Uncertainty Analysis of the Time-Varying Response of Centroid Displacement of a High-Precision Inertial Device Under Vibration

Time-varying uncertainty in the assembly process of precision instruments and devices, particularly its complex effects on product stability and consistency, poses significant challenges compared to traditional parameter uncertainty. This article investigates the time-varying uncertainty response of centroid displacement in a high-precision inertial instrument during vibration stability treatment. First, interval process modeling combined with Karhunen–Loève (K–L) expansion is employed to characterize vibration uncertainty. Second, centroidal displacement responses are determined through finite-element analysis, while a hybrid approach integrating time–frequency conversion and long short-term memory (LSTM) neural network establishes the uncertainty propagation mechanism from vibration inputs to displacement outputs. Third, an interval process sampling inverse method is proposed to determine the time-varying uncertainty boundaries of centroid displacement. Results demonstrate that the centroid displacement uncertainty intensifies with increasing magnitude of the vibration radius function and longer temporal correlation durations. Finally, the envelope boundary of centroid displacement under vibration stability treatment with a specific power spectral density (PSD) profile (0.04 g²/Hz in 80–350 Hz, +3 dB/Oct slope for 20–80 Hz, and −3 dB/Oct slope for 350–2000 Hz) is determined and validated numerically. This framework enables the quantitative evaluation of inertial device stability and consistency under vibrational conditions, thereby addressing a critical gap in precision assembly research. The proposed methodology advances time-varying uncertainty analysis by integrating interval process theory, surrogate modeling, and inverse sampling techniques, offering a systematic solution applicable to nonlinear systems in high-precision manufacturing scenarios.