Axial vibrations of U-shaped bellows with elastically restrained end conditions

Previous work by Li et al. in the area of axial vibrations of bellows dealt with fixed end conditions. However, it is seen on several occasions that bellow ends are welded to a small pipe spool that has a lumped mass such as a valve or an instrument. Hence, the present paper aims at finding out the effect of elastically restrained ends on the axial natural frequencies. The analysis considers finite stiffness axial restraints on the bellows, i.e. solving the set of equations with non-homogeneous boundary conditions. Two bellow specimens are considered for comparison having the same dimensions as taken by Li in his analysis. The transcendental frequency equation deduced is accurate as the first, second and third mode frequencies computed are in close agreement to the ones obtained by Li.