UNIFORM 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

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_] := Pi r^4/4

Clear[Jc];
Jc[r_] := Pi 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/L^3 Ic[r]
kcthetaz[L_,r_,Em_] := 2 Em/L^2 Ic[r]

Rectangular Cantilever

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

Force in the $x$ Direction: Bending

Circular Cantilever

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

Rectangular Cantilever

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

Force in the $y$ Direction: Twisting

Circular Cantilever

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

Rectangular Cantilever

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

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_]    := Pi r^2;
Ar[w_,d_] := w d;

f1C = 1/(2 Pi) (rt1/L)^2*
Sqrt[Em Ic[r]/(rho Ac[r])]/.geoC;
f2C = 1/(2 Pi) (rt2/L)^2*
Sqrt[Em Ic[r]/(rho Ac[r])]/.geoC;
f3C = 1/(2 Pi) (rt3/L)^2*
Sqrt[Em Ic[r]/(rho Ac[r])]/.geoC;

f1R = 1/(2 Pi) (rt1/L)^2*
Sqrt[Em Ir[w,d]/(rho Ar[w,d])]/.geoR;
f2R = 1/(2 Pi) (rt2/L)^2*
Sqrt[Em Ir[w,d]/(rho Ar[w,d])]/.geoR;
f3R = 1/(2 Pi) (rt3/L)^2*
Sqrt[Em Ir[w,d]/(rho Ar[w,d])]/.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 [um]","dz [a.u.]"},
Evaluate[opt]];

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