Stability of free rotation of a rigid body
Rigid body free rotational stability, discussing initial conditions excursion
Vinti, J. P.↗
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Kinematics of finite, rigid-body displacements.
Rigid body kinematics for finite displacements, proving several theorems analytically by matrix algebraic methods using mass density description
Thurnauer, P. G.↗
Stiffness-generated rigid-body mode shapes for Lanczos eigensolution with SUPORT DOF by way of a MSC/NASTRAN DMAP alter
When using all MSC/NASTRAN eigensolution methods except Lanczos, the analyst can replace the coupled system rigid-body modes calculated within DMAP module READ with mass orthogonalized and normalized rigid-body modes generated from the system stiffness. This option is invoked by defining MSC/NASTRAN r-set degrees of freedom via the SUPORT bulk data card. The newly calculated modes are required if the rigid-body modes calculated by the eigensolver are not 'clean' due to numerical roundoffs in the solution. When performing transient structural dynamic load analysis, the numerical roundoffs can result in inaccurate rigid-body accelerations which affect steady-state responses. Unfortunately, when using the Lanczos method and defining r-set degrees of freedom, the rigid-body modes calculated within DMAP module REIGL are retained. To overcome this limitation and to allow MSC/NASTRAN to handle SUPORT degrees of freedom identically for all eigensolvers, a DMAP Alter has been written which replaces Lanczos-calculated rigid-body modes with stiffness-generated rigid-body modes. The newly generated rigid-body modes are normalized with respect to the system mass and orthogonalized using the Gram-Schmidt technique. This algorithm has been implemented as an enhancement to an existing coupled loads methodology.
Abdallah, Ayman A.↗
The relative motion of two spheroidal rigid bodies. II
Two spheroid rigid bodies rotational and translational motion, using linear and Hill type differential equations for angular variables and coordinates
Lanzano, P.↗
Kinematics of rigid bodies in spaceflight
Rigid rotating body kinematic equations for use in space flight mechanics
Kane, T. R.↗
Elimination of uncontrollable rigid body mode, appendix C
The planar system model is examined which reveals two rigid body modes. One is due to the pure translation of the whole experiment assembly and is uncontrollable because the only control input is a torque at the gimbal. The other rigid body mode is due to the pointing of the gimbal and is the one for which control is sort. The problem arises from the Vector V2 which relates the control input to the modal coordinates. Since its fifth and sixth elements are both nonzero, neither body mode can be thrown away as is. The two rigid body modes must be transformed so that one is unaffected by the control input and one is left controllable. The uncontrollable mode can then be eliminated from the model for design purposes. A process is described to accomplish the elimination of the uncontrollable rigid body mode.
Source record↗
Interactions between rigid-body and flexible-body motions in maneuvering spacecraft
The present consideration of the significant interactions between rigid-body and flexible-body motions in maneuvering spacecraft proceeds by distinguishing between the two types of motion on the basis of a tracking coordinate system which coincides with the rigid-body component of the motion, as well as by maintaining the motion relative to the tracking coordinate as orthogonal to the rigid-body motion. The elastic motion is excited by the rigid-body motion via Coriolis terms, angular acceleration terms, and centrifugal terms. These interactions are illustrated for spacecraft undergoing bidirectional elastic motions via the dynamics of constantly rotating free-free beams subject to combined bending and longitudinal vibration.
Silverberg, Larry M.↗
An Interface for Specifying Rigid-Body Motions for CFD Applications
An interface for specifying rigid-body motions for CFD applications is presented. This interface provides a means of describing a component hierarchy in a geometric configuration, as well as the motion (prescribed or six-degree-of-freedom) associated with any component. The interface consists of a general set of datatypes, along with rules for their interaction, and is designed to be flexible in order to evolve as future needs dictate. The specification is currently implemented with an XML file format which is portable across platforms and applications. The motion specification is capable of describing general rigid body motions, and eliminates the need to write and compile new code within the application software for each dynamic configuration, allowing client software to automate dynamic simulations. The interface is integrated with a GUI tool which allows rigid body motions to be prescribed and verified interactively, promoting access to non-expert users. Illustrative examples, as well as the raw XML source of the file specifications, are included.
Murman, Scott M.↗
A Novel Approach for Computing Rigid Body Motion Using Linear Accelerations
Here, a novel approach is presented for computing general rigid body motion based on a few known linear accelerations. This method utilizes linear acceleration data obtained from three distinct points on the body, all within a body-fixed reference frame. The only requirement is that the three chosen points must not be collinear. A system of differential-algebraic equations is derived, combining principles of rigid body kinematics with theory of the rotation group SO(3). These equations provide a framework for numerically computing various motion parameters, including angular velocity, angular acceleration, body orientation, velocity field, acceleration field, and displacement field. By numerically solving this system of equations, we can fully characterize rigid body motion in three-dimensional space. A numerical example is provided to demonstrate the practical implementation and efficacy of the proposed technique, illustrating its potential for accurate motion computation in various applications.
42 ENGINEERING↗
Analytic Theory and Control of the Motion of Spinning Rigid Bodies
Numerical simulations are often resorted to, in order to understand the attitude response and control characteristics of a rigid body. However, this approach in performing sensitivity and/or error analyses may be prohibitively expensive and time consuming, especially when a large number of problem parameters are involved. Thus, there is an important role for analytical models in obtaining an understanding of the complex dynamical behavior. In this dissertation, new analytic solutions are derived for the complete attitude motion of spinning rigid bodies, under minimal assumptions. Hence, we obtain the most general solutions reported in the literature so far. Specifically, large external torques and large asymmetries are included in the problem statement. Moreover, problems involving large angular excursions are treated in detail. A new tractable formulation of the kinematics is introduced which proves to be extremely helpful in the search for analytic solutions of the attitude history of such kinds of problems. The main utility of the new formulation becomes apparent however, when searching for feedback control laws for stabilization and/or reorientation of spinning spacecraft. This is an inherently nonlinear problem, where standard linear control techniques fail. We derive a class of control laws for spin axis stabilization of symmetric spacecraft using only two pairs of gas jet actuators. Practically, this could correspond to a spacecraft operating in failure mode, for example. Theoretically, it is also an important control problem which, because of its difficulty, has received little, if any, attention in the literature. The proposed control laws are especially simple and elegant. A feedback control law that achieves arbitrary reorientation of the spacecraft is also derived, using ideas from invariant manifold theory. The significance of this research is twofold. First, it provides a deeper understanding of the fundamental behavior of rigid bodies subject to body-fixed torques. Assessment of the analytic solutions reveals that they are very accurate; for symmetric bodies the solutions of Euler's equations of motion are, in fact, exact. Second, the results of this research have a fundamental impact on practical scientific and mechanical applications in terms of the analysis and control of all finite-sized rigid bodies ranging from nanomachines to very large bodies, both man made and natural. After all, Euler's equations of motion apply to all physical bodies, barring only the extreme limits of quantum mechanics and relativity.
Tsiotras, Panagiotis↗
Planar dynamics of a uniform beam with rigid bodies affixed to the ends
The planar dynamics of a uniform elastic beam subject to a variety of geometric and natural boundary conditions and external excitations were analyzed. The beams are inextensible and capable of small transverse bending deformations only. Classical beam vibration eigenvalue problems for a cantilever with tip mass, a cantilever with tip body and an unconstrained beam with rigid bodies at each are examined. The characteristic equations, eigenfunctions and orthogonality relations for each are derived. The forced vibration of a cantilever with tip body subject to base acceleration is analyzed. The exact solution of the governing nonhomogeneous partial differential equation with time dependent boundary conditions is presented and compared with a Rayleigh-Ritz approximate solution. The arbitrary planar motion of an elastic beam with rigid bodies at the ends is addressed. Equations of motion are derived for two modal expansions of the beam deflection. The motion equations are cast in a first order form suitable for numerical integration. Selected FORTRAN programs are provided.
Storch, J.↗
Gravitational Equilibria of a Rigid Body Containing Symmetric Rotors
Gravitational equilibria of rigid body containing symmetric constant speed fixed axis rotor
ROTOR↗
A frequency independent technique for extracting the rigid-body motion from the total motion of a large flexible launch vehicle
Mathematical model to compute rigid body motion independently of flexible body motion in launch vehicles
FLEXIBLE BODY↗
A boundary integral approach to the scattering of nonplanar acoustic waves by rigid bodies
The acoustic scattering of an incident wave by a rigid body can be described by a singular Fredholm integral equation of the second kind. This equation is derived by solving the wave equation using generalized function theory, Green's function for the wave equation in unbounded space, and the acoustic boundary condition for a perfectly rigid body. This paper will discuss the derivation of the wave equation, its reformulation as a boundary integral equation, and the solution of the integral equation by the Galerkin method. The accuracy of the Galerkin method can be assessed by applying the technique outlined in the paper to reproduce the known pressure fields that are due to various point sources. From the analysis of these simpler cases, the accuracy of the Galerkin solution can be inferred for the scattered pressure field caused by the incidence of a dipole field on a rigid sphere. The solution by the Galerkin technique can then be applied to such problems as a dipole model of a propeller whose pressure field is incident on a rigid cylinder. This is the groundwork for modeling the scattering of rotating blade noise by airplane fuselages.
Gallman, Judith M.↗
Attitude dynamics simulation subroutines for systems of hinge-connected rigid bodies with nonrigid appendages
Three computer subroutines designed to solve the vector-dyadic differential equations of rotational motion for systems that may be idealized as a collection of hinge-connected rigid bodies assembled in a tree topology, with an optional flexible appendage attached to each body are reported. Deformations of the appendages are mathematically represented by modal coordinates and are assumed small. Within these constraints, the subroutines provide equation solutions for (1) the most general case of unrestricted hinge rotations, with appendage base bodies nominally rotating at a constant speed, (2) the case of unrestricted hinge rotations between rigid bodies, with the restriction that those rigid bodies carrying appendages are nominally nonspinning, and (3) the case of small hinge rotations and nominally nonrotating appendages. Sample problems and their solutions are presented to illustrate the utility of the computer programs.
Fleischer, G. E.↗
Development of an SE(3)-based Rigid Body Pose Estimation Scheme for Unknown Moments of Inertia in a Spacecraft
In real-world applications, the consideration of state and parameter uncertainties is an important part of any navigation and control system. Uncertainties can arise because of the instruments, communication systems, or external disturbances. Parameter uncertainty can be due to uncertainties in the data or the calibration process used. The objective of this research is to develop a rigid body pose estimation scheme on nonlinear manifolds of rigid body motion groups for unknown mass properties moments of inertia with application in space launch. An unscented Kalman filter is developed on special Euclidean space and their tangent bundle to address uncertainties in the states and inertia properties. This estimator accounts for uncertainties, while considering rotational-translational coupling, and avoids singularity or non-uniqueness issues. Since this estimator is based on the nonlinear manifolds on which rigid body motion evolves, it is reliable and is expected to result in high level of accuracy. The estimators are validated on a generic model of Logistic Module (LM) and a cuboid spacecraft model. The algorithms can then be tested on Dragon XL, HTV-X, and possibly other spacecraft.
Moments of Inertia↗
Coupled motion of rigid bodies about their center of mass
Nontrivial analytical solutions for the coupled motion of two rigid bodies about their center of mass are obtained on the assumptions that the rigid bodies are coupled by a massless rigid boom and that no external forces are acting on the system. Both relative rotational and translational motions of the two bodies are considered. General equations of motion are derived by regarding the two bodies as consisting of two distinct systems of particles and by applying the principle of conservation of angular momentum. It is shown that a basic nontrivial solution can be obtained for the translational problem if an assumption is made concerning the relative orientation of one principal axis of inertia of each body and that fundamental nontrivial solutions are readily obtained for the rotational problem if an additional assumption is made with respect to the symmetry of one body. Certain stability criteria are found for some of these motions by defining regions of constraint for the relative translational and rotational elements.
Jezewski, D. J.↗
A general purpose nonlinear rigid body mass finite element for application to rotary wing dynamics
The Second Generation Comprehensive Helicopter Analysis System employs the present formulation of the general-purpose nonlinear rigid body mass finite element, which represents the hub masses, blade tip masses, and pendulum vibration absorbers. The rigid body mass element has six degrees of freedom, and accounts for gravitational as well as dynamic effects. A consequence of deriving the element's equations from various physical principles is that, prior to the transformation which couples the rigid body mass element to the rotor blade finite element, the forces obtained for each element are fundamentally different; this is true notwithstanding the degrees-of-freedom of each element are parameterized using the same coordinates.
Hamilton, B. K.↗