Angular momentum of free variable mass systems is partially conserved

Abstract

Variable mass systems are a classic example of open systems in classical mechanics with rockets being a standard practical example. Due to the changing mass, the angular momentum of these systems is not generally conserved. Here, we show that the angular momentum vector of a free variable mass system is fixed in inertial space and, thus, is a partially conserved quantity. It is well known that such conservation rules allow simpler approaches to solving the equations of motion. This is demonstrated by using a graphical technique to obtain an analytic solution for the second Euler angle that characterizes nutation in spinning bodies.

Geometry of motion and nutation stability of free axisymmetric variable mass systems

“…whereχ has the units of angular speed. From Equations (21) and (19), the transverse speeds are given by ω 1 = Γ (ω 10 cos χ + ω 20 sin χ) (25) ω 2 = Γ (−ω 10 sin χ + ω 20 cos χ) (26) and are oscillatory in nature.…”

“…These developments are then used to study the variable mass cylinder problem in the following section. Exploiting the fact that the variable mass system's angular momentum vector is fixed in inertial space [25,26] enables us to visually study the angular velocity evolution from an inertial frame. The angular momentum of a general variable mass system can be written as…”

“…First, the scalar equations of attitude motion for a general axisymmetric system are formulated and analytic solutions to these equations are obtained. Then, based on a recent result concerning the inertial fixedness of the central angular momentum vector of such systems [25,26], the approach to visualize the rotations of these systems from both body-fixed and inertial frames is developed. Finally, this theory is used to evaluate the motion of axisymmetric rigid cylinders subject to three idealized models of mass loss.…”

Vibrations of a nonlinear stochastic system with a varying mass under near resonant excitation

“…The analysis of linear and nonlinear dynamics of systems with varying masses is a popular research subject, cf., e.g. Feynman et al (1964), Irschik and Holl (2004), Li et al (2016), Nanjangud and Eke (2018), van Horssen et al (2010), and Wang and Pumera (2015). Such systems appear in many different areas, from micro- and nano-applications (Li et al, 2016; Wang and Pumera, 2015) to space industry (Nanjangud and Eke, 2018).…”

“…Feynman et al (1964), Irschik and Holl (2004), Li et al (2016), Nanjangud and Eke (2018), van Horssen et al (2010), and Wang and Pumera (2015). Such systems appear in many different areas, from micro- and nano-applications (Li et al, 2016; Wang and Pumera, 2015) to space industry (Nanjangud and Eke, 2018). Although the effects of deterministic mass variations on the dynamic behaviour of linear systems are relatively well understood (Bogoliubov, 1969; van Horssen et al, 2010), it is not the case for stochastic variations, especially in the nonlinear context.…”