Symbolic equations of motion (EOMs) for multibody systems are desirable for simulation, stability analyses, control system design, and parameter studies. Despite this, the majority of engineering software designed to analyze multibody systems are numeric in nature (or present a purely numeric user interface). To our knowledge, none of the existing software packages are 1) fully symbolic, 2) open source, and 3) implemented in a popular, general, purpose high level programming language. In response, we extended SymPy (an existing computer algebra system implemented in Python) with functionality for derivation of symbolic EOMs for constrained multibody systems with many degrees of freedom. We present the design and implementation of the software and cover the basic usage and workflow for solving and analyzing problems. The intended audience is the academic research community, graduate and advanced undergraduate students, and those in industry analyzing multibody systems. We demonstrate the software by deriving the EOMs of a N-link pendulum, show its capabilities for LATEX output, and how it integrates with other Python scientific libraries — allowing for numerical simulation, publication quality plotting, animation, and online notebooks designed for sharing results. This software fills a unique role in dynamics and is attractive to academics and industry because of its BSD open source license which permits open source or commercial use of the code.
It is shown that any finite number of plants that belong to certain classes of multi-input multi-output systems with no zeros in the region of instability can be simultaneously stabilized using linear, time-invariant integral-action controllers. These plants may be stable or unstable and their poles are not restricted; they may also have any number of zeros in the stable region of the complex plane. The classes of systems under consideration include plants with blocking or transmission zeros at infinity. The common controller achieves asymptotic tracking of step-input references with zero steady-state error and has a low order transfer-function. Systematic synthesis methods are presented, and a parametrization of all simultaneously stabilizing controllers with integral-action is also provided.
This paper presents a derivation of the equations of motion of variable mass systems based on Lagrange's equations. The derivation makes use of the control volume concept and exploits Reynolds Transport Theorem to generate equations that are reasonably compact, yet general enough to capture the dynamical behavior of variable mass systems of any shape and configuration. The only restriction is that the system should include a solid base. The equations are thus very well suited for the study of the translational as well as rotational motions of rockets and similar systems.
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.
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