format rat

%
% 'kuupvõrrandi 1 * x^3 - 18 x^2 - 405 x + 4374 = 0 lahend'
%
roots([1 -18 -405 4374])
ans =
    -18      
     27      
      9      

pause

% 1. peasuund: sigma1=27  
A=[-14 -8; -8 -11], B=[1;7]
A =
    -14           -8      
     -8          -11      
B =
      1      
      7      
N1=[inv(A)*B;1]
N1 =
     1/2     
     -1      
      1      
% N1 pikkus ja normeerimine
norm(N1)
ans =
     3/2     
N1=N1/norm(N1)
N1 =
     1/3     
    -2/3     
     2/3     
norm(N1)
ans =
      1      
pause

% 2. peasuund: sigma2=9
A=[-5 -8; -8 -2], B=[1;7]
A =
     -5           -8      
     -8           -2      
B =
      1      
      7      
N2=[inv(A)*B;1]
N2 =
     -1      
     1/2     
      1      
% N2 pikkus ja normeerimine
norm(N2)
ans =
     3/2     
N2=N2/norm(N2)
N2 =
    -2/3     
     1/3     
     2/3     
norm(N2)
ans =
      1      
pause

% 3. peasuund: sigma3=-18
A=[17 -16; -16 23], B=[2;14]
A =
     17          -16      
    -16           23      
B =
      2      
     14      
N3=[inv(A)*B;1]
N3 =
      2      
      2      
      1      
% N3 pikkus ja normeerimine
norm(N3)
ans =
      3      
N3=N3/norm(N3)
N3 =
     2/3     
     2/3     
     1/3     
norm(N3)
ans =
      1      

% Kontroll: N3=N1xN2
N3=cross(N1,N2)
N3 =
    -2/3     
    -2/3     
    -1/3     
pause

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% OMAVÄÄRTUSED JA OMAVEKTORID:  eig
%
%
% Pingetensor S
S=[-1 -16 -2; -16 5 -14; -2 -14 14]
S =
     -1          -16           -2      
    -16            5          -14      
     -2          -14           14      
% omaväärtused
eig(S)
ans =
    -18      
      9      
     27      
pause

% omaväärtused ja omavektorid
[V,D]=eig(S)
V =
    -2/3         -2/3          1/3     
    -2/3          1/3         -2/3     
    -1/3          2/3          2/3     
D =
    -18            0            0      
      0            9            0      
      0            0           27      

% omaväärtuste ja omavektorite ümberjärjestamine, antud juhul 3<->1
D(1,1)=27;  D(3,3)=-18;
NN=V(:,1);  V(:,1)=V(:,3);  V(:,3)=NN;
D,V
D =
     27            0            0      
      0            9            0      
      0            0          -18      
V =
     1/3         -2/3         -2/3     
    -2/3          1/3         -2/3     
     2/3          2/3         -1/3     

% Kontroll: N3=N1xN2
cross(V(:,1),V(:,2))
ans =
    -2/3     
    -2/3     
    -1/3     

pause

%joonistamine
h3=plot3([0,V(1,1)],[0,V(2,1)],[0,V(3,1)],'g'); set(h3,'linewidth',2)
text(V(1,1),V(2,1),V(3,1),'   N1')
hold on
h3=plot3([0,V(1,2)],[0,V(2,2)],[0,V(3,2)],'b'); set(h3,'linewidth',2)
text(V(1,2),V(2,2),V(3,2),'   N2')
h3=plot3([0,V(1,3)],[0,V(2,3)],[0,V(3,3)],'r'); set(h3,'linewidth',2)
text(V(1,3),V(2,3),V(3,3),'   N3')
plot3([0,0],[0,0],[-1,1],'k')
plot3([0,0],[-1,1],[0,0],'k')
plot3([-1,1],[0,0],[0,0],'k')
xlabel('x'),ylabel('y'),zlabel('z')
text(1,0,0,'  x'),text(0,1,0,'  y'),text(0,0,1,'  z')
times12, view(110,45)


grid on, hold off
print -deps2c yl1

pause
format short
% Pinged kaldpinnal 
% 1) võrdselt kallutatud kõigi telgede suhtes
% pinnanormaal 
nu1=[1 1 1]/sqrt(3)
nu1 =
    0.5774    0.5774    0.5774

p1=S*nu1'
p1 =
  -10.9697
  -14.4338
   -1.1547
p1/sqrt(3)
ans =
   -6.3333
   -8.3333
   -0.6667
norm(p1)
ans =
   18.1659
pause

% 2) pinnanormaal 
nu1=[-1 1 1]/sqrt(3)
nu1 =
   -0.5774    0.5774    0.5774

p1=S*nu1'
p1 =
   -9.8150
    4.0415
    1.1547
p1/sqrt(3)
ans =
   -5.6667
    2.3333
    0.6667
norm(p1)
ans =
   10.6771
pause

% 3) pinnanormaal 
nu1=[1 -1 1]/sqrt(3)
nu1 =
    0.5774   -0.5774    0.5774

p1=S*nu1'
p1 =
    7.5056
  -20.2073
   15.0111
p1/sqrt(3)
ans =
    4.3333
  -11.6667
    8.6667
norm(p1)
ans =
   26.2679
pause

% 4) pinnanormaal 
nu1=[-1 -1 1]/sqrt(3)
nu1 =
   -0.5774   -0.5774    0.5774

p1=S*nu1'
p1 =
    8.6603
   -1.7321
   17.3205
p1/sqrt(3)
ans =
    5.0000
   -1.0000
   10.0000
norm(p1)
ans =
   19.4422
pause

echo off