ComputerAided_Design_Engineering_amp_Manufactur.pdf

classified into two major groups by their relationships to datum: single features and related features.
The single features are defined by the single datum; related features are defined by the relationship
between the features. Form tolerances state how far the actual features are permitted to vary from the
designed nominal form and consist of standard form features and non-standard form features. Standard
form features include lines, planes, circles, and cylinders, while non-standard form features include curves
and surfaces. Thus the corresponding form tolerances are defined as straightness, flatness, circularity,
cylindricity, and profile of curves or surfaces. The form tolerances are described by tolerance zones, which
set the limits of the extreme boundaries of features. The orientation tolerances specify the geometrical
relationships to datum. Three types of orientation tolerances are (1) parallelism, (2) perpendicularity, and
(3) angularity. Location tolerances state the permissible variation in the specified position of a feature in
relation to some other features or data: true position, concentricity (coaxiality), and symmetry. Run-out
tolerance is the deviation from the perfect form of a part surface of revolution directed by full rotation of
the part on a datum axis. 7
Sculptured Surfaces: Mathematical Description
There are several ways of describing sculptured surfaces mathematically. It is usual to consider a sculptured
surface as consisting of several patches of cubic polynomial surfaces. Parametric polynomial surfaces and
B-spline surfaces are considered in this section. 2,4
Parametric Polynomial Surface
When u,
v are the principal parameters defining the principal direction in a surface patch, the points
data, P(
u,
v),
on the surface can be expressed as the sum of parametric polynomials of the third order.
That is,
where
Aij
is the coefficient matrix of the column vector defining the polynomial.
B-Spline Surface
The points data, P(
u,
v),
on the surface can be described in the B-spline form as follows:
(4.1)
(4.2)
where Qij(
i � 1, 2,…, n;
j � 1, 2,…, m)
are the control points data, ( m � 1) and ( n � 1) are the number
of control points, and ( u),
( v)
are the cubic blending functions for the u,
v directions, respectively. 6
Ni,3
Nj,3
P( u,v)
Aiju i v j
�
i�0 j�0
Normal Vector to the Sculptured Surface
The normal vector, N(
u,
v),
at ( u,
v)
location on the surface can be evaluated from the partial derivatives
of the surface along the u,
v directions. Thus,
where the normal vector is divided by the absolute value to make the unit normal vector.
n
3
3
� �
m
� �
P( u,v)
� QijNi,3( u)Nj,3(
v)
i�0 j�0
N( u, v)
�
( � P( u,v)��u
� �P( u,v)��v
)� �P( u,v)��u
��P( u,v)
��v
(4.3)