The 2nd International Conference
Computational Mechanics
and
Virtual Engineering
COMEC 2007
11 – 13 OCTOBER 2007, Brasov, Romania
MathCAD APPLICATION FOR STRUCTURES FUNDAMENTAL
CONTOURS BASE GENERATION
Mihai C. TOFAN, Mihai ULEA, Sorin VLASE
Transilvania University of Brasov, ROMANIA, e-mail: [email protected]
Abstract: The paper presents MathCAD design for fundamental contours of structure graph using two kinds of addressing
matrix for sequence chain of going over it.
Keywords: graph, fundamental cycles, elastic structures
1. INTRODUCTION
A graph is associated to an elastic mechanical structure. This graph can be built on the base of the fundamental
contours.
Consider a structure with three fundamental contours.
As defines in classical graph theory, for this application are involved equations defined by the nodal junction
matrix
I  2  3 X Nel  ,where Nel is the number of structure elements
start node


, u = 0 1,
I u, j = 
arrival node

binodal element typ (optionally)

j=0
Nel ,
( 1)

and the extended incidence matrix SE N + 1 X Nel , where one node is fixed, all the sides or couplings have
between 1 to 6 degrees of freedom and N+ 1 is the number of the structure nodes.
1 start node


SEi, j = 0 the element not incident in node , i = 0


N,
j=0
Nel ,
( 2)
-1 arrival
The matrix SE is defined by :
u
SE I
= -1 ,
u, j
 
( 3)
The extended incidence matrix is built on regular tree cell Si,j and represents the Hamilton’s way on the graph
nodes and junctions Ji.jo :


S i, j ,i  0;N, i  0;N,

(4)

J i, jo i  0,N, jo  0, C , C = Nel - N,
521
(5)
2. STRUCTURE WITH THREE FUNDAMENTAL CONTOURS
The application from Figure 1 has N=6 and Nel=8. Consider C+1, the number of the contours, ta, the line of the
tree, r, the nodal position, tr, the line of junctions.
Elements of the MathCAD program are below:
( N Nel C )  ( 6 8 Nel  N ) ( i j jo )  ( 0  N  1 0  N  1 0  C )
( i j )  ( 0  N 0  Nel ) ( N Nel C )  ( 6 8 2 )
col ( A)  cols ( A) lin ( A)  rows ( A)
EXT( A  B)  augment ( A  B) unu ( n)  identity ( n)
EXT3 ( A  B  C)  augment3 ( A  B  C)
T
DIM ( A)  ( col ( A) lin ( A) )
 0 0 1 1 1 1 1 
r  
 ta  ( 0 1
0 1 1 0 1 2 2 
( N Nel C )  ( 6 8 Nel  N )
(i j
 i  i 
( N Nel C )  ( 6 8 2 )
I  

i  1 
0 1 2 1 4 5 

1 2 3 4 5 6 
T
2 3 2 4 5 6)
II  
I  EXT( I  II)
T
tr  ( 3 0 4 1 2 6 )
qr  0  5
2
2
r1  ta
u  0  1
jo )  ( 0  N  1 0  N  1 0  C ) ( i j )  ( 0  N 0  Nel )
 3  1 
I   
4 
0 0 2 

3 4 6 
I
q  0  7
r1  tr
qr
1
q
r1  ta
q
0
1
r0  ta
0
r0  tr  r0  ta
q
qr
1
q
Figure 1
 1
 1
0

SE   0
0

0
0

0
0
0
0
0
1
1
1
0
1
0
0
0
0
1
1
0
0
0
0
0
0
1
0
0
0
1
0
0
0
1
1
0
0
1
0
0
0
1
1
0
0
0
0
0
0
1
0
0
Si  j  SEi 1  j T  S
1

0 

1

0 

0

0 

1 
0
 1 1 1 0 0 0 
3 
T
BF   1 0 0 1 0 0  DIM BF   
6 
 0 1 0 1 1 1 


 
Ji  jo  SEi 1  N jo
522
T
B  ( T  J)
Bjo  N jo  1
 1

0
0
S
0
0

0
1
0
1
0
1
1
0
0
0
1
0
0
0
0
1
1
0
0
0
1
0
0
0
0
 1 1 1 0
B 1 0 0 1
 0 1 0 1

0 0

0 0
 1 0
J
 0 1
0 0

0 0
 1



0 
0
0
0 
 T
0 
0
0
1 


1 
0
0
1 1 1 1 1 
0 0 0 0 0
1 1
0
0
1 0 0 0 0
0
1
0
0
0
0
1 1
0
0
0
1
0
0
0
0

0 0 0 1 0 

1 1 0 0 1 
0 

1 
0 
S 1

0 
0 

1 
0 0 1
0
1


0 
0
0
0 
 T S  
1 
0
0
1 


1 
0

0 1 0 0 0

0 0 1 0 0
0 0 0 1 0
 1 1 1 0 0 0
Bf   1 0 0 1 0 0
 0 1 0 1 1 1

0 0

0 0
 1 0
T
S  BF  
 0 1
0 0

0 0
0

0 0 0 0 1

0 1 0 

0 0 1 
0 

1 
0 

0 
0 

1 
1
0
0
The fundamental contours matrix was built by classical sequence, on the lines matrix T oriented to the fixed
node 0 and on the junctions:
T = S1 ,
( 6)
The fundamental contours are built on the tree branches by the application:
B =  T  J ,
T
( 7)
extended with the close end by the junctions:
B jo, jo+ N  1.
( 8)
The series of the fundamental contours is not directly addressed by two index matrix both on the same
dimension, k jo, j  jo, . The first matrix has a void elements column, which are addressed to the closing
element of the incidence junction jo contour. In the matrix klu, jo and κl u, jo , u  0 1, respectively, are defined
the border of the rows of the index matrix. The graph sides which belong to each contour are stored in the matrix
D.
In the paper are described two metods. The first one used the sequence building of the contours defined on the
matrix B rows.
The second one used the simple enumeration of the nodes sequence which defines the contour, in the C matrix.
3. FIRST METHOD
The used nodal coordinates calculus is made on the fundamental contours:
T
(  1 )  ( 0  Nel  1 1  Nel  1 ) qo  ( 0 0 2 )
k jo  j  ( Nel  2)  jo  j  1
klu  jo  k jo  Nel u
T
ns  ( 0 10 20 )
523
 jo    ( Nel  2)  jo  
 1 2 3 4 5 6 7 8 9 
k   11 12 13 14 15 16 17 18 19 
 21 22 23 24 25 26 27 28 29 


 0 10 20 

 9 19 29 
l  
 1 11 21 

 9 19 29 
0
kl  
Sa j  SE0  j

1
T
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
1
10
11
12
13
14
15
16
17
18
19
2
20
21
22
23
24
25
26
27
28
29
SJ jo  SE0  N jo
T
Sa  ( 1 0 0 0 0 0 )
2
0
SJ  ( 1 1 0 )
DIM ( S)  ( 6 6 )
T
DIM ( J)  ( 6 3 )
T
T
DIM ( I)  ( 2 9 )
T
DIM ( Bf)  ( 3 9 )
 0 1 2 1 4 5 0 0 2   j  I1  j  I0  j
r
 D  r
1 2 3 4 5 6 3 4 6 
 0 1 0 1 0 2 1 1 0 
T
D
 qo  ( 0 0 2 )
 1 0 1 0 1 0 0 1 1 
 1 
 
 1 
 1 
T
T  Sa  

 1 
 1 
 
 1 
I
0
  0  
 
f0  f
  1  
 
f1  f
1
2
3
4
5
6
7
8
9
f0  0
0
0
1
1
1
1
1
0
0
0
1
0
1
1
0
0
0
0
0
0
0
0
1
8
9
2
3
f1  0
0
0
0
1
0
1
1
  2  
 
f2  f
4
5
6
7
0
-1
-1
-1
-1
0
0
1
1
1
1
1
0
0
0
1
2
3
4
5
6
7
8
9
f2  0
1
1
0
0
-1
-1
1
1
1
1
1
1
1
1
1
1
2
2
2
2
1
In the Figures 2, 3 and 4 are presented respectively the three fundamental contours.
2
r1  ta
f1  
2
r1  ta
q
1
f1  
0
1
0
r0  ta  f0  
q
1
q
1
1
1
0
r0  ta  f0  
0
q
Figure 1
Figure 2
524
1
1
2
r1  ta
f1  
q
1
2
1
0
1
r0  ta  f0  
q
2
Figure 3
4. THE SECOND METHOD
In this case is defined directly the used nodes succession of the contour by the C matrix.
j
B jo  j 

Bf jo  p
2
p 0
 1
B  1
0

 0
C   0
2

r1  ta
2 3 3 3 3 4 4 4
1 1 2 2 2 2 3
1 1 2 3 4 4 4
1 2 3 3 3 0 0
1 1 1 4 4 4 0
2 2 2 4 5 6 6

3

5
0

0

2
q
1
r1  C
0j
1
0
1
r0  ta  r0  C
0j
q
Figure 4
In the figures 5, 6 and 7 are presented the three fundamental contours obtained by this method.
2
r1  ta
2
r1  ta
q
1
r1  C
q
1
r1  C
1j
2j
1
0
1
1
r0  ta  r0  C
q
0
1
r0  ta  r0  C
1j
q
Figure 5
Figure 6
525
2j
The figurile 8 and 9 present a n iterative buiding of the fundamental contours.
J  0
j 1
  jo  j  1
  jo  0
f
 f


 z1
D
 B jo  z1
kz  0  kl1  C
kl1  C  29
z1
2
2
r1  ta
f1  
r1  ta
q
1
f1  
J 
1
0
r0  ta  f0  
q
q
1
J 
1
1
0
r0  ta  f0  
J 
q
1
J 
Figure 8
Figure 7
REFERENCES
[1] Wittenburg, J.: Dynamics of Systems of Rigid Bodies, B.G. Teubner, Stuttgart, 1977
[2] Tofan, M.C., Goia, I., Ţierean, M., Ulea, M.: Deformatele Structurilor, Ed. Lux Libris, Braşov, 1995
526