V-SHAPED 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

geo
       = {rho -> rhoSi, Em -> ESi, L -> 190 micro, 
        L1 -> 87.5 micro, B -> 165 micro, w -> 33.3 micro, 
        d -> 0.6 micro, del -> 7.1 micro, 
        htip -> 3.5 micro, Fx -> 1 nano, Fz -> 1 nano};

Spring Constant in the $z$ Direction

General Solution

Clear[My, I1, I2]; Mz[x_] := (L - x) Fz; I1[x_] := ((2 w (d^3))/12); I2[x_] := (1/12) (1 - (x/L)) B (d^3);

Clear[d2Z1, d2Z2]; d2Z1[x_] := Mz[x]/(Em I1[x]); d2Z2[x_] := Mz[x]/(Em I2[x]);

Clear[d1Z1, d1Z2]; \(d1Z1[x1_] := ∫\_0\%x1 d2Z1[x] \[DifferentialD]x; 

    d1Z2[x1_]\  := \ 
      Integrate[d2Z2[x]\ , {x, L1, x1}, \ GenerateConditions -> False]\  + 

        \ \ \ \ \ \ \ \ \ \ \ \ \ d1Z1[L1]; 
\)

Clear[Z1, Z2]; Z1[x2_] := (∫_0^x2) d1Z1[x1] \[DifferentialD]x1; Z2[x2_] := (∫_L1^x2) d1Z2[x1] \[DifferentialD]x1 + Z1[L1];

kz = Re[(Fz/Z2)[L - del]]; ZMax = (1/kz); ktheta = d1Z2[L - del]/Fz;

Print["kz     = ", N[kz /. geo], " N/m"]Print["Z Max  = ", N[ZMax /. geo], " nm/nN"]Print["ktheta = ", N[((ktheta nano)/milli) /. geo], 
      " mrad/nN"]

Special Cases

V-Shaped Cantilever with a Triangular Recess

Clear[kz1, ZMax, ktheta]; kz1 = (Fz/Z2)[L]; ZMax = (1/kz1); ktheta = d1Z2[L]/Fz;

Print["kz1    = ", 
      Simplify[kz1 /. L1 -> L (1 - ((2 w)/B))]]Print["ZMax   = ", 
      Simplify[ZMax /. L1 -> L (1 - ((2 w)/B))]]Print["ktheta = ", 
      Simplify[ktheta /. L1 -> L (1 - ((2 w)/B))]]

V-Shaped Cantilever without a Recess

Print["kz1    = ", Simplify[kz1 /. L1 -> 0]]Print["ZMax   = ", Simplify[ZMax /. L1 -> 0]]Print["ktheta = ", Simplify[ktheta /. L1 -> 0]]

Trapezoidal Cantilever without a Recess

m = ((1 - (B1/B))/L); g = 
      m L - (((m^2) (L^2))/(2 (1 - m L))) + 
        (1 - m L) Log[1 - m L]; kz2 = Simplify[((Em B (d^3) (m^3))/(12 g (m L - 1)))]; 
    Print["kz2 = ", kz2]

Spring Constant in the $x$ Direction

General Solution

Clear[Mx]; Mx[x_] := htip Fx;

Clear[d2Z1, d2Z2, d1Z1, d1Z2, Z1, Z2]; d2Z1[x_] := Mx[x]/(Em I1[x]); d2Z2[x_] := Mx[x]/(Em I2[x]);

d1Z1[x1_] := (∫_0^x1) d2Z1[x] \[DifferentialD]x; d1Z2[x1_] := 
      (∫_L1^x1) d2Z2[x] \[DifferentialD]x + d1Z1[L1];

Z1[x2_] := (∫_0^x2) d1Z1[x1] \[DifferentialD]x1; Z2[x2_] := (∫_L1^x2) d1Z2[x1] \[DifferentialD]x1 + Z1[L1];

Clear[kx, ZMax, ktheta]; kx = Re[(Fx/Z2)[L - del]]; ZMax = (1/kx); ktheta = d1Z2[L - del];

Print["kx     = ", Re[N[kx /. geo]], " N/m"]Print["Z Max  = ", Re[N[ZMax /. geo]], " nm/nN"]Print["ktheta = ", Re[N[(ktheta/milli) /. geo]], " mrad/nN"]

Special Cases

V-Shaped Cantilever with a Triangular Recess

Clear[kx1, ZMax, ktheta]; kx1 = (Fx/Z2)[L]; ZMax = (1/kx1); ktheta = d1Z2[L - del];

Print["kx1      = ", 
      Simplify[kx1 /. L1 -> L (1 - ((2 w)/B))]]Print["ZMax     = ", 
      Simplify[ZMax /. L1 -> L (1 - ((2 w)/B))]]Print["ktheta = ", 
      Simplify[ktheta /. L1 -> L (1 - ((2 w)/B))]]

V-Shaped Cantilever without a Recess

Print["kx2    = ", Simplify[kx1 /. L1 -> 0]]Print["ZMax   = ", Simplify[ZMax /. L1 -> 0]]Print["ktheta = ", Simplify[ktheta /. L1 -> 0]]

Resonance Frequencies

\(\(a = 
      Table[Em\ \((
              ∫\_0\%L1
                    \( 2\/12\ w\ d\^3\ ∂\_{x, 2}x\^\(n + 1\)\ 
                      ∂\_{x, 2}x\^\(m + 1\)\) \[DifferentialD]x + 
                ∫\_L1\%L
                    \( 1\/12\ B\ d\^3\ \((1 - x\/L)\)\ 
                      ∂\_{x, 2}x\^\(n + 1\)\ 
                      ∂\_{x, 2}x\^\(m + 1\)\) \[DifferentialD]x)
              \), {n, 1, 3}, {m, 1, 3}] /. geo; \)\)

b = 
      Table[rho 
            ((∫_0^L1) 2 w d (x^(n + 1)) (x^(m + 1)) 
                    \[DifferentialD]x + 
                (∫_L1^L)
                     B d (1 - (x/L)) (x^(n + 1)) (x^(m + 1)) 
                    \[DifferentialD]x), {n, 1, 3}, {m, 1, 3}] /. geo;

ddt = NRoots[Det[a - ww b] == 0, ww];

Print["    L = ", N[(L/micro)] /. geo, " um"]Print["   L1 = ", (L1/micro) /. geo, " um"]Print["    B = ", (B/micro) /. geo, " um"]Print["    w = ", (w/micro) /. geo, " um"]Print["    d = ", (d/micro) /. geo, " um"]Print["  del = ", (del/micro) /. geo, " um"]Print[" htip = ", (htip/micro) /. geo, " um"]Print["   kz = ", N[kz /. geo], " N/m"]Print["   kx = ", N[kx /. geo], " N/m"]Print["   f1 = ", 
      N[(Sqrt[ddt])[[1, 2]]/((2 π) 
            kilo)], " KHz"]Print["   f2 = ", 
      N[(Sqrt[ddt])[[2, 2]]/((2 π) 
            kilo)], " KHz"]Print["   f3 = ", 
      N[(Sqrt[ddt])[[3, 2]]/((2 π) 
            kilo)], " KHz"]

Characteristic Functions

m1 = a - ddt[[1, 2]] b /. geo; 
    m2 = a - ddt[[2, 2]] b /. geo; 
    m3 = a - ddt[[3, 2]] b /. geo;

bv1 = NullSpace[m1] /. geo; bv2 = NullSpace[m2] /. geo; bv3 = NullSpace[m3] /. geo;

Clear[poly1, poly2, poly3]; poly1[x_] := 
      bv1[[1, 1]] (x^2) + 
        bv1[[1, 2]] (x^3) + 
        bv1[[1, 3]] (x^4); poly2[x_] := 
      bv2[[1, 1]] (x^2) + 
        bv2[[1, 2]] (x^3) + 
        bv2[[1, 3]] (x^4); poly3[x_] := 
      bv3[[1, 1]] (x^2) + 
        bv3[[1, 2]] (x^3) + 
        bv3[[1, 3]] (x^4);

Fig4x2 = 
      Plot[{(10^14) poly1[x micro], (10^15) poly2[x micro], 
          (10^15) poly3[x micro]}, {x, 0, (L/micro) /. geo}, 
        FrameLabel -> {"x in um", "Z in a.u."}, Evaluate[opt]];