DEFAULT_ARGUMENT_VALUE 1 hex # mesh, either hex or unstruct

DEFAULT_ARGUMENT_VALUE 2 copper # material, either copper or aluminum

l = 0.5*303e-3 # cantilever wire length [ m ]

d = 1.948e-3 # wire diameter [ m ]

# material properties for copper

m_copper = 0.5*8.02e-3 # total mass (half the measured because of the experimental disposition) [ kg ]

E_copper = 2*66.2e9 # [ Pa ] Young modulus (twice because the factor-two error)

# material properties for aluminum

m_aluminum = 0.5*2.67e-3

E_aluminum = 2*40.2e9

# problem’s properties

E = E_$2 # [ MPa ]

rho = m_$2/(pi*(0.5*d)^2*l) # [ kg / m^3 ] density = mass (measured) / volume

nu = 0 # Poisson’s ratio (does not appear in Euler-Bernoulli)

# analytical solution

VECTOR kl[5]

VECTOR f_euler[5]

# first compute the first five roots ok cosh(kl)*cos(kl)+1

kl[i] = root(cosh(t)*cos(t)+1, t, 3*i-2,3*i+1)

# then compute the frequencies according to Euler-Bernoulli

# note that we need to use SI inside the square root

A = pi * (d/2)^2

I = pi/4 * (d/2)^4

f_euler[i] = 1/(2*pi) * kl(i)^2 * sqrt((E * I)/(rho * A * l^4))

# now compute the modes numerically with FEM

# note that each mode is duplicated as it is degenerated

READ_MESH wire-$1.msh DIMENSIONS 3

PROBLEM modal MODES 10

BC fixed fixed

SOLVE_PROBLEM

# github-formatted markdown table

# compare the frequencies

PRINT " \$n\$ | FEM [Hz] | Euler [Hz] | Relative difference [%]"

PRINT ":------:|:-------------:|:-------------:|:--------------:"

PRINT_VECTOR SEP "\t|\t" i %.4g f(2*i-1) f_euler %.2f 100*(f_euler(i)-f(2*i-1))/f_euler(i)

PRINT

PRINT ": $2 wire over $1 mesh"

# commonmark table

PRINT

PRINT " \$n\$ | \$L\$ | \$\\Gamma\$ | \$\\mu\$ | \$M\$"

PRINT ":------:+:---------------------:+:---------------------:+:-------------:+:--------------:"

PRINT_VECTOR SEP "\t|\t" i "%+.1e" L Gamma "%.4f" mu Mu

PRINT

PRINT ": $2 wire over $1 mesh, participation and excitation factors \$L\$ and \$\\Gamma\$, effective per-mode and cumulative mass fractions \$\\mu\$ and \$M\$"

# write the modes into a vtu file

WRITE_MESH wire-$1-$2.vtu \

VECTOR u1 v1 w1 VECTOR u2 v2 w2 VECTOR u3 v3 w3 \

VECTOR u4 v4 w4 VECTOR u5 v5 w5 VECTOR u6 v6 w6 \

VECTOR u7 v7 w7 VECTOR u8 v8 w8 VECTOR u9 v9 w9 VECTOR u10 v10 w10

# and into a msh file

WRITE_MESH wire-$1-$2.msh {

u1 v1 w1

u2 v2 w2

u3 v3 w3

u4 v4 w4

u5 v5 w5

u6 v6 w6

u7 v7 w7

u8 v8 w8

u9 v9 w9

u10 v10 w10

}