V-SHAPED CANTILEVERS

Remove["Global`*"]

Highlights

Abstract

Introduction

Off[General::spell,General::spell1];

Needs["Graphics`Arrow`"];

tera  = 10^12  ; micro = 10^-6;
giga  = 10^9   ; nano  = 10^-9; 
mega  = 10^6   ; pico  = 10^-12;
kilo  = 10^3   ; femto = 10^-15; 
milli = 10.^-3 ; atto  = 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. Pi 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[{.0200}]}};

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-x/L) B d^3/12;

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

Clear[d1Z1,d1Z2];
d1Z1[x1_] := Integrate[d2Z1[x],{x,0,x1}];
d1Z2[x1_] := Integrate[d2Z2[x],{x,L1,x1}]+
d1Z1[L1];

Clear[Z1,Z2];
Z1[x2_] := Integrate[d1Z1[x1],{x1,0,x2}];
Z2[x2_] := Integrate[d1Z2[x1],{x1,L1,x2}]+
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_] := Integrate[d2Z1[x],{x,0,x1}];
d1Z2[x1_] := Integrate[d2Z2[x],{x,L1,x1}]+
d1Z1[L1];

Z1[x2_] := Integrate[d1Z1[x1],{x1,0,x2}];
Z2[x2_] := Integrate[d1Z2[x1],{x1,L1,x2}]+
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(Integrate[
2 w d^3/12 D[x^(n+1),{x,2}]*
D[x^(m+1),{x,2}],{x,0,L1}]+
Integrate[
B d^3/12 (1-x/L) D[x^(n+1),{x,2}]*
D[x^(m+1),{x,2}],{x,L1,L}]),
{n,1,3},{m,1,3}]/.geo;

b =
Table[
rho(Integrate[
2 w d x^(n+1) x^(m+1),{x,0,L1}]+
Integrate[
B d(1-x/L) x^(n+1) x^(m+1),{x,L1,L}]),
{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[1/(2 Pi) Sqrt[ddt[[1,2]]]/kilo]," KHz"]
Print["   f2 = ",N[1/(2 Pi) Sqrt[ddt[[2,2]]]/kilo]," KHz"]
Print["   f3 = ",N[1/(2 Pi) Sqrt[ddt[[3,2]]]/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 [um]","Z(x) [a.u.]"},
Evaluate[opt]];