Left boundary condition Fixed end Simply supported end Free end

    

Right boundary condition Fixed end Simply supported end Free end

 

Mode Shape: 1st

 

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

$$\lambda_n L = 4.71$$ $$\omega_n = 2.29$$

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