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Nonlinear_Cantilever_Vibration_varying_input.asv
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Nonlinear_Cantilever_Vibration_varying_input.asv
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%Nonlinear Vibration of a Cantilever Beam
L= 0.662; W= 0.01271; tk= 5.5e-4; %beam dimensions
rho = 7400; E = 165.5e9; %Material properties
Alpha1= 0.4; %0.064;
Alpha2= 0.004;%4.72e-5; %proportional damping coefficients
N= 20; h= L/N; numX=10; deltax= h/numX; %elements variables
TOL=0.01; %convergence criterion for nonlinear loop
deltat= 0.001; %tf=3;
k= tf/deltat; %time variables
g=9.81; ab= 2.97*g; %OMEGA= 17.547; %force variables
Area= W*tk; I= 1/12*W*tk^3; %section properties
%f_constant = 0.0001;
%shape functions and derivatives
xs=0:deltax:h;
PSI(:,1)= 1 - (3/h^2)*xs.^2 + (2/h^3)*xs.^3;
PSI(:,2)= h*((xs/h) -(2/h^2)*xs.^2 + (1/h^3)*xs.^3);
PSI(:,3)= (3/h^2)*xs.^2 - (2/h^3)*xs.^3;
PSI(:,4)= h*(-(1/h^2)*xs.^2 + (1/h^3)*xs.^3);
PSIP(:,1)= -6*(xs/h^2) + (6/h^3)*xs.^2;
PSIP(:,2)= 1 - 4*(xs/h) + (3/h^2)*xs.^2;
PSIP(:,3)= 6*(xs/h^2) - (6/h^3)*xs.^2;
PSIP(:,4)= - 2*(xs/h) + (3/h^2)*xs.^2;
xs4=0:deltax/4:h;
PSIP_f2_4(:,1)= -6*(xs4/h^2) + (6/h^3)*xs4.^2; %FOR EVALUATION OF f2 ONLY
PSIP_f2_4(:,2)= 1 - 4*(xs4/h) + (3/h^2)*xs4.^2;
PSIP_f2_4(:,3)= 6*(xs4/h^2) - (6/h^3)*xs4.^2;
PSIP_f2_4(:,4)= - 2*(xs4/h) + (3/h^2)*xs4.^2;
PSIPP(:,1)= -6/h^2 + (12/h^3)*xs;
PSIPP(:,2)= -4/h + (6/h^2)*xs;
PSIPP(:,3)= 6/h^2 - (12/h^3)*xs;
PSIPP(:,4)= -2/h + (6/h^2)*xs;
PSIPPP(:,1)= (12/h^3)*ones(1,(h/deltax)+1);
PSIPPP(:,2)= (6/h^2)*ones(1,(h/deltax)+1);
PSIPPP(:,3)= -(12/h^3)*ones(1,(h/deltax)+1);
PSIPPP(:,4)= (6/h^2)*ones(1,(h/deltax)+1);
Me= (rho*Area*h/420)*[156 22*h 54 -13*h; 22*h 4*h*h 13*h -3*h*h; 54 13*h 156 -22*h; -13*h -3*h*h -22*h 4*h*h];
Ke= (E*I/h^3)*[12 6*h -12 6*h; 6*h 4*h*h -6*h 2*h*h;-12 -6*h 12 -6*h; 6*h 2*h*h -6*h 4*h*h];
Ce= Alpha1*Me + Alpha2*Ke; %element matrices - [M], [KL], [C]
Gamma= 0.5; Beta= 0.25; %Newmark coefficients
epsilon= 0.5-2*Beta + Gamma;
PHI= rho*Area*ab; delta= 0.5 + Beta - Gamma;
THETA= 2*pi*OMEGA;
%global matrices - [M], [KL], [C]
z= 2*(N+1);
Mg= zeros(z);
Kg= zeros(z);
Cg= zeros(z);
x1= 1;
for x=1:N
a=1;
for i=x1:x1+3
b=1;
for j=x1:x1+3
Mg(i,j)=Mg(i,j)+ Me(a,b);
Kg(i,j)=Kg(i,j)+ Ke(a,b);
Cg(i,j)=Cg(i,j)+ Ce(a,b);
b= b+1;
end
a= a+1;
end
x1= x1 +2;
end
%reduced global matrices
for i=1:z-2
for j=1:z-2
MgR(i,j)= Mg(i+2,j+2);
KgR(i,j)= Kg(i+2,j+2);
CgR(i,j)= Cg(i+2,j+2);
end
end
%Newmark matrices - LINEAR
A1= MgR + Gamma*deltat*CgR + Beta*deltat^2*KgR;
A2= -2*MgR + (1-2*Gamma)*deltat*CgR + epsilon*deltat^2*KgR;
A3= MgR - (1-Gamma)*deltat*CgR + delta*deltat^2*KgR;
%Force discretization vector
FF= zeros(z,1);
a=0;
for ll=1:N
for aa=1:4
S1=0;
S2=0;
for l=1:(h/deltax +1)
alpha(l)= PSI(l,aa);
end
for i1=2:2:h/deltax
S1= S1 + alpha(i1);
end
for j1=3:2:h/deltax -1
S2= S2 + alpha(j1);
end
SS(aa)=(deltax/3)*(alpha(1)+ 4*S1+2*S2+ alpha(h/deltax+1));
end
for i1=1:4
FF(i1+a)= FF(i1+a) + SS(i1);
end
a= a+2;
for i1=1:z-2
FD(i1,1)= FF(i1+2);
end
end
UL= zeros(z,k); %from IC
%UL(:,1) = linspace(0,0.01,z);
ULin = zeros(z,k);
FLin = zeros(z-2,k+1);
FNLin = zeros(z-2,k+1);
UNL = zeros(z,k);
UNLin = zeros(z,k);
time= 0:deltat:tf; %time vector
%Main loop
for j=1:k
%Fg = PHI*FD*cos(THETA*time(j));
Fg = ones(z-2,1)*f_constant; Fg
Fg = zeros(z-2,1); Fg(end,1)
if mod(j,100)==0
disp(j);
end
%Linear loop
for o=1:z-2
for p=1:j
ULr(o,p)= UL(o+2,p);
end
end
if j==1
U0= ULr(:,1);
U1= ULr(:,1);
F0 = Fg;
FLin(:,j) = Fg;
F1= FLin(:,j);
else
U0= ULr(:,j-1);
U1= ULr(:,j);
F0= FLin(:,j-1);
F1= FLin(:,j);
end
FLin(:,j+1) = Fg;
F2= FLin(:,j+1);
F= F2; %Beta*F2 + epsilon*F1 + delta*F0;
U2= -inv(A1)*A2*U1 - inv(A1)*A3*U0 + deltat^2*inv(A1)*F;
ULr(:,j+1)= U2;
for i=1:z
for p=1:j+1
if i<= 2
UL(i,p)=0;
%ULin(i,p)=0;
else
UL(i,p)= ULr(i-2,p);
%ULin(i,p) = ULr(i-2,p);
end
end
end
%Nonlinear loop
eps= 10^5;
counter= 0;
while eps > TOL
if counter == 0
ULin(3:end,j) = U2;
end
%first nonlinear stiffness matrix
a1= 1;
b0= 0;
for o=1:N
for p= 1:(h/deltax)+1
WWP(p)= PSIP(p,1)*UL(a1,j+1) + PSIP(p,2)*UL(a1+1,j+1)+...
PSIP(p,3)*UL(a1+2,j+1) + PSIP(p,4)*UL(a1+3,j+1);
WWPP(p)= PSIPP(p,1)*UL(a1,j+1)+ PSIPP(p,2)*UL(a1+1,j+1)+...
PSIPP(p,3)*UL(a1+2,j+1) + PSIPP(p,4)*UL(a1+3,j+1);
WWPPP(p)= PSIPPP(p,1)*UL(a1,j+1)+ PSIPPP(p,2)*...
UL(a1+1,j+1)+PSIPPP(p,3)*UL(a1+2,j+1) + ...
PSIPPP(p,4)*UL(a1+3,j+1);
WP(p+b0)= WWP(p);
WPP(p+b0)= WWPP(p);
WPPP(p+b0)= WWPPP(p);
end
a1= a1+2;
b0= b0 + h/deltax;
end
AA= N*(h/deltax + 1)- N +1;
AA_f2_2= N*(2*h/deltax + 1)- N +1;
AA_f2_4= N*(4*h/deltax + 1)- N +1;
f1= WP.*WPPP + WPP.^2;
f1(AA)= 0; %from BC
aaa=0;
bbb=0;
mmm=0;
KNL1=zeros(z);
for o=1:N
for pp=1:4
for qq=1:4
S1=0;
S2=0;
for l=1:(h/deltax)+1
PsiPijF1(l)= PSIP(l,pp)*PSIP(l,qq)*f1(mmm+l);
end
for ii=2:2:(h/deltax)
S1= S1 + PsiPijF1(ii);
end
for ii=3:2:(h/deltax - 1)
S2= S2 + PsiPijF1(ii);
end
KKNL(pp,qq)= (deltax/3)*(PsiPijF1(1)+ 4*S1+2*S2 + ...
PsiPijF1(h/deltax+1));
KNL1(pp+aaa,qq+bbb)= KNL1(pp+aaa,qq+bbb)+ KKNL(pp,qq);
end
end
aaa= aaa+2;
bbb= bbb+2;
mmm= mmm + h/deltax;
end
KNL1= E*I*KNL1; %first nonlinear stiffness matrix
for ii=1:z-2
for jj=1:z-2
KNL1r(ii,jj)= KNL1(ii+2,jj+2); %reduced KNL1
end
end
%second nonlinear stiffness matrix
if j == 1
WPsq2Dot= zeros(1000,1); %approx. with lin. disp.
else
a1= 1;
b0= 0;
for o=1:N
for p= 1:(4*h/deltax)+1
wWP(p,3)= PSIP_f2_4(p,1)*UL(a1,j+1)+PSIP_f2_4(p,2)*...
UL(a1+1,j+1)+PSIP_f2_4(p,3)*UL(a1+2,j+1) +...
PSIP_f2_4(p,4)*UL(a1+3,j+1);
wWP(p,2)= PSIP_f2_4(p,1)*UL(a1,j) + PSIP_f2_4(p,2)*...
UL(a1+1,j)+PSIP_f2_4(p,3)*UL(a1+2,j) +...
PSIP_f2_4(p,4)*UL(a1+3,j);
wWP(p,1)= PSIP_f2_4(p,1)*UL(a1,j-1)+PSIP_f2_4(p,2)*...
UL(a1+1,j-1)+PSIP_f2_4(p,3)*UL(a1+2,j-1) +...
PSIP_f2_4(p,4)*UL(a1+3,j-1);
wP(p+b0,3)= wWP(p,3); wP(p+b0,2)= wWP(p,2);
wP(p+b0,1)= wWP(p,1);
end
a1= a1+2;
b0= b0 + 4*h/deltax;
end
for ii=1:AA_f2_4
for jj=1:3
WPsq(ii,jj)= wP(ii,jj)*wP(ii,jj);
end
end
for ii=1:AA_f2_4
WPsqDot(ii,1)= (WPsq(ii,2)-WPsq(ii,1))/deltat;
WPsqDot(ii,2)= (WPsq(ii,3)-WPsq(ii,2))/deltat;
WPsq2Dot(ii)= (WPsqDot(ii,2)-WPsqDot(ii,1))/deltat;
end
end
xx1=1;
for xx=1:2:AA_f2_4
S1=0; S2=0; i1=2; j1=3;
while i1<= xx-1
S1= S1 + WPsq2Dot(i1);
i1= i1 + 2;
end
while j1<= xx-2
S2= S2 + WPsq2Dot(j1);
j1= j1 + 2;
end
SS1(xx1)=(deltax/12)*(WPsq2Dot(1)+4*S1 + 2*S2 + WPsq2Dot(xx));
xx1=xx1+1;
end
SS1(1)=0;
xx2=1;
for xx=1:2:AA_f2_2
S1=0; S2=0; i1=xx+1; j1=xx+2;
while i1<= AA_f2_2 - 1
S1= S1 + SS1(i1);
i1= i1 + 2;
end
while j1<= AA_f2_2 - 2
S2= S2 + SS1(j1);
j1= j1 + 2;
end
f2(xx2)=(-deltax/6)*(SS1(xx) + 4*S1 + 2*S2 + SS1(AA_f2_2));
xx2=xx2+1;
end
f2(AA)= 0;
aaa=0;
bbb=0;
mmm=0;
KNL2=zeros(z);
for o=1:N
for pp=1:4
for qq=1:4
S1=0;
S2=0;
for l=1:(h/deltax)+1
PsiPijF2(l)= PSIP(l,pp)*PSIP(l,qq)*f2(mmm+l);
end
for ii=2:2:(h/deltax)
S1= S1 + PsiPijF2(ii);
end
for ii=3:2:(h/deltax - 1)
S2= S2 + PsiPijF2(ii);
end
KKnl(pp,qq)= (deltax/3)*(PsiPijF2(1)+ 4*S1+2*S2 + ...
PsiPijF2(h/deltax+1));
KNL2(pp+aaa,qq+bbb)= KNL2(pp+aaa,qq+bbb)+ KKnl(pp,qq);
end
end
aaa= aaa+2;
bbb= bbb+2;
mmm= mmm + h/deltax;
end
KNL2= 0.5*rho*Area*KNL2; %second nonlinear stiffness matrix
for ii=1:z-2
for jj=1:z-2
KNL2r(ii,jj)= KNL2(ii+2,jj+2); %reduced KNL2
end
end
KTotal= KgR-KNL1r-KNL2r; %total stiffness matrix
CNL= Alpha1*MgR + Alpha2*KTotal; %nonlinear damping matrix
%nonlinear displacement - Newmark technique
for o=1:z-2
for p=1:j
UNLr(o,p)= UL(o+2,p);
end
end
if j==1
U0= UNLr(:,1); U1= UNLr(:,1);
F0= FLin(:,j);
FNLin(:,j)= FLin(:,j);
F1= FNLin(:,j);
else
U0= UNLr(:,j-1);
U1= UNLr(:,j);
F0= FNLin(:,j-1);
F1= FNLin(:,j);
end
%Newmark matrices - NONLINEAR
A1NL= MgR + Gamma*deltat*CNL + Beta*deltat^2*KTotal;
A2NL= -2*MgR + (1-2*Gamma)*deltat*CNL + epsilon*deltat^2*KTotal;
A3NL= MgR - (1-Gamma)*deltat*CNL + delta*deltat^2*KTotal;
%total force vector
X= 0:deltax:L; %global X vector
for ii=1:AA
f3(ii)= (X(ii)*deltax - L)*WPP(ii) + WP(ii);
end
G= zeros(z,1);
a=0; m=0;
for ll=1:N
for aa=1:4
S1=0;
S2=0;
for l=1:(h/deltax +1)
alpha(l)= PSI(l,aa)*f3(m+l);
end
for i1=2:2:h/deltax
S1= S1 + alpha(i1);
end
for j1=3:2:h/deltax -1
S2= S2 + alpha(j1);
end
SS(aa)=(deltax/3)*(alpha(1)+ 4*S1+2*S2 + ...
alpha(h/deltax+1));
end
for i1=1:4
G(i1+a)= G(i1+a) + SS(i1);
end
a= a+2;
m= m+h/deltax;
end
G= rho*Area*g*G; %gravity vector
for i1=1:z-2
Gr(i1)= G(i1+2);
end
FNLin(:,j+1)= FLin(:,j+1);%+ Gr'; %nonlinear force vector
F2= FNLin(:,j+1);
%F= PHI*FD*cos(THETA*time(j));
F = Fg; %f_constant*ones(z-2,1);%FNLin(:,j+1);%Beta*F2 + epsilon*F1 + delta*F0;
U2= -inv(A1NL)*A2NL*U1 - inv(A1NL)*A3NL*U0 + deltat^2*inv(A1NL)*F;
UNLr(:,j+1)= U2;
for i=1:z
for p=1:j+1
if i<= 2
UNL(i,p)=0;
else
UNL(i,p)= UNLr(i-2,p);
end
end
end
for ii=1:z
DELTA(ii)= abs(UL(ii,j+1) - UNL(ii,j+1));
end
eps= sum(DELTA);
UL(:,j+1)= UNL(:,j+1);
counter=counter +1;
%if mod(counter,1000)
% disp(eps)
%end
end
kounter(j)=counter; %number of iterations in NL loop
EPS(j)= eps; %convergence variable for each time step
end