Natural frequency of aluminum bar given by consistent-mass matrices

a. \(58554 \mathrm{rad} / \mathrm{s}\) lumped-mass matrices

b. \(33806 \mathrm{rad} / \mathrm{s}\) consistent-mass matrices

c. \(33758 \mathrm{rad} / \mathrm{s}\)

d. \(58471 \mathrm{rad} / \mathrm{s}\)

Assume a fixed-fixed bar with one middle node:

Element matrices: \([k]=\frac{A E}{l}\left[\begin{array}{rr}1 & -1 \\ -1 & 1\end{array}\right],[m]_{c}=\frac{ho A l}{6}\left[\begin{array}{ll}2 & 1 \\ 1 & 2\end{array}\right],[m]_{l}=\frac{ho A l}{2}\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]\)

Steel bar: \(E=200 \times 10^{9} \mathrm{~Pa}, ho=7800 \mathrm{~kg} / \mathrm{m}^{3}, l=0.3 \mathrm{~m}\)

Aluminum bar: \(E=72 \times 10^{9} \mathrm{~Pa}, ho=2800 \mathrm{~kg} / \mathrm{m}^{3}, l=0.3 \mathrm{~m}\)