Hydrodynamic Interpretation of the Euler Equations of Motion of a Classical Gyroscope and Their Invariants

2013 ◽  
pp. 217-224
Author(s):  
Felix V. Dolzhansky
Keyword(s):  
Euler Equations ◽  
Equations Of Motion

Cayley kinematics and the Cayley form of dynamic equations

2005 ◽  
Vol 461 (2055) ◽  
pp. 761-781 ◽  
Author(s):  
Andrew J. Sinclair ◽  
John E. Hurtado
Keyword(s):  
Euler Equations ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Rigid Bodies ◽  
Matrix Form ◽  
Dynamic Equations ◽  
Cayley Transform ◽  
Higher Dimensional ◽  
The Matrix ◽  
General Motion

The Cayley transform and the Cayley–transform kinematic relationships are an important and fascinating set of results that have relevance in N –dimensional orientations and rotations. In this paper these results are used in two significant ways. First, they are used in a new derivation of the matrix form of the generalized Euler equations of motion for N –dimensional rigid bodies. Second, they are used to intimately relate the motion of general mechanical systems to the motion of higher–dimensional rigid bodies. This approach can be used to describe an enormous variety of systems, one example being the representation of general motion of an N –dimensional body as pure rotations of an ( N + 1)–dimensional body.

Equimomental System and Its Applications

2006 ◽  
Author(s):  
Himanshu Chaudhary ◽  
Subir Kumar Saha
Keyword(s):  
Rigid Body ◽  
Euler Equations ◽  
Dynamic Performance ◽  
Equations Of Motion ◽  
Performance Characteristics ◽  
Spatial Motion ◽  
Space System ◽  
Optimization Scheme ◽  
Equimomental System ◽  
Dynamic Performances

This paper presents a study of an equimomental system and its application. The equimomental system of point-masses for a rigid body moving in plane and space system is studied. Sets of three and seven equimomental point-masses are proposed for a planar and spatial motion, respectively. The set of equimomental point-masses is then applied for the optimization of dynamic performance characteristics of a mechanism, e.g, shaking forces and moments, driving torques, bearing reactions, etc. The Newton-Euler equations of motion of a link undergoing planar motion are formulated systematically in the parameters related to the point-masses, which leads to an optimization scheme for the mass distribution of the links to improve the dynamic performances. The effectiveness of the proposed methodology is shown by applying it to a multiloop mechanism of carpet scrapping machine. A significant improvement in all the dynamic performance characteristics is obtained compared to those of the original mechanism.

scholarly journals Inverse Method for Turbomachine Blades Using Existing Time-Marching Techniques

1994 ◽  
Author(s):  
T. Dang ◽  
V. Isgro
Keyword(s):  
Euler Equations ◽  
Inverse Method ◽  
Numerical Solutions ◽  
Equations Of Motion ◽  
Tangential Velocity ◽  
Steady State Solution ◽  
Slip Boundary ◽  
Time Stepping ◽  
Time Marching ◽  
Blade Geometry

A newly-developed inverse method for the design of turbomachine blades using existing time-marching techniques for the numerical solutions of the unsteady Euler equations is proposed. In this inverse method, the pitch-averaged tangential velocity (or the blade loading) is the specified quantity, and the corresponding blade geometry is Iteratively sought after. The presence of the blades are represented by a periodic array of discrete body forces which are included in the equations of motion. A four-stage Runge-Kutta time-stepping scheme is used to march a finite-volume formulation, of the unsteady Euler equations to a steady-state solution. Modification of the blade geometry during this time marching process is achieved using the slip boundary conditions on the blade surfaces. This method is demonstrated for the design of infinitely-thin cascaded blades in the subsonic, transonic, and supersonic flow regimes. Results are validated using an Euler analysis method and are compared against those obtained using a similar inverse method. Excellent agreement in the results are obtained between these different approaches.

scholarly journals Equivalence of Euler equations and torque-angular momentum relation

2021 ◽  
Vol 18 (1) ◽  
pp. 136
Author(s):  
V. Tanriverdi
Keyword(s):  
Angular Momentum ◽  
Rigid Body ◽  
Reference Frame ◽  
Euler Equations ◽  
Equations Of Motion ◽  
The Body ◽  
Moments Of Inertia ◽  
Body Reference ◽  
Stationary Reference Frame ◽  
Momentum Relation

Euler derived equations for rigid body rotations in the body reference frame and in the stationary reference frame by considering an infinitesimal part of the rigid body.Another derivation is possible, and it is widely used: transforming torque-angular momentum relation to the body reference frame.However, their equivalence is not shown explicitly.In this work, for a rigid body with different moments of inertia, we calculated Euler equations explicitly in the body reference frame and in the stationary reference frame and torque-angular momentum relation.We also calculated equations of motion from Lagrangian.These calculations show that all four of them are equivalent.

scholarly journals On rheonomic nonholonomic deformations of the Euler equations proposed by Bilimovich

2020 ◽  
pp. 9-9 ◽  
Author(s):  
A.V. Borisov ◽  
A.V. Tsiganov
Keyword(s):  
Euler Equations ◽  
Equations Of Motion ◽  
System Equations

In 1913 A. D. Bilimovich observed that rheonomic constraints which are linear and homogeneous in generalized velocities are ideal. As a typical example, he considered rheonomic nonholonomic deformation of the Euler equations whose scleronomic version is equivalent to the nonholonomic Suslov system. For the Bilimovithch system, equations of motion are reduced to quadrature, which is discussed in rheonomic and scleronomic cases.

Application of Generalized Newton-Euler Equations and Recursive Projection Methods to Dynamics of Deformable Multibody Systems

1989 ◽  
Author(s):  
R. A. Wehage ◽  
A. A. Shabana
Keyword(s):  
Euler Equations ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Coupled System ◽  
Projection Methods ◽  
Open Loop ◽  
System Of Equations ◽  
Kinematic Chains ◽  
Deformable Bodies ◽  
Generalized Newton

Abstract A general symbolic-based method is presented for solving equations of motion for open-loop kinematic chains consisting of interconnected rigid and deformable bodies. The method utilizes matrix partitioning, recursive projection based on optimal block U-L factorization and generalized Newton-Euler equations to obtain an order n solution for the constrained equations of motion. Kinematic relationships between the absolute reference, joint and elastic coordinates are used with the generalized Newton-Euler equations for deformable bodies to obtain a large, loosely coupled system of equations. Taking advantage of the inertia matrix structure associated with elastic coordinates yields a recursive solution algorithm whose dimension is independent of the elastic degrees of freedom. The above solution techniques applied to this system of equations yield a much smaller operations count and can more effectively exploit vectorization and parallel processing. The algorithms presented in this paper are illustrated with the aid of cylindrical joints which are easily extended to revolute, prismatic, rigid and other joint types.

scholarly journals Non-metric fluid dynamics and cosmology on Finsler spacetimes

2016 ◽  
Vol 31 (02n03) ◽  
pp. 1641012 ◽  
Author(s):  
Manuel Hohmann
Keyword(s):  
Fluid Dynamics ◽  
Kinetic Theory ◽  
Euler Equations ◽  
Liouville Equation ◽  
Equations Of Motion ◽  
Finsler Geometry ◽  
Equation Of Motion ◽  
Geodesic Motion ◽  
Simple Generalization ◽  
Dust Fluid

We generalize the kinetic theory of fluids, in which the description of fluids is based on the geodesic motion of particles, to spacetimes modeled by Finsler geometry. Our results show that Finsler spacetimes are a suitable background for fluid dynamics and that the equation of motion for a collisionless fluid is given by the Liouville equation, as it is also the case for a metric background geometry. We finally apply this model to collisionless dust and a general fluid with cosmological symmetry and derive the corresponding equations of motion. It turns out that the equation of motion for a dust fluid is a simple generalization of the well-known Euler equations.

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