scholarly journals Extended Lagrangian formalism for rheonomic systems with variable mass

2017 ◽  
Vol 44 (1) ◽  
pp. 115-132 ◽  
Author(s):  
Djordje Musicki
Keyword(s):  
Energy Conservation ◽  
Variable Mass ◽  
Frame Of Reference ◽  
Lagrangian Formalism ◽  
Extended System ◽  
Generalized Coordinates ◽  
Lagrangian Equations ◽  
Straight Line ◽  
The Masses ◽  
Energy Relations

In this paper the extended Lagrangian formalism for the rheonomic systems (Dj. Musicki, 2004), which began with the modification of the mechanics of such systems (V. Vujicic, 1987), is extended to the systems with variable mass, with emphasis on the corresponding energy relations. This extended Lagrangian formalism is based on the extension of the set of chosen generalized coordinates by new quantities, suggested by the form of nonstationary constraints, which determine the position of the frame of reference in respect to which these generalized coordinates refer. As a consequence, an extended system of the Lagrangian equations is formulated, accommodated to the variability of the masses of particles, where the additional ones correspond to the additional generalized coordinates. By means of these equations, the energy relations of such systems have been studied, where it is demonstrated that here there are four types of energy conservation laws. The obtained energy laws are more complete and natural than the corresponding ones in the usual Lagrangian formulation for such systems. It is demonstrated that the obtained energy laws, are in full accordance with the energy laws in the corresponding vector formulation, if they are expressed in terms of the quantities introduced in this formulation of mechanics. The obtained results are illustrated by an example: the motion of a rocket, which ejects the gasses backwards, while this rocket moves up a straight line on an oblique plane, which glides uniformly in a horizontal direction.

scholarly journals A contribution to the theory of the extended Lagrangian formalism for rheonomic systems

2009 ◽  
Vol 36 (1) ◽  
pp. 47-83 ◽  
Author(s):  
Djordje Musicki
Keyword(s):  
Conservation Laws ◽  
Conservation Law ◽  
Mechanical Systems ◽  
Frame Of Reference ◽  
Horizontal Axis ◽  
Energy Change ◽  
Lagrangian Formalism ◽  
Inclined Plane ◽  
Nonconservative Systems ◽  
Generalized Coordinates

In this paper the generalization of the notion of variation in the extended Lagrangian formalism for the rheonomic mechanical systems (Dj. Musicki, 2004) is formulated and analyzed in details. This formalism is based on the extension of a set of generalized coordinates by new quantities, which determine the position of the frame of reference to which the chosen generalized coordinates refer. In the process of varying, the notion of variation is extended so that these introduced quantities, being additional generalized coordinates, must also to be varied, since the position of each particle of this system is completely determined only by all these generalized coordinates. With the consistent utilization of this notion of variation, the main results of this extended Lagrangian formalism are systematically presented, with the emphasis on the corresponding energy laws, first examined by V. Vujicic (1987), where there are two types of the energy change laws dE/dt and the corresponding conservation laws. Furthermore, the generalized Noether's theorem for the nonconservative systems with the associated Killing's equations (B. Vujanovic, 1978) is extended to this formulation of mechanics, and applied for obtaining the corresponding energy laws. It is demonstrated that these energy laws, which are more general and more natural than the usual ones, are in full accordance with the corresponding ones in the vector formulation of mechanics, if they are expressed in terms of quantities introduced in this extended Lagrangian formalism. Finally, the obtained results are illustrated by an example: the motion of a damped linear harmonious oscillator on an inclined plane, which moves along a horizontal axis, where it is demonstrated that there is valid an energy-like conservation law of Vujanovic's type.

scholarly journals Central Configurations for NewtonianN+2p+1-Body Problems

2013 ◽  
Vol 2013 ◽  
pp. 1-5
Author(s):  
Furong Zhao ◽  
Jian Chen
Keyword(s):  
Central Configurations ◽  
Straight Line ◽  
Spatial Central Configurations ◽  
The Masses

We show the existence of spatial central configurations for theN+2p+1-body problems. In theN+2p+1-body problems,Nbodies are at the vertices of a regularN-gonT;2pbodies are symmetric with respect to the center ofT, and located on the straight line which is perpendicular to the regularN-gonTand passes through the center ofT; theN+2p+1th is located at the the center ofT. The masses located on the vertices of the regularN-gon are assumed to be equal; the masses located on the same line and symmetric with respect to the center ofTare equal.

Dynamics of a Self-Propelling Wheel With Three Eccentric Masses

2000 ◽  
Author(s):  
Tuhin K. Das ◽  
Ranjan Mukherjee
Keyword(s):  
Potential Energy ◽  
Constant Velocity ◽  
Simple Approach ◽  
Constant Acceleration ◽  
Physical Constraints ◽  
Straight Line ◽  
Radial Spokes ◽  
Simulation Results ◽  
The Masses ◽  
Variable Potential

Abstract This paper investigates the dynamics of a rolling disk with three unbalance masses that can slide along radial spokes equispaced in angular orientation. The objective is to design trajectories for the masses that satisfy physical constraints and enable the disk to accelerate or move with constant velocity. The disk is designed to remain vertically upright and is constrained to move along a straight line. We design trajectories for constant acceleration through detailed analysis using a dynamic model. The analysis considers two separate cases; one where the potential energy of the system is conserved, and the other where it continually varies. Whereas trajectories conserving potential energy are limacons, the variable potential energy trajectories are the most general and allow greater acceleration. Following the strategy for constant acceleration maneuvers, we give a simple approach to tracking an acceleration profile and provide simulation results.

scholarly journals On Robe's Circular Restricted Problem of Three Variable Mass Bodies

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Jagadish Singh ◽  
Oni Leke
Keyword(s):  
Equilibrium Points ◽  
Mass Parameter ◽  
Variable Mass ◽  
Density Ratio ◽  
Density Parameter ◽  
Out Of Plane ◽  
Nonautonomous Equations ◽  
The Masses ◽  
Three Body ◽  
Plane Equilibrium

This paper investigates the motion of a test particle around the equilibrium points under the setup of the Robe’s circular restricted three-body problem in which the masses of the three bodies vary arbitrarily with time at the same rate. The first primary is assumed to be a fluid in the shape of a sphere whose density also varies with time. The nonautonomous equations are derived and transformed to the autonomized form. Two collinear equilibrium points exist, with one positioned at the center of the fluid while the other exists for the mass ratio and density parameter provided the density parameter assumes value greater than one. Further, circular equilibrium points exist and pairs of out-of-plane equilibrium points forming triangles with the centers of the primaries are found. The out-of-plane points depend on the arbitrary constant , of the motion of the primaries, density ratio, and mass parameter. The linear stability of the equilibrium points is studied and it is seen that the circular and out-of-plane equilibrium points are unstable while the collinear equilibrium points are stable under some conditions. A numerical example regarding out-of-plane points is given in the case of the Earth, Moon, and submarine system. This study may be useful in the investigations of dynamic problem of the “ocean planets” Kepler-62e and Kepler-62f orbiting the star Kepler-62.

Step lengths and efficiencies of serial-operated and parallel-operated mechanical legs

Robotica ◽  
1990 ◽  
Vol 8 (1) ◽  
pp. 23-29
Author(s):  
D McCloy
Keyword(s):  
Mechanical Efficiency ◽  
Step Length ◽  
Power Source ◽  
Parallel Operation ◽  
Large Power ◽  
Wind Resistance ◽  
Walking Machine ◽  
Straight Line ◽  
The Masses ◽  
Free Ranging

SUMMARYWhen a free-ranging walking machine carries its own fuel and power source as well as a payload, it is important to maintain high mechanical efficiency. If the masses of the legs are negligible then, in a straight and level walk, the only power to be generated would be that needed to overcome ground and wind resistance and internal frictions. The paper shows this to be the case when one-input straight-line generating mechanisms are used. However, when two-input mechanisms are used the actuators can work against each other resulting in finite, sometimes large, power requirements. The paper determines the effects of limb lengths and ground clearance on step length and quantifies the power requirements of serial-operated and parallel-operated legs. It shows that parallel-operation offers greater efficiency.

A unified tensor approach to the analysis of mechanical systems

1997 ◽  
Vol 211 (4) ◽  
pp. 313-322 ◽  
Author(s):  
A A Fogarasy ◽  
M R Smith
Keyword(s):  
Dynamic Analysis ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Equation Of Motion ◽  
Newtonian Mechanics ◽  
Kinematic Constraint ◽  
Lagrangian Equation ◽  
Generalized Coordinates ◽  
Lagrangian Equations

It is shown in this paper that all methods of dynamic analysis of mechanisms used in practice can be derived from an invariant formed from the Lagrangian equation of motion. For the dynamic analysis of mechanisms subjected to kinematic constraint conditions, the Lagrangian equations of motion are far more suitable than the Newtonian approach. Since the Lagrangian equations are tensor equations, they are valid irrespective of what kind of generalized coordinates are used. This is not so, however, when the Newtonian approach is used. It is demonstrated by a simple example that a careless use of Newtonian mechanics can lead to erroneous results.

The Effect of Visual Frame of Reference on a Judgement of Plane Stimulus Orientation by Children

Perception ◽  
1978 ◽  
Vol 7 (2) ◽  
pp. 139-149 ◽  
Author(s):  
John M Elliott ◽  
Kevin J Connolly
Keyword(s):  
Frame Of Reference ◽  
Stimulus Orientation ◽  
Contour Matching ◽  
Straight Line ◽  
Visual Frame ◽  
Selection Of

Three studies are reported with children aged [Formula: see text] to [Formula: see text] years and also with educationally subnormal children, in which it was found that up to about 7 years children's selection of one of a pair of line stimuli as ‘falling over’ is affected by the nature of the visual surround contours provided. Both the form of the stimulus surround and the presence or absence of a baseline tilted 45° relative to the horizontal were found to affect the children's judgements. The plane (horizontal or vertical) in which the stimuli were presented had no effect on the results. It is argued that while the results show the influence of the visual surround on children's comprehension of ‘falling over’, this may not be wholly explicable in terms of surround contour matching, as conventionally correct judgements were obtained in the absence of all straight line contours in the immediate surround.

Dynamic Analysis of Rectilinear Motion of a Self-Propelling Disk With Unbalance Masses

2000 ◽  
Vol 68 (1) ◽  
pp. 58-66 ◽  
Author(s):  
T. Das ◽  
R. Mukherjee
Keyword(s):  
Potential Energy ◽  
Dynamic Model ◽  
Center Of Mass ◽  
Static Model ◽  
Rectilinear Motion ◽  
Constant Acceleration ◽  
Physical Constraints ◽  
Straight Line ◽  
Radial Spokes ◽  
The Masses

This paper investigates the dynamics of a rolling disk with three unbalance masses that can slide along radial spokes equispaced in angular orientation. The objective is to design trajectories for the masses that satisfy physical constraints and enable the disk to accelerate or move with constant velocity. The disk is designed to remain vertically upright and is constrained to move along a straight line. We design trajectories for constant acceleration, first using a static model, and then through detailed analysis using a dynamic model. The analysis based on the dynamic model considers two separate cases; one where the potential energy of the system is conserved, and the other where it continually varies. Whereas trajectories conserving potential energy are quite similar to those obtained from the static model, the variable potential energy trajectories are the most general. A number of observations related to the system center-of-mass are made with respect to both trajectories. Following the strategy for constant acceleration maneuvers, we give a simple approach to tracking an acceleration profile and provide some simulation results.

scholarly journals Violation of the Relativistic Energy Conservation Law and Einstein’s Principle of Relativity Caused by the Generation of Mechanical Transverse Waves in a Moving Medium

2021 ◽  
Author(s):  
Kenji Kawashima
Keyword(s):  
Energy Conservation ◽  
Conservation Law ◽  
Rest Mass ◽  
Frame Of Reference ◽  
Relativistic Energy ◽  
Moving Medium ◽  
Wave Source ◽  
Principle Of Relativity ◽  
Moving Direction ◽  
Energy Conservation Law

We study the effect of the generation of the mechanical transverse wave (MTW)　travelling in the opposite direction (OD) to a moving medium (MM) on the relativistic energy conservation law (RECL). From the viewpoint of the relativity of simultaneity (RS), the time on the coordinate coinciding with the advance end of the wave (AEW) travelling toward the rear of the MM passes faster than that on the coordinate coinciding with the wave source (WS). Then the AEW in the MM travels forward compared to that in the rest frame of reference (RFR) which is stationary relative to the medium when the time on the coordinate coinciding with the WS is same for each inertial frame of reference (IFR). Hence, the coordinate interval (CI) between the AEW and WS in the MM is observed to be larger than that between them in the RFR. We show that this difference holds true for the CI of any portion having transverse velocities mutually converted by the Lorentz transformation (LT). This difference in the CI leads to that in the rest mass (RM). We demonstrate that the RM included in wave motion (WM) in the MM is larger than one included in WM in the RFR when comparing the portions having transverse velocities mutually converted by the LT. This relation holds true for all portions in WM. Therefore, the total coordinate interval of the portion (CIP) and total RM (TRM) included in WM in the MM (WMMM) are large compared to them included in WM in the RFR. Furthermore, we compare the relativistic kinetic energy (RKE) of the MTW travelling in the OD to the MM (ODMM) with that of the MTW propagating in the direction vertical to the moving direction of the medium. We prove that the CIP and RM included in the former MTW are larger than them included in the latter MTW when comparing each portion with the same transverse velocity (TV). Moreover, the total CIP and TRM included in the former MTW are also large compared to them included in the latter MTW. The reason for these is that the latter CIP and RM are equal to them in the RFR when comparing the portions having transverse velocities mutually converted by the LT. On the other hand, the energy supplied to generate each MTW is the same. From these, we demonstrate that the RKE of the MTW travelling in the ODMM can be larger than the total relativistic energy (TRE) of the MTW propagating in the direction vertical to the moving direction of the medium. Consequently, we propose a violation of the RECL and Einstein’s principle of relativity (EPR) because the TRE is not necessarily conserved in the IFR in which the medium is moving.

Sign in / Sign up

Export Citation Format

Share Document