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 regimeseach 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.

Angular momentum of free variable mass systems is partially conserved

“…In the above equation, N d/dt is a time derivative taken in the inertial reference frame N. Any vector can be expressed as a combination of a scalar and a unit vector directed along the vector itself. So, h is rewritten as h = hn h , where n h is a unit vector directed along h whose magnitude is h. As a result, Equation 12can be written as (14) and Equation 13as (15) Equation (14) asserts that H * o is generally not of constant magnitude while Equations (14) and (15) assert that it is always directed along the n h vector, which it will now be proved is an inertially fixed vector.…”

“…A control volume approach [8,9] to account for continuous mass variation subsequently emerged which has since become the modeling standard amongst the community of researchers on rocket flight dynamics. Recent work using the control volume formulation has focused on equation of motion formulation for general variable mass systems [10,11], modeling and analysis of rocket-type systems [12][13][14][15] and an abstraction of the rocket problem [16], and stability of transverse rotational motion in solid rocket motors [17]. The developments presented here on angular momentum also utilize this control volume formulation.…”

Time-dependent inertia of self-propelled particles: the Langevin rocket

“…The effect of noise on rocket motion has been considered previously, see e.g. [99], but a simple basis reference model for that is missing. We therefore simplify the general Eqs.…”