I have modified the spring functions to oscillate in time, instead of manually adjusting the spring. The problem is that they do not compress and contract properly. Based on the oscillation functions defined, does anyone know how to use them so that the springs compress and contract along with the movement of the masses?

\[Theta]1[t_] = 
  1/72 (40 + E^(-I Sqrt[3/10] t) + E^(I Sqrt[3/10] t) + 
     3 E^(-((I t)/Sqrt[10])) + 3 E^((I t)/Sqrt[10])) \[Pi];

\[Theta]2[t_] = 
  1/72 (88 + E^(-I Sqrt[3/10] t) + E^(I Sqrt[3/10] t) - 
     3 E^(-((I t)/Sqrt[10])) - 3 E^((I t)/Sqrt[10])) \[Pi];

\[Theta]3[t_] = 
  1/36 (68 - E^(-I Sqrt[3/10] t) - E^(I Sqrt[3/10] t)) \[Pi];

Manipulate[
 Show[
  PolarPlot[
   {
    (1 + 0.2*
       TriangleWave[(3*x)/
         2*( \[Theta]2[time] - \[Theta]1[time] - \[Theta]3[time] ) ])
    },
   {x, \[Theta]2[time], \[Theta]1[time]},
   AspectRatio -> Automatic,
   Axes -> None,
   PlotRange -> {  -1.2*1, 1.2*1  },
   PerformanceGoal -> "Quality",
   ImageSize -> Medium
   ],
  PolarPlot[
   {
    (1 + 0.2*
       TriangleWave[(3*x)/
         2*(\[Theta]2[time] - \[Theta]1[time] - \[Theta]3[time])])
    },
   {x, \[Theta]2[time], \[Theta]3[time]},
   AspectRatio -> Automatic,
   Axes -> None,
   PlotRange -> {  -1.2*1, 1.2*1  },
   PerformanceGoal -> "Quality",
   ImageSize -> Medium
   ],
  PolarPlot[
   {
    (1 + 0.2*
       TriangleWave[(3*x)/
         2*(\[Theta]2[time] - \[Theta]1[time] - \[Theta]3[time] )])
    },
   {x, \[Theta]1[time], \[Theta]3[time] - 2 \[Pi]},
   AspectRatio -> Automatic,
   Axes -> None,
   PlotRange -> {  -1.2*1, 1.2*1  },
   PerformanceGoal -> "Quality",
   ImageSize -> Medium
   ],

  Graphics[{
    {(* Mass-1 *)
     Directive[  RGBColor[0, 0, 1], Opacity[0.75]  ],
     Disk[
      {
       1*Cos[  \[Theta]1[time]  ],
       1*Sin[  \[Theta]1[time]  ]
       },
      1/15
      ]
     (* Mass-1 *)},
    {(* Mass-2 *)
     Directive[  RGBColor[0, 0, 1], Opacity[0.75]  ],
     Disk[
      {
       1*Cos[  \[Theta]2[time]  ],
       1*Sin[  \[Theta]2[time]  ]
       },
      1/15
      ]
     (* Mass-2 *)},
    {(* Mass-3 *)
     Directive[  RGBColor[0, 0, 1], Opacity[0.75]  ],
     Disk[
      {
       1*Cos[  \[Theta]3[time]  ],
       1*Sin[  \[Theta]3[time]  ]
       },
      1/15
      ]
     (* Mass-3 *)}
    }]
  ],

 {
  {time, 1 E - 20},
  1 E - 20,
  120,
  ControlType -> Trigger,
  DefaultDuration -> 120.0,
  DisplayAllSteps -> True,
  AnimationRate -> 2.0
  },
 TrackedSymbols :> {time}
 ]

Hi Tim,

is it this you are having in mind(?):

sp[a_, t_] := {(r0 + r1 Cos[a t]) Cos[t], (r0 + r1 Cos[a t]) Sin[t], r1 Sin[a t]};

{r0, r1} = {1, .4};
Manipulate[ParametricPlot3D[sp[200/tt, t], {t, 0, tt}, AspectRatio -> Automatic,
   Boxed -> False, Axes -> None, PlotRange -> {-r0 - r1, r0 + r1}, 
  PerformanceGoal -> "Quality"], {{tt, Pi}, 1, 2 Pi}]

Regards -- Henrik

If you strip the 3rd dimension from Henrik's example you get:

sp[a_, t_] := {(r0 + r1 Cos[a t]) Cos[t], (r0 + r1 Cos[a t]) Sin[t]};

{r0, r1} = {1, .4};
Manipulate[
 ParametricPlot[sp[200/tt, t], {t, 0, tt}, AspectRatio -> Automatic, 
  Axes -> None, PlotRange -> {-r0 - r1, r0 + r1}, 
  PerformanceGoal -> "Quality"], {{tt, Pi}, 1, 2 Pi}]

I agree; that solution works. I had tunnel vision focusing on using the TriangleWave function to define the spring. I am not certain the the TriangleWave function is robust enough to handle 2D oscillation though, so defining the spring as above is a good solution. Thanks!