Left boundary condition Fixed end Simply supported end Free end

    

Right boundary condition Fixed end Simply supported end Free end

 

Mode Shape: 1st

 

$$ V(0) = 0 $$ $$ \frac{\mathrm{d} V}{\mathrm{d} z}(0) = 0 $$ $$ V(0) = 0 $$ $$ \frac{\mathrm{d} ^2 V}{\mathrm{d} z^2}(0) = 0 $$ $$ \frac{\mathrm{d} ^2 V}{\mathrm{d} z^2}(0) = 0 $$ $$ \frac{\mathrm{d} ^3 V}{\mathrm{d} z^3}(0) = 0 $$

$$ \lambda_n L = 1 $$ $$ \omega_n = 1 $$

$$ V(L) = 0 $$ $$ \frac{\mathrm{d} V}{\mathrm{d} z}(L) = 0 $$ $$ V(L) = 0 $$ $$ \frac{\mathrm{d} ^2 V}{\mathrm{d} z^2}(L) = 0 $$ $$ \frac{\mathrm{d} ^2 V}{\mathrm{d} z^2}(L) = 0 $$ $$ \frac{\mathrm{d} ^3 V}{\mathrm{d} z^3}(L) = 0 $$