Abstract: The scope of the present article, motivated by the case of the composite wooden propeller of an airplane, is to deal tentatively with the longitudinal free vibrationproblem of an elastic straight bar with a more general mathematical treatment.In this analysis, we have assigned to the modulus of elasticity, the bar cross section as well as the mass per unit length of the bar an exponential function variation, and then found a general solution, wherein three parameters were considered as the main factors to affect the longitudinal free vibration of the inhomogeneous elastic straight bar with a variable cross section.

Key words: precise integration, Riccati equation, optimal control