Evaluation of path-dependent work and internal energy change for hysteretic mechanical systems

The accuracy of procedures for the analysis and design of hysteretic mechanical systems, in which both generalized rate-dependent and rate-independent hysteretic forces have a non-conservative nature, can be significantly improved by a straightforward computation of work and energy terms. Thus, with the aim of shedding light on the definition and evaluation of work and energy components, we first derive the nonlinear equilibrium equations of a conventional family of MDOF hysteretic mechanical systems and, subsequently, we introduce the modified work-energy theorem. In addition, to allow for a forthright evaluation of path-dependent work and internal energy change associated with generalized rate-dependent (rate-independent) hysteretic forces, we derive the related (closed form) expressions by adopting the Seleemah and Constantinou (Vaiana and Rosati) model. The former is capable of simulating both linear and nonlinear rate-dependent hysteretic responses; the latter can simulate a great variety of rate-independent hysteretic behavior, characterized by symmetric, asymmetric, pinched, S-shaped, flag-shaped hysteresis loops or by an arbitrary combination of them. Finally, several nonlinear dynamic analyses are performed on a MDOF hysteretic mechanical system to clearly describe its free vibration and forced harmonic vibration responses in terms of work and energy quantities.