CANTILEVER CONVERSION TABLES

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

Circular Cantilever

Conversion Table

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

f = 1/(2 Pi) Sqrt[4 k/(rho Pi r^2 L)];

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}][[5]];
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 = 1/(2 Pi) Sqrt[3 Em/rho];
c2 = 3 Pi/4 Em;

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 = 1/4 Em w^4/L^3;

f = 1/(2 Pi) Sqrt[4 k/(rho Pi w^2 L)];

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}][[5]];
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[1/(2 Pi) Sqrt[Em/(Pi rho)]];
c2 = N[1/4 Em];

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];