Starrk\303\266rper mit DrehfederErstellt: 31.5.2020
Ersteller: W. Kemmetm\303\274ller
\303\204nderungen: restart;
with(LinearAlgebra):
with(VectorCalculus,Jacobian):
with(VectorCalculus,SpaceCurve):
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wie im SkriptumFreiheitsgrade, zeitlich Ableitungenqq:=<varphi>;
qqp:=<omega>;
qqpp:=<alpha>;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(qqp,list)[i],i=1..Dimension(qq))};
set_qp2qpt:={seq(convert(qqp,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(qqpp,list)[i],i=1..Dimension(qq))};
set_qpp2qppt:={seq(convert(qqpp,list)[i]=diff(convert(qq,list)[i](t),t$2),i=1..Dimension(qq))};Lage der Schwerpunkte f\303\274r varphi=0pp_0_s1_I:=<0,Ls/2,0>;
pp_0_s2_I:=<Ls/4,Ls,0>;Gesamter Schwerpunkt f\303\274r varphi=0pp_0_s_I:=(m*pp_0_s1_I+m/2*pp_0_s2_I)/(m+m/2);Tr\303\244gheitsmomente um z-Achse f\303\274r die beiden St\303\244beIzz_s1:=m*Ls^2/12;
Izz_s2:=m/2*(Ls/2)^2/12;Abstand der Schwerpunkte der Teilk\303\266per zum gesamten Schwerpunktpp_ss1:=pp_0_s_I-pp_0_s1_I;
pp_ss2:=pp_0_s_I-pp_0_s2_I;Gesamtes Tr\303\244gheitsmomentIzz_s:=Izz_s1+Izz_s2+m*(pp_ss1^+.pp_ss1)+m/2*(pp_ss2^+.pp_ss2);Lage des Schwerpunkts als Funktion von varphiRR_z:=phi-><<cos(phi),sin(phi),0>|<-sin(phi),cos(phi),0>|<0,0,1>>;RR_0_1:=RR_z(-varphi);pp_0_s:=RR_0_1.pp_0_s_I;Translatorische Geschwindigkeiten, Manipulator Jacobi-MatrizenJJv_0_s:=Jacobian(pp_0_s,convert(qq,list));
vv_0_s:=JJv_0_s.qqp;Kinetische EnergieTt:=simplify(1/2*3*m/2*vv_0_s^+ . vv_0_s);Tr:=1/2*Izz_s*omega^2;T:=Tr+Tt:Potentielle Energie der FederVF:=1/2*cF*varphi^2;Potentielle EnergieVg:=simplify(3*m/2*g*pp_0_s[2]);V:=VF+Vg:Vektor der verallgemeinerten Kr\303\244fte und Momenteff:=<f*cos(varphi),-f*sin(varphi),0>;pp_0_f:=Ls*<sin(varphi),cos(varphi),0>;ffq_f:=map(simplify,Jacobian(pp_0_f,convert(qq,list))^+ .ff)[1];Terme der BewegungsgleichungL:=T-V;T1:=diff(L,omega);
T2:=eval(eval(eval(diff(eval(T1,set_q2qt union set_qp2qpt),t),set_qppt2qpp),set_qpt2qp),set_qt2q);
T3:=diff(L,varphi);Bewegungsgleichungenode_varphi:=T2-T3=ffq_f;Berechnung mittels homogener TransformationenDefinition der elementaren DrehmatrizenRR_z:=phi-><<cos(phi),sin(phi),0>|<-sin(phi),cos(phi),0>|<0,0,1>>;DrehmatrizenRR_0_1:=RR_z(-varphi);Hinweis zur Nomenklatur: RR_j_k bezeichnet die Matrix R_j^k.Verschiebungsvektoren der Koordinatensystemedd_0_1:=<0,0,0>;Vektoren zu den Schwerpunkte und zum Angriffspunkten der Reibkraft in den jeweiligen k\303\266rperfesten Koordinatensystemenpp_1_s1:=<0,Ls/2,0>;
pp_1_s2:=<Ls/4,Ls,0>;
pp_1_f:=<0,Ls,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);Lage der relevanten Punkte im InertialkoordinatensystemPP_0_s1:=HH_0_1.PP(pp_1_s1);
PP_0_s2:=HH_0_1.PP(pp_1_s2);
PP_0_f:=HH_0_1.PP(pp_1_f);Jacobi-MatrizenJJv_0_s1:=Jacobian(convert(PP_0_s1[1..3],Vector),convert(qq,list));
JJv_0_s2:=Jacobian(convert(PP_0_s2[1..3],Vector),convert(qq,list));
JJv_0_f:=Jacobian(convert(PP_0_f[1..3],Vector),convert(qq,list));
ww:=SS-><SS[3,2],SS[1,3],SS[2,1]>:
SS_0_1:=simplify((map(diff,RR_0_1,varphi)*omega).Transpose(RR_0_1)):
ww_0_1:=ww(SS_0_1);
JJw_0_1:=Jacobian(convert(ww_0_1,Vector),convert(qqp,list));MassenmatrixMMt_s1:=m*combine(JJv_0_s1^+.JJv_0_s1,trig);
MMr_s1:=Izz_s1*combine(JJw_0_1^+.JJw_0_1,trig);MMt_s2:=m/2*combine(JJv_0_s2^+.JJv_0_s2,trig);
MMr_s2:=Izz_s2*combine(JJw_0_1^+.JJw_0_1,trig);MM:=simplify((MMt_s1+MMr_s1+MMt_s2+MMr_s2));Vektor der Potentialkr\303\244fte (Potentielle Energie wird von oben \303\274bernommen)gg:=Jacobian(<V>,convert(qq,list))^+;Vektor der verallgemeinerten Kr\303\244fteffq_f:=simplify(JJv_0_f^+. ff)[1];CoriolismatrixCalc_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:=simplify(Calc_CC(MM,qq,qqp));Damit ergeben sich die gleichen Bewegungsgleichung wie nach dem L\303\266sungsweg aus dem Skriptum