We will use the workflow as it is industry-standard for complex shapes and fast results. Workflow Summary:
% Cantilever Beam Dynamic Analysis Script % Required: Partial Differential Equation Toolbox %% 1. Geometry & Model Creation L = 0.5; W = 0.02; H = 0.02; % Dimensions in meters gm = multicuboid(L, W, H); model = createpde('structural','transient-solid'); model.Geometry = gm; generateMesh(model, 'Hmax', H/2); %% 2. Material Properties (Steel) E = 210e9; % Young's Modulus (Pa) nu = 0.3; % Poisson's Ratio rho = 7800; % Density (kg/m^3) structuralProperties(model, 'YoungsModulus', E, 'PoissonsRatio', nu, 'MassDensity', rho); %% 3. Boundary Conditions % Fix the face at x = 0 (Assuming Face 5 is the start of the beam) structuralBC(model, 'Face', 5, 'Constraint', 'fixed'); %% 4. Modal Analysis (Natural Frequencies) modalResults = solve(model, 'FrequencyRange', [0, 1e4]); fprintf('First Natural Frequency: %.2f Hz\n', modalResults.NaturalFrequencies(1)/(2*pi)); %% 5. Transient Analysis (Impulse Response) % Apply a tip load for a short duration tipVertex = 1; % Vertex ID at the free end structuralBoundaryLoad(model, 'Vertex', tipVertex, 'Force', [0; 0; -100], 'EndTime', 0.005); % Solve for 0.5 seconds tlist = linspace(0, 0.5, 100); res = solve(model, tlist); %% 6. Plotting Results plot(res.SolutionTimes, squeeze(res.Displacement.uz(tipVertex, :))); title('Tip Displacement vs Time'); xlabel('Time (s)'); ylabel('Vertical Displacement (m)'); grid on; Use code with caution. Copied to clipboard 4. Key Takeaways Dynamic Analysis Cantilever Beam Matlab Code
The first few ( \beta L ) values: 1.8751, 4.6941, 7.8548, etc. We will use the workflow as it is
A typical MATLAB code for this purpose employs the Finite Difference Method or, more commonly, the Finite Element Method (FEM). A well-structured code follows a logical sequence. First, the user defines the beam's physical and material properties: length (( L )), Young's modulus (( E )), moment of inertia (( I )), mass per unit length (( m )), and the number of elements (( n )). The code then assembles the global mass matrix (( [M] )) and stiffness matrix (( [K] )) for the beam. For a cantilever, boundary conditions are applied by eliminating the degrees of freedom (displacement and rotation) at the fixed node. Material Properties (Steel) E = 210e9; % Young's
% Define beam properties L = 10; % length of the beam (m) b = 0.1; % width of the beam (m) h = 0.2; % height of the beam (m) rho = 7850; % density of the beam material (kg/m³) E = 200e9; % modulus of elasticity of the beam material (Pa) I = b*h^3/12; % moment of inertia of the beam cross-section (m⁴)