Transverse vibration of free–free beams carrying two unequal end masses

Author-Highlights
• Exact frequency equation of free–free beams carrying two unequal end masses is derived.
• Vibration mode shapes of free–free beams with unequal end masses are obtained.
• Simple approximate expressions for the fundamental frequencies with high accuracy are given.
• Approximate fundamental frequency for equal end masses in open literature has been modified.

Transverse vibration of free–free slender beams carrying two tip masses is studied. An exact frequency equation is derived and the natural frequencies are calculated. An alternative Fredholm integral equation approach is presented to determine resonant frequencies of a vibrating mass–beam system. Simple approximate expressions for fundamental frequencies with high accuracy are obtained. Analytical and approximate results of the frequencies are computed and compared. The effects of concentrated masses at two ends on the natural frequencies and mode shapes are discussed. The results are helpful to design micromechanical sensors and energy harvesters.