Free Vibration of a Simply Supported Bar With a Linearly Variable Height of Cross Section

Abstract

The problem of vibration of nonhomogeneous bars or bars of variable cross section1 leads to differential equations, which are generally unsolvable by formal integration. It is known that functional coefficients occur in these equations, which make it difficult, if not impossible, to obtain exact solutions by simple integration. Several exact solutions obtained for a few special cases and also some interesting approximate solutions are mentioned in the paper.

Effect of Thrust on the Structural Vibrations of a Nonuniform Slender Rocket

“…The development method of solution was in part validated with published results. In Table 1, a comparison is made with the results of Krynicki and Mazurkiewicz [26] for a solid cylindrical beam with free-free boundary conditions and with the thrust set at zero. It can be found that there is reasonable agreement between the two sets of results, although, generally, the current method gives slightly higher solutions for all three natural frequencies.…”

“…The coefficients C m can be found from the recurrent Equations ( 32) and (33) and the solution for ϕ(η) subsequently calculated from Equation (26). The series solution is ϕ(η) = ∞ m=0 C m η m , although all of the coefficients C m cannot be determined, and thus the solutions must be approximated by the truncated series n−1 m=0 C m η m .…”

Vibration of structures with beams of variable cross section. I

“…pro výpočet vlastních kmitočtů krakorce nebo tyče proměnného průřezu s volnými konci) bylo nalezeno exaktní řešení pro zvláštní průběh tuhosti a hmoty po délce prutu [4]. V ostatních případech bylo po užito jen přibližných metod [4], [6], [7], [8], [9], [10], [11].…”