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On the equations of motion of mechanical systems subject to nonlinear nonholonomic constraints Ghori, Qamaruddin Khan
Abstract
Suppose q₁,q₂,…,qn are the generalised coordinates of a mechanical system moving with constraints expressed by r nonintegrable equations (r〈n) (1) [equation omitted] where the dots denote differentiation with respect to the time t, and fα are nonlinear in the q’s. The equations (1) are said to represent nonlinear nonholonomic constraints and the system moving with such constraints is called nonlinear nonholonomic. From a purely analytical point of view, the author has obtained the equations of motion for a nonlinear nonholonomic mechanical system in many a different form. The importance of these forms lies in their simplicity and novelty. Some of these forms are deduced from the principle of d'AlembertLagrange using the definition of virtual (possible) displacements due to Četaev [ll] . The others are obtained as a result of certain transformations. Moreover, these different forms of equations of motion are written either in terms of the generalised coordinates or in terms of nonlinear nonholonomic coordinates introduced by V.S. Novoselov [23]. These forms involve the energy of acceleration of the system or the kinetic energy or some new functions depending upon the kinetic energy of the system. Two of these new functions, denoted by R (Sec. 2.3) and K (Sec. 2.4), can be identified, to a certain approximation, with the energy of acceleration of the system and the Gaussian constraint, respectively. An alternative proof (Sec.2.5) is given to the fact that, if virtual displacements are defined in the sense of N.G. Četaev [ll], the two fundamental principles of analytical dynamics  the principle of d'AlembertLagrange and the principle of least constraint of Gauss are consistent. If the1 constraints are rheonomic but linear, a generalisation of the classical theorem of Poisson is obtained in terms of quasicoordinates and the generalised Poisson's brackets introduced by V.V. Dobronravov [17]. The advantage of the various novel forms for the equations of motion is illustrated by solving a few problems.
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Title 
On the equations of motion of mechanical systems subject to nonlinear nonholonomic constraints

Creator  
Publisher 
University of British Columbia

Date Issued 
1960

Description 
Suppose q₁,q₂,…,qn are the generalised coordinates of a mechanical system moving with constraints expressed by r nonintegrable equations (r〈n) (1) [equation omitted] where the dots denote differentiation with respect to the time t, and fα are nonlinear in the q’s. The equations (1) are said to represent nonlinear nonholonomic constraints and the system moving with such constraints is called nonlinear nonholonomic.
From a purely analytical point of view, the author has obtained the equations of motion for a nonlinear nonholonomic mechanical system in many a different form. The importance of these forms lies in their simplicity and novelty. Some of these forms are deduced from the principle of d'AlembertLagrange using the definition of virtual (possible) displacements due to Četaev [ll] . The others are obtained as a result of certain transformations. Moreover, these different forms of equations of motion are written either in terms of the generalised coordinates or in terms of nonlinear nonholonomic coordinates introduced by V.S. Novoselov [23].
These forms involve the energy of acceleration of the system or the kinetic energy or some new functions depending upon the kinetic energy of the system. Two of these new functions, denoted by R (Sec. 2.3) and K (Sec. 2.4), can be identified, to a certain approximation, with the energy of acceleration of the system and the Gaussian constraint, respectively.
An alternative proof (Sec.2.5) is given to the fact that, if virtual displacements are defined in the sense of N.G. Četaev [ll], the two fundamental principles of analytical dynamics  the principle of d'AlembertLagrange and the principle of least constraint of Gauss are consistent.
If the1 constraints are rheonomic but linear, a generalisation of the classical theorem of Poisson is obtained in terms of quasicoordinates and the generalised Poisson's brackets introduced by V.V. Dobronravov [17].
The advantage of the various novel forms for the equations of motion is illustrated by solving a few problems.

Subject  
Genre  
Type  
Language 
eng

Date Available 
20111209

Provider 
Vancouver : University of British Columbia Library

Rights 
For noncommercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.

DOI 
10.14288/1.0080604

URI  
Degree  
Program  
Affiliation  
Degree Grantor 
University of British Columbia

Campus  
Scholarly Level 
Graduate

Aggregated Source Repository 
DSpace

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Rights
For noncommercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.