Add example on amplitude spectrum for free vibration case.

examples/dynamics/beams/example_dynamic_beam1b_free_spectrum.m

+%% Example Beam1b
+% A one-dimensional cantilever beam composed of N finite beam elements with spring at tip
+addpath([ '..' filesep '..' filesep '..' filesep ]);
+%% General properties
+L = 1.0;  % length of the cantilever
+N = 8;    % number of finite elements
+%% Definition of sections
+s1 = mafe.Section1D(); % beam element section
+s1.E   = 2.0e11; % [N/m^2]
+s1.Iy  = 1.0e-7; % [m^4] = 0.01m * (0.05m)^3 / 12
+s1.A   = 5.0e-4; % [m^2] = 0.01m * 0.05m
+s1.rho = 8.0e+3; % [kg/m^3]
+%% Definition of nodes in the system
+nodes = mafe.Node.empty(0,0);
+for ii = 1:N+1
+    % create node
+    nodes(end+1) = mafe.Node1D( [L/N*(ii-1)] );
+end
+%% Definition of elements in the system
+elems = mafe.Element.empty(0,0);
+for ii = 1:N
+    % create element
+    elems(end+1) = mafe.Beam1D( [nodes(ii), nodes(ii+1)], s1 );
+end
+%% Definition of constraints in the system
+const = mafe.Constraint.empty(0,0);
+% Constraint 1 and 2 (clamped at left end)
+const(end+1) = mafe.ConstraintNode( nodes(1), mafe.DofType.Disp2, mafe.ConstraintType.Dirichlet, 0.0 );
+const(end+1) = mafe.ConstraintNode( nodes(1), mafe.DofType.Rota3, mafe.ConstraintType.Dirichlet, 0.0 );
+%% Setup of the finite element problem
+fep = mafe.FeProblem( nodes, elems, const );
+%% Select an analysis type: dynamic
+ana = mafe.FeAnalysisDynamicFD( fep );
+%% Calculate response
+ana.analyse();
+%% Visualise system response
+cla; clf;
+fig = figure(1); subplot('Position',[0.05 0.05 0.90 0.90]); hold on;
+% apply initial conditions
+modeshape = real(ana.X(:,1:2:end)); % get some selected (or all) distinct modes
+u0 = 0.20 * rand(fep.ndofs,1);
+v0 = 0.05 * rand(fep.ndofs,1);
+% u0 = 0.2 * modeshape(:,1) + 0.10 * modeshape(:,2);
+% v0 = 0.0 * modeshape(:,1) + 0.00 * modeshape(:,2);
+ana.applyInitialConditions( u0, v0 );
+% obtain and plot the amplitude spectrum
+omega = abs(imag(ana.L(1:2:end)));
+ampli = abs(ana.A(1:2:end)) * 2.; % complex-conjugate contributes
+plot(omega,ampli,'o');
+bar(omega,ampli);
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