Ascending rockets as macroscopic self-propelled Brownian oscillators

Abstract

High-fidelity numerical experiments and theoretical modelling are used to study the dynamics of a sounding-rocket-scale rocket, subject to altitude-dependent random wind and nozzle side loads and deterministic aerodynamic loading. This paper completes a series of studies that showed that Ornstein–Uhlenbeck (OU) rotational dynamics arise when random nozzle side loads dominate wind and aerodynamic loading. In contrast to the earlier work, this paper elucidates that under conditions where aerodynamic, wind and nozzle side loads are comparable, the rocket behaves as stochastic Brownian oscillator. The Brownian oscillator model allows straightforward interpretation of the complex rotational dynamics observed: three dynamical regimes—each characterized by differing balances between nozzle-side-load-induced torques, spring-like aerodynamic torques and mass flux damping torques—characterize rocket ascent. Further, the paper illuminates that in the limit where wind and aerodynamic loads are small, random mass flux variations exponentially amplify side-load-induced rotational stochasticity. In this practical limit, pitch/yaw dynamics are described by a randomly damped OU process; an exact solution of the associated Fokker–Planck equation can be obtained and used to compute, e.g. time-dependent pitch/yaw rate means and variances.