Study on the Exact Solution For Natural Frequencies and Mode Shapes of the Longitudinal-Vibration Conic Rod Carrying Arbitrary Concentrated Elements

Jia-Jang Wu

In this paper, a conic rod carrying arbitrary concentrated elements is called the conic rod system. First of all, the equation of motion for the longitudinal free vibration of a conic rod is transformed into a Bessel equation, and then the exact displacement function in terms of the Bessel functions is obtained. Next, based on the equations for compatibility of deformations and those for equilibrium of longitudinal forces at each attaching point (including the two ends of entire bar) between the concentrated elements and the conic rod, a characteristic equation of the form [H]{C}= {0} is obtained. Now, the natural frequencies of the conic rod system can be determined from the determinant equation |H| = 0, and the associated column vector for the integration constants, {C}, corresponding to each natural frequency, can be obtained from the simultaneous equation [H]{C}= {0}. The substitution of the last integration constants into the displacement functions of all the associated rod segments will produce the corresponding mode shape of the entire conic rod system. Finally, the important factors affecting the longitudinal vibration characteristics of a conic rod system will be investigated. To confirm the reliability of the presented technique, in this research, the exact solutions obtained from the presented technique were compared with the numerical solutions obtained from the conventional finite element method (FEM). Good agreement is achieved.