Naloga 9

Napetosti so podane v pascalih [Pa].

Podatki

In[63]:=

σij = {{1, -1, 0}, {-1, 1, 0}, {0, 0, 1}} ; σαβ = {{1, 1, 0}, {1, 1, 0}, {0, 0, 1}} ;

Matriki [σij] in [σαβ] imata enaka karakteristična polinoma

In[65]:=

CharacteristicPolynomial[σij, λ] CharacteristicPolynomial[σαβ, λ]

Out[65]=

-2 λ + 3 λ^2 - λ^3

Out[66]=

-2 λ + 3 λ^2 - λ^3

in zato tudi enake lastne vrednosti

In[67]:=

Reduce[% 0, λ, Reals]

Out[67]=

λ0 || λ1 || λ2

Resitev

Potemtakem obstaja takšna ortogonalna matrika [T], da velja [σαβ] = [T]^T [σij] [T]. Z Mathematico lahko poiščemo vse takšne matrike [T].
Z ukazom Det[T]==1 zahtevamo rotacije.

In[68]:=

t1 = {t11, t12, t13} ; t2 = {t21, t22, t23} ; t3 = {t31, t32, t33} ;  T = Transpose[{t ...         }, {t11, t12, t13, t21, t22, t23, t31, t32, t33}]

Out[72]=

t110&& (t12 -1 || t121) &&t130&&t21 ... &&t230&&t310&&t320&&t331 - 2 t11^2

In[73]:=

Clear[σij, σαβ, t1, t2, t3, t11, t12, t13, t21, t22, t23, t31, t32, t33] ;

Created by Mathematica  (October 30, 2003)