<p>The accuracy, functionality quality, structural integrity, or data integrity due to in-plane vibrations of the rotary storage, power transmission, or cutting macro/micro-discs may be significantly reduced by increasing structural damping, which can be modeled using integer- or, much more accurately, fractional-order viscoelastic models. The current article investigates, for the first time, the nonlinear in-plane time-dependent vibration of rotating integer and fractional-order viscoelastic micro-discs that are shrink-fitted to micro-shafts and subjected to external pressure shocks. Since the Eringen and nonlocal strain gradient theories yield unreliable displacement and stress results, a new energy-based fractional-order modified strain gradient theory (FOMSGT) that accounts for the relaxation of the higher-order stress tensors as well is developed. The presence of the nonlinear shrink-fit contact condition between the viscoelastic disc and the shaft significantly increases the complexity of the problem. The nonlinear coupled integrodifferential system of the governing finite element equations is derived by employing the principle of minimum total potential energy and either a hierarchical three-element Prony-series or a Caputo-kernel fractional viscoelastic model, and solved using a combination of iterative Runge–Kutta and trapezoidal time-expansion/integration schemes. Using distinct micro-dimension coefficients is another novelty. Results reveal that the dilatation-gradient-relevant micro-dimension coefficient has the dominant effect. The conventional viscoelasticity models lead to transient oscillations with time-varying natural frequencies (in contrast to the fractional-order model), the shrink-fit stress is time-dependent and is larger for viscoelastic materials and larger micro-dimension coefficients.</p>