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Computer aided development of surfaces for intersecting cylinders and cones
Compurers d Srrucrures Vol. 33, No. 3, pp. 729-733. 1989 Pnnted m Great Britain.
0
COMPUTER AIDED DEVELOPMENT INTERSECTING CYLINDERS
OF SURFACES AND CONES
0045-7949189 $3.00 + 0.00 1989 Pergamon Press plc
FOR
N. SIVA F’RASAD and K. V. R. VARDHANI Department of Mechanical Engineering, IIT, Madras 600 036, India (Received 24 August 1988) Abstract-The intersection of cylindrical and conical tubes is very common in process plants. The intersection of nozzles or inlet/outlet pipes with a pressure vessel falls into this category. These intersecting pieces may have their axes intersecting or offset, with any angle of inclination between them. An algorithm has been presented to obtain the developed view of intersection joints when the geometries and the position of the axes of the intersecting pieces are known. The intersection boundaries on the developed surfaces are calculated. A methodology to obtain the required dimensions for developing the surfaces has been presented. Results of an illustrative problem have been presented and discussed.
INTRODUCTION
ematical modelling needed for computerized surface development. a mathematical procedure is In this paper, presented which generates the development of intersection surfaces.
To branch or to combine flow direction in the process industry, intersecting joints are used. These intersecting pieces may be of cylindrical or conical section, with their axes intersecting or non-intersecting. Such types of joints are usually fabricated by developing MATHEMATICAL FORMULATION the surfaces and folding them appropriately, without producing significant plastic deformation in the In the development of intersection surfaces, the metallic sheet used for this purpose. important step is the determination of the interSurfaces are mainly classified as either ruled or section curve between the two pieces which is double curved surfaces [l]. In the present paper, the common to both of them. After determining the development of ruled surfaces is considered. Ruled intersection curve, development of the components surfaces may further be classified as plane, single involves the determination and plotting of the curved and warped vrfaces. Any plane surface is appropriate position and lengths of the parallel or already developed in itself. Single curved surfaces are radial lines in the developed plane, corresponding to generated by moving a straight line generatrix in the points on the intersection curve. contact with a curved line such that any two conThe intersection of an inclined conical cylinder secutive positions of it are either parallel or intersectwith a straight cylinder is considered for illustration ing. The cone and cylinder come under this category as shown in Fig. 1. Dimensions of both the and are developable surfaces. components are given below. In the design of developable surfaces, it is necessary not only to define the geometry of the surface, but Vertical cylinder Conical cylinder also to prescribe how the surface is to be formed from R, = radius Rb = base circle radius its developed shape into the specified design shape. H, = height Development is the process of laying out or unfoldR2 = truncated radius Hz = height ing a surface into a plane [2]. The three methods available for the development of surfaces are: the Consider a ‘global’ coordinate system (O-X, Y, Z) parallel line method, the radial line method and the Y,, Z,) as triangulation method. The parallel line method is and a ‘local’ coordinate system (0,-X,, used to develop surfaces of cylinders and prisms, shown in Fig. 1. The coordinates of 0, with respect since in these cases the successive positions of the to the global system are (TX, 0, T,) where T, is the generatrix form a coplanar set of parallel lines. The offset measured parallel to the X-axis and Tz is the radial line method is used to develop the surfaces of height of intersection measured parallel to the Z-axis. cones and pyramids. Here, successive positions of the ‘A’ is the angle of inclination between the axes of the generatrix form a set of intersecting straight lines. components either in the Y-Z plane or in a plane Graphical methods for developing such surfaces are parallel to it. If the penetrating part is cylindrical in described in the literature on engineering graphics geometry, then R, = R,. [3-51. Although these methods are excellent as graphIf xl, y, and zI are the coordinates of a point in the ical procedures, they are not adaptable to the mathlocal coordinate system, then the corresponding 129
730
and K.V. R. VARDHANI
N.SIVAPRASAD
where RI is the radius of the penetratmg cylinder. These local coordinates can be transformed into the global system using eqn (1). In the case of a penetrating cone, the unknown term in the expressions for x, y and : is the radius R, and in the case of the cylinder, the unknown term is z,. The necessary condition to be satisfied for a point to lie on the intersection curve is
2
t
.yyl+ ,‘? = R:.
(3)
In eqn (3), x and J’ in terms of R or 2, are obtained from eqn (1). Solution of eqn (3) gives two values for R (in the case of a cone) or z, (in the case of a cylinder). For these two values of R or 2, there correspond two points, one on either side of the Z-axis on the intersection curve. The local coordinates of these points for the cone and cylinder can be obtained as given below.
For a penetrating cone Point 1. R = larger of the two values obtained from eqn (3).
Fig. I. Intersection of a cylinder and a cone
global coordinates X, y, z can be calculated using the following transformation: 1
y, =
0
L
cos A sin A TY 0
Lx Y zl = b, )‘I ZI 11 “0
_x,= R
0
-sin A (1) cos A ’ TZ I
Along the intersection curve, the coordinates of any point in the local coordinate system can be written as
COS o
R sin 0
CR,- R)Hz =I= (Rb-Rz)
(4)
Point 2. R = smaller of the two values obtained from eqn (3). X, = R cos(-e)
x, =
R cos 8
(24 y, = R sin(-0)
p, = R sin 0
(2b)
(R, - RW,
‘I=
(Rb- R?)
UC)
For 0 < 0 < 2n in radians, R is a particular radius of the cone for a given value of 6 at the intersection point and R, is its radius at a distance H, from its base given by the equation R,
=
R,,
_
”=
Point 1. z, = positive of the two values obtained from eqn (3).
y, =
R2 cos 0
(5)
For a penetrating cylinder
2
x, =
(R, - R2)
x, = &
Hc’R;-R2).
If the penetrating part is a cylinder, (2a)-(2c) can be written as
(4 - RW,
CO5 o
R2 sin 0.
(6)
then eqns
(24
y, = R2 sin 0
(2e)
z,= z,,
m
Point 2. z, = negative of the two values obtained from eqn (3). x,= R,cos(-B) y, = R2 sin( -8).
(7)
731
Intersecting cylinders and cones Development of the vertical cylinder The development of a cylinder is a rectangular plane figure with a length and height equal to its perimeter and height respectively. Knowing the x, y coordinates of a point on the intersection curve, its angular position can be calculated in the range 0 < 6 < 27r as follows:
Step 8.
Step 9. 6 = cos-‘(x/R,)
radians.
(8)
Equation (8) gives a result only for 6 in the range O-X A modification can be made for the value of 8 in order to make its range 0-2n, i.e. if y is less than zero then 6 = 6 + I[. After determining the 6 values for all the points along the intersection curve, the developed surface can be plotted with 26R, radians along the X-axis and a corresponding z coordinate along Y-axis. Development of the penetrating cone The development of a cone is a sector with a radius equal to its slant height, and an included angle equal to 2nr/s where r and s are its base radius and slant height respectively. The development of the conical portion can be drawn by knowing the angular position and length of each radial line from its base. The coordinates x,, y,, z, and the value of 8 (0 < 0 < 2n) are known for each point on the intersection curve. The corresponding angular position of each radial line in the developed sector is given by R,B/s. The length of a particular radial line from the base is given by the expression (s - I) where 1 is the distance between the apex and the corresponding point on the intersection curve. The procedure explained above for generating the intersection curve and developing the intersecting pieces can be briefly written in the form of an algorithm as follows. Algorithm Read the values of R,, R,, R1, T,, Tz, A, H,, Hz, H, and N,,,. Initial 8 = 0. Step 2a. Corresponding to 6, calculate xi, y, and z, in terms of R (if the penetrating part is conical in geometry). Step 2b. If the penetrating part is cylindrical in geometry, then calculate x, and y, in terms of z,. Step 3. Transform x,, y,, z, into x, y, z using eqn (1). Step 4. Using eqn (3), calculate the numerical values of X, y and z. Step 5. Repeat Steps 2-4 for 0 Q 8 < 27r,incrementing 6 at steps equal to NItp. Step 6. Compute 2R for the development of the vertical cylinder and store them as z values for all the points on the intersection curve. Step 7. Plot 2R,6’ along the X-axis and z along Step 1.
the Y-axis of the plotter to obtain the developed view of the vertical cylinder. For developing the penetrating conical part, compute the included angle of the sector. Corresponding to the points on the intersection curve, determine the angular position and length for each radial line. Draw the development of the conical portion by properly positioning the radial lines and joining the successive end points on either end by two polylines. ILLUSTRATIVE EXAMPLE
Based on the present algorithm, a computer program has been developed in FORTRAN-77 on an IBM compatible PC. An intersection joint used in a process industry (shown in Fig. 2) has been solved as an example. The details of the problem are as follows. Radius of the vertical cylinder, R, = 393.0 mm. Base circle radius of the penetrating cone, R, = 522.0 mm. Truncated radius of the cone, R, = 0.0 mm. Offset along the X-axis, T, = 0.0 mm. Height of intersection, Tz = 1309.0 mm. Angle of intersection, A = 60.0”. Height of the cone, H, = 2430.0 mm. Distance between origin of the local coordinate system and base centre of the cone, H, = 1172.0 mm. The value at which 6 should be incremented, N,,, = 30.0”. A comparison between the results obtained from the graphical procedures shown in Fig. 3 (drawn to a scale of 1 : 10) and those obtained using the present program has been made in Tables 1 and 2.
CONCLUSIONS
In the present paper, an algorithm has been presented for obtaining the development of intersection pieces which are developable separately. The illustrative example shows such a development for a cone and cylinder intersection. The results obtained from the computer program indicate a maximum deviation of 20 mm from the graphical solution. Graphical procedures, particularly for large size structures, gave erroneous dimensions, leading to fabrication difficulties. Hence, it can be said that graphical methods are not suitable for large problems. The present example was actually implemented at the site and it was found that the computer results led to an economical fabrication by way of exact dimensions. From field experience the method is found to be reliable and adaptable to any CAD system for the development of surfaces for sheet metal or structural work. This method can also be used for determining the
732
N.%VA&ASAD
and K.V. R.
VARDHANI
Fig. 2. Illustrative example.
Cylinder Cone Fig. 3. Development of conical and cylindrical components.
733
Intersecting cylinders and cones Table 2. Development of the cylinder
Table 1. Development of conical portion Length of the radial line (mm) Computed Graphical
Radial line No. L
3 4 5 6 1 8
9 10 II 12 Included angle for the sector(in degrees)
472.0 511.0 780.0 930.0 960.0 930.0 915.0 930.0 960.0 930.0 780.0 577.0 80.0
1
488.8 596.3 800.8 933.0 959.8 939.6 926.3 939.6 959.6 933.0 800.8 596.3
2 3 4
5 6 7
Length of the radial line (mm) Computed Graphical 585.0 700.0 945.0
601.0 719.0 963.0
1170.0 1320.0 1410.0 1448.0
963.0 1329.0 1423.0 1456.0
M. C. Hawk, Theory and Problems of Descriptive Geometry, 2nd edn. McGraw-Hill, New York (1963). M. L. Betterley, Sheet Metal Drafiing. McGraw-Hill,
77.31
intersection curves of two intersecting cylinders addition to the development of surfaces.
Radial line No.
in
REFERENCES
1. T. E. French and C. J. Vierk, Graphic Science, 2nd edn. McGraw-Hill, New York (1958). 2. C. E. Rowe and J. D. McFarland, Engineering Descriptive Geometry. Van Nostrand, London (1961). 3. W. E. Street, Technical Descriptive Geometry, 6th edn. Van Nostrand, London (1959).
New York (1961). W. J. Luzzadder, Graphics for Engineers. Prentice-Hall, Englewood Cliffs, NJ (1958). W. M. Newman and R. F. Sproul, Principles of Interactive Graphics, 2nd edn. McGraw-Hill, New York (1979). 8. I. D. Faux and M. D. Pratt Comoutational Geometrv for Design and Manufacture. Ellis -Hotwood, Chichesier, U.K. (1979). 9. S. G: Ddande and P. B. Ramulu, Computeraided methods for development of transition sections. ASME J. Mech. Transmissions Automation 106,401-408, September (1984).
Design
10. K. V. R. Vardhani, Mesh generation using transfinite mapping. M.S. thesis, Department of Mechanical Engineering, IIT, Madras (1988).
0
COMPUTER AIDED DEVELOPMENT INTERSECTING CYLINDERS
OF SURFACES AND CONES
0045-7949189 $3.00 + 0.00 1989 Pergamon Press plc
FOR
N. SIVA F’RASAD and K. V. R. VARDHANI Department of Mechanical Engineering, IIT, Madras 600 036, India (Received 24 August 1988) Abstract-The intersection of cylindrical and conical tubes is very common in process plants. The intersection of nozzles or inlet/outlet pipes with a pressure vessel falls into this category. These intersecting pieces may have their axes intersecting or offset, with any angle of inclination between them. An algorithm has been presented to obtain the developed view of intersection joints when the geometries and the position of the axes of the intersecting pieces are known. The intersection boundaries on the developed surfaces are calculated. A methodology to obtain the required dimensions for developing the surfaces has been presented. Results of an illustrative problem have been presented and discussed.
INTRODUCTION
ematical modelling needed for computerized surface development. a mathematical procedure is In this paper, presented which generates the development of intersection surfaces.
To branch or to combine flow direction in the process industry, intersecting joints are used. These intersecting pieces may be of cylindrical or conical section, with their axes intersecting or non-intersecting. Such types of joints are usually fabricated by developing MATHEMATICAL FORMULATION the surfaces and folding them appropriately, without producing significant plastic deformation in the In the development of intersection surfaces, the metallic sheet used for this purpose. important step is the determination of the interSurfaces are mainly classified as either ruled or section curve between the two pieces which is double curved surfaces [l]. In the present paper, the common to both of them. After determining the development of ruled surfaces is considered. Ruled intersection curve, development of the components surfaces may further be classified as plane, single involves the determination and plotting of the curved and warped vrfaces. Any plane surface is appropriate position and lengths of the parallel or already developed in itself. Single curved surfaces are radial lines in the developed plane, corresponding to generated by moving a straight line generatrix in the points on the intersection curve. contact with a curved line such that any two conThe intersection of an inclined conical cylinder secutive positions of it are either parallel or intersectwith a straight cylinder is considered for illustration ing. The cone and cylinder come under this category as shown in Fig. 1. Dimensions of both the and are developable surfaces. components are given below. In the design of developable surfaces, it is necessary not only to define the geometry of the surface, but Vertical cylinder Conical cylinder also to prescribe how the surface is to be formed from R, = radius Rb = base circle radius its developed shape into the specified design shape. H, = height Development is the process of laying out or unfoldR2 = truncated radius Hz = height ing a surface into a plane [2]. The three methods available for the development of surfaces are: the Consider a ‘global’ coordinate system (O-X, Y, Z) parallel line method, the radial line method and the Y,, Z,) as triangulation method. The parallel line method is and a ‘local’ coordinate system (0,-X,, used to develop surfaces of cylinders and prisms, shown in Fig. 1. The coordinates of 0, with respect since in these cases the successive positions of the to the global system are (TX, 0, T,) where T, is the generatrix form a coplanar set of parallel lines. The offset measured parallel to the X-axis and Tz is the radial line method is used to develop the surfaces of height of intersection measured parallel to the Z-axis. cones and pyramids. Here, successive positions of the ‘A’ is the angle of inclination between the axes of the generatrix form a set of intersecting straight lines. components either in the Y-Z plane or in a plane Graphical methods for developing such surfaces are parallel to it. If the penetrating part is cylindrical in described in the literature on engineering graphics geometry, then R, = R,. [3-51. Although these methods are excellent as graphIf xl, y, and zI are the coordinates of a point in the ical procedures, they are not adaptable to the mathlocal coordinate system, then the corresponding 129
730
and K.V. R. VARDHANI
N.SIVAPRASAD
where RI is the radius of the penetratmg cylinder. These local coordinates can be transformed into the global system using eqn (1). In the case of a penetrating cone, the unknown term in the expressions for x, y and : is the radius R, and in the case of the cylinder, the unknown term is z,. The necessary condition to be satisfied for a point to lie on the intersection curve is
2
t
.yyl+ ,‘? = R:.
(3)
In eqn (3), x and J’ in terms of R or 2, are obtained from eqn (1). Solution of eqn (3) gives two values for R (in the case of a cone) or z, (in the case of a cylinder). For these two values of R or 2, there correspond two points, one on either side of the Z-axis on the intersection curve. The local coordinates of these points for the cone and cylinder can be obtained as given below.
For a penetrating cone Point 1. R = larger of the two values obtained from eqn (3).
Fig. I. Intersection of a cylinder and a cone
global coordinates X, y, z can be calculated using the following transformation: 1
y, =
0
L
cos A sin A TY 0
Lx Y zl = b, )‘I ZI 11 “0
_x,= R
0
-sin A (1) cos A ’ TZ I
Along the intersection curve, the coordinates of any point in the local coordinate system can be written as
COS o
R sin 0
CR,- R)Hz =I= (Rb-Rz)
(4)
Point 2. R = smaller of the two values obtained from eqn (3). X, = R cos(-e)
x, =
R cos 8
(24 y, = R sin(-0)
p, = R sin 0
(2b)
(R, - RW,
‘I=
(Rb- R?)
UC)
For 0 < 0 < 2n in radians, R is a particular radius of the cone for a given value of 6 at the intersection point and R, is its radius at a distance H, from its base given by the equation R,
=
R,,
_
”=
Point 1. z, = positive of the two values obtained from eqn (3).
y, =
R2 cos 0
(5)
For a penetrating cylinder
2
x, =
(R, - R2)
x, = &
Hc’R;-R2).
If the penetrating part is a cylinder, (2a)-(2c) can be written as
(4 - RW,
CO5 o
R2 sin 0.
(6)
then eqns
(24
y, = R2 sin 0
(2e)
z,= z,,
m
Point 2. z, = negative of the two values obtained from eqn (3). x,= R,cos(-B) y, = R2 sin( -8).
(7)
731
Intersecting cylinders and cones Development of the vertical cylinder The development of a cylinder is a rectangular plane figure with a length and height equal to its perimeter and height respectively. Knowing the x, y coordinates of a point on the intersection curve, its angular position can be calculated in the range 0 < 6 < 27r as follows:
Step 8.
Step 9. 6 = cos-‘(x/R,)
radians.
(8)
Equation (8) gives a result only for 6 in the range O-X A modification can be made for the value of 8 in order to make its range 0-2n, i.e. if y is less than zero then 6 = 6 + I[. After determining the 6 values for all the points along the intersection curve, the developed surface can be plotted with 26R, radians along the X-axis and a corresponding z coordinate along Y-axis. Development of the penetrating cone The development of a cone is a sector with a radius equal to its slant height, and an included angle equal to 2nr/s where r and s are its base radius and slant height respectively. The development of the conical portion can be drawn by knowing the angular position and length of each radial line from its base. The coordinates x,, y,, z, and the value of 8 (0 < 0 < 2n) are known for each point on the intersection curve. The corresponding angular position of each radial line in the developed sector is given by R,B/s. The length of a particular radial line from the base is given by the expression (s - I) where 1 is the distance between the apex and the corresponding point on the intersection curve. The procedure explained above for generating the intersection curve and developing the intersecting pieces can be briefly written in the form of an algorithm as follows. Algorithm Read the values of R,, R,, R1, T,, Tz, A, H,, Hz, H, and N,,,. Initial 8 = 0. Step 2a. Corresponding to 6, calculate xi, y, and z, in terms of R (if the penetrating part is conical in geometry). Step 2b. If the penetrating part is cylindrical in geometry, then calculate x, and y, in terms of z,. Step 3. Transform x,, y,, z, into x, y, z using eqn (1). Step 4. Using eqn (3), calculate the numerical values of X, y and z. Step 5. Repeat Steps 2-4 for 0 Q 8 < 27r,incrementing 6 at steps equal to NItp. Step 6. Compute 2R for the development of the vertical cylinder and store them as z values for all the points on the intersection curve. Step 7. Plot 2R,6’ along the X-axis and z along Step 1.
the Y-axis of the plotter to obtain the developed view of the vertical cylinder. For developing the penetrating conical part, compute the included angle of the sector. Corresponding to the points on the intersection curve, determine the angular position and length for each radial line. Draw the development of the conical portion by properly positioning the radial lines and joining the successive end points on either end by two polylines. ILLUSTRATIVE EXAMPLE
Based on the present algorithm, a computer program has been developed in FORTRAN-77 on an IBM compatible PC. An intersection joint used in a process industry (shown in Fig. 2) has been solved as an example. The details of the problem are as follows. Radius of the vertical cylinder, R, = 393.0 mm. Base circle radius of the penetrating cone, R, = 522.0 mm. Truncated radius of the cone, R, = 0.0 mm. Offset along the X-axis, T, = 0.0 mm. Height of intersection, Tz = 1309.0 mm. Angle of intersection, A = 60.0”. Height of the cone, H, = 2430.0 mm. Distance between origin of the local coordinate system and base centre of the cone, H, = 1172.0 mm. The value at which 6 should be incremented, N,,, = 30.0”. A comparison between the results obtained from the graphical procedures shown in Fig. 3 (drawn to a scale of 1 : 10) and those obtained using the present program has been made in Tables 1 and 2.
CONCLUSIONS
In the present paper, an algorithm has been presented for obtaining the development of intersection pieces which are developable separately. The illustrative example shows such a development for a cone and cylinder intersection. The results obtained from the computer program indicate a maximum deviation of 20 mm from the graphical solution. Graphical procedures, particularly for large size structures, gave erroneous dimensions, leading to fabrication difficulties. Hence, it can be said that graphical methods are not suitable for large problems. The present example was actually implemented at the site and it was found that the computer results led to an economical fabrication by way of exact dimensions. From field experience the method is found to be reliable and adaptable to any CAD system for the development of surfaces for sheet metal or structural work. This method can also be used for determining the
732
N.%VA&ASAD
and K.V. R.
VARDHANI
Fig. 2. Illustrative example.
Cylinder Cone Fig. 3. Development of conical and cylindrical components.
733
Intersecting cylinders and cones Table 2. Development of the cylinder
Table 1. Development of conical portion Length of the radial line (mm) Computed Graphical
Radial line No. L
3 4 5 6 1 8
9 10 II 12 Included angle for the sector(in degrees)
472.0 511.0 780.0 930.0 960.0 930.0 915.0 930.0 960.0 930.0 780.0 577.0 80.0
1
488.8 596.3 800.8 933.0 959.8 939.6 926.3 939.6 959.6 933.0 800.8 596.3
2 3 4
5 6 7
Length of the radial line (mm) Computed Graphical 585.0 700.0 945.0
601.0 719.0 963.0
1170.0 1320.0 1410.0 1448.0
963.0 1329.0 1423.0 1456.0
M. C. Hawk, Theory and Problems of Descriptive Geometry, 2nd edn. McGraw-Hill, New York (1963). M. L. Betterley, Sheet Metal Drafiing. McGraw-Hill,
77.31
intersection curves of two intersecting cylinders addition to the development of surfaces.
Radial line No.
in
REFERENCES
1. T. E. French and C. J. Vierk, Graphic Science, 2nd edn. McGraw-Hill, New York (1958). 2. C. E. Rowe and J. D. McFarland, Engineering Descriptive Geometry. Van Nostrand, London (1961). 3. W. E. Street, Technical Descriptive Geometry, 6th edn. Van Nostrand, London (1959).
New York (1961). W. J. Luzzadder, Graphics for Engineers. Prentice-Hall, Englewood Cliffs, NJ (1958). W. M. Newman and R. F. Sproul, Principles of Interactive Graphics, 2nd edn. McGraw-Hill, New York (1979). 8. I. D. Faux and M. D. Pratt Comoutational Geometrv for Design and Manufacture. Ellis -Hotwood, Chichesier, U.K. (1979). 9. S. G: Ddande and P. B. Ramulu, Computeraided methods for development of transition sections. ASME J. Mech. Transmissions Automation 106,401-408, September (1984).
Design
10. K. V. R. Vardhani, Mesh generation using transfinite mapping. M.S. thesis, Department of Mechanical Engineering, IIT, Madras (1988).