Manipulate[
If[on1flag,on1=1,on1=0];
If[on2flag,on2=1,on2=0];
If[on3flag,on3=1,on3=0];
If[on4flag,on4=1,on4=0];
ParametricPlot[{on1*Sin[(a+afine) t+phase1]Exp[-xdamp1 t] +on2*Sin[(b+bfine) t+phase2]Exp[-xdamp2 t],on3*Sin[(c+cfine) t+phase3]Exp[-ydamp1 t]+on4*Sin[(d+dfine) t+phase4]Exp[-ydamp2 t]},{t,0,tmax},PlotRange->All,ImageSize->{500,500},MaxRecursion->RecMax,Axes->ShowAxis]
,Style["Pendulum 1 - x axis",Bold]
,{{a,1,"Frequency - coarse tuning"},1,10,0.1,Appearance->"Labeled"}
,{{afine,0,"Frequency - fine tuning"},0,0.099,0.001,Appearance->"Labeled"}
,{{phase1,Pi/2,"Phase"},0,2Pi,Pi/16,Appearance->"Labeled"}
,{{xdamp1,0.02,"Damping factor"},0,0.25,Appearance->"Labeled"}
,{{on1flag,True,"Activate"},{True,False}}
,Delimiter
,Style["Pendulum 2 - x axis",Bold]
,{{b,6,"Frequency"},1,10,0.1,Appearance->"Labeled"}
,{{bfine,0,"Frequency - fine tuning"},0,0.099,0.001,Appearance->"Labeled"}
,{{phase2,3Pi/2,"Phase"},0,2Pi,Pi/16,Appearance->"Labeled"}
,{{xdamp2,0.02,"Damping factor"},0,0.25,Appearance->"Labeled"}
,{{on2flag,True,"Activate"},{True,False}}
,Delimiter
,Style["Pendulum 3 - y axis",Bold]
,{{c,1,"Frequency"},1,10,0.1,Appearance->"Labeled"}
,{{cfine,0,"Frequency - fine tuning"},0,0.099,0.001,Appearance->"Labeled"}
,{{phase3,0,"Phase"},0,2Pi,Pi/16,Appearance->"Labeled"}
,{{ydamp1,0.02,"Damping factor"},0,0.25,Appearance->"Labeled"}
,{{on3flag,True,"Activate"},{True,False}}
,Delimiter
,Style["Pendulum 4 - y Axis",Bold]
,{{d,6,"Frequency"},1,10,0.1,Appearance->"Labeled"}
,{{dfine,0,"Frequency - fine tuning"},0,0.099,0.001,Appearance->"Labeled"}
,{{phase4,0,"Phase"},0,2Pi,Pi/16,Appearance->"Labeled"}
,{{ydamp2,0.02,"Damping factor"},0,0.25,Appearance->"Labeled"}
,{{on4flag,True,"Activate"},{True,False}}
,Delimiter
,{{tmax,100,"time"},1,1000,Appearance->"Labeled",ControlPlacement->Left}
,{{RecMax,10,"MaxRecursion"},1,15,1,Appearance->"Labeled"}
,{{ShowAxis,False,"Show Axes"},{True,False}}
,ControlPlacement->Left
]