JUNE 2012  VOLUME 7 ISSUE 1 JIDEG 41 Abstract : The compound surfaces which form the technical parts of metal woks are very diverse. Their practical… [600277]

JUNE 2012  VOLUME 7 ISSUE 1 JIDEG 41
Abstract : The compound surfaces which form the technical parts of metal woks are very diverse. Their practical
production involves solving certain geometrical problems through the methods of descriptive geometry.
This article wishes to emphasize the importance of knowing and applying descriptive geometry notions by
presenting the surface developments of some technical parts formed of combined surfaces.
Key words : surface developments, transformed through unrolling, compound surfaces, reductions
1. INTRODUCTION
A large variaty of technical parts are made from
sheets of metal. The objects are laid out, cut, formed into
the required form and fastened together. The fastening is
done by welding, soldering, riverting or seaming.
In the making of reservoirs, pipe fittings and metal
works one can frequently see parts formed out of
cylindrical, conical and polyhedral surfaces which are
intersecting.
For the practical making of these compound surfaces,
the unfinished goods for each surface are drawn and
discharged separately. This is followed by the individual
processing of these surfaces and their assembling.
For the drawing and discharging operations of
unfinished goods, it is necessary to know the unrolling
manner of the surfaces, to determine their intersection
curve and its transposition on the surface development.
During these operations, patterns are built, taking into
account the fact that to a curve drawn on the lateral side
o f a cy l in d e r , a c o n e o r a p o ly h e d r o n , an o t h e r c u rv e o f
the same length has to correspond on the surface
development, called modified curve through the
unrolling of the given curve.
The construction of the surface developments of
technical parts comprises a big diversity of cases which
can be solved through the methods of descriptive
geometry. Thus, revolving, rotation and changing of
plans are applied in order to find out the real sizes of the
segments, angles and surfaces which form the proper
surfaces.
This article wishes to exemplify the above mentioned
in order to unroll certain technical parts made of
combined surfaces.
2. ASPECTS REGARDING THE CONSTRUCTION
OF SURFACE DEVELOPMENTS MADE OF
SECOND DEGREE QUADRICS
Generally, when intersecting surfaces are second
degree quadrics, the intersection curve is a fourth degree
curve which can have one or two inflection points, as
surfaces have one or two common tangent planes.
In the following applications, the cylinders are
considered second degree quadrics, due to their highest
technical applicability. We can see the surface
developments of two cylinders with perpendicular,
respectively concurrent axes at a given angle. In the first case there are two right circular cylinders,
with perpendicular axes and equal diameters (Fig. 1).
The first cylinder is vertical, having as base the C1 (c1,
c’1, c’’ 1) centre circle, and the second cylinder is frontal
horizontal, having as base the C2 (c2, c’ 2, c’’ 2) centre
circle.
I n o r d e r t o d e t e r m in e t h e c o m m on cu r v e of th e tw o
cylinders, the intersection points of the coplanar
generatrices are determined. For this, the two cylinders
are sectioned with frontal planes. Thus, the frontal plane
F1 sections the C1 cylinder after the generatrices from
points A (a, a’, a”) a n d B (b, b’, b”) and the C2 cylinder
after the generatrices from points 2 a n d 3. The needed
curve intersection points are found at the intersection of
the segments:
AA 1 ∩ 24 = A 2 ; a’a 1’ ∩ 2’4’ = a 2’ (1)
AA 1 ∩ 35 = A 3 ; a’a 1’ ∩ 3’5’ = a 3’ (2)
BB 1 ∩ 24 = B 2 ; b’b 1’ ∩ 2’4’ = b 2’ (3)
BB 1 ∩ 35 = B 3 ; b’b 1’ ∩ 3’5’ = b 3’ (4)
The other intersection points are analogically
determined.
The analysed intersection curve is a continuous,
breaking, simple tangent curve and it divides in two
identical branches. This curve degenerates into two plane
curves, called ellipses, situated on end planes. In the
horizontal and lateral projections, the ellipses’
projections coincide with the bases of the projecting
cylinders.
The surface developments of cylinders C1 and C2 are
built and the modified curves through unrolling of the
intersection curve are transposed.
According to Oliver’s Theorem, the modified curve
through unrolling of the section made by a plane into a
cylinder presents inflections in the points where the
tangent plane to the cylinder is perpendicular to the
secant plane. If one of the cylinder’s generatrices is
perpendicular on the secant plane, then the modified
curve through unrolling of the section has no inflection
p o i n t . A l s o , t h e a n g l e o f t w o c u r v e s o f t h e s u r f a c e i s
equal to the angle of their modified curves through
unrolling.
ASPECTS REGARDING THE SURFACE DEVELOPMENTS OF
COMPOUND SURFACES
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Aspects regarding the Surface Developments of Compo und Surfaces

JUNE 2012  VOLUME 7 ISSUE 1 JIDEG 42

Fig.1

Fig.2

Aspects regarding the Surface Developments of Comp ound Surfaces
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Fig.3

Fig.4

Aspects regarding the Surface Developments of Compo und Surfaces

JUNE 2012  VOLUME 7 ISSUE 1 JIDEG 44

Fig.5

Fig.6

Aspects regarding the Surface Developments of Comp ound Surfaces
JUNE 2012  VOLUME 7 ISSUE 1 JIDEG 45

Fig.7

Fig.8

Aspects regarding the Surface Developments of Compound Surfaces
JUNE 2012  VOLUME 7 ISSUE 1 JIDEG 46In figure 2 we can see the surface development of the
cylinder with a vertical axis and in figure 3, of the one
with a horizontal axis [1].
To apply Oliver’s Theorem to the analysed modified
curves through unrolling leads to the F 10 a n d G 10
inflection points, common to both geometrical solids. In
these points, the tangent plane to the C1 and C2 cylinders
is perpendicular on the secant plane.
Another frequently encountered technical case is the
one of the metal parts made of two circular right
cylinders, with equal diameters and concurrent axes at an
angle α, figure 4, [2].
For the drawing and discharging operations of
unfinished goods, it is necessary to build certain patterns.
Flat patterns are designed, printed, cut, creased, folded
on machines made for these purpose. The patterns
contain the surface developments of the cylinders and the
transposition of their intersection curve through
unrolling.
The bases of the two cylinders are revolved on the
vertical plane (figures 5, 6). Fourteen generatices are
built on the surface of the cylinders, with a uniform
distribution and noted a’1, b’2, etc. The two cylinders are
sectioned with frontal planes which contain these
generatrices and it is analogically made with the previous
case. The normal sections in the cylinders transform into
segments with the length equal to their circumference.
The intersection curve is a penetration with a double
tangent.
3. ASPECTS REGARDING THE CONSTRUCTION
OF SURFACE DEVELOPMENTS MADE OF
SECOND DEGREE QUADRICS AND
POLYHEDRAL SURFACES
The connection of second degree quadrics with the
polyhedral surfaces can be made through link surfaces,
called reductions [3].
In figure 7 we can see the reduction which represents
the link between a prismatic surface, with a square base
and a right circular cylinder [4].
The reduction’s surface is composed of four conic
surfaces and four triangular surfaces.
The tips of the cones are the points S1, S2, S3 and S4
and their bases represent the circle of the superior base.
The lengths of the generatrices of the cone with the tip in
S1 are determined through the rotation around the D (d,
d’) axis, which contains the S1 tip.
In figure 8 the surface development of the reduction
connection surface is drawn.
The surface development of the cone with the centre
in S1 is com pleted w ith th e on e of th e S10110S20 triangle
and the result repeats itself for four times, because on the
connection surface there are four cones and four
triangles. 4. CONCLUSIONS
The surface developments of compound surfaces
which form metal works are frequently encountered in
technique. Also, in the construction of pipes there are
used non-standardised fittings and which need to be
produced in workshops, such as: leg-pipes made of
segments, wrinkling leg-pipes, ramifications made by
fittings production, etc.
Solving the graphic representations is very different
and particularises from case to case the applying of
methods from the descriptive geometry.
This subject is very interesting due to the fact that the
accuracy and precision of geometrical constructions are
determined in the production of parts which have to fully
respect the geometrical and functional parameters.
These actual topics represent the source of future
research based on the drawing and discharging
operations of metal parts, in order to increase precision
and to reduce production costs.

REFERENCES
[1] Precupetu, Paul, Dale, Constantin. (1987) . Probleme
de geometrie descriptiva cu aplicatii in tehnica, Ed.
Tehnica, p. 302-317, Bucuresti.
[2] Moncea, Jean. (1982) .Geometrie descriptiva si desen
tehnic , Ed. Didactica si Pedagogica, p. 262-265,
Bucuresti.
[3]http://www.manandmachine.ro/RO-Calculul-lungimii-
desfasurate-pentru-unghiuri-de-indoire-mai-mari-de-
90_.CAD
[4]http://www.practicapentrusucces.ro/workspace/uploa
ds/biblioteca/practica-inginereasca-extras.
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