UNIFORM CANTILEVERS

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[{.0500}], 
            Dashing[{.0200}], Dashing[{.0200}], Dashing[{.0001}]}};

Operating Conditions

Circular Cantilever

geoC
       = {Em -> EPtIr, nu -> nuPtIr, rho -> rhoPtIr, 
        L -> 5 milli, r -> 25 micro, h -> 1 milli};

Rectangular Cantilever

geoR
       = {Em -> ESi, nu -> nuSi, rho -> rhoSi, 
        L -> 225 micro, w -> 35 micro, d -> 5 micro, 
        h -> 5 micro};

Area Moments of Inertia

Circular Cantilever

Clear[Ic]; Ic[r_] := ((π (r^4))/4)

Clear[Jc]; Jc[r_] := ((π (r^4))/2)

Rectangular Cantilever

Clear[Ir]; Ir[w_, d_] := ((w (d^3))/12)

Clear[Jr]; Jr[w_, d_] := Ir[w, d] + Ir[d, w]

Clear[G]; G[Em_, nu_] := (Em/(2 (1 + nu)))

Force in the $z$ Direction: Bending

Circular Cantilever

Clear[kcz, kcthetaz]; kcz[L_, r_, Em_] := ((3 Em Ic[r])/(L^3))kcthetaz[L_, r_, Em_] := ((2 Em Ic[r])/(L^2))

Rectangular Cantilever

Clear[krz, krthetaz]; krz[L_, w_, d_, Em_] := ((3 Em Ir[w, d])/(L^3))krthetaz[L_, w_, d_, Em_] := ((2 Em Ir[w, d])/(L^2))

Force in the $x$ Direction: Bending

Circular Cantilever

Clear[kcx, kcthetax]; kcx[L_, r_, h_, Em_] := ((2 Em Ic[r])/(h (L^2)))kcthetax[L_, r_, h_, Em_] := ((2 Em Ic[r])/(h L))

Rectangular Cantilever

Clear[krx, krthetax]; krx[L_, r_, d_, h_, Em_] := ((2 Em Ir[w, d])/(h (L^2)))krthetax[L_, w_, d_, h_, Em_] := ((2 Em Ir[w, d])/(h L))

Force in the $y$ Direction: Twisting

Circular Cantilever

Clear[kcthetay]; kcthetay[L_, r_, h_, Em_, nu_] := ((G[Em, nu] Jc[r])/(h L))

Rectangular Cantilever

Clear[krthetay]; krthetay[L_, w_, d_, h_, Em_, nu_] := 
      ((G[Em, nu] Jr[w, d])/(h L))

Summary of Results

Circular Cantilever

Print["kcz = ", kcz[L, r, Em]]Print["kcx = ", kcx[L, r, h, Em]]Print["kcthetaz = ", kcthetaz[L, r, Em]]Print["kcthetax = ", kcthetax[L, r, h, Em]]Print["kcthetay = ", kcthetay[L, r, h, Em, nu]]

Rectangular Cantilever

Print["krz = ", krz[L, w, d, Em]]Print["krx = ", krx[L, w, d, h, Em]]Print["krthetaz = ", krthetaz[L, w, d, Em]]Print["krthetax = ", krthetax[L, w, d, h, Em]]Print["krthetay = ", krthetay[L, w, d, h, Em, nu]]

Numerical Examples

Spring Constant

Circular Cantilever

Print["bending kcz       = ", N[kcz[L, r, Em] /. geoC], 
      " [N/m]"]Print["bending kcx       = ", N[kcx[L, r, h, Em] /. geoC], 
      " [N/m]"]Print["bending kcthetaz  = ", N[kcthetaz[L, r, Em] /. geoC], 
      " [rad/N]"]Print["bending kcthetax  = ", N[kcthetax[L, r, h, Em] /. geoC], 
      " [rad/N]"]Print["twisting kcthetay = ", N[kcthetay[L, r, h, Em, nu] /. geoC], 
      " [rad/N]"]

Rectangular Cantilever

Print["bending krz       = ", N[krz[L, w, d, Em] /. geoR], 
      " [N/m]"]Print["bending krx       = ", N[krx[L, w, d, h, Em] /. geoR], 
      " [N/m]"]Print["bending krthetaz  = ", N[krthetaz[L, w, d, Em] /. geoR], 
      " [rad/N]"]Print["bending krthetax  = ", N[krthetax[L, w, d, h, Em] /. geoR], 
      " [rad/N]"]Print["twisting krthetay = ", 
      N[krthetay[L, w, d, h, Em, nu] /. geoR], " [rad/N]"]

Bending Resonance Frequencies

rt1 = 
      FindRoot[Cos[xi] Cosh[xi] + 1 == 0, {xi, 1}][[1, 
        2]]; rt2 = 
      FindRoot[Cos[xi] Cosh[xi] + 1 == 0, {xi, 5}][[1, 
        2]]; rt3 = 
      FindRoot[Cos[xi] Cosh[xi] + 1 == 0, {xi, 8}][[1, 
        2]];

Clear[Ac, Ar]; Ac[r_] := π (r^2); Ar[w_, d_] := w d;

f1C = 
      ((((((rt1/L)))^2) (Sqrt[((Em Ic[r])/(rho Ac[r]))]))/(2 
            π)) /. geoC; f2C = 
      ((((((rt2/L)))^2) (Sqrt[((Em Ic[r])/(rho Ac[r]))]))/(2 
            π)) /. geoC; f3C = 
      ((((((rt3/L)))^2) (Sqrt[((Em Ic[r])/(rho Ac[r]))]))/(2 
            π)) /. geoC;

f1R = 
      ((((((rt1/L)))^2) (Sqrt[((Em Ir[w, d])/(rho Ar[w, d]))]))/(2 
            π)) /. geoR; f2R = 
      ((((((rt2/L)))^2) (Sqrt[((Em Ir[w, d])/(rho Ar[w, d]))]))/(2 
            π)) /. geoR; f3R = 
      ((((((rt3/L)))^2) (Sqrt[((Em Ir[w, d])/(rho Ar[w, d]))]))/(2 
            π)) /. geoR;

Print[" f01C = ", N[(f1C/kilo)], " KHz"]Print[" f02C = ", N[(f2C/kilo)], " KHz"]Print[" f03C = ", N[(f3C/kilo)], " KHz"]

Print[" f1R = ", N[(f1R/kilo)], " KHz"]Print[" f2R = ", N[(f2R/kilo)], " KHz"]Print[" f3R = ", N[(f3R/kilo)], " KHz"]

Characteristic Functions

c11 = ((Sin[rt1] - Sinh[rt1])/(Cos[rt1] + Cosh[rt1])); c12 = ((Sin[rt2] - Sinh[rt2])/(Cos[rt2] + Cosh[rt2])); c13 = ((Sin[rt3] - Sinh[rt3])/(Cos[rt3] + Cosh[rt3]));

c21c = (rt1/L) /. geoC; c22c = (rt2/L) /. geoC; c23c = (rt3/L) /. geoC;

c21r = (rt1/L) /. geoR; c22r = (rt2/L) /. geoR; c23r = (rt3/L) /. geoR;

Clear[dz1c, dz2c, dz3c]; dz1c[x_] := 
      (Cos[c21c x] - Cosh[c21c x]) + 
        c11 (Sin[c21c x] - Sinh[c21c x]); dz2c[x_] := 
      (Cos[c22c x] - Cosh[c22c x]) + 
        c12 (Sin[c22c x] - Sinh[c22c x]); dz3c[x_] := 
      (Cos[c23c x] - Cosh[c23c x]) + 
        c13 (Sin[c23c x] - Sinh[c23c x]);

Clear[dz1r, dz2r, dz3r]; dz1r[x_] := 
      (Cos[c21r x] - Cosh[c21r x]) + 
        c11 (Sin[c21r x] - Sinh[c21r x]); dz2r[x_] := 
      (Cos[c22r x] - Cosh[c22r x]) + 
        c12 (Sin[c22r x] - Sinh[c22r x]); dz3r[x_] := 
      (Cos[c23r x] - Cosh[c23r x]) + 
        c13 (Sin[c23r x] - Sinh[c23r x]);

Fig2x2 = 
      Plot[{dz1c[x milli], dz2c[x milli], dz3c[x milli]}, {x, 0.001, 
          (L/milli) /. geoC}, PlotRange -> All, 
        FrameLabel -> {"x in um", "dz in a.u."}, Evaluate[opt]];

Fig2x3 = 
      Plot[{dz1r[x milli], dz2r[x milli], dz3r[x milli]}, {x, 0.001, 
          (L/milli) /. geoR}, 
        FrameLabel -> {"x in um", "dz in a.u."}, Evaluate[opt]];