ClearAll[eq1, eq2, eq3, eq4, eq5, eq6, eq7, eq8, eq9, θ1, θ2, θ3, Fn10, Ft10, Fn12, Ft12, Fn23, Ft23, Fn30, Ft30, yc1, xc1, yc2, xc2, yc3, xc3, sol1, sol2, T1, T2, T3]


ics = {θ1[0] == 2, θ2[0] == 3, θ3[0] == 1, θ1'[0] == 0, θ2'[0] == 0, θ3'[0] == 0};
sol1 = NDSolve[{θ1''[t] + 2 θ1[t] == 0, θ2''[t] + 3 θ2[t] == 0, θ3''[t] + 5 θ3[t] == 0, ics}, {θ1, θ2, θ3,θ1', θ2', θ3',θ1'', θ2'', θ3''}, {t, 0, 20}];
(*---Geometry------------------------------------------------*)
xc1[t_] := -Sin[θ1[t]];
yc1[t_] := -Cos[θ1[t]];

xc2[t_] := -2 Sin[θ1[t]] - Cos[θ2[t]];
yc2[t_] := -2 Cos[θ1[t]] - Sin[θ2[t]];

xc3[t_] := -2 - Sin[θ1[t]];
yc3[t_] := -Cos[θ1[t]];
 (*---------------------------------------------------------*)
T1[t_] := 2 + 2 Sin[t];
T2[t_] := 1 + Cos[t];
T3[t_] := Sin[2 t];

(*---Dynamics---------------------*)
eq1 = (Fn10[t] - Fn12[t])*Sin[θ1[t]] + (Ft12[t] - Ft10[t])*Cos[θ1[t]] == D[xc1[t], {t, 2}];
eq2 = (Fn10[t] - Fn12[t])*Cos[θ1[t]] - (Ft12[t] - Ft10[t])*Sin[θ1[t]] == D[yc1[t], {t, 2}];
eq3 = -(Ft12[t] + Ft10[t]) + T1[t] == D[θ1[t], {t, 2}];
(*--------------------------------------*)
eq4 = Fn12[t]*Sin[θ1[t]] - Ft12[t]*Cos[θ1[t]] + Ft23[t]*Cos[θ3[t]] + Fn23[t]*Sin[θ3[t]] == D[xc2[t], {t, 2}];
eq5 = Fn12[t]*Cos[θ1[t]] + Ft12[t]*Sin[θ1[t]] - Ft23[t]*Sin[θ3[t]] + Fn23[t]*Cos[θ3[t]] == D[yc2[t], {t, 2}];
eq6 = Fn12[t]*Cos[θ1[t] + θ2[t]] + Ft12[t]*Sin[θ1[t] + θ2[t]] + Ft23[t]*Sin[θ2[t] + θ3[t]] - Fn23[t]*Cos[θ2[t] + θ3[t]] - T2[t] == D[θ2[t], {t, 2}];
(*------------------------------------*)
eq7 = (Fn30[t] - Fn23[t])*Sin[θ3[t]] + (Ft30[t] - Ft23[t])*Cos[θ3[t]] == D[xc3[t], {t, 2}];
eq8 = (Fn30[t] - Fn23[t])*Cos[θ3[t]] - (Ft30[t] - Ft23[t])*Sin[θ3[t]] == D[yc3[t], {t, 2}];
eq9 = +(Ft30[t] + Ft23[t]) + T3[t] == D[θ3[t], {t, 2}];

(*Solve eq1,eq2,eq3,eq4,eq5,eq6,eq7,eq8,eq9*)
eqns = {eq1, eq2, eq3, eq4, eq5, eq6, eq7, eq8, eq9} /. sol1;
sol2 = NSolve[eqns, {Fn10, Ft10, Fn12, Ft12, Fn23, Ft23, Fn30, Ft30}];


Plot[{Evaluate[θ1[t] /. sol1], Evaluate[θ2[t] /. sol1], Evaluate[θ3[t] /.sol1]]}, {t, 0, 20}]
Plot[{Evaluate[(Ft10[t]^2 + Fn10[t]^2) /. sol2]}, {t, 0, 20}]