Euler Winkel

Individuelle Rotationen Z, X’, Z’’

(rotZϕ={{Cos[ϕ],Sin[ϕ],0},{-Sin[ϕ],Cos[ϕ],0},{0,0,1}})//MatrixForm

(rotXθ={{1,0,0},{0,Cos[θ],Sin[θ]},{0,-Sin[θ],Cos[θ]}})//MatrixForm

(rotZψ=rotZϕ/.ϕ->ψ)//MatrixForm

Produkt der Rotationen

(Euler=rotZψ.rotXθ.rotZϕ);

(Euler/.Cos[ϕ_]->C[ϕ]/.Sin[ϕ_]->S[ϕ])//MatrixForm

Plot

origin={0,0,0};

axes={Thick,Darker[Blue],Arrow[{origin,Euler[[1]]}],Darker[Red],Arrow[{origin,Euler[[2]]}],Darker[Green],Arrow[{origin,Euler[[3]]}]}/.ϕ->α/.θ->β/.ψ->γ;

lines={Black,Arrow[{{-1,0,0},{1,0,0}}],Arrow[{{0,-1,0},{0,1,0}}],Arrow[{{0,0,-1},{0,0,1}}]};

AxesRot[ϕ_,θ_,ψ_]:=Graphics3D[{lines,axes}/.α->ϕ/.β->θ/.γ->ψ,ViewPoint->{Pi,Pi/2,2}];

Manipulate[AxesRot[ϕ,θ,ψ],{ϕ,0,2π},{θ,0,π},{ψ,0,2π}]

Winkelgeschwindigkeit

zeitabh={ϕ->ϕ[t],θ->θ[t],ψ->ψ[t]};

EulerT=Euler/.zeitabh;

(Subscript[ω, 1])'= ((∂_t)(Subscript[(OverVector[e]), 2])')·((Subscript[(OverVector[e]), 3])')

w1p=D[EulerT[[2]],t].(EulerT[[3]])//Simplify

w2p=D[EulerT[[3]],t].(EulerT[[1]])//Simplify

w3p=D[EulerT[[1]],t].(EulerT[[2]])//Simplify

len=(Sqrt[{w1p,w2p,w3p}.{w1p,w2p,w3p}])//Simplify//PowerExpand

(w1p^2)+(w2p^2)//Simplify

Übung 9, Aufgabe 2

((Transpose[Euler/.zeitabh]).{w1p,w2p,w3p})/.ϕ->Function[t,0]/.θ->Function[t,α t]/.ψ->Function[t,α t]//Simplify