Calculations of suspended mirror frequencies

Shanti Rao, rao_s@ligo.caltech.edu

Standoffs

All dimensions in SI

D1  =  0.0254 * 0.025;  
     
L0  =  9.78* .0254;     
L8  =   1.502*0.0254;  
L9  =  0.438 * 0.0254;  
L4  = 0.0254 * (3/16);    
L5  = 2*0.0254;    
L6  =  .1016;    
D2  = 30;

Dimensions

L1  =  
        Sqrt[(((L0 - L4))^(2 )) +  (L3^2)];  
L3  =  L5 Cos[ArcTan[L4/L5]] +  std; (* 
      1/2 distance between lower contact points *) 
 (* standoff diameter :  std  =  
          0.001;*)  

L2  =   (L8/2)  +  (((2 L3  - 
                  L8))/2) ((L9/(L0 - L4)))   
  
g  =  9.81;   
ρ  =  2.2* (10^3);   
L7  =  wedge * ((2 Pi)/(360*60));   
MagnetMass  =.      
M1  =  0.001;   
MirrorMass  =  ρ π (L5^2) L6;   
MagnetRadius  =  0.0247;
StandardDimensions  =  {std -> D1,  
          wedge ->  D2};
TCDimensions  =  
        Append[StandardDimensions, MagnetMass -> M1];

Inertia

Inertia  =  (((L5^2)/4) + ((L6^2)/12)) ;  
ϕInertia  =  (((MirrorMass*Inertia  +  4 (MagnetRadius^2) MagnetMass))/(MirrorMass  +  4 MagnetMass));
θInertia  =  (MirrorMass*Inertia  +  
              4 ((L6^2)/2  +  
                    (MagnetRadius^2)) MagnetMass)/(MirrorMass  + 
              4  MagnetMass);
CenterOfMass  =  ((L7 (L5^2))/(4 L6));

Frequencies

ωz  =  
        Sqrt[(g/(L0-CenterOfMass ))];
ωθ  =  
        Sqrt[ ((g (L4 - CenterOfMass))/θ)Inertia  ];
ωϕ  =  
        Sqrt [ ((g (L3 L3  -  L2 L3))/(4 (((L1 L1  -  2 L3 L3  +  2 L2 L3  -  L2 L2))^(1/2)) ϕInertia))];
\(ωϕ2\  = \ 
      Sqrt[\((g\/\(2\ Inertia\))\) \((\(2\ L2\ L3\)\/\(L0\  - \ L4\))\)]\ ;  (*\ susres\ calculation\ method\ *) 
νz\  = \ ωz\/\(2\ π\);\)
νθ  =  ω(θ/(2 π));
νϕ  =  ω(ϕ/(2 π));
kz  =  ω(z^2) MirrorMass;
kθ  =  ω(θ^(2 )) 
          Inertia;
kϕ  =  ω(ϕ^2) Inertia;

Numerical values of the angular frequencies

CenterOfMass /. TCDimensions
N[MirrorMass  /. TCDimensions]
N[{νz, νϕ, νθ} /. TCDimensions]
{ωz,  ωϕ, ωθ}  /. 
      TCDimensions

What-if?

Effect of magnet mass

Plot[{ϕInertia, θInertia}, {MagnetMass, 0, M1}, 
        PlotStyle -> {RGBColor[1, 0, 0], RGBColor[0, 1, 0]}, 
        PlotRange -> All];

Plot[{νϕ /. StandardDimensions, νθ /. 
            StandardDimensions}, {MagnetMass, 0, M1}, 
        PlotStyle -> {RGBColor[1, 0, 0], RGBColor[0, 1, 0]}, 
        PlotRange -> All];

pitch-yaw as a function of wedge and standoff size

Plot3D[νθ  -  νϕ /. 
          MagnetMass -> M1, {std, 0, D1}, {wedge, 0, D2}, 
        AxesLabel -> {"standoff", "wedge", "νθ - \
νϕ"}];

Three frequencies as a function of radius, for zero wedge and a standard standoff, for constant thickness

Plot[{νz , νθ /. TCDimensions , νϕ /. 
            TCDimensions }, {wedge, -D2, D2}, 
        PlotStyle -> {RGBColor[1, 0, 0], RGBColor[0, 1, 0], 
            RGBColor[0, 0, 1]}];

Z - Pitch frequency as a function of wedge angle (degrees)

Plot[νz  -  νθ  /.  {MagnetMass -> M1, 
            std -> D1}, {wedge, -D1,  D2}];

Yaw frequency as a function of standoff size, in meters

Plot[νϕ /. MagnetMass -> M1 , {std, 0, 
          D1}];