Problem $1$

For the bar shown with length $2$ L $,$ modulus of elasticity $\backslash ($ E $\backslash ),$ mass density $\backslash (\backslash$rho $\backslash ),$ and cross$-$sectional area A $,$ determine the first two natural frequencies. $($Discretize into two elements$-$using lumped mass$)$

Problem $2$

For the bar shown with length $2$ L $,$ modulus of elasticity E $,$ mass density $\backslash (\backslash$rho $\backslash ),$

and cross$-$sectional area A $,$ determine the first two natural frequencies. $($Discretize into two elements$-$using consistent mass$).$ Compare the results with problem $1$ F$,$

Figure $1$

Figure $2$