Improving the linear stability of the Beck׳s beam by added dashpots

The effects of lumped dashpots on the linear stability of the Beck׳s beam, internally and externally damped, are studied, with the aim to improve the mechanical performances of the system. A perturbation analysis is carried out, in which the sensitivities of the simple eigenvalues of the undamped and sub-critically loaded beam are evaluated. The analysis provides a simple and very efficient tool for damping design. It is proved that there exists a region along the beam axis assuring that the added dashpots, of extensional and flexural type, increase the critical load of the beam under distributed damping, thus mitigating the detrimental effect of the destabilization paradox. It is also shown that, if the flexural dashpot is located at the tip of the beam, it has a beneficial effect at low level of load, but, counter-intuitively, it is detrimental when the load approaches the critical value of the undamped beam, thus extending analogous findings of the literature concerning the extensional dashpot.