CANTILEVER CONVERSION TABLES

Remove["Global`*"]

Highlights

Abstract

Introduction

Off[General::"spell", General::"spell1"]; Needs["Graphics`Arrow`"]; tera = (10^12); micro = (1/(10^6)); giga = (10^9); nano = (1/(10^9)); mega = (10^6); pico = (1/(10^12)); kilo = (10^3); femto = (1/(10^15)); milli = (1/(10.^3)); atto = (1/(10^18)); NSub
       = {c -> 2.99792458 (10^8), e -> (1.6021773/(10^19)), 
        h -> (6.6260755/(10.^34)), hbar -> (1.05457/(10.^34)), 
        K -> (1.380658/(10^23)), me -> (9.109389/(10^31)), 
        mu0 -> ((4. π)/(10^7)), eps0 -> (8.85419/(10^12))}; ESi = 0.179 tera; rhoSi = 2330; nuSi = 0.28; ESiO2 = 0.06 tera; rhoSiO2 = 2200; nuSiO2 = 0.3; ESi3N4 = 0.15 tera; rhoSi3N4 = 3100; nuSi3N4 = 0.24; EAu = 0.08 tera; rhoAu = 19300; nuAu = 0.42; ENi = 0.207 tera; rhoNi = 8900; nuNi = 0.41; EW = 0.345 tera; rhoW = 19300; nuW = 0.28; EPt = 0.17 tera; rhoPt = 21400; nuPt = 0.39; EIr = 0.52 tera; rhoIr = 22500; nuIr = 0.3; EHOPG = 0.2 tera; rhoHOPG = 2250; nuHOPG = 0.3; EPol = 0.001 tera; rhoPol = 1000; nuPol = 0.3; xx = 0.8; EPtIr = xx 0.8 EPt + (1 - xx) EIr; rhoPtIr = xx rhoPt + (1 - xx) rhoIr; nuPtIr = 0.3; opt
       = {PlotRange -> All, AxesOrigin -> {0, 0}, 
        Frame -> True, GridLines -> Automatic, 
        DefaultFont -> {"Times-Roman", 8}, 
        PlotStyle 
          -> {Dashing[{.0001}], Dashing[{.0200}], Dashing[{.0200}], 
            Dashing[{.0200}], Dashing[{.0200}], Dashing[{.0001}]}};

Circular Cantilever

Conversion Table

Clear[rho, Em, L, r, k, f, c1, c2]; k = ((3 π Em (r^4))/(4 (L^3)));

f = ((Sqrt[((4 k)/(rho π (r^2) L))])/(2 π));

Print["f = ", f]

Clear[eq1, eq2, rho, Em, L, r, k, f, c1, c2]; eq1 = k (L^3) == c2 (r^4); eq2 = f (L^2) == c1 r;

fk = 
      Solve[{eq1, eq2}, {f, k}][[1
        ]]; fL = 
      Solve[{eq1, eq2}, {f, L}][[1
        ]]; fr = 
      Solve[{eq1, eq2}, {f, r}][[4
        ]]; \(Lr = 
      \(Solve[{eq1, eq2}, {L, r}]\)[[3
        ]]; 

    Lk = \(Solve[{eq1, eq2}, {k, L}]\)[[2
        ]]; 

    \(kr = \(Solve[{eq1, eq2}, {k, r}]\)[[1
        ]]; \)\)

lst1
       = {"f(L,r)=", fk[[1, 2]], 
        "k(L,r)=", fk[[2, 2]]}; lst2
       = {"f(k,r)=", fL[[1, 2]], 
        "L(k,r)=", fL[[2, 2]]}; lst3
       = {"f(L,k)=", fr[[1, 2]], 
        "r(L,k)=", fr[[2, 2]]}; lst4
       = {"L(f,k)=", Lr[[1, 2]], 
        "r(f,k)=", Lr[[2, 2]]}; lst5
       = {"k(f,r)=", Lk[[1, 2]], 
        "L(f,r)=", Lk[[2, 2]]}; lst6
       = {"k(f,L)=", kr[[1, 2]], 
        "r(f,L)=", kr[[2, 2]]};

TableForm[{lst1, lst2, lst3}]

TableForm[{lst4, lst5, lst6}]

Automatic Conversion

Choosing Two Parameters

c1 = ((Sqrt[((3 Em)/rho)])/(2 π)); c2 = ((3 π Em)/4);

L0 = 0.644933 milli; r0 = 4.80939 micro; k0 = 1; f0 = 10 kilo;

Clear[L, r, k, f]; geo1 = {EPtIr, rhoPtIr, L0, r0, k, f}; geo2 = {EPtIr, rhoPtIr, L0, r, k, f0}; geo3 = {EPtIr, rhoPtIr, L0, r, k0, f}; geo4 = {EPtIr, rhoPtIr, L, r0, k, f0}; geo5 = {EPtIr, rhoPtIr, L, r, k0, f0}; geo6 = {EPtIr, rhoPtIr, L, r0, k0, f};

Clear[L, r, k, f]; geo = geo1;

Em = geo[[1]]; rho = geo[[2]];

Calculation of the Two Other Parameters

Clear[L, r, k, f]; L = geo[[3]]; 
    r = geo[[4]]; test1 = 
      If[geo[[5]] == k && 
          geo[[6]] == f, 
        Print["f = ", 
            N[fk[[1, 2]]]/kilo, 
            " kHz"] && 
          Print["k = ", 
            N[fk[[2, 2]]], " N/m"], 
        0];

Clear[L, r, k, f]; L = geo[[3]]; 
    f = geo[[6]]; test2 = 
      If[geo[[4]] == r && 
          geo[[5]] == k, 
        Print["k = ", N[kr[[1, 2]]], 
            " N/m"] && 
          Print["r = ", 
            N[kr[[2, 2]]]/micro, 
            " um"], 0];

Clear[L, r, k, f]; L = geo[[3]]; 
    k = geo[[5]]; test3 = 
      If[geo[[4]] == r && 
          geo[[6]] == f, 
        Print["f = ", 
            N[fr[[1, 2]]]/kilo, 
            " kHz"] && 
          Print["r = ", 
            N[fr[[2, 2]]]/micro, 
            " um"], 0];

Clear[L, r, k, f]; r = geo[[4]]; 
    f = geo[[6]]; test4 = 
      If[geo[[3]] == L && 
          geo[[5]] == k, 
        Print["k = ", N[Lk[[1, 2]]], 
            " N/m"] && 
          Print["L = ", 
            N[Lk[[2, 2]]]/milli, 
            " mm"], 0];

Clear[L, r, k, f]; k = geo[[5]]; 
    f = geo[[6]]; test5 = 
      If[geo[[3]] == L && 
          geo[[4]] == r, 
        Print["L = ", 
            N[Lr[[1, 2]]]/milli, 
            " mm"] && 
          Print["r = ", 
            N[Lr[[2, 2]]]/micro, 
            " um"], 0];

Clear[l, r, k, f]; r = geo[[4]]; 
    k = geo[[5]]; test6 = 
      If[geo[[3]] == L && 
          geo[[6]] == f, 
        Print["f = ", 
            N[fL[[1, 2]]]/kilo, 
            " kHz"] && 
          Print["L = ", 
            N[fL[[2, 2]]]/milli, 
            " mm"], 0];

Square Cantilever

Conversion Table

Clear[rho, Em, L, w, k, f, c1, c2]; k = ((Em (w^4))/(4 (L^3)));

f = ((Sqrt[((4 k)/(rho π (w^2) L))])/(2 π));

Print["f = ", f]

Clear[rho, Em, L, w, k, f, c1, c2]; eq1 = k (L^3) == c2 (w^4); eq2 = f (L^2) == c1 w;

fk = 
      Solve[{eq1, eq2}, {f, k}][[1
        ]]; fL = 
      Solve[{eq1, eq2}, {f, L}][[1
        ]]; fw = 
      Solve[{eq1, eq2}, {f, w}][[4
        ]]; \(Lw = 
      \(Solve[{eq1, eq2}, {L, w}]\)[[3
        ]]; 

    Lk = \(Solve[{eq1, eq2}, {k, L}]\)[[2
        ]]; 

    \(kw = \(Solve[{eq1, eq2}, {k, w}]\)[[1
        ]]; \)\)

lst1
       = {"f(L,w)=", fk[[1, 2]], 
        "k(L,w)=", fk[[2, 2]]}; lst2
       = {"f(k,w)=", fL[[1, 2]], 
        "L(k,w)=", fL[[2, 2]]}; lst3
       = {"f(L,k)=", fw[[1, 2]], 
        "w(L,k)=", fw[[2, 2]]}; lst4
       = {"L(f,k)=", Lw[[1, 2]], 
        "w(f,k)=", Lw[[2, 2]]}; lst5
       = {"k(f,w)=", Lk[[1, 2]], 
        "L(f,w)=", Lk[[2, 2]]}; lst6
       = {"k(f,L)=", kw[[1, 2]], 
        "w(f,L)=", kw[[2, 2]]};

TableForm[{lst1, lst2, lst3}]

TableForm[{lst4, lst5, lst6}]

Automatic Conversion

Choosing Two Parameters

c1 = N[((Sqrt[(Em/(π rho))])/(2 π))]; c2 = N[(Em/4)];

L0 = 0.411775 milli; w0 = 6.0189 micro; k0 = 1; f0 = 10 kilo;

Clear[geo, L, w, k, f]; geo1 = {EPtIr, rhoPtIr, L0, w0, k, f}; geo2 = {EPtIr, rhoPtIr, L0, w, k, f0}; geo3 = {EPtIr, rhoPtIr, L0, w, k0, f}; geo4 = {EPtIr, rhoPtIr, L, w0, k, f0}; geo5 = {EPtIr, rhoPtIr, L, w, k0, f0}; geo6 = {EPtIr, rhoPtIr, L, w0, k0, f};

Clear[geo, L, w, k, f]; geo = geo1;

Em = geo[[1]]; rho = geo[[2]];

Calculation of the Two Other Parameters

Clear[L, w, k, f]; L = geo[[3]]; 
    w = geo[[4]]; test1 = 
      If[geo[[5]] == k && 
          geo[[6]] == f, 
        Print["f = ", 
            N[fk[[1, 2]]]/kilo, 
            " kHz"] && 
          Print["k = ", 
            N[fk[[2, 2]]], " N/m"], 
        0];

Clear[L, w, k, f]; L = geo[[3]]; 
    f = geo[[6]]; test2 = 
      If[geo[[4]] == w && 
          geo[[5]] == k, 
        Print["k = ", N[kw[[1, 2]]], 
            " N/m"] && 
          Print["w = ", 
            N[kw[[2, 2]]]/micro, 
            " um"], 0];

Clear[L, w, k, f]; L = geo[[3]]; 
    k = geo[[5]]; test3 = 
      If[geo[[4]] == w && 
          geo[[6]] == f, 
        Print["f = ", 
            N[fw[[1, 2]]]/kilo, 
            " kHz"] && 
          Print["w = ", 
            N[fw[[2, 2]]]/micro, 
            " um"], 0];

Clear[L, w, k, f]; w = geo[[4]]; 
    f = geo[[6]]; test4 = 
      If[geo[[3]] == L && 
          geo[[5]] == k, 
        Print["k = ", N[Lk[[1, 2]]], 
            " N/m"] && 
          Print["L = ", 
            N[Lk[[2, 2]]]/milli, 
            " N/m"], 0];

Clear[L, w, k, f]; k = geo[[5]]; 
    f = geo[[6]]; test5 = 
      If[geo[[3]] == L && 
          geo[[4]] == w, 
        Print["L = ", 
            N[Lw[[1, 2]]]/milli, 
            " mm"] && 
          Print["w = ", 
            N[Lw[[2, 2]]]/micro, 
            " um"], 0];

Clear[L, w, k, f]; w = geo[[4]]; 
    k = geo[[5]]; test6 = 
      If[geo[[3]] == L && 
          geo[[6]] == f, 
        Print["f = ", 
            N[fL[[1, 2]]]/kilo, 
            " kHz"] && 
          Print["L = ", 
            N[fL[[2, 2]]]/milli, 
            " mm"], 0];