An interval algorithm for uncertain dynamic stability analysis

Analysis on the stability of dynamic systems is a very important field in structural elasticity theory. For possible engineering situations where the system parameters are uncertain‑but‑bounded, the propagation from the initial uncertainties of parameters to the consequent uncertainty of dynamic stability of structure is elaborately investigated by interval analysis for the first time. The interval dynamic stability issue is studied through the column structure as a generalized interval eigenvalue problem, and an analytic theorem is rigorously derived and numerically verified for solving the generalized interval eigenvalue problem. Moreover, to avoid the exaggeration on actual bounds of solution, an effective approach is established to limit the dependency phenomenon in the interval dynamic stability analysis. The interval relationships of the excitation parameter and the critical load frequency with five types of system parameters are further bridged according to the interval arithmetic to propose a three-stage interval scheme for evaluating the effects of interval system parameters on the dynamic stability of structures. Numerical studies demonstrate that the uncertainty of the constant part of load influences the boundaries of principal instability region much more than that of the variable part of load. Within all structural parameters, the uncertainty of column length has the most effect on the boundaries, which could make the critical frequency of load be magnified about ten times the uncertainty of parameter. In particular, if all the system parameters are interval with an identical uncertainty degree, the uncertainty of parameter would be propagated in the system of dynamic stability and enlarged as high as 20 times the uncertainty of parameter. Consequently, the impacts of parameter uncertainties on the dynamic stability of structures are fairly significant.