Generalized Vector Derivatives for Systems With Multiple Relative Motion

1968 ◽  
Vol 35 (1) ◽  
pp. 20-24 ◽  
Author(s):  
T. A. Sherby ◽  
J. F. Chmielewski
Keyword(s):  
Reference Frame ◽  
Relative Motion ◽  
Reference Frames ◽  
Equations Of Motion ◽  
Rigid Bodies ◽  
Main Body ◽  
Angular Velocity Vector ◽  
Relative Coordinates ◽  
Moving Body ◽  
Classical Vector

For the analysis of relative motion, classical vector mathematics is limited to the use of one moving reference frame when taking vector derivatives. However, many dynamical systems consist of a number of rigid bodies in motion relative to one another. The classical procedure requires the specification of the position of each body relative to a single “main body.” The use of relative coordinates allows a natural specification of the position of one moving body relative to another moving body in network fashion. To use relative coordinates in dynamic and kinematic analyses, it is necessary to use relative vector derivatives involving more than one moving reference frame. This paper presents general expressions for the kth-order derivative of a relative position and angular velocity vector measured in any moving reference frame in a system of m reference frames with multiple relative motion. These expressions are used to develop a procedure which generates the differential equations of motion for the system by routine substitution of the relative coordinates and their scalar derivatives. This procedure offers promise as an algorithm for machine generation of the system equations and eliminates the possibility of neglecting subtle accelerations due to relative motion. The use of the procedure is demonstrated by generating the equations of motion of an offset unsymmetrical gyroscope.

Explaining Ground Effect Aerodynamics via a Real-Life Reference Frame

2014 ◽  
Vol 553 ◽  
pp. 229-234
Author(s):  
Philip Close ◽  
Tracie J. Barber
Keyword(s):  
Reference Frame ◽  
Reference Frames ◽  
Computational Simulation ◽  
Real Life ◽  
Computational Modelling ◽  
Ground Effect ◽  
The Body ◽  
Moving Body ◽  
Set Up ◽  
Insight Into

The principle of relative motion as the cause of forces on a body submersedin a uid is foundational in the study of uid mechanics. In aerodynamics the wind tunnelis used as a convenient and safe method by which to test the aerodynamic performance ofbodies. This body-stationary convention has carried over into the computational world withthe development of CFD, though there is no practical reason why the moving body/stationaryuid set-up that corresponds to reality cannot be used for computational modelling. This pointbecomes particularly important as the concept of ground e ect is introduced. With an extraboundary nearby it becomes harder to appropriatel y match the experimental set-up with reality,and the extra boundary condition also adds complexity to computational simulation. A studywas undertaken to compare the body-stationary and body-moving reference frames in grounde ect. The moving reference frame velocity elds allowed new insight into the aerodynamics ofground e ect.

Dynamics of Multirigid-Body Systems

1978 ◽  
Vol 45 (4) ◽  
pp. 889-894 ◽  
Author(s):  
R. L. Huston ◽  
C. E. Passerello ◽  
M. W. Harlow
Keyword(s):  
Equations Of Motion ◽  
Mechanical Systems ◽  
Rigid Bodies ◽  
Computer Algorithms ◽  
Euler Parameters ◽  
Closed Loops ◽  
Explicit Formulation ◽  
Relative Coordinates ◽  
D'alembert's Principle

New and recently developed concepts and ideas useful in obtaining efficient computer algorithms for solving the equations of motion of multibody mechanical systems are presented and discussed. These ideas include the use of Euler parameters, Lagrange’s form of d’Alembert’s principle, quasi-coordinates, relative coordinates, and body connection arrays. The mechanical systems considered are linked rigid bodies with adjoining bodies sharing at least one point, and with no “closed loops” permitted. An explicit formulation of the equations of motion is presented.

Dynamic Analysis of Multi-Rigid-Body Systems

1974 ◽  
Vol 96 (3) ◽  
pp. 886-892 ◽  
Author(s):  
V. K. Gupta
Keyword(s):  
Digital Simulation ◽  
Equations Of Motion ◽  
Rigid Bodies ◽  
Mechanism Analysis ◽  
Spatial Mechanism ◽  
Higher Derivatives ◽  
Reaction Forces ◽  
Representative Problem ◽  
Classical Vector ◽  
Vector Approach

A method is presented for formulating and solving the Newton-Euler equations of motion of a system of interconnected rigid bodies. The digital simulation may involve numerical integration of the kinematic equations as well as the dynamic equations. The reaction forces and torques resulting from rigid constraints imposed at the connecting joints are also determined. The derivation of kinematic expressions for first and higher derivatives is demonstrated based on direct differentiation of the rotation matrix in the spirit of the classical vector approach. A representative problem in spatial mechanism analysis is solved and illustrated with numerical results.

Idealisation and Formulation in Structural Dynamics Using Modal Analysis

2011 ◽  
Vol 418-420 ◽  
pp. 1022-1025
Author(s):  
Muhammad Danish ◽  
Vinay Kumar Pingali ◽  
Somnath Chattopadhyaya ◽  
N.K. Singh ◽  
A.K. Ray
Keyword(s):  
Reference Frame ◽  
Structural Dynamics ◽  
Reference Frames ◽  
Dynamic Equilibrium ◽  
Equations Of Motion ◽  
Complex Structure ◽  
Complicated Structure ◽  
Multiple Reference Frames ◽  
Lumped Masses ◽  
Multiple Reference

The crux feature of this paper is the equations of motion in a structural dynamics with respect to single reference frame that can be easily derived, and the results are well defined and converged. However, problem occurs, when the analysis of any complex, complicated structure is considered and its equation of motion is extracted with respect to single reference frame. The results are indecipherable, ambiguous and less converged. Thus, for such a complex structure, the results obtain with respect to multiple reference frames. In present study, an approximated model with a set of lumped masses, properly interconnected, along with discrete spring and damper elements are in consideration for continuous vibrating system. This results in dynamic equilibrium, which in turn results in formulation and idealization. As, rightly said by scientist Steve Lacy- “To me, there is spirit in a reed. It is a living thing, a weed, really and it does not contain spirit of sort. It’s really an ancient vibration”

Contact Kinematics between Three-Dimensional Rigid Bodies with General Surface Parameterization

2020 ◽  
pp. 1-28
Author(s):  
Mubang Xiao ◽  
Ye Ding
Keyword(s):  
Reference Frame ◽  
Relative Motion ◽  
Screw Theory ◽  
Dimensional Space ◽  
Contact Point ◽  
Three Dimensional ◽  
Rigid Bodies ◽  
Frame Field ◽  
Contact Kinematics ◽  
Acceleration Analysis

Abstract This paper provides an improvement of the classic Montana's contact kinematics equations considering non-orthogonal object parameterizations. In Montana's model, the reference frame used to define the relative motion between two rigid bodies in three-dimensional space is chosen as the Gauss frame, assuming there is an orthogonal coordinate system on the object surface. To achieve global orthogonal parameterizations on arbitrarily shaped object surfaces, we define the relative motion based on the reference frame field, which is the orthogonalization of the surface natural basis at every contact point. The first- and second-order contact kinematics, including the velocity and acceleration analysis of the relative rolling, sliding, and spinning motion, are reformulated based on the reference frame field and the screw theory. We use two simulation examples to illustrate the proposed method. The examples are based on simple non-orthogonal surface parameterizations, instead of seeking for global orthogonal parameterizations on the surfaces.

Curvature Theory on Contact and Transfer Characteristics of Enveloping Curves

2018 ◽  
Author(s):  
Kwun-Lon Ting ◽  
Cody Leeheng Chan
Keyword(s):  
Relative Motion ◽  
Contact Point ◽  
Rigid Bodies ◽  
Body Motion ◽  
Transfer Characteristics ◽  
Moving Body ◽  
Curvature Theory ◽  
The Veil ◽  
Motion Characteristics ◽  
Definition Of

In differential geometry, a curve is characterized by the curvature properties and so is a point trajectory in curvature theory. However, due to the rolling and sliding between contact curves, the characterization of enveloping curves embedded on rigid bodies in relative motion is not complete without the transfer (or shifting) characteristics of the contact point. This paper presents the new perspectives and the first comprehensive theory on not only the curvature characteristics but also the transfer characteristics between enveloping curves embedded on rigid bodies. The paper contains three parts. In the first part, a point traces a curve on the moving body and consequently traces a curve on the fixed body. Both generated curves form a pair of enveloping curves. This part establishes the foundation of the paper. Because each enveloping curve is treated as a point trajectory. One may examine all aspects of the enveloping process. Essentially this unmasks the veil that has hindered further understanding and observation of the enveloping behavior beyond the fundamental curvature. It represents a significant advancement on envelope theory. In the second part, the moving point is the instant center, which traces the moving centrode on the moving body and the fixed centrode on the fixed body. It characterizes the rolling between centrodes and the transfer characteristics of the instant center on each centrode. It not only offers a simple way to treat the instant center transfer (shifting) velocity but also successfully extends it to any order of motion. The third part is about the rolling and sliding of between enveloping curves embedded on rigid bodies in relative motion. It addresses the transfer characteristics of the contact on each of the contact curves for the first time. The transfer characteristics are functions of the rigid body motion characteristics. This part offers the vital kinematic aspect of enveloping curves distinctly different from the conventional curvature theory that addresses an individual curve. The proposed enveloping curvature theory offers an important model to account for all aspects of the contact and removes the veil that blurs the contact behavior caused by the traditional envelope definition of Fλxy=∂F∂λλxy=0. This is a kinematic solution for envelopes. The proposed theory is illustrated with an example of two rolling cylinders.

On the Euler–Lagrange Equation for Planar Systems of Rigid Bodies or Lumped Masses

2019 ◽  
Vol 14 (9) ◽  
Author(s):  
R. Wiebe ◽  
P. S. Harvey
Keyword(s):  
Rigid Body ◽  
Reference Frames ◽  
Dynamic Equilibrium ◽  
Lagrange Equation ◽  
Equations Of Motion ◽  
Rigid Bodies ◽  
Double Pendulum ◽  
Governing Equations ◽  
Lumped Mass ◽  
Planar Systems

The Euler–Lagrange equation is frequently used to develop the governing dynamic equilibrium expressions for rigid-body or lumped-mass systems. In many cases, however, the rectangular coordinates are constrained, necessitating either the use of Lagrange multipliers or the introduction of generalized coordinates that are consistent with the kinematic constraints. For such cases, evaluating the derivatives needed to obtain the governing equations can become a very laborious process. Motivated by several relevant problems related to rigid-body structures under seismic motions, this paper focuses on extending the elegant equations of motion developed by Greenwood in the 1970s, for the special case of planar systems of rigid bodies, to include rigid-body rotations and accelerating reference frames. The derived form of the Euler–Lagrange equation is then demonstrated with two examples: the double pendulum and a rocking object on a double rolling isolation system. The work herein uses an approach that is used by many analysts to derive governing equations for planar systems in translating reference frames (in particular, ground motions), but effectively precalculates some of the important stages of the analysis. It is hoped that beyond re-emphasizing the work by Greenwood, the specific form developed herein may help researchers save a significant amount of time, reduce the potential for errors in the formulation of the equations of motion for dynamical systems, and help introduce more researchers to the Euler–Lagrange equation.

scholarly journals Robustness of reference-frame-independent quantum key distribution against the relative motion of the reference frames

2017 ◽  
Vol 381 (31) ◽  
pp. 2497-2501 ◽  
Author(s):  
Tanumoy Pramanik ◽  
Byung Kwon Park ◽  
Young-Wook Cho ◽  
Sang-Wook Han ◽  
Yong-Su Kim ◽  
...  
Keyword(s):  
Reference Frame ◽  
Relative Motion ◽  
Quantum Key Distribution ◽  
Reference Frames ◽  
Key Distribution

A General and Efficient Method for Dynamic Analysis of Mechanical Systems Using Velocity Transformations

1986 ◽  
Vol 108 (2) ◽  
pp. 176-182 ◽  
Author(s):  
S. S. Kim ◽  
M. J. Vanderploeg
Keyword(s):  
Equations Of Motion ◽  
General Purpose ◽  
Rigid Bodies ◽  
Transformation Matrix ◽  
Cartesian Coordinates ◽  
Velocity Transformation ◽  
Joint Coordinates ◽  
Relative Coordinates ◽  
General Purpose Computer ◽  
New Formulation

This paper presents a new formulation for the equations of motion of interconnected rigid bodies. This formulation initially uses Cartesian coordinates to define the position of the system, the kinematic joints between bodies, and forcing functions on and between bodies. This makes initial system definition straightforward. The equations of motion are then derived in terms of relative joint coordinates through the use of a velocity transformation matrix. The velocity transformation matrix relates relative coordinates to Cartesian coordinates. It is derived using kinematic relationships for each joint type and graph theory for identifying the system topology. By using relative coordinates, the equations of motion are efficiently integrated. Use of both Cartesian and relative coordinates produces an efficient set of equations without loss of generality. The algorithm just described is implemented in a general purpose computer program. Examples are used to demonstrate the generality and efficiency of the algorithms.

Allocentric to Egocentric Spatial Switching: Impairment in aMCI and Alzheimer's Disease Patients?

2018 ◽  
Vol 15 (3) ◽  
pp. 229-236 ◽  
Author(s):  
Gennaro Ruggiero ◽  
Alessandro Iavarone ◽  
Tina Iachini
Keyword(s):  
Alzheimer’S Disease ◽  
Alzheimer's Disease ◽  
Reference Frame ◽  
Reference Frames ◽  
Memory Task ◽  
Spatial Representations ◽  
The Novel ◽  
Spatial Memory Task ◽  
Switching Condition ◽  
Normal Controls

Objective: Deficits in egocentric (subject-to-object) and allocentric (object-to-object) spatial representations, with a mainly allocentric impairment, characterize the first stages of the Alzheimer's disease (AD). Methods: To identify early cognitive signs of AD conversion, some studies focused on amnestic-Mild Cognitive Impairment (aMCI) by reporting alterations in both reference frames, especially the allocentric ones. However, spatial environments in which we move need the cooperation of both reference frames. Such cooperating processes imply that we constantly switch from allocentric to egocentric frames and vice versa. This raises the question of whether alterations of switching abilities might also characterize an early cognitive marker of AD, potentially suitable to detect the conversion from aMCI to dementia. Here, we compared AD and aMCI patients with Normal Controls (NC) on the Ego-Allo- Switching spatial memory task. The task assessed the capacity to use switching (Ego-Allo, Allo-Ego) and non-switching (Ego-Ego, Allo-Allo) verbal judgments about relative distances between memorized stimuli. Results: The novel finding of this study is the neat impairment shown by aMCI and AD in switching from allocentric to egocentric reference frames. Interestingly, in aMCI when the first reference frame was egocentric, the allocentric deficit appeared attenuated. Conclusion: This led us to conclude that allocentric deficits are not always clinically detectable in aMCI since the impairments could be masked when the first reference frame was body-centred. Alongside, AD and aMCI also revealed allocentric deficits in the non-switching condition. These findings suggest that switching alterations would emerge from impairments in hippocampal and posteromedial areas and from concurrent dysregulations in the locus coeruleus-noradrenaline system or pre-frontal cortex.

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