Equivalent damping and closed-form response statistics of stochastic oscillators

Damped oscillators with uncertain natural frequency exhibit fundamentally different dynamic behaviour compared to their deterministic counterparts. This paper develops exact closed-form solutions for the statistical characterisation of damped stochastic single-degree-of-freedom systems. In particular, we derive explicit closed-form expressions for the mean response across the entire frequency spectrum, an elementary-function expression for the equivalent damping, and the complete frequency-dependent variance. These results reveal that parametric uncertainty manifests physically as enhanced effective damping in ensemble-averaged responses. The expressions involve only elementary transcendental functions and require no numerical integration. The central theoretical contribution is the equivalent damping formula, demonstrating that uncertainty in natural frequency increases apparent energy dissipation monotonically with randomness magnitude. Remarkably, these solutions remain valid precisely at resonance frequencies where conventional perturbation methods fail, and polynomial chaos expansions require prohibitively high-order approximations. The analytical framework is found to be numerically robust across the probability distributions tested: expressions derived assuming uniform uncertainty provide good accuracy when validated against Gaussian, lognormal, and chi-square probability models. This is presented as an empirical observation over the four distributions examined rather than as a general theoretical result, and it suggests that, within this range, the second-order response statistics are controlled primarily by the variance rather than by higher-order moments of the uncertain parameter. These results transform stochastic frequency response analysis from a computationally intensive numerical exercise into an analytically transparent process, enabling physics-informed robust design strategies and providing the theoretical foundation for understanding how parametric uncertainty fundamentally alters dynamic response characteristics in vibration-critical systems. Beyond their interpretive value, the closed-form expressions are intended to serve as exact benchmarks for validating numerical uncertainty-quantification schemes and as ready design references that expose, in transparent form, how the randomness level and baseline damping jointly govern the mean response, the effective damping, the resonance shift, and the response variance.