127exam.mws

> restart:

Ejercicio 6 - Examen final 11 septiembre 2000

José Mª Goicolea, septiembre 2000

Problema de cinemática 3D

> with(linalg):

Warning, the protected names norm and trace have been redefined and unprotected

> u:=vector([-cos(theta),0,sin(theta)]);

u := vector([-cos(theta), 0, sin(theta)])

> j:=vector([0,1,0]);

j := vector([0, 1, 0])

> ju:=crossprod(j,u);

ju := vector([sin(theta), 0, cos(theta)])

> AD:=evalm(a*cos(theta)*j+a*sin(theta)*ju);

AD := vector([a*sin(theta)^2, a*cos(theta), a*sin(t...

> Omega:=vector([-omega/tan(theta),omega/sin(theta),omega]);

Omega := vector([-omega/tan(theta), omega/sin(theta...

> thetad:=omega/sin(theta);

thetad := omega/sin(theta)

> Omegad:=evalm(map(diff,Omega,theta)*thetad);

Omegad := vector([omega^2*(1+tan(theta)^2)/(sin(the...

> Omegad:=simplify(Omegad);

Omegad := vector([omega^2*sin(theta)/((-1+cos(theta...

> map(simplify,Omegad,{1-cos(theta)^2=sin(theta)^2});

vector([omega^2/(sin(theta)^3), -omega^2*cos(theta)...

> v[AD]:=crossprod(Omega,AD,trig);

v[AD] := vector([0, omega*a*sin(theta)^2+omega*a*si...

> v[AD]:=simplify(v[AD],trig);

v[AD] := vector([0, omega*a, -omega*a/sin(theta)])

> v[A]:=vector([0,0,diff(a*sin(theta),theta)*thetad]);

v[A] := vector([0, 0, a*cos(theta)*omega/sin(theta)...

> v[D]:=evalm(v[A]+v[AD]);

v[D] := vector([0, omega*a, a*cos(theta)*omega/sin(...

> Omegan2:=simplify((Omega[1]^2+Omega[2]^2+Omega[3]^2));

Omegan2 := -2*omega^2/(-1+cos(theta)^2)

> Omegan2:=simplify(Omegan2,{1-cos(theta)^2=sin(theta)^2});

Omegan2 := 2*omega^2/(sin(theta)^2)

> Omegan:=sqrt(Omegan2,symbolic);

Omegan := sqrt(2)*omega/sin(theta)

> Omegan1:=evalm(Omega/Omegan);

Omegan1 := vector([-1/2*sqrt(2)*sin(theta)/tan(thet...

> Omegan1:=map(simplify,Omegan1);

Omegan1 := vector([-1/2*sqrt(2)*cos(theta), 1/2*sqr...

> v[min]:=simplify(dotprod(v[A],Omegan1),symbolic);

v[min] := 1/2*a*cos(theta)*omega*sqrt(2)