vector of instant rotation of the surface T about an axis that
is perpendicular to the normal plane through the vector VΣ; and R P.Σ is the
radius of normal curvature of the surface P in the direction of VΣ
Trang 8Kinematics of Surface Generation 39
The relative motion V z of the cutting tool (Figure 2.1) is not completely
eliminated from further analysis Taking into consideration the tolerance on
accuracy of machining of the surface P (Figure 1.5), the motion V z along the
unit normal vector nP is feasible if it is performing within the tolerance d =
d++d- Moreover, due to deviations of the desired cutting tool motion from
the actual cutting tool motion, the motion V z always observed is the actual
machining operation If necessary, the motion V z can be incorporated into
the principal instant kinematics of surface generation
This is one more example of the difference between the classical
differen-tial geometry of surfaces and the kinematic geometry of surface generation
The principal instant kinematics of surface generation includes five
elemen-tary relative motions Thus, the surface P can be represented as an
envelop-ing surface to consecutive positions of not more than five-parametric motion
of the surface T of the cutting tool.
2.3 Motions of Orientation of the Cutting Tool
As mentioned above, the machining of a sculptured surface on a multi-axis
NC machine is the most general case of surface generation This is because
two surfaces, the sculptured surface P and the generating surface T of the
cutting tool make point contact at every instant of machining
Among various kinds of relative motions of the cutting tool, one more
motion can be distinguished When performing relative motion of this kind,
the CC-point does not change its position on the sculptured surface P being
machined This motion changes only the orientation of the cutting tool
rela-tive to the work Motions of this kind are referred to as orientational motions
of the cutting tool
When performing the orientational motion, the CC-point can remain in
location on both, on the surface P as well as on the generating surface T of
the cutting tool Orientational motion of this kind is referred to as the
orien-tational motion of the first kind
When machining a sculptured surface, the CC-point can remain in
loca-tion on the surface P and change its localoca-tion on the generating surface T of
the cutting tool Orientational motion of this kind is referred to as the
orien-tational motion of the second kind
Speed of the orientational motions of the cutting tool is a function of
varia-tion of the principal curvatures of the surface P at the current CC-point, and
of speed of the generating motion Orientational motions of the cutting tool
do not directly affect the stock removal capability of the cutting tool or the
generation of the surface P These motions change orientation of the cutting
tool relative to the work as well as the relative direction of the generating
motion of the cutting tool
In order to identify all feasible orientational motions of the cutting
tool, it is helpful to consider all feasible groups of relative motions of the
Trang 940 Kinematic Geometry of Surface Machining
cutting tool All groups of the relative motions are represented with the
singular relative motions and with the combined relative motions The
sin-gular relative motions are composed of one elementary relative motion of
the cutting tool The combined relative motions are composed of two or
more elementary relative motions of the cutting tool
There are only five groups of elementary relative motions of the cutting
tool The number of elementary relative motions at every group of relative
motions is equal to the number of combinations of five elementary motions
by i elementary motions Here i = 1, 2, …, 5 The total number N of relative
motions in the principal instant kinematics of surface generation can be
com-puted from the following equation:
i
i i
Analysis of all 31 relative motions reveals that only a few elementary
rela-tive motions and their combinations can be distinguished as the
orienta-tional motions of the cutting tool They are as follows [30, 31, 33]:
The first group of the motions: {n}
The second group of the motions: {u , V v}, {v , V u}
The third group of the motions: {u, n , V v}, {v, n , V u}
The fourth group of the motions: {u , V v, v , V u}
The fifth group of the motions: {u , V v, n, v , V u}
Ultimately, one can come up with a set of the orientational motions of the
cutting tool One is the singular orientational motion, and six others are the
combined orientational motions of the cutting tool
The orientational motions of the first kind are represented with the only
singular orientational motion {n} All other orientational motions are the
orientational motions of the second kind The orientational motion {u , V v,
n, v , V u} is the most general Other orientational motions can be
consid-ered as a particular motion of that one
Elementary motions that include a combined orientational motion of the
cutting tool are timed (synchronized) with one another The rotational
ele-mentary motion u about the x P axis is timed with the translational motion
V v along the y P axis Similarly, the rotational elementary motion v about
the y P axis is timed with the translational motion V u along the x P axis The
timing of the elementary motions results in the surface T sliding over the
sculptured surface P The timing of that kind of elementary motions can be
achieved when the following condition is satisfied
Orientational motion of the second kind can be considered as a superposition
of the instant translational motion with a certain instant speed VT −P, and of the
instant rotation T −P of the cutting tool (Figure 2.5) In order to be an orientational
Trang 1042 Kinematic Geometry of Surface Machining
Equation (2.21) yields a representation of the combined orientational
motion of the cutting tool Vorient in the form
ω
u v v u
u v u
On the premises of the performed analysis, the following classification of
the orientational motions of the cutting tool is developed (Figure 2.6)
When representing a sculptured surface P as an enveloping surface to
con-secutive positions of the five-parametric motion of the generating surface T
(see Section 2.2), the orientational motions of the cutting tool can be omitted
from consideration
Orientational motions of the cutting tool make the machining of a
sculp-tured surface more agile The orientational motions of the cutting tool could
increase the performance of the machining operation
The developed approach yields computation of the optimal parameters
of all the motions of the cutting tool relative to the work The solution to
the problem of synthesis of the optimal kinematics of generation of a
sculp-tured surface on a multi-axis NC machine can be drawn up from the analysis
of kinematics of multiparametric motion of the cutting tool relative to the
work
2.4 Relative Motions Causing Sliding of a Surface over Itself
Surfaces that allow sliding over themselves are convenient for many
applica-tions in mechanical and manufacturing engineering Surfaces of this kind
can be generated by corresponding motion of a curve of an appropriate
shape The necessary motion can be easily performed on a machine tool
Relative motions causing sliding of a surface over itself are investigated in
[30,32,35] and others Reshetov and Portman [37] introduced the so-called
hidden connections that must be performed on the machine tool of a motion
that results in sliding of the surface over itself For the analytical
descrip-tion of the hidden connecdescrip-tions, it is necessary to compute the derivatives of
rP with respect to the elementary motions Ωi (here i = 1, 2, …, n designates
an integer number) The hidden connections exist if and only if pairs of
colinear or triples of coplanar vectors are available among the derivatives
∂rP/∂Ωi of rP
Surfaces that allow sliding over themselves can be considered as the
sur-faces for which a resultant relative motion of a special kind is feasible
Rel-ative motion of this kind results in the enveloping surface to consecutive
positions of the moving surface P being congruent to the surface P The same
is true for the generating surface T of the cutting tool.
Trang 1144 Kinematic Geometry of Surface Machining
of revolution is considered as a parametric motion — that is, as a
two-degrees-of-freedom relative motion
A smooth, regular surface P or T can perform three elementary relative
motions that are not timed (synchronized) with one another Only two
three-parametric motions are feasible
The first three-parametric motion includes three independent rotations
about axes of a certain Cartesian coordinate system Three independent
rota-tions cause the sphere to slide over itself The second three-parametric motion
includes two translations and one rotation All of these elementary relative
motions are independent of each other Two translations and one rotation cause
the plane to slide over itself
Three rotations in the first example as well as two translations and one
rota-tion in the second example represent the three parametric morota-tions — that is,
the three-degrees-of-freedom relative motions
The performed analysis allows for another interpretation of surfaces that
allow sliding over themselves The surfaces of that kind can also be considered
as the surfaces that are invariant with respect to a certain group of elementary
Two Degrees of Freedom
Cylinder of Revolution
X
Y
General Cylinder
X
Y
Surface of Revolution
P
X
Y
X
Y
Z Plane
Three Nonsynchronized Elementary Motions
Three Degrees of Freedom
Trang 12Kinematics of Surface Generation 45
According to the Bonnet theorem (see Section 1.1), the specification of the first
and second fundamental forms determines a unique surface, and those two
surfaces that have identical first and second fundamental forms must be
con-gruent Six fundamental magnitudes determine a surface uniquely, except as to
position and orientation in space This is the main theorem in surface theory
It is natural to assume that the property of surfaces that allows for sliding
over can be interpreted in terms of six fundamental magnitudes of the first
and the second fundamental forms if Gauss’ characteristic equation and the
Codazzi-Mainardi’s relationships of compatibility are satisfied
The property of surfaces to allow sliding over themselves imposes an
additional restriction onto relationships among the fundamental
magni-tudes Thus, a representation of a surface in natural form will possess certain
features Comprehensive analysis and solution to the problem of analytical
representation of surfaces that allow sliding over themselves can be found
in [30–33,35] For this purpose, analytical criteria expressed in terms of the
fundamental magnitudes E P , F P , G P , and L P , M P , N P were composed for all
fea-sible kinds of surfaces that allow for sliding over themselves These criteria
enable one to identify whether a certain surface allows for sliding over itself
or not, and if it does, what type of surface it represents The criteria reflect the
following properties of a surface:
For a screw surface: axial pitch is constant (p x= Const)
For a surface of revolution: axial pitch is equal to zero (p x= 0)
For a general cylinder: axial pitch is equal to infinity (p x= ∞)
For all the above-listed surfaces, the generating line is of nonchangeable
shape, and its orientation with respect to the directing line of the surface
remains the same
More particular cases of surfaces that allow sliding over themselves require
additional restrictions to be imposed The interested reader may wish to see
References [32,35] for details on analytical descriptions of surfaces that allow
sliding over themselves
2.5 Feasible Kinematic Schemes of Surface Generation
Kinematics of surface generation is a cornerstone of an infinite variety of
various machining operations Together with the shape and parameters
of geometry of the surface P, the kinematic schemes of surface generation
specify the generating surface of the cutting tool, as well as the principal part
of the kinematic structure of a machine tool
Nonagile kinematics of surface generation is usually performed on
con-ventional machine tools Kinematics of this kind features relative motion
of the cutting tool with constant parameters It is used for the generation of
surfaces P that allow for sliding over themselves Various feasible versions of
Trang 1346 Kinematic Geometry of Surface Machining
nonagile kinematics of surface generation can be considered as a particular
(degenerated) case of agile kinematics of surface generation, for example, as
the kinematics of sculptured surface machining on a multi-axis NC machine
(Figure 2.1)
Only relative motions are investigated in kinematics of surface generation
A certain combination of elementary relative motions of the work and of the
cutting tool makes up the corresponding nonagile kinematics of surface
gen-eration Translational and rotational motions of the work and of the cutting
tool along and about certain axes are implemented as the elementary relative
motions A combination of translations and of rotations of the cutting tool is
referred to as the kinematic scheme of surface generation.
Kinematics of surface generation that allows sliding of one or both
sur-faces P and T over themselves is usually referred to as the rigid kinematics of
surface generation Rigid kinematics of surface generation usually features
constant relative translation motions Vi and rotations w i No acceleration or
deceleration usually occurs Due to this, the kinematic schemes of surface
generation can be composed
Similar to the multi-axis sculptured surface generation, the principle of
inversion is used for the investigation of the nonagile kinematics of surface
generation In order to apply this fundamental principle of mechanics, both
the work and the cutting tool are moved with the motions directed
oppo-site to the motions that the work is performing in the machining operation
Ultimately, this results in the work becoming motionless, and the cutting
tool performs all the required relative motions Under such an assumption, a
Cartesian coordinate system X P Y P Z P associated with the work is considered
as the stationary coordinate system
Two principal problems are tightly connected with the kinematics of
sur-face generation One is referred to as the direct principal problem of sursur-face
generation, and the other is referred to as the inverse principal problem of
sur-face generation
The direct principal problem of surface generation relates to the
determi-nation of the shape and parameters of the generating surface of the cutting
tool for machining a given part surface Therefore, the generation surface
T of the cutting tool is the solution to the first principal problem of surface
generation The cutting tool surface T is expressed in terms of shape and
parameters of the surface P and of parameters of the kinematic scheme of
the machining operation
The inverse principal problem of surface generation relates to the
deter-mination of the shape and parameters of the actual machined surface P
The set of necessary and sufficient conditions of proper surface generation*
(further, conditions of proper PSG) are not always satisfied [29,31,33] This
causes unavoidable deviations of the actual machined surface P act from the
desired nominal surface P des Therefore, the actual generated surface P act is
Chapter 7.
Trang 14Kinematics of Surface Generation 47
the solution to the second principal problem of surface generation The
sur-face P act is expressed in terms of parameters of the generating surface T of
the cutting tool and of parameters of the inverse kinematics scheme of the
machining operation
On the premises of the performed analysis, it is convenient to distinguish
between two different kinematic schemes First, when a surface P is
consid-ered as the stationary surface, then the combination of elementary motions
of the cutting tool relative to the work represents a true kinematic scheme of
surface generation Second, in another case, the cutting tool (or a coordinate
system to which the cutting tool will be associated) can be considered as the
stationary element Then the combination of elementary motions of the work
relative to the coordinate system of the cutting tool can be considered as the
kinematic scheme of the cutting tool profiling
Kinematic schemes of cutting tool profiling are used for solving the direct
principal problem of surface generation The application of it yields the
deter-mination of the generating surface T of the cutting tool for the machining of
a given surface P In order to solve the inverse principal problem of surface
generation, the true kinematic schemes of surface machining are used This
yields determination of the actual machined part surface P act
Machining operations of actual surfaces often include motions that cause
sliding of the surface P or the surface T over them Motions of this kind
simplify obtaining the required speed of cutting, make possible the
genera-tion of the entire surface P, and so forth Relative mogenera-tions that cause sliding
of the surface P or the surface T over them are not a part of the kinematic
schemes of surface generation They are considered separate from the
kine-matic schemes of surface generation
Various kinematic schemes of surface generation can be composed
Aim-ing for the development of the easiest possible machinAim-ing operations, it
is recommended that a limited number of elementary motions of simple
kinds (translations and rotations) be used Usually, the total number of
elementary relative motions in a kinematic scheme does not exceed five
motions Due to that, the kinematic schemes of surface generation
com-posed of three or less translations and rotations are the most implemented
in practice Because of this, the total number of kinematic schemes of
sur-face generation is not infinite and is relatively small Kinematic schemes
of surface generation having more than three elementary motions have
limited implementation in industry Therefore, the consideration below is
limited to the analysis of kinematic schemes of surface generation having
three or less elementary motions Kinematic schemes with more complex
structures are considered briefly
Instant motion of a rigid body in E3 space can be represented as a
combi-nation of an instant rotation and an instant translation along and about a
particular line or axis The combination of the translation and rotation of the
rigid body in E3 space is referred to as an instant screw motion Apparently,
this is because the instant screw motion of the rigid body resembles in part
the motion that is performed by a bolt or a screw Consideration of instant
Trang 1548 Kinematic Geometry of Surface Machining
motion of a rigid body in E3 space is an appropriate starting point at which to
begin the investigation of the kinematic schemes of surface generation
Relative motion of the cutting tool can be reduced to a corresponding
kinematic scheme of surface generation only for nonagile kinematics of
sur-face machining The nonagile kinematics of sursur-face machining features the
elementary relative motions of the cutting tool with constant parameters
of the motions Under such a scenario, for the investigation of all feasible
kinematic schemes of surface generation, axodes that are associated with
the work and with the cutting tool can be employed Implementation of the
concept of axodes is useful for the analysis and visualization of mating
sur-faces in a certain surface machining operation An axode associated with
the work, and the axode associated with the cutting tool make line contact
with one another and roll over each other without sliding In many
practi-cal cases of surface generation (e.g., in gear-shaping operation), the axodes
are congruent to the pitch surfaces In the above example, the axodes are
congruent to the pitch surfaces of the gear being machined and of the gear
shaper However, the congruence of the axodes and of the pitch surface is
not the mandatory requirement for the surface generation In a gear hobbing
operation when axes of rotation of the gear and of the gear hob cross each
other, the axodes do not coincide with the corresponding pitch surfaces In
the kinematic schemes of surface generation with more complex structure,
axodes yield their sliding in their axial direction However, there must exist
at least one point within the line of the axodes’ contact at which there is no
sliding in transverse direction This point is usually referred to as the mean
point of contact of axodes
It is reasonable to begin the analysis of kinematic schemes from the most
general kinematic scheme of surface generation The most general kinematic
scheme of surface generation includes five elementary relative rotations
Kinematic schemes of surface generation with more general structure are
theoretically feasible However, they are not used in practice and therefore
are not considered here
Various combinations of five or less elementary translations and rotations
make up different kinematic schemes of surface generation A detailed
anal-ysis of all feasible combinations of five elementary motions reveals that
with-out loss of generality, it is sufficient to consider just one kinematic scheme of
surface generation This combination of five elementary motions is referred
to as the kinematic scheme of the fifth 50 class
The kinematic scheme of the 50 class is made up of one translation and four
rotations (Figure 2.8) For convenience, axodes of the different elementary
motions in Figure2.8 are shown separately
Axes of the two rotations are parallel to each other The axis of the third
rota-tion crosses the first two axes at a certain crossed-axis angle Σ1−2 (Figure2.8)
The axis of the fourth rotation aligns with the shortest distance of approach
of the above axes of rotation The translational motion is performed along
the fourth axis of rotation