Processing math: 100%

NS_Truss_1_(v0.6)

906 days ago by fresl

 

Jednostavni(ji) sistemi

 

       

 

Primjer 1.

 

Opis rešetke:

  • čvorovi, štapovi, ležajevi, opterećenja u čvorovima

Zadavanje čvorova:

  • rječnik čvorova:

{ oznaka0: (x0, y0, z0), oznaka1: (x1, y1, z1), oznaka2: (x2, y2, z2), ... }

  • oznakaioznaka (indeks) čvora i
  • (xi, yi, zi) koordinate čvora (os z je orijentirana „prema gore”)
  • općenito:  rječnik (dictionary) — niz uređenih parova  ključ: vrijednost  navedenih unutar vitičastih zagrada {}

Zadavanje štapova:

  • rječnik štapova:

{ oznaka0: (čvor00, čvor10), oznaka1: (čvor01, čvor11), oznaka2: (čvor02, čvor12), ... }

  • oznakaioznaka (indeks) štapa i
  • (čvor0i, čvor1i)početni i krajnji čvor štapa i

Zadavanje ležajeva:

  • lista oznaka ležajnih čvorova:

[ oznaka0, oznaka1, ... ]

Zadavanje opterećenja u čvorovima:

  • rječnik opterećenja:

{ oznaka0: (Fx0, Fy0, Fz0), oznaka1: (Fx1, Fy1, Fz1), oznaka2: (Fx2, Fy2, Fz2), ... }

  • oznakaioznaka čvora u kojem djeluje opterećenje
  • (Fxi, Fyi, Fzi)komponente čvornoga opterećenja
       
joints:
   0:   (0.000000000000000, 0.000000000000000, 0.000000000000000)
   1:   (4.00000000000000, -1.50000000000000, 0.000000000000000)
   2:   (2.00000000000000, 3.50000000000000, 0.000000000000000)
   3:   (1.50000000000000, 2.00000000000000, 3.00000000000000)

bars:
   0:   (0, 3)
   1:   (1, 3)
   2:   (2, 3)

supports: [0, 1, 2]

loads:
   3:   (75.0000000000000, 100.000000000000, -25.0000000000000)
joints:
   0:   (0.000000000000000, 0.000000000000000, 0.000000000000000)
   1:   (4.00000000000000, -1.50000000000000, 0.000000000000000)
   2:   (2.00000000000000, 3.50000000000000, 0.000000000000000)
   3:   (1.50000000000000, 2.00000000000000, 3.00000000000000)

bars:
   0:   (0, 3)
   1:   (1, 3)
   2:   (2, 3)

supports: [0, 1, 2]

loads:
   3:   (75.0000000000000, 100.000000000000, -25.0000000000000)

Grafički prikaz rešetke:

  • crno — ležajni čvorovi
  • bijelo — slobodni čvorovi
       

Izračunavanje vrijednosti sila u štapovima:

truss3D (joints, bars, supports, loads) 

rezultat: rječnik s komponentama  oznakai: silai

    • oznakai — oznaka štapa i (iz rječnika bars)
    • silaivrijednost sile u štapu i
       
{0: 121.55658196570405, 1: -22.492836817374567, 2: -119.27259222775442}
{0: 121.55658196570405, 1: -22.492836817374567, 2: -119.27259222775442}
       
0:   121.55658196570405
1:   -22.492836817374567
2:   -119.27259222775442
0:   121.55658196570405
1:   -22.492836817374567
2:   -119.27259222775442

Grafički prikaz rješenja:

  • plavo — štapovi s vlačnim silama
  • crveno — štapovi s tlačnim silama
  • žuto — štapovi bez sila
       

 

Rješavanje u koracima:

  •  slobodni čvorovi:

others (a1, a2) — rezultat: komponente liste ili ključevi komponenata rječnika a2 koje nisu u listi ili rječniku a1

       
[3]
[3]
  •  provjera zadovoljenja Maxwellova „pravila”

3nj=nb

nj — broj slobodnih čvorova

nb — broj štapova

       
3 * 1 - 3  ==  0
3 * 1 - 3  ==  0
  •  ravnotežna matrica:
       
(0.38411063979868790.47673129462279620.147441956154897140.51214751973158390.66742381247191460.44232586846469140.76822127959737590.57207755354735530.8846517369293828)

                                
  •  vektor vanjskih sila:
       
(75.0,100.0,25.0)

                                
  •  rješavanje matričnoga sustava:
       
(121.55658196570404,22.492836817374574,119.2725922277544)

                                
  •  pridruživanje rješenja štapovima:

dict_bar_force (bars, forces)rezultat: rječnik s komponentama  oznakai: silai

  •  oznakaioznaka štapa i u rječniku bars
  •  silai vrijednost uzdužne sile u štapu i u listi forces
       
0:   121.55658196570404
1:   -22.492836817374574
2:   -119.2725922277544
0:   121.55658196570404
1:   -22.492836817374574
2:   -119.2725922277544


Rješavanje u koracima (drugi način):

  • proširena ravnotežna matrica — posljednji stupac sadrži vektor vanjskih sila:
       
(0.38411063979868790.47673129462279620.1474419561548971475.00.51214751973158390.66742381247191460.4423258684646914100.00.76822127959737590.57207755354735530.884651736929382825.0)

                                
  • matrica u gornjem stepeničastom obliku:

rezultat funkcije ref_wpp(): matrica u gornjem stepeničastom obliku i lista indeksa baznih/uporišnih stupaca

       
(1.00.74467808786445061.151558490273828532.5427069829443950.00.99999999999999990.7731921410338582114.713467768610330.05.551115123125783×10171.0119.27259222775442)

                                
       
(1.00000000.744678091.151558532.5427070.000000001.00000000.77319214114.713470.000000000.000000001.0000000119.27259)

                                
       
[0, 1, 2]
[0, 1, 2]
  • iz posljednjega stupca proširene matrice u stepeničastomu obliku može se uvrštavanjem unazad izračunati vektor rješenja:
       
(121.55658196570405, -22.492836817374567, -119.27259222775442)
(121.55658196570405, -22.492836817374567, -119.27259222775442)
       
0:   121.55658196570405
1:   -22.492836817374567
2:   -119.27259222775442
0:   121.55658196570405
1:   -22.492836817374567
2:   -119.27259222775442

 

Potprostori ravnotežne matrice:

  • rezultat funkcije subspaces(): rječnici s komponentama
    • 'basis' lista vektorâ baze potprostora
    • 'degree' broj skalarnih komponenata vektorâ baze
    • 'dimension'dimenzija potprostora (broj vektorâ baze)
    • 'name'naziv potprostora
       
  • (pot)prostor stupaca:
       
Column space:
   Vector space of degree 3 and dimension 3
   Basis:
     (-0.3841106397986879, -0.5121475197315839, -0.7682212795973759)
     (0.4767312946227962, -0.6674238124719146, -0.5720775535473553)
     (0.14744195615489714, 0.4423258684646914, -0.8846517369293828)
Column space:
   Vector space of degree 3 and dimension 3
   Basis:
     (-0.3841106397986879, -0.5121475197315839, -0.7682212795973759)
     (0.4767312946227962, -0.6674238124719146, -0.5720775535473553)
     (0.14744195615489714, 0.4423258684646914, -0.8846517369293828)
       
(0.38411063979868790.47673129462279620.147441956154897140.51214751973158390.66742381247191460.44232586846469140.76822127959737590.57207755354735530.8846517369293828)

                                
  • (pot)prostor redaka:
       
Row space:
   Vector space of degree 3 and dimension 3
   Basis:
     (1.0, 0.7446780878644506, 1.1515584902738285)
     (0.0, 0.9999999999999999, 0.7731921410338582)
     (0.0, -5.551115123125783e-17, 1.0)
Row space:
   Vector space of degree 3 and dimension 3
   Basis:
     (1.0, 0.7446780878644506, 1.1515584902738285)
     (0.0, 0.9999999999999999, 0.7731921410338582)
     (0.0, -5.551115123125783e-17, 1.0)
       
(1.00.74467808786445061.15155849027382850.00.99999999999999990.77319214103385820.05.551115123125783×10171.0)

                                
  • jezgra (nul-potprostor):
       
Kernel:
   Vector space of degree 3 and dimension 0
Kernel:
   Vector space of degree 3 and dimension 0
  • lijeva jezgra:
       
Left kernel:
   Vector space of degree 3 and dimension 0
Left kernel:
   Vector space of degree 3 and dimension 0


Rješavanje u koracima (treći način):

  • matrica u reduciranom stepeničastom obliku (Hermiteov normalni/reducirani oblik):

 rezultat funkcije rref_wpp(): reducirana matrica i lista indeksa baznih/uporišnih stupaca

       
(1.01.1102230246251565×10160.0121.556581965704050.00.99999999999999990.022.4928368173745670.05.551115123125783×10171.0119.27259222775442)

                                
       
(1.00000000.000000000.00000000121.556580.000000001.00000000.0000000022.4928370.000000000.000000001.0000000119.27259)

                                
  • posljednji stupac reducirane proširene matrice sadrži vektor rješenja:
       
0:   121.55658196570405
1:   -22.492836817374567
2:   -119.27259222775442
0:   121.55658196570405
1:   -22.492836817374567
2:   -119.27259222775442

 

Primjer 2.

 

       
joints:
   0:   (-2.00000000000000, -1.00000000000000, 0.000000000000000)
   1:   (0.000000000000000, 0.000000000000000, 0.000000000000000)
   2:   (2.00000000000000, 1.00000000000000, 0.000000000000000)
   3:   (0.000000000000000, 0.000000000000000, 2.00000000000000)

bars:
   0:   (0, 3)
   1:   (1, 3)
   2:   (2, 3)

supports: [0, 1, 2]

loads:
   3:   (100.000000000000, 50.0000000000000, 0.000000000000000)
joints:
   0:   (-2.00000000000000, -1.00000000000000, 0.000000000000000)
   1:   (0.000000000000000, 0.000000000000000, 0.000000000000000)
   2:   (2.00000000000000, 1.00000000000000, 0.000000000000000)
   3:   (0.000000000000000, 0.000000000000000, 2.00000000000000)

bars:
   0:   (0, 3)
   1:   (1, 3)
   2:   (2, 3)

supports: [0, 1, 2]

loads:
   3:   (100.000000000000, 50.0000000000000, 0.000000000000000)
       


Rješavanje sustava:

       
[3]
[3]
       
3 * 1 - 3  ==  0
3 * 1 - 3  ==  0
       
(0.66666666666666660.00.6666666666666666100.00.33333333333333330.00.333333333333333350.00.66666666666666661.00.66666666666666660.0)

                                
       
(1.00.01.0150.00.01.01.3333333333333333100.00.00.00.00.0)

                                
       
[0, 1]
[0, 1]
       
[0, 1]
[0, 1]
       
[2]
[2]
       
0:   150.0
1:   -100.0
2:   0.0
0:   150.0
1:   -100.0
2:   0.0
       


Potprostori ravnotežne matrice:

       
       
Column space:
   Vector space of degree 3 and dimension 2
   Basis:
     (-0.6666666666666666, -0.3333333333333333, -0.6666666666666666)
     (0.0, 0.0, -1.0)
Column space:
   Vector space of degree 3 and dimension 2
   Basis:
     (-0.6666666666666666, -0.3333333333333333, -0.6666666666666666)
     (0.0, 0.0, -1.0)
       
(0.66666666666666660.00.66666666666666660.33333333333333330.00.33333333333333330.66666666666666661.00.6666666666666666)

                                
       
[0, 1]
[0, 1]
       
Row space:
   Vector space of degree 3 and dimension 2
   Basis:
     (1.0, -0.0, -1.0)
     (0.0, 1.0, 1.3333333333333333)
Row space:
   Vector space of degree 3 and dimension 2
   Basis:
     (1.0, -0.0, -1.0)
     (0.0, 1.0, 1.3333333333333333)
       
(1.00.01.00.01.01.33333333333333330.00.00.0)

                                
       
Kernel:
   Vector space of degree 3 and dimension 1
   Basis:
     (1.0, -1.3333333333333333, 1.0)
Kernel:
   Vector space of degree 3 and dimension 1
   Basis:
     (1.0, -1.3333333333333333, 1.0)
       
0:   1.0
1:   -1.3333333333333333
2:   1.0
0:   1.0
1:   -1.3333333333333333
2:   1.0
       
       
Left kernel:
   Vector space of degree 3 and dimension 1
   Basis:
     (-0.5, 1.0, 0.0)
Left kernel:
   Vector space of degree 3 and dimension 1
   Basis:
     (-0.5, 1.0, 0.0)
       
       
(100.0, 50.0, 0.0)
(100.0, 50.0, 0.0)
       
True
True
       
(-50, 100, 0)
(-50, 100, 0)
       
False
False
       
True
True
       
(100, 0, 0)
(100, 0, 0)
       
False
False
       
False
False
       
True
True

 

„Zabranjena” opterećenja:

       
       
       
(0.66666666666666660.00.666666666666666650.00.33333333333333330.00.3333333333333333100.00.66666666666666661.00.66666666666666660.0)

                                
       
(1.00.01.00.00.01.01.33333333333333330.00.00.00.01.0)

                                
       
[0, 1, 3]
[0, 1, 3]
       
       
       
(0.66666666666666660.00.6666666666666666100.00.33333333333333330.00.33333333333333330.00.66666666666666661.00.66666666666666660.0)

                                
       
(1.00.01.00.00.01.01.33333333333333330.00.00.00.01.0)

                                
       
[0, 1, 3]
[0, 1, 3]

 

Varijacija — promjena redoslijeda štapova:

       
bars:
   0:   (0, 3)
   1:   (2, 3)
   2:   (1, 3)
bars:
   0:   (0, 3)
   1:   (2, 3)
   2:   (1, 3)
       
(0.66666666666666660.66666666666666660.0100.00.33333333333333330.33333333333333330.050.00.66666666666666660.66666666666666661.00.0)

                                
       
(0.66666666666666660.00.6666666666666666100.00.33333333333333330.00.333333333333333350.00.66666666666666661.00.66666666666666660.0)

                                
       
(1.01.00.0150.00.01.00.7575.00.00.00.00.0)

                                
       
0:   75.0
1:   -75.0
2:   0.0
0:   75.0
1:   -75.0
2:   0.0
       
       
       
Column space:
   Vector space of degree 3 and dimension 2
   Basis:
     (-0.6666666666666666, -0.3333333333333333, -0.6666666666666666)
     (0.6666666666666666, 0.3333333333333333, -0.6666666666666666)
Column space:
   Vector space of degree 3 and dimension 2
   Basis:
     (-0.6666666666666666, -0.3333333333333333, -0.6666666666666666)
     (0.6666666666666666, 0.3333333333333333, -0.6666666666666666)
       
(0.66666666666666660.66666666666666660.00.33333333333333330.33333333333333330.00.66666666666666660.66666666666666661.0)

                                
       
Row space:
   Vector space of degree 3 and dimension 2
   Basis:
     (1.0, -1.0, -0.0)
     (0.0, 1.0, 0.75)
Row space:
   Vector space of degree 3 and dimension 2
   Basis:
     (1.0, -1.0, -0.0)
     (0.0, 1.0, 0.75)
       
(1.01.00.00.01.00.750.00.00.0)

                                
       
0:   -0.75
1:   -0.75
2:   1.0
0:   -0.75
1:   -0.75
2:   1.0
       

 

Primjer 3.

 

       
joints:
   0:   (0.000000000000000, 0.000000000000000, 0.000000000000000)
   1:   (2.00000000000000, 0.000000000000000, 0.000000000000000)
   2:   (0.000000000000000, 0.000000000000000, 2.00000000000000)

bars:
   0:   (0, 2)
   1:   (1, 2)

supports: [0, 1]

loads:
   2:   (100.000000000000, 0.000000000000000, 0.000000000000000)
joints:
   0:   (0.000000000000000, 0.000000000000000, 0.000000000000000)
   1:   (2.00000000000000, 0.000000000000000, 0.000000000000000)
   2:   (0.000000000000000, 0.000000000000000, 2.00000000000000)

bars:
   0:   (0, 2)
   1:   (1, 2)

supports: [0, 1]

loads:
   2:   (100.000000000000, 0.000000000000000, 0.000000000000000)
       


Rješavanje sustava:

       
[2]
[2]
       
3 * 1 - 2  ==  1  !=  0
3 * 1 - 2  ==  1  !=  0
       
(0.00.7071067811865475100.00.00.00.01.00.70710678118654750.0)

                                
       
(1.00.70710678118654750.00.01.0141.42135623730950.00.00.0)

                                
       
0:   100.0
1:   -141.4213562373095
0:   100.0
1:   -141.4213562373095
       
[0, 1]
[0, 1]
       


Potprostori ravnotežne matrice:

       
       
Column space:
   Vector space of degree 3 and dimension 2
   Basis:
     (0.0, 0.0, -1.0)
     (0.7071067811865475, 0.0, -0.7071067811865475)
Column space:
   Vector space of degree 3 and dimension 2
   Basis:
     (0.0, 0.0, -1.0)
     (0.7071067811865475, 0.0, -0.7071067811865475)
       
(0.00.70710678118654750.00.01.00.7071067811865475)

                                
       
(100.0,0.0,0.0)

                                
       
True
True
       
Row space:
   Vector space of degree 2 and dimension 2
   Basis:
     (1.0, 0.7071067811865475)
     (0.0, 1.0)
Row space:
   Vector space of degree 2 and dimension 2
   Basis:
     (1.0, 0.7071067811865475)
     (0.0, 1.0)
       
(1.00.70710678118654750.01.00.00.0)

                                
       
Kernel:
   Vector space of degree 2 and dimension 0
Kernel:
   Vector space of degree 2 and dimension 0
       
Left kernel:
   Vector space of degree 3 and dimension 1
   Basis:
     (0.0, 1.0, 0.0)
Left kernel:
   Vector space of degree 3 and dimension 1
   Basis:
     (0.0, 1.0, 0.0)
       


Primjer 4.


       
joints:
   0:   (-1.00000000000000, 0.000000000000000)
   1:   (0.000000000000000, 0.000000000000000)
   2:   (4.00000000000000, 0.000000000000000)
   3:   (0.000000000000000, 2.00000000000000)

bars:
   0:   (0, 3)
   1:   (1, 3)
   2:   (2, 3)

supports: [0, 1, 2]

loads:
   3:   (100.000000000000, 0.000000000000000)
joints:
   0:   (-1.00000000000000, 0.000000000000000)
   1:   (0.000000000000000, 0.000000000000000)
   2:   (4.00000000000000, 0.000000000000000)
   3:   (0.000000000000000, 2.00000000000000)

bars:
   0:   (0, 3)
   1:   (1, 3)
   2:   (2, 3)

supports: [0, 1, 2]

loads:
   3:   (100.000000000000, 0.000000000000000)
       
       
[3]
[3]
       
2 * 1 - 3  ==  -1  !=  0
2 * 1 - 3  ==  -1  !=  0
       
(0.44721359549995790.00.8944271909999159100.00.89442719099991591.00.44721359549995790.0)

                                
       
(1.01.1180339887498950.50.00.01.02.23606797749979200.0)

                                
       
[0, 1]
[0, 1]
       
[2]
[2]
       
(223.60679774997897, -200.0, 0.0)
(223.60679774997897, -200.0, 0.0)
       
0:   223.60679774997897
1:   -200.0
2:   0.0
0:   223.60679774997897
1:   -200.0
2:   0.0
       
       
       
Column space:
   Vector space of degree 2 and dimension 2
   Basis:
     (-0.4472135954999579, -0.8944271909999159)
     (0.0, -1.0)
Column space:
   Vector space of degree 2 and dimension 2
   Basis:
     (-0.4472135954999579, -0.8944271909999159)
     (0.0, -1.0)
       
(0.44721359549995790.00.89442719099991590.89442719099991591.00.4472135954999579)

                                
       
Row space:
   Vector space of degree 3 and dimension 2
   Basis:
     (1.0, 1.118033988749895, 0.5)
     (0.0, 1.0, 2.23606797749979)
Row space:
   Vector space of degree 3 and dimension 2
   Basis:
     (1.0, 1.118033988749895, 0.5)
     (0.0, 1.0, 2.23606797749979)
       
(1.01.1180339887498950.50.01.02.23606797749979)

                                
       
Kernel:
   Vector space of degree 3 and dimension 1
   Basis:
     (2.0000000000000004, -2.23606797749979, 1.0)
Kernel:
   Vector space of degree 3 and dimension 1
   Basis:
     (2.0000000000000004, -2.23606797749979, 1.0)
       
0:   2.0000000000000004
1:   -2.23606797749979
2:   1.0
0:   2.0000000000000004
1:   -2.23606797749979
2:   1.0
       
       
Left kernel:
   Vector space of degree 2 and dimension 0
   Basis:
     []
Left kernel:
   Vector space of degree 2 and dimension 0
   Basis:
     []

 

Primjer 5.

 

       
joints:
   0:   (-2.00000000000000, 0.000000000000000, 0.000000000000000)
   1:   (0.000000000000000, 0.000000000000000, 0.000000000000000)
   2:   (2.00000000000000, 0.000000000000000, 0.000000000000000)
   3:   (0.000000000000000, -2.00000000000000, 0.000000000000000)
   4:   (0.000000000000000, 2.00000000000000, 0.000000000000000)
   5:   (0.000000000000000, 0.000000000000000, 2.00000000000000)

bars:
   0:   (0, 5)
   1:   (1, 5)
   2:   (2, 5)
   3:   (3, 5)
   4:   (4, 5)

supports: [0, 1, 2, 3, 4]

loads:
   5:   (100.000000000000, 100.000000000000, 0.000000000000000)
joints:
   0:   (-2.00000000000000, 0.000000000000000, 0.000000000000000)
   1:   (0.000000000000000, 0.000000000000000, 0.000000000000000)
   2:   (2.00000000000000, 0.000000000000000, 0.000000000000000)
   3:   (0.000000000000000, -2.00000000000000, 0.000000000000000)
   4:   (0.000000000000000, 2.00000000000000, 0.000000000000000)
   5:   (0.000000000000000, 0.000000000000000, 2.00000000000000)

bars:
   0:   (0, 5)
   1:   (1, 5)
   2:   (2, 5)
   3:   (3, 5)
   4:   (4, 5)

supports: [0, 1, 2, 3, 4]

loads:
   5:   (100.000000000000, 100.000000000000, 0.000000000000000)
       
       
[5]
[5]
       
3 * 1 - 5  ==  -2  !=  0
3 * 1 - 5  ==  -2  !=  0
       
(0.70710678118654750.00.70710678118654750.00.0100.00.00.00.00.70710678118654750.7071067811865475100.00.70710678118654751.00.70710678118654750.70710678118654750.70710678118654750.0)

                                
       
(1.00.01.00.00.0141.42135623730950.01.01.4142135623730950.70710678118654750.7071067811865475100.00.00.00.01.01.0141.4213562373095)

                                
       
[0, 1, 3]
[0, 1, 3]
       
[2, 4]
[2, 4]
       
0:   141.4213562373095
1:   -200.0
2:   0.0
3:   141.4213562373095
4:   0.0
0:   141.4213562373095
1:   -200.0
2:   0.0
3:   141.4213562373095
4:   0.0
       
       
(0.70710678118654750.00.70710678118654750.00.00.00.00.00.70710678118654750.70710678118654750.70710678118654751.00.70710678118654750.70710678118654750.7071067811865475)

                                
       
       
Row space:
   Vector space of degree 5 and dimension 3
   Basis:
     (1.0, -0.0, -1.0, -0.0, -0.0)
     (0.0, 1.0, 1.414213562373095, 0.7071067811865475, 0.7071067811865475)
     (0.0, 0.0, 0.0, 1.0, -1.0)
Row space:
   Vector space of degree 5 and dimension 3
   Basis:
     (1.0, -0.0, -1.0, -0.0, -0.0)
     (0.0, 1.0, 1.414213562373095, 0.7071067811865475, 0.7071067811865475)
     (0.0, 0.0, 0.0, 1.0, -1.0)
       
Column space:
   Vector space of degree 3 and dimension 3
   Basis:
     (-0.7071067811865475, 0.0, -0.7071067811865475)
     (0.0, 0.0, -1.0)
     (0.0, -0.7071067811865475, -0.7071067811865475)
Column space:
   Vector space of degree 3 and dimension 3
   Basis:
     (-0.7071067811865475, 0.0, -0.7071067811865475)
     (0.0, 0.0, -1.0)
     (0.0, -0.7071067811865475, -0.7071067811865475)
       
Kernel:
   Vector space of degree 5 and dimension 2
   Basis:
     (1.0, -1.414213562373095, 1.0, -0.0, 0.0)
     (0.0, -1.414213562373095, 0.0, 1.0, 1.0)
Kernel:
   Vector space of degree 5 and dimension 2
   Basis:
     (1.0, -1.414213562373095, 1.0, -0.0, 0.0)
     (0.0, -1.414213562373095, 0.0, 1.0, 1.0)
       
0:   1.0
1:   -1.414213562373095
2:   1.0
3:   -0.0
4:   0.0
0:   1.0
1:   -1.414213562373095
2:   1.0
3:   -0.0
4:   0.0
       
       
0:   0.0
1:   -1.414213562373095
2:   0.0
3:   1.0
4:   1.0
0:   0.0
1:   -1.414213562373095
2:   0.0
3:   1.0
4:   1.0
       
       
Left kernel:
   Vector space of degree 3 and dimension 0
   Basis:
     []
Left kernel:
   Vector space of degree 3 and dimension 0
   Basis:
     []

 

Primjer 6. — Slobodni tetraedar

 

       
joints:
   0:   (0.000000000000000, 0.000000000000000, 0.000000000000000)
   1:   (2.00000000000000, 0.000000000000000, 0.000000000000000)
   2:   (0.000000000000000, 2.00000000000000, 0.000000000000000)
   3:   (0.000000000000000, 0.000000000000000, 2.00000000000000)

bars:
   0:   (0, 3)
   1:   (1, 3)
   2:   (2, 3)
   3:   (0, 1)
   4:   (1, 2)
   5:   (2, 0)
joints:
   0:   (0.000000000000000, 0.000000000000000, 0.000000000000000)
   1:   (2.00000000000000, 0.000000000000000, 0.000000000000000)
   2:   (0.000000000000000, 2.00000000000000, 0.000000000000000)
   3:   (0.000000000000000, 0.000000000000000, 2.00000000000000)

bars:
   0:   (0, 3)
   1:   (1, 3)
   2:   (2, 3)
   3:   (0, 1)
   4:   (1, 2)
   5:   (2, 0)
       
3 * 4 - 6  ==  6  !=  0
3 * 4 - 6  ==  6  !=  0
       
[0, 1, 2, 3]
[0, 1, 2, 3]
       
       
(0.00000.00000.00001.0000.00000.00000.00000.00000.00000.00000.00001.0001.0000.00000.00000.00000.00000.00000.00000.70710.00001.0000.70710.00000.00000.00000.00000.00000.70710.00000.00000.70710.00000.00000.00000.00000.00000.00000.00000.00000.70710.00000.00000.00000.70710.00000.70711.0000.00000.00000.70710.00000.00000.00000.00000.70710.00000.00000.00000.00000.00000.00000.70710.00000.00000.00001.0000.70710.70710.00000.00000.0000)

                                
       
       
Row space:
   Vector space of degree 6 and dimension 6
   Basis:
     (1.0, 0.0, 0.0, 0.0, 0.0, 0.0)
     (0.0, 1.0, -0.0, 1.4142135623730951, 1.0, -0.0)
     (0.0, 0.0, 1.0, -0.0, 1.0, 1.4142135623730951)
     (0.0, 0.0, 0.0, 1.0, 0.7071067811865475, -0.0)
     (0.0, 0.0, 0.0, 0.0, 1.0, 0.0)
     (0.0, 0.0, 0.0, 0.0, 0.0, 1.0)
Row space:
   Vector space of degree 6 and dimension 6
   Basis:
     (1.0, 0.0, 0.0, 0.0, 0.0, 0.0)
     (0.0, 1.0, -0.0, 1.4142135623730951, 1.0, -0.0)
     (0.0, 0.0, 1.0, -0.0, 1.0, 1.4142135623730951)
     (0.0, 0.0, 0.0, 1.0, 0.7071067811865475, -0.0)
     (0.0, 0.0, 0.0, 0.0, 1.0, 0.0)
     (0.0, 0.0, 0.0, 0.0, 0.0, 1.0)
       
Column space:
   Vector space of degree 12 and dimension 6
   Basis:
     (0.0, 0.0, 1.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, -1.0)
     (0.0, 0.0, 0.0, -0.7071067811865475, 0.0, 0.7071067811865475, 0.0, 0.0,
0.0, 0.7071067811865475, 0.0, -0.7071067811865475)
     (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, -0.7071067811865475,
0.7071067811865475, 0.0, 0.7071067811865475, -0.7071067811865475)
     (1.0, 0.0, 0.0, -1.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0)
     (0.0, 0.0, 0.0, -0.7071067811865475, 0.7071067811865475, 0.0,
0.7071067811865475, -0.7071067811865475, 0.0, 0.0, 0.0, 0.0)
     (0.0, 1.0, 0.0, 0.0, 0.0, 0.0, 0.0, -1.0, 0.0, 0.0, 0.0, 0.0)
Column space:
   Vector space of degree 12 and dimension 6
   Basis:
     (0.0, 0.0, 1.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, -1.0)
     (0.0, 0.0, 0.0, -0.7071067811865475, 0.0, 0.7071067811865475, 0.0, 0.0, 0.0, 0.7071067811865475, 0.0, -0.7071067811865475)
     (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, -0.7071067811865475, 0.7071067811865475, 0.0, 0.7071067811865475, -0.7071067811865475)
     (1.0, 0.0, 0.0, -1.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0)
     (0.0, 0.0, 0.0, -0.7071067811865475, 0.7071067811865475, 0.0, 0.7071067811865475, -0.7071067811865475, 0.0, 0.0, 0.0, 0.0)
     (0.0, 1.0, 0.0, 0.0, 0.0, 0.0, 0.0, -1.0, 0.0, 0.0, 0.0, 0.0)
       
Kernel:
   Vector space of degree 6 and dimension 0
   Basis:
     []
Kernel:
   Vector space of degree 6 and dimension 0
   Basis:
     []
       
Left kernel:
   Vector space of degree 12 and dimension 6
   Basis:
     (0.0, 0.0, 0.0, 0.0, -1.0, 0.0, 1.0, 0.0, 0.0, 0.0, 0.0, 0.0)
     (1.0, 0.0, 0.0, 1.0, 1.0, 1.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0)
     (0.0, 1.0, 0.0, 0.0, 1.0, 0.0, 0.0, 1.0, 1.0, 0.0, 0.0, 0.0)
     (0.0, 0.0, 0.0, 0.0, 0.0, -1.0, 0.0, 0.0, 0.0, 1.0, 0.0, 0.0)
     (0.0, 1.0, 0.0, 0.0, 1.0, 0.0, 0.0, 1.0, 0.0, 0.0, 1.0, 0.0)
     (0.0, -1.0, 1.0, 0.0, -1.0, 1.0, 0.0, -1.0, 0.0, 0.0, 0.0, 1.0)
Left kernel:
   Vector space of degree 12 and dimension 6
   Basis:
     (0.0, 0.0, 0.0, 0.0, -1.0, 0.0, 1.0, 0.0, 0.0, 0.0, 0.0, 0.0)
     (1.0, 0.0, 0.0, 1.0, 1.0, 1.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0)
     (0.0, 1.0, 0.0, 0.0, 1.0, 0.0, 0.0, 1.0, 1.0, 0.0, 0.0, 0.0)
     (0.0, 0.0, 0.0, 0.0, 0.0, -1.0, 0.0, 0.0, 0.0, 1.0, 0.0, 0.0)
     (0.0, 1.0, 0.0, 0.0, 1.0, 0.0, 0.0, 1.0, 0.0, 0.0, 1.0, 0.0)
     (0.0, -1.0, 1.0, 0.0, -1.0, 1.0, 0.0, -1.0, 0.0, 0.0, 0.0, 1.0)