Equations of Motion in Moving Coordinate Systems

2017 ◽  
pp. 357-382
Author(s):  
Carmen Chicone
Keyword(s):  
Equations Of Motion ◽  
Coordinate Systems ◽  
Moving Coordinate

scholarly journals Method of applied direction of mathematics in the aviation institution of higher education

2021 ◽  
Vol 10 ◽  
pp. 13-19
Author(s):  
Olga BONDAR ◽  
◽  
Oksana ZADOROZHNA ◽  
Irina YAKUNINA ◽  
◽  
...  
Keyword(s):  
Higher Education ◽  
Equations Of Motion ◽  
Center Of Mass ◽  
Professional Activity ◽  
Coordinate Systems ◽  
Mathematical Concepts ◽  
Parallel Transfer ◽  
Moving Coordinate ◽  
Rotation Of Axes

The elements of the method of applied direction of mathematics in the aviation institution of higher education developed by us are considered. We use this technique in higher mathematics classes at the Flight Academy of the National Aviation University. Examples of a combination of fundamental mathematical concepts and practical methods of their application are given. We illustrate in detail the coordinate systems used in aviation. Among these systems are mobile and fixed coordinate systems. Problems of higher mathematics related to one or another coordinate system are indicated. For example, to record vector equations of motion in projections, moving coordinate systems are used, the beginning of which is located in the center of mass of the aircraft. Therefore, the study of the topic "Reflection of linear (vector) spaces" acquires a professional orientation. In particular, we present the formulas of coordinate transformations for parallel transfer and rotation of the axes. Note that the transformations of rectangular coordinate systems are used in aviation. Having considered aviation coordinate systems, the teacher is interested in students in the study of equations of motion, determination of accelerations, velocities and displacements. Methodical methods of formation of practical skills and abilities of future aviation specialists contribute to the implementation of the applied direction of mathematics. We have given some examples of methods of applied direction of mathematics in aviation in the sources that are currently published. The prospect of our research is to further improve practical approaches to solving problems of mathematical training of students of aviation institutions of higher education. This should help increase the level of methodological training of scientific and pedagogical staff of higher education institutions. At the same time, it should contribute to the improvement of methods of teaching mathematics in terms of its application. As a result, the graduate of the aviation institution of higher education must be ready for successful professional activity. Key words: applied direction of mathematics, coordinate systems in aviation, parallel transfer, rotation of axes, fundamental mathematical concepts in aviation.

Inverse Dynamics of Flexible Robot Arms: Modeling and Computation for Trajectory Control

1990 ◽  
Vol 112 (2) ◽  
pp. 177-185 ◽  
Author(s):  
H. Asada ◽  
Z.-D. Ma ◽  
H. Tokumaru
Keyword(s):  
Efficient Algorithm ◽  
Inverse Dynamics ◽  
Equations Of Motion ◽  
Compact Form ◽  
Coordinate Systems ◽  
Flexible Robot ◽  
Rigid Link ◽  
Flexible Arm ◽  
Moving Coordinate ◽  
Virtual Coordinate Systems

The inverse dynamics of robot manipulators based on flexible arm models are considered. Actuator torques required for a flexible arm to track a given trajectory are formulated and computed by using special moving coordinate systems, called virtual rigid link coordinates. Dynamic deformations of the flexible arm can be represented in a simple and compact form with use of the virtual coordinate systems. This eliminates a number of terms involved in the equations of motion and significantly reduces complexity in the inverse dynamics computation. An efficient algorithm for computing the actuator torques is then presented on the basis of the simplified formulation, and applied to a two-link arm problem.

scholarly journals The (A)symmetry between the Exterior and Interior of a Schwarzschild Black Hole

2018 ◽  
Author(s):  
Pawel Gusin ◽  
Andy Augousti ◽  
Filip Formalik ◽  
Andrzej Radosz
Keyword(s):  
Black Hole ◽  
Event Horizon ◽  
Equations Of Motion ◽  
Coordinate Systems ◽  
Schwarzschild Black Hole ◽  
Schwarzschild Spacetime ◽  
The Relationship

A black hole in a Schwarzschild spacetime is considered. A transformation is proposed that describes the relationship between the coordinate systems exterior and interior to an event horizon. Application of this transformation permits considerations of the (a)symmetry of a range of phenomena taking place on both sides of the event horizon. The paper investigates two distinct problems of a uniformly accelerated particle. In one of these, although the equations of motion are the same in the regions on both sides, the solutions turn out to be very different. This manifests the differences of the properties of these two ranges.

Experimental and numerical investigation of railroad vehicle braking dynamics

2009 ◽  
Vol 223 (3) ◽  
pp. 255-267
Author(s):  
C Mellace ◽  
A P Lai ◽  
A Gugliotta ◽  
N Bosso ◽  
T Sinokrot ◽  
...  
Keyword(s):  
Equations Of Motion ◽  
The Body ◽  
Computer Algorithms ◽  
Coordinate Systems ◽  
State Equations ◽  
Relative Coordinates ◽  
Linear State ◽  
Cartesian Coordinate ◽  
Railroad Vehicle ◽  
Braking Dynamics

One of the important issues associated with the use of trajectory coordinates in railroad vehicle dynamic algorithms is the ability of such coordinates to deal with braking and traction scenarios. In these algorithms, track coordinate systems that travel with constant speeds are introduced. As a result of using a prescribed motion for these track coordinate systems, the simulation of braking and/or traction scenarios becomes difficult or even impossible. The assumption of the prescribed motion of the track coordinate systems can be relaxed, thereby allowing the trajectory coordinates to be effectively used in modelling braking and traction dynamics. One of the objectives of this investigation is to demonstrate that by using track coordinate systems that can have an arbitrary motion, the trajectory coordinates can be used as the basis for developing computer algorithms for modelling braking and traction conditions. To this end, a set of six generalized trajectory coordinates is used to define the configuration of each rigid body in the railroad vehicle system. This set of coordinates consists of an arc length that represents the distance travelled by the body, and five relative coordinates that define the configuration of the body with respect to its track coordinate system. The independent non-linear state equations of motion associated with the trajectory coordinates are identified and integrated forward in time in order to determine the trajectory coordinates and velocities. The results obtained in this study show that when the track coordinate systems are allowed to have an arbitrary motion, the resulting set of trajectory coordinates can be used effectively in the study of braking and traction conditions. The results obtained using the trajectory coordinates are compared with the results obtained using the absolute Cartesian-coordinate-based formulations, which allow modelling braking and traction dynamics. In addition to this numerical validation of the trajectory coordinate formulation in braking scenarios, an experimental validation is also conducted using a roller test rig. The comparison presented in this study shows a good agreement between the obtained experimental and numerical results.

The Use of Vector Techniques in Variational Problems

1986 ◽  
Vol 108 (2) ◽  
pp. 141-145 ◽  
Author(s):  
L. J. Everett ◽  
M. McDermott
Keyword(s):  
Rigid Body ◽  
Equations Of Motion ◽  
Variational Problems ◽  
Double Pendulum ◽  
Coordinate Transformations ◽  
Elastic Deformations ◽  
Variational Techniques ◽  
Convenient Means ◽  
Moving Coordinate ◽  
Rigid Body Motions

A convenient means for applying vector mathematics to variational problems is presented. The total and relative variations of a vector are defined and results which follow from these definitions are developed and proved. These results are then used to express the variation of a functional using vector techniques rather than the classical scalar or matrix techniques. The simple problems of deriving equations of motion for a rigid body and for a rigid double pendulum are presented as examples of the technique. The key advantages of the method are that (1) it allows the investigator who is familiar and proficient with vector techniques to apply these skills to variational problems and (2) it greatly simplifies the application of variational techniques to problems which include both rigid body motions and elastic deformations. This is accomplished by providing the techniques necessary for computing the variation of a vector defined in a moving coordinate system without using coordinate transformations.

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