Computer Aided Engineering Design

44 COMPUTER AIDED ENGINEERING DESIGN

2.6 Computer Aided Assembly of Rigid Bodies

Transformations can be used to position CAD primitives created separately and then to manipulate
these using solid modeling Boolean operations like join,cut and intersect. Such operations are
discussed in Chapter 8 in detail. Here, however, discussion shall be restricted to relative positioning.
Consider a triangular rigid-body S 1 (P 1 P 2 P 3 ) to be joined to another rigid body S 2 (Q 1 Q 2 Q 3 ) such that
P 1 coincides with Q 1 and the edge P 1 P 2 is colinear with Q 1 Q 2.
The first objective is to have both triangles in the same plane after assembly. Two local coordinate
systems are constructed at the corner points P 1 and Q 1 with unit vectors (p 1 , p 2 , p 3 ) and (q 1 , q 2 , q 3 ).
Here, unit vectors p 1 and p 2 are along the sides P 1 P 2 and P 1 P 3 respectively, and p 3 is perpendicular
to the plane P 1 P 2 P 3. Unit vectors q 1 and q 2 are along Q 1 Q 2 and Q 1 Q 3 , respectively, and q 3 is
perpendicular to the plane Q 1 Q 2 Q 3. Thus

p

PP

PP

p

PP

PP

1 21 ppp

21

2

31

31

= 312

• |– |
  , =


• 
  

  |– |
  , = ×

  
  

Vectors q 1 , q 2 , q 3 can be determined in a similar way. Note that each of the unit vectors p (p 1 , p 2 , p 3 )

wherePi = [1]xyzPPPiii and Qi = [1]xyzQQQiii,i = 1, 2, 3.
(b) At this stage, the two planes PPP 1 2 3 and Q 1 Q 2 Q 3 are joined together at Q 1. The edge PP 1 2
may not be in line with Q 1 Q 2. Let p 1 be the unit vector along PP 1 2 . Then

p 1 *^2

*

1

*

2

*

1

= *

• |– |

PP
PP
Angleα between p 1 andq 1 can be found using cos α = p 1 ·q 1. Let u=p 1 ×q 1 = [uxuyuz 0]
be a unit vector passing through P 1 (which is coincident with Q 1 ) and perpendicular to the plane
containingp 1 and q 1. Rotating PP 1 2 to coincide with Q 1 Q 2 involves rotating P 2 about u
through an angle α for P 2 * to finally lie on Q 1 Q 2. Let the new position of P 1 P 2 P 3 be PP P 123 ′′ ′.
(c) At this time, the two edges PP 12 ′′ and Q 1 Q 2 are coincident. However, angle between the triangular
planes may not be the desired angle. To rotate PP P 123 ′′ ′ about Q 1 Q 2 would require knowing the
angle between the planes PP P 123 ′′ ′ and Q 1 Q 2 Q 3. This is given by the angle between the normal
vectors to the two planes. The unit normal to Q 1 Q 2 Q 3 is q 3. Forp′ 3 , the unit normal toPP P 123 ′′ ′,

we compute the unit vectors along PP 12 ′′ and PP 13 ′′. With p 1 ′known (as q 1 ),p′ 2 as

′′

′′

PP

PP

31

31

• |– |

,p′ 3

Figure 2.19(a) Assembly of two triangular laminae

andq (q 1 , q 2 , q 3 ) are 4 × 3 matrices, the last row
entries being zeros. The transformations can be
constructed in the following steps:

(a) Translate P 1 to Q 1. The new set of co-ordinates

forP 1 ,P 2 and P 3 are now PP 1 *, 2 * and P 3 *res-

pectively, given by

P

P

P

xx

yy

zz

P

P

P

T qp

qp

qp

T

1 *

2 *

3 *

11

11

11

1

2

3

=

100 –

010 –

001 –

000 1

⎡

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎡

⎣

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

⎡

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

Q 3

Q 1

q (^1) Q 2
q 2
p 2
p 1
P 1
P 3
P 2