An Analytical and Geometrical Study of the Dzhanibekov and Tennis Racket Phenomena

2016 ◽  
Author(s):  
Oscar Rios ◽  
Takeyuki Ono ◽  
Hidenori Murakami ◽  
Thomas J. Impelluso
Keyword(s):  
Rigid Body ◽  
Complete Solution ◽  
Satellite System ◽  
Moment Of Inertia ◽  
Moving Frame ◽  
Solar Panels ◽  
Tennis Racket ◽  
Principal Moments Of Inertia ◽  
Moving Frame Method ◽  
Analytical Expressions

We consider the torque-free rotation of a rigid body with three distinct moment of inertia values and angular velocity components. As can be seen in the Dzhanibekov and tennis racket phenomena, rotations about the largest and smallest principal moments of inertia create stable rotations. However, when rotating about the principal intermediate moment of inertia, an unstable rotation is produced that leads to the basis of this phenomena. In this publication, the above phenomena are examined and explained analytically and applied to a satellite system to observe the change in trajectory as the solar panels and reflectors are deployed. To begin the derivation, the Euler torque-free equations for a rotating rigid body are formulated using the moving frame method. The derived equations are then non-dimensionalized and a complete analytical solution, including an expression for the non-dimensional period, is presented. Second, we further look at the limiting axisymmetric cases and examine the effect as the intermediate moment of inertia is varied. Lastly, the analytical expressions are compared with the numerical simulation to validate the results. The complete solution is then summarized and shown to clearly prove that the conservation of angular momentum is indeed preserved in the phenomena.

Stability of a Rigid Body With an Oscillating Particle: An Application of MACSYMA

1985 ◽  
Vol 52 (3) ◽  
pp. 686-692 ◽  
Author(s):  
L. A. Month ◽  
R. H. Rand
Keyword(s):  
Angular Velocity ◽  
Rigid Body ◽  
Classical Problem ◽  
Moment Of Inertia ◽  
Model Parameters ◽  
Stability Chart ◽  
The Stability ◽  
Transition Curves ◽  
Minimum Moment ◽  
Small Angular Velocity

This problem is a generalization of the classical problem of the stability of a spinning rigid body. We obtain the stability chart by using: (i) the computer algebra system MACSYMA in conjunction with a perturbation method, and (ii) numerical integration based on Floquet theory. We show that the form of the stability chart is different for each of the three cases in which the spin axis is the minimum, maximum, or middle principal moment of inertia axis. In particular, a rotation with arbitrarily small angular velocity about the maximum moment of inertia axis can be made unstable by appropriately choosing the model parameters. In contrast, a rotation about the minimum moment of inertia axis is always stable for a sufficiently small angular velocity. The MACSYMA program, which we used to obtain the transition curves, is included in the Appendix.

Finding Principal Axes of Complex Plane Rigid Body with Rending-Image of MATLAB

2012 ◽  
Vol 490-495 ◽  
pp. 2156-2159
Author(s):  
Wu Gang Li
Keyword(s):  
Rigid Body ◽  
Complex Plane ◽  
Principal Axis ◽  
Flat Surface ◽  
Moment Of Inertia ◽  
Principal Axes ◽  
The Moment

In order to find the principal axes of inertia and calculate their moment of inertia to any plane homogeneous rigid body for calculating easily the moment of inertia to any axis of this rigid body, the principal axes could be found and their moment of inertia could be calculated automatically by using the reading-image of MATLAB to read the image messages about the flat surface of the rigid body and by the procedures which ware made according to the logic relation about the principal axis and the moment of inertia of the rigid body. Applying this method in a homogeneous cube, a result was acquired, error of which is small compared with the theoretical value. So this method is reliable, convenient and practical

scholarly journals Stabilizing Ship Motion With a Dual System Inertial Disk

2019 ◽  
Author(s):  
Torstein R. Storaas ◽  
Kasper Virkesdal ◽  
Gitle S. Brekke ◽  
Thorstein Rykkje ◽  
Thomas Impelluso
Keyword(s):  
Oil And Gas ◽  
New Technologies ◽  
Equations Of Motion ◽  
Moving Frame ◽  
Moving Frames ◽  
New Approach ◽  
Lie Group Theory ◽  
Moving Frame Method ◽  
Frame Method ◽  
Modern Mathematics

Abstract Norwegian industries are constantly assessing new technologies and methods for more efficient and safer maintenance in the aqua cultural, renewable energy, and oil and gas industries. These Norwegian offshore industries share a common challenge: to install new equipment and transport personnel in a safe and controllable way between ships, farms and platforms. This paper deploys the Moving Frame Method (MFM) to analyze ship stability moderated by a dual gyroscopic inertial device. The MFM describes the dynamics of the system using modern mathematics. Lie group theory and Cartan’s moving frames are the foundation of this new approach to engineering dynamics. This, together with a restriction on the variation of the angular velocity used in Hamilton’s principle, enables an effective way of extracting the equations of motion. This project extends previous work. It accounts for the dual effect of two inertial disk devices, it accounts for the prescribed spin of the disks. It separates out the prescribed variables. This work displays the results in 3D on cell phones. It represents a prelude to testing in a wave tank.

scholarly journals Modeling a Knuckle-Boom Crane Control to Reduce Pendulum Motion Using the Moving Frame Method

2019 ◽  
Author(s):  
Maren Eriksen Eia ◽  
Elise Mari Vigre ◽  
Thorstein Ravneberg Rykkje
Keyword(s):  
Equations Of Motion ◽  
Moving Frame ◽  
Web Pages ◽  
Moving Frames ◽  
Pendulum Motion ◽  
Moving Frame Method ◽  
Frame Method ◽  
The Masses ◽  
And Control ◽  
Boom Crane

Abstract A Knuckle Boom Crane is a pedestal-mounted, slew-bearing crane with a joint in the middle of the distal arm; i.e. boom. This distal boom articulates at the ‘knuckle (i.e.: joint)’ and that allows it to fold back like a finger. This is an ideal configuration for a crane on a ship where storage space is a premium. This project researches the motion and control of a ship mounted knuckle boom crane to minimize the pendulum motion of a hanging load. To do this, the project leverages the Moving Frame Method (MFM). The MFM draws upon Lie group theory — SO(3) and SE(3) — and Cartan’s Moving Frames. This, together with a compact notation from geometrical physics, makes it possible to extract the equations of motion, expeditiously. The work reported here accounts for the masses and geometry of all components, interactive motor couples and prepares for buoyancy forces and added mass on the ship. The equations of motion are solved numerically using a 4th order Runge Kutta (RK4), while solving for the rotation matrix for the ship using the Cayley-Hamilton theorem and Rodriguez’s formula for each timestep. This work displays the motion on 3D web pages, viewable on mobile devices.

scholarly journals GNSS-Acoustic Observations of Seafloor Crustal Deformation Using a Wave Glider

2021 ◽  
Vol 9 ◽  
Author(s):  
Takeshi Iinuma ◽  
Motoyuki Kido ◽  
Yusaku Ohta ◽  
Tatsuya Fukuda ◽  
Fumiaki Tomita ◽  
...  
Keyword(s):  
Data Acquisition ◽  
Crustal Deformation ◽  
Satellite Communication ◽  
Satellite System ◽  
Japan Trench ◽  
Research Vessel ◽  
Solar Panels ◽  
Wave Glider ◽  
Human Costs ◽  
Global Navigation Satellite

Crustal deformation of the seafloor is difficult to observe solely using global navigation satellite system (GNSS). The GNSS-acoustic (GNSS-A) technique was developed to observe seafloor crustal deformation, and it has produced a steady series of successful observations with remarkable results related to crustal deformation associated with huge earthquakes around the Japanese Islands. However, utilizing GNSS-A incurs very large financial and human costs as it requires the use of a research vessel as a surface platform and has a limited observation frequency, which is less than once a year at seafloor stations along the Japan Trench. To conduct frequent observations, an automatic GNSS-A data acquisition system was developed that operates via an unmanned surface vehicle (wave glider). The first observations using this system were performed at a seafloor station off Aomori Prefecture in July 2019. The wave glider was equipped with two GNSS antennas, an acoustic transducer, a microelectromechanical system gyroscope, and associated control and logging units. Data acquisition and autonomous activation of the seafloor stations were successfully executed by controlling the power supply to the payload via satellite communication with the wave glider. The glider rarely strayed off the configured course and the solar panels generated sufficient power to perform the observations although the weather was mostly cloudy. The GNSS-A data processing results show that the position of the station was determined with the same accuracy and precision as in previous observations performed using a research vessel.

scholarly journals A teaching proposal for the study of Eigenvectors and Eigenvalues

2017 ◽  
Vol 7 (1) ◽  
pp. 100 ◽  
Author(s):  
María José Beltrán Meneu ◽  
Marina Murillo Arcila ◽  
Enrique Jordá Mora
Keyword(s):  
Rigid Body ◽  
Mathematical Thinking ◽  
Point Of View ◽  
Moment Of Inertia ◽  
Geometric Point ◽  
The Moment ◽  
Eigenvectors And Eigenvalues

In this work, we present a teaching proposal which emphasizes on visualization and physical applications in the study of eigenvectors and eigenvalues. These concepts are introduced using the notion of the moment of inertia of a rigid body and the GeoGebra software. The proposal was motivated after observing students’ difficulties when treating eigenvectors and eigenvalues from a geometric point of view. It was designed following a particular sequence of activities with the schema: exploration, introduction of concepts, structuring of knowledge and application, and considering the three worlds of mathematical thinking provided by Tall: embodied, symbolic and formal.

Development of Equations of Motion for a Stewart Platform Under Prescribed Mount Motion

2018 ◽  
Author(s):  
Takeyuki Ono ◽  
Ryosuke Eto ◽  
Junya Yamakawa ◽  
Hidenori Murakami
Keyword(s):  
Rigid Body ◽  
Virtual Work ◽  
Equations Of Motion ◽  
Base Plate ◽  
Stewart Platform ◽  
Euler Method ◽  
Moving Frame ◽  
Real Time Control ◽  
Precise Positioning ◽  
Time Control

Analytical equations of motion are critical for real-time control of translating manipulators, which require precise positioning of various tools for their mission. Specifically, when manipulators mounted on moving robots or vehicles perform precise positioning of their tools, it becomes economical to develop a Stewart platform, whose sole task is stabilizing the orientation and crude position of its top table, onto which various precision tools are attached. In this paper, analytical equations of motion are developed for a Stewart platform whose motion of the base plate is prescribed. To describe the kinematics of the platform, the moving frame method, presented by one of authors [1,2], is employed. In the method the coordinates of the origin of a body attached coordinate system and vector basis are expressed by using 4 × 4 frame connection matrices, which form the special Euclidean group, SE(3). The use of SE(3) allows accurate description of kinematics of each rigid body using (relative) joint coordinates. In kinetics, the principle of virtual work is employed, in which system virtual displacements are expressed through B-matrix by essential virtual displacements, reflecting the connection of the rigid body system [2]. The resulting equations for fixed base plate reduce to those for the top plate, obtained by the Newton-Euler method. A main result of the paper is the analytical equations of motion in matrix form for dynamics analyses of a Stewart platform whose base plate moves. The control applications of those equations will be deferred to subsequent publications.

Modeling Stablization of Crane and Ship by Gyroscopic Control Using the Moving Frame Method

2018 ◽  
Author(s):  
Josef Flatlandsmo ◽  
Torbjørn Smith ◽  
Ørjan O. Halvorsen ◽  
Johnny Vinje ◽  
Thomas J. Impelluso
Keyword(s):  
Oil And Gas ◽  
New Technologies ◽  
Equations Of Motion ◽  
Long Term Results ◽  
Moving Frame ◽  
Moving Frames ◽  
Learning Modules ◽  
Moving Frame Method ◽  
Frame Method

Norwegian industries are constantly assessing new technologies and methods for more efficient and safer production in the aqua cultural, renewable energy, and oil and gas industries. These Norwegian offshore industries share a common challenge: to install new equipment and transport personnel in a safe and controllable way between ships, farms and platforms. This paper deploys the Moving Frame Method (MFM) to analyze the motion induced by a crane and controlled by a gyroscopic inertial device mounted on a ship. The crane is a simple two-link system that transfers produce and equipment to and from barges. An inertial flywheel — a gyroscope — is used to stabilize the barge during transfer. The MFM describes the dynamics of the system using modern mathematics. Lie group theory and Cartan’s moving frames are the foundation of this new approach to engineering dynamics. This, together with a restriction on the variation of the angular velocity used in Hamilton’s principle, enables an effective way of extracting the equations of motion. This project extends previous work. It accounts for the dual effect of both the crane and the stabilizing inertial device. Furthermore, this work allows for buoyancy and motor induced torques. Furthermore, this work displays the results in 3D on cell phones. The long-term results of this work leads to a robust 3D active compensation method for loading/unloading operations offshore. Finally, the interactivity between the crane and the stabilizing gyro anticipates the impending time of artificial intelligence when machines, equipped with on-board CPU’s and IP addresses, are empowered with learning modules to conduct their operations.

Comparison of Molecular Simulation and Pseudo-Rigid-Body Model Predictions for a Carbon Nanotube–Based Compliant Parallel-Guiding Mechanism

2008 ◽  
Vol 130 (4) ◽  
Author(s):  
Christopher M. DiBiasio ◽  
Martin L. Culpepper ◽  
Robert Panas ◽  
Larry L. Howell ◽  
Spencer P. Magleby
Keyword(s):  
Rigid Body ◽  
Molecular Simulation ◽  
Moment Of Inertia ◽  
Bulk Deformation ◽  
Body Model ◽  
Kinematic Behavior ◽  
Model Predictions ◽  
Rigid Body Model ◽  
Simulation Results ◽  
Selection Of

We report on the accuracy of the pseudo-rigid-body model (PRBM) in predicting the behavior of a nanoscale parallel-guiding mechanism (nPGM) that uses two single-walled (5,5) carbon nanotubes (CNTs) as the flexural guiding elements. The nPGM has two regions of behavior: region 1 is governed by the bulk deformation of the nanotubes, and region 2 is characterized by hingelike flexing of four “kinks” that occur due to buckling of the nanotube walls. PRBM parameters for (5,5) CNTs are proposed. Molecular simulation results of region 1 behavior match PRBM predictions of (1) kinematic behavior with less than 7.3% error and (2) elastomechanic behavior with less than 5.7% error. Although region 1 is of more interest because of its well-defined and stable nature, region 2 motion is also investigated. We show that the PRBM parameters are dependent on the selection of the effective tube thickness and moment of inertia, the lesson being that designers must take care to consider the thickness and moment of inertia values when deriving PRBM constants.

Sign in / Sign up

Export Citation Format

Share Document