My name is Miguel Pinto Charro and I'm a student at the Engineering faculty of University of Porto and in order to complete the course of Plates and Shells, it was proposed a pratical assignment that I explain on the question. If someone could help me I would really appreciate it.
# Set up problem parameters
E1, E2, G12, nu12, nu21, rho, h, R := 6e10, 2.25e10, 1.15e10, 0.3, 0.3, 7850, 0.01, 0.5:
# Define the displacement field
w := (a, b, theta) -> a*sin(b*theta) + b*cos(a*theta):
# Define the strain-displacement matrix
B := (a, b, theta) -> matrix(
[diff(w(a, b, theta), theta), 0, 0,
diff(w(a, b, theta), theta, 2), 0, 0]
):
# Define the stiffness matrix
D := (E1, E2, G12, nu12, nu21) -> matrix(
[E1/(1-nu12*nu21), E1*nu12/(1-nu12*nu21), G12,
E1*nu21/(1-nu12*nu21), E2/(1-nu12*nu21), G12]
):
# Define the mass matrix
M := rho*h -> matrix(
[2*rho*h/R^2, 0, 0, 0, rho*h/R^2, 0]
):
# Define the function to be minimized
f := (a, b) -> (B(a, b, Pi/2).D(E1, E2, G12, nu12, nu21).B(a, b, Pi/2) - M).det:
# Use the Rayleigh-Ritz method to find the natural frequencies
rayleigh_ritz(f, [a = 1, b = 1])
This code defines the displacement field, strain-displacement matrix, stiffness matrix, and mass matrix for a circular plate made of composite laminated materials. It then uses the Rayleigh-Ritz method to minimize a function defined using these matrices and find the natural frequencies of the plate
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