Equilibria and vibration of a buckled beam with attached masses or spring-mass systems

A buckled beam with immovable pinned ends is considered. Attached to the beam are either one concentrated mass, two concentrated masses, a spring-mass system (that could model a human, robot, or passive vibration absorber), or a horizontal rigid bar with two vertical end springs (a “bounce-pitch” system that could model an animal or a vehicle). In the theoretical analysis, the beam is modeled as an inextensible elastica. Equilibrium configurations are determined first. Then small free vibrations about equilibrium are examined, and the lowest frequencies and corresponding modes are computed. The effects of various parameters are investigated, such as the ratio of the span to the total arc length of the beam, the locations and weights of the attached masses and systems, and the stiffnesses of the springs. For the case of a single attached mass, experiments are conducted and the results are compared to the theoretical ones.