Exact Damped Bessel Dynamics and Inverse Identification of Linear Mass Depletion in a Time-varying Oscillator

Background: Variable-mass oscillators combine changing inertia with momentum transfer. Interpreting transient responses therefore depends on modelling assumptions, particularly when spring stiffness is uncertain. Aims: To derive an exact analytical representation for a viscously damped oscillator with linearly decreasing mass and to determine whether mass-depletion rate and damping can be identified reliably from transient displacement records when spring stiffness is independently constrained. Study Design: Analytical and computational method-validation study using reproducible stochastic computational experiments. Place and Duration of Study: Department of Education in Science and Mathematics, Regional Institute of Education, Bhopal (NCERT), India; computational work conducted during 2026. Methodology: The open-system equation for a linearly depleting damped oscillator was transformed to a Bessel equation of order . Twenty-four independent computational runs were generated at nominal depletion rates of , and with run-to-run variability, 0.50 mm displacement noise and 0.40 g gravimetric noise. Constant-mass damped, undamped variable-mass and damped variable-mass models were compared using residual error and information criteria. Identifiability, observation-window sensitivity, and noise/sampling robustness were also evaluated. Results: Mean inverse depletion estimates were , and , with recovered damping near the prescribed . Direct and inverse rate estimates showed . The damped variable-mass model retained trajectory RMSE near the imposed noise level (0.490-0.505 mm), whereas incomplete models produced multi-millimetre errors. Joint estimation of spring stiffness and depletion rate produced strong negative correlation ( over the full record). Conclusion: The non-zero-order Bessel formulation provides an efficient forward model for inverse identification. Independent stiffness calibration and explicit model discrimination are essential for defensible depletion-rate inference.