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| clear all
close all
clc
%% properties of the beams
N=2;
A =1e-3 ; % Cross section [m^2]
D =sqrt(4*A/pi); % Cross section diameter [m]
rho = 2700; % density of material [kg/m^3]
E = 69e9; % Young modulus [Pa]
mu = 0.33; % Poisson coefficient [-]
G = E/(2+2*mu); % Shear modulus [Pa]
I_y = (pi*D^4)/64; % Cross section moment of inertia along z [m^4]
I_z = I_y; % Cross section moment of inertia along y [m^4]
J_x = 2*I_y;% Cross section torsion c [m^4]
r=sqrt(I_z/A); %giration radius of circular cross section
%% lumped mass:
m=20; % kg
%% coordinates of principal nodes and beam list
%coord=zeros(18,3);
for i=0:5
coord(3*i+1,:)=[i*tand(4) 0 i];
coord(3*i+2,:)=[0.6 -0.3+i*tand(2) i];
coord(3*i+3,:)=[0.6 0.3-i*tand(2) i];
end
Coord=coord
elementNodes=zeros(48,2);
elementNodes(1:9,:)=[1 2;...
2 3;...
3 1;...
3 6;...
1 4;...
2 5;...
1 5;...
3 5;...
1 6];
for i=1:4
elementNodes(i*9+1:i*9+9,:)=elementNodes(1:9,:)+3*i;
end
elementNodes(end-2:end,:)=[16 17;...
17 18;...
16 18];
numberNodes=size(coord,1);
numberElements=size(elementNodes,1);
hold off
for i=1:numberElements
element1=elementNodes(i,1);
element2=elementNodes(i,2);
x=[coord(element1,1) coord(element2,1)];
y=[coord(element1,2) coord(element2,2)];
z=[coord(element1,3) coord(element2,3)];
plot3(x,y,z,'-ok');hold on
end
axis equal
subNO
%% List of subelements of structure
subelement=zeros(numberElements*N,2);
index = 19;
k=1;
if N==1
disp('Please N must be upper than one or equal to zero in case of no subelement in a beam in the subNO file; thanks')
elseif N==0
subelement=elementNodes;
else
for i=1:numberElements
subelement(k,:)=[elementNodes(i,1) index];
k=k+1;
index= index+1;
for j=2:N-1
subelement(k,:) = [index-1 index+1-1];
index = index+1;
k=k+1;
end
subelement(k,:) = [index-1 elementNodes(i,2)];
k =k+1;
end
end
%% list of DOFs of structure
t= ((N-1)*length(elementNodes) + numberNodes);
dofList=zeros(t,6);
for i=1:t
for j=1:6
dofList(i,j)=(i-1)*6+j;
end
end
%%
%%Localisation Matrix
Locel=zeros(N*numberElements,12);
for i=1:N*numberElements
dof_1=dofList(subelement(i,1),:); % dofs of 1st node
dof_2=dofList(subelement(i,2),:); % dofs of 2nd node
Locel(i,:) = [dof_1 dof_2];
end
%%
% Kglobal(locel,locel)=Kglobal(locel,locel)+T(:,:,s)'*Kel*T(:,:,s);
% Mglobal(locel,locel)=Mglobal(locel,locel)+T(:,:,s)'*Mel*T(:,:,s);
p3=[0 -0.3 0];
X3=0; Y3=-0.3; Z3=0;
eX = [1 0 0];
eY = [0 1 0];
eZ = [0 0 1]; % Global axis (structure)
k=1;
n=N*numberElements;
T=zeros(12,12,n)
l=zeros(N*numberElements,1)
for i=1:numberElements
L(i,:)=norm(coord(elementNodes(i,2),:)-coord(elementNodes(i,1),:)); % length of each beam
l=L(i,:)/N %length of each subelement(discretized beam)
ex(i,:)=(coord(elementNodes(i,2),:)-coord(elementNodes(i,1),:))/norm(coord(elementNodes(i,2),:)-coord(elementNodes(i,1),:));
d2(i,:)=(coord(elementNodes(i,2),:)-coord(elementNodes(i,1),:));
d3(i,:)=p3-coord(elementNodes(i,1),:);
ey(i,:)=cross(d3(i,:),d2(i,:))./norm(cross(d3(i,:),d2(i,:)));
ez(i,:)=cross(ex(i,:),ey(i,:));
R(:,:,i)=[dot(eX,ex(i,:)) dot(eY,ex(i,:)) dot(eZ,ex(i,:));...
dot(eX,ey(i,:)) dot(eY,ey(i,:)) dot(eZ,ey(i,:));...
dot(eX,ez(i,:)) dot(eY,ez(i,:)) dot(eZ,ez(i,:)) ]
Te=[R zeros(3) zeros(3) zeros(3);...
zeros(3) R zeros(3) zeros(3);...
zeros(3) zeros(3) R zeros(3);...
zeros(3) zeros(3) zeros(3) R];
end
%%Assembly
Kel=[(E*A/l) 0 0 0 0 0 -(E*A/l) 0 0 0 0 0 ;
0 ((12*E*I_z)/l^3) 0 0 0 (6*E*I_z)/(l^2) 0 (-12*E*I_z)/(l^3) 0 0 0 (6*E*I_z)/(l^2) ;
0 0 (12*E*I_y)/(l^3) 0 (-6*E*I_y)/(l^2) 0 0 0 (-12*E*I_y)/(l^3) 0 (-6*E*I_y)/(l^2) 0 ;
0 0 0 (G*J_x)/(l) 0 0 0 0 0 (-G*J_x)/(l) 0 0 ;
0 0 (-6*E*I_y)/(l^2) 0 (4*E*I_y)/(l) 0 0 0 (6*E*I_y)/(l^2) 0 (2*E*I_y)/(l) 0 ;
0 (6*E*I_z)/(l^2) 0 0 0 (4*E*I_z)/(l) 0 (-6*E*I_z)/(l^2) 0 0 0 (2*E*I_z)/(l) ;
(-E*A)/(l) 0 0 0 0 0 (E*A)/(l) 0 0 0 0 0 ;
0 (-12*E*I_z)/(l^3) 0 0 0 (-6*E*I_z)/(l^2) 0 (12*E*I_z)/(l^3) 0 0 0 (-6*E*I_z)/(l^2);
0 0 (-12*E*I_y)/(l^3) 0 (6*E*I_y)/(l^2) 0 0 0 (12*E*I_y)/(l^3) 0 (6*E*I_y)/(l^2) 0 ;
0 0 0 (-G*J_x)/(l) 0 0 0 0 0 (G*J_x)/(l) 0 0 ;
0 0 (-6*E*I_y)/(l^2) 0 (2*E*I_y)/(l) 0 0 0 (6*E*I_y)/(l^2) 0 (4*E*I_y)/(l) 0 ;
0 (6*E*I_z)/(l^2) 0 0 0 (2*E*I_z)/(l) 0 (-6*E*I_z)/(l^2) 0 0 0 (4*E*I_z)/(l)] ;
% Elementary mass matrix
Mel = rho*A*l* ...
[ 1/3 0 0 0 0 0 1/6 0 0 0 0 0;
0 13/35 0 0 0 11*l./210 0 9/70 0 0 0 -13*l./420;
0 0 13/35 0 -11*l./210 0 0 0 9/70 0 13*l./420 0;
0 0 0 (r^2)/3 0 0 0 0 0 (r^2)/6 0 0;
0 0 -11*l./210 0 (l)^2/105 0 0 0 -13*l./420 0 -(l)^2/140 0;
0 11*l./210 0 0 0 l.^2/105 0 13*l/420 0 0 0 -l.^2/140;
1/6 0 0 0 0 0 1/3 0 0 0 0 0;
0 9/70 0 0 0 13*l./420 0 13/35 0 0 0 -11*l./210;
0 0 9/70 0 -13*l./420 0 0 0 13/35 0 11*l/210 0;
0 0 0 (r^2)/6 0 0 0 0 0 r^2/3 0 0;
0 0 13*l./420 0 -((l)^2)/140 0 0 0 11*l./210 0 (l^2)/105 0;
0 -13*l./420 0 0 0 -(l^2)/140 0 -11*l./210 0 0 0 ((l)^2)/105];
%% assembly process:
Nnode_tot = (N-1)*length(elementNodes) + numberNodes; % Number of nodes of structure
Ndof = Nnode_tot*6;
K_global=zeros(Ndof,Ndof); %reset the stiffness matrix of the element in global reference frame
M_global=zeros(Ndof,Ndof); %reset the stiffness matrix of the element in global reference frame
for k=1:numberElements
for s=1:N*numberElements
K_global(Locel(i,:),Locel(i,:))=K_global(Locel(i,:),Locel(i,:))+T(:,:,s)'*Kel*T(:,:,s);
M_global(Locel(i,:),Locel(i,:))=M_global(Locel(i,:),Locel(i,:))+T(:,:,s)'*Mel*T(:,:,s);
end
end
M_global(91,91)=M_global(91,91)+ m; % lumped mass is located at the 16th node
M_global(92,92)=M_global(92,92)+ m;
M_global(93,93)=M_global(93,93)+ m;
% Boundary conditions.
% nodes 1,2 and 3 are fixed.
fixNode_1 = [doflist(1,:)];
fixNode_2 = [doflist(2,:)];
fixNode_3 = [doflist(3,:)];
clampedDOF = [fixNode_1 fixNode_2 fixNode_3 ];
K_global(clampedDOF, :) = [];
K_global(:, clampedDOF) = [];
M_global(clampedDOF, :) = [];
M_global(:, clampedDOF) = [];
[V,D]=eigs(K_global,M_global,5,'sm');
omega = diag(D.^(0.5));
f = omega/(2*pi);
%% Damping matrix
Coeff = 2*[omega(1) 1/omega(1); omega(2) 1/omega(2)]^(-1)*[0.01 0.01]'; % Eq. 3.20 p.156
a = Coeff(1);
b = Coeff(2);
CGlobal = a*KGlobal + b*MGlobal; % Eq. 3.19 p.156
eps = 1/2*(a.*omega + b./omega); |
