root/OptimizingMATLABCode/Truss/trussCantilever.m @ 10
| 10 | anderm8 | function [Yr,bars,L,N,H]=trussCantilever(N,Aval)
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% Yr = trussCantilever(N,Aval);
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%
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% Usage:
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% Y=trussCantilever(N,Aval)
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% [Yr,bars,L,N,H]=trussCantilever(N,Aval)
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%
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% Inputs:
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%
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% N: number of horizontal elements (recommend range of [1,400])
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%
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% Aval: cross section area for each rod (recommend range of [10 1000])
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%
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% Outputs:
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%
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% Yr:
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%
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%
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% Copyright 2015 The MathWorks, Inc
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%% Defaults
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if nargin < 2
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% Ten horizontal truss elements in bottom cord
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N = 10;
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% Cross sectional area of 100
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Aval = 100;
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end
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%% Truss Parameters
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% Physical parameters of the material and truss
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% Horizontal Length of the truss
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L = 1;
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% Vertical Height of the truss
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H = 0.025;
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% Density of truss bar material
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rho = 1;
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% Modulus of Elasticity of truss bar material
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Eval = 1e-1;
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% Rayleigh, "alpha,beta" damping coefficients
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dampingCoeffAlpha = 0.01;
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dampingCoeffBeta = 0.01;
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%% Force Parameters
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%
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% The following downward force is applied to the truss
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%
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% max(0,(applyTime-t)*downwardForceMag/applyTime)*sin(2*pi*freq*t)
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%
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% The frequency of the force is 1. The magnitude is linearly lowered from
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% _downwardForceMag_ to zero over the interval |[0,applyTime]|. The
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% simulation time is simTime. Because of the damping the vibration will die
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% out.
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downwardForceMag = 0.02;
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freq = 1;
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applyTime = 20;
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simTime = 100;
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%% Map reduced Dofs to actual Dofs
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% The truss is fixed to the wall on the left hand side. Therefore the two
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% nodes there are not mobile and can be eliminated. Create a map between
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% the "reduced" degrees of freedom and the actual degrees of freedom
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numDofs = 2*2*(N+1)-2; % -2 because the cantilever is pointed upwards at the end
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groundDofs = [1,2,2*(N+1)+1,2*(N+1)+2]; % Degrees of Freedom that will be eliminated
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%% Add bars along with geometry
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% bars are #of Bars X [Area,E,length of bar,angle of bar w.r.t. horizontal
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% axis, node1 (from node), node2 (to node)]
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bars = zeros(2*N+2*(N-1),6);
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for n = 1:N
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% upper bars
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lelem = L/N;
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bars(n,:) = [Aval,Eval,lelem,0,n,n+1];
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% diagonal bars
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lelem = sqrt((L/N)^2+H^2);
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bars(N+n,:) = [Aval,Eval,lelem,pi/4,N+1+n,n+1];
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end
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for n = 1:N-1
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% lower bars
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lelem = L/N;
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bars(2*N+n,:) = [Aval,Eval,lelem,0,N+1+n,N+1+n+1];
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% vertical bars
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lelem = H;
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bars(2*N+N-1+n,:) = [Aval,Eval,lelem,pi/2,N+1+n+1,n+1];
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end
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%% Assemble all bars into global stiffness and mass matrices
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% Use a dense matrix for insertion efficiency
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K = zeros(numDofs);
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M = zeros(numDofs);
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% Mass element matrix
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% the mass matrix is a lumped matrix with half the mass going to one
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% node and half to the other node
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unitMassMatrix = diag([1/2 1/2 1/2 1/2]);
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melem = rho*lelem*unitMassMatrix;
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for ii=1:size(bars,1)
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% extract parameters for stiffness and mass matrices
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Aelem = bars(ii,1);
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Eelem = bars(ii,2);
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lelem = bars(ii,3);
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telem = bars(ii,4);
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kelem = localStiffness(Aelem,Eelem,lelem,telem); % stiffness matrix
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n1 = bars(ii,5);
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n2 = bars(ii,6);
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% convert to reduced dofs
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ix = [n1*2-1,2*n1,n2*2-1,n2*2];
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% element "stamping"
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K(ix,ix) = K(ix,ix) + kelem; % Add in element's contribution to K
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M(ix,ix) = M(ix,ix) + melem; % Add in element's contribution to M
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end
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% Zero out rows and columns of ground DOFs in stiffness and mass matrix
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K(groundDofs,:) = 0;
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K(:,groundDofs) = 0;
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M(groundDofs,:) = 0;
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M(:,groundDofs) = 0;
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% Add ones to diagonal of ground DOFs to freeze them
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groundDofDiag = sub2ind(size(M),groundDofs,groundDofs);
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K(groundDofDiag) = 1;
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M(groundDofDiag) = 1;
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% convert stiffness and mass matrices to sparse
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Kr = sparse(K);
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Mr = sparse(M);
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% Force vector iz zeros with magnitude on end Y Dof in Y
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F = zeros(size(Kr,1),1);
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F(2*(N+1)) = downwardForceMag;
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%% Solve the system
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% Transform M*d2x/dt2 + damping*dx/dt + K*x = F into dydt = A*y + f form
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% Convert 2nd order ODE to first order ODE with y = dx/dt transformation.
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% The state space is therefore [position,velocity] and in that order
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[LM,UM] = lu(Mr);
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A = [-UM\(LM\Kr),-UM\(LM\(dampingCoeffAlpha*Kr+dampingCoeffBeta*Mr));sparse(numDofs,numDofs),speye(numDofs)];
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f = [UM\(LM\F);zeros(numDofs,1)];
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myEvaluator = @(t,y) A*y + max(0,(applyTime-t)*f/applyTime)*sin(2*pi*freq*t);
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initialCondition = zeros(numDofs*2,1); % initial conditions for ode23t
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% Yr is #Time points X [position, velocity]
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[~,Yr] = ode23t(myEvaluator,[0 simTime],initialCondition);
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% Remove the velocity components from Yr
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Yr = Yr(:,1:numDofs);
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end
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