Rotatorischer ZweimassenschwingerErstellt: 7.5.2020
Ersteller: W. Kemmetm\303\274ller
\303\204nderungen: restart;
with(LinearAlgebra):
with(VectorCalculus,Jacobian):
with(VectorCalculus,SpaceCurve):
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entsprechend handschriftlichem RechenwegFreiheitsgradeqq:=<varphi1, varphi2>;
qq_p:=<varphi1_p, varphi2_p>;
qq_pp:=<varphi1_pp, varphi2_pp>;Substitutionslisten (Zeitabh\303\244ngigkeit)
Hinweis: Diese Substitutionslisten sind notwendig, da Maple nicht nach der Zeitableitung dx(t)/dt einer Gr\303\266\303\237e ableiten kann.set_q2qt:={seq(s=s(t),s in convert(qq,list))};
set_qt2q:={seq(s(t)=s,s in convert(qq,list))};
set_qpt2qp:={seq(diff(convert(qq,list)[i](t),t)=convert(qq_p,list)[i],i=1..Dimension(qq))};
set_qp2qpt:={seq(convert(qq_p,list)[i]=diff(convert(qq,list)[i](t),t),i=1..Dimension(qq))};
set_qppt2qpp:={seq(diff(convert(qq,list)[i](t),t$2)=convert(qq_pp,list)[i],i=1..Dimension(qq))};
set_qpp2qppt:={seq(convert(qq_pp,list)[i]=diff(convert(qq,list)[i](t),t$2),i=1..Dimension(qq))};Kinetischen Energie der beiden Starrk\303\266rperT1:=1/2*Ixx1_s1*varphi1_p^2;
T2:=1/2*Ixx2_s2*varphi2_p^2;
T:=T1+T2:Potentielle Energie der linearen Federn (entspannte Winkel 0)Vc1:=1/2*c1*varphi1^2;
Vc12:=1/2*c12*(varphi2-varphi1)^2;
V:=Vc1+Vc12:Externe Kr\303\244fte / MomenteIm System treten dissipative Momente auf (Drehd\303\244mpfer), die ebenfalls im Vektor der externen (nicht Potentialkr\303\244fte) zusammengefasst werdenMomente der Drehd\303\244mpfertau_d1:=-d1*varphi1_p;
tau_d12:=-d12*(varphi1_p-varphi2_p);Momente auf Starrk\303\266rper 1tau_e1:=-tau_1+tau_d1+tau_d12;tau_e2:=-tau_2-tau_d12;Lagrange-FunktionL:=T-V;H1:=map(diff,eval(Jacobian(<L>,convert(qq_p,list)),set_qp2qpt),t);
H2:=eval(Jacobian(<L>,convert(qq,list)),set_q2qt);Bewegungsgleichungenode1:=H1[1,1]-H2[1,1]=tau_e1;
ode2:=H1[1,2]-H2[1,2]=tau_e2;L\303\266sung mittels homogener TransformationenAuch wenn dieser L\303\266sungsweg f\303\274r das relative einfache Beispiel umst\303\244ndlich erscheint, wird hier angegeben, wie man die Bewegungsgleichung systematisch mit Hilfe von homogenen Transformationen, der Massenmatrix, der Coriolismatrix sowie dem Vektor der Potentialkr\303\244fte herleiten kannDefinition der elementaren DrehmatrixRR_x:=phi-><<1,0,0>|<0,cos(phi),sin(phi)>|<0,-sin(phi),cos(phi)>>;Drehmatrizen (Winkel varphi1 und varphi2 sind relativ zum Inertialkoordinatensystem definiert)RR_0_1:=RR_x(varphi1);
RR_0_2:=RR_x(varphi2);
RR_1_2:=map(combine,RR_0_1^+. RR_0_2,trig);Verschiebungsvektoren der Koordinatensysteme (diese sind f\303\274r die weitere Rechnung irrelevant und werden daher zu 0 gesetzt)dd_0_1:=<0,0,0>;
dd_0_2:=<0,0,0>;Homogene TransformationenHH:=(RR,dd)-> <<RR|dd>,<0|0|0|1>>;
PP:=pp-> <pp,1>;HH_0_1:=HH(RR_0_1,dd_0_1);
HH_0_2:=HH(RR_0_2,dd_0_2);Berechnung der Manipulator-Jacobi Matrizen der beiden Starrk\303\266rperww:=SS-><SS[3,2],SS[1,3],SS[2,1]>:
SS_0_1:=simplify((map(diff,RR_0_1,varphi1)*varphi1_p+map(diff,RR_0_1,varphi2)*varphi2_p).Transpose(RR_0_1));
ww_0_1:=ww(SS_0_1);
JJw_0_1:=Jacobian(ww_0_1,convert(qq_p,list));
SS_0_2:=simplify((map(diff,RR_0_2,varphi1)*varphi1_p+map(diff,RR_0_2,varphi2)*varphi2_p).Transpose(RR_0_2));
ww_0_2:=ww(SS_0_2);
JJw_0_2:=Jacobian(ww_0_2,convert(qq_p,list));Da es keine translatorische Bewegungen gibt, sind die entsprechenden Jacobi-Matrizen 0Tr\303\244gheitsmatrizen in den jeweiligen k\303\266rperfesten Koordinatensystemen sowie im InertialsystemII_1_s1:=DiagonalMatrix([I_xx_s1,I_yy_s1,I_zz_s1]);
II_0_s1:=simplify(simplify(RR_0_1.II_1_s1.RR_0_1^+)):II_2_s2:=DiagonalMatrix([I_xx_s2,I_yy_s2,I_zz_s2]);
II_0_s2:=simplify(simplify(RR_0_2.II_2_s2.RR_0_2^+)):Berechnung der Teile der MassenmatrixRotation Starrk\303\266rper 1:MMr_1:=simplify(JJw_0_1^+.II_0_s1.JJw_0_1,trig);Rotation Starrk\303\266per 2:MMr_2:=simplify(JJw_0_2^+.II_0_s2.JJw_0_2,trig);Gesamte Massenmatrix:MM:=MMr_1+MMr_2;Potentielle Energie der Federn (entspannte Lagen bei varphi1=0 bzw. varphi1-varphi2=0)V1:=1/2*c1*varphi1^2;
V2:=1/2*c12*(varphi2-varphi1)^2;
V:=V1+V2:Verallgemeinerte Kr\303\244fte und Momentettau_1:=<-tau1,0,0>;
ttau_2:=<-tau2,0,0>;
ttau_d1:=<-d1*varphi1_p,0,0>;
ttau_d12:=<-d12*(varphi1_p-varphi2_p),0,0>;
ffq:=JJw_0_1^+.(ttau_1+ttau_d1+ttau_d12)+JJw_0_2^+.(ttau_2-ttau_d12);Berechnung des Vektors der Potentialkr\303\244fte g(q)gg(q):=Jacobian(<V>,convert(qq,list))^+;Berechnung der Coriolismatrix C(q,qp)Calc_CC := proc(MM::Matrix,qq::Vector,qqp::Vector)::Matrix;
local i,j,k,qql,qqlp,CCh,N,ccijk;
description "Calculate SS(omega) from given rotation matrix RR, vector qq of generalized coordinates and velocities qqp";
qql:=convert(qq,list);
qqlp:=convert(qqp,list);
N:=numelems(qql);
CCh:=Matrix(N,N);
for j from 1 to N do
for k from 1 to N do
for i from 1 to N do
ccijk:=1/2*(diff(MM[k,j],qql[i])+diff(MM[k,i],qql[j])-diff(MM[i,j],qql[k]));
CCh[k,j]:=CCh[k,j]+ ccijk*qqlp[i];
od;
od;
od;
CCh:=simplify(CCh);
end proc:CC:=Calc_CC(MM,qq,qq_p);Damit folgen die Bewegungsgleichungen in der Darstellung von (5.98) des Skriptums