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 The motion of a selfexcited rigid body
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The motion of a selfexcited rigid body Lee, Richard Way Mah
Abstract
This thesis discusses the motion of a rigid body about a fixed point subject to a torque arising from internal reaction. Such a body is called selfexcited, and its motion is governed by Euler’s dynamical equations. First, in Section 2 we consider the case of a torque vector which is fixed in direction along the largest or smallest principal axis of the body, and has a component in the chosen axis equal to a constant plus a perturbation term that is proportional to the square of the modulus of the spin vector [character omitted] (p,q,r). It is shown that Euler's equations can be integrated in terms of a variable φ, introduced by means of a differential relation. Further quadrature and inversion yield p,q, and r as functions of the time t. Using the method of phaseplane analysis, we show that the spin vector can perform a variety of motions with respect to the bodyfixed trihedral. In particular, when the perturbation is zero, we infer from the corresponding phaseplane trajectories that the spin vector can perform asymptotic motions of the first and second kinds and periodic motions about permanent axes lying in the principal plane perpendicular to the torque vector. Some of the results for this case were also obtained by Grammel, using different method. In the general case, when the perturbation is not zero, these motions are preserved. However, a second type of periodic motion exists; it occurs about the principal axis containing the torque vector, the principal axis itself being a direction of stable permanent rotation. In Section 3 we consider the same problem with the torque vector acting along the middle principal axis. Using the methods of the previous Section, we show that [character omitted] can assume periodic motions as well as asymptotic motions of various kinds. The periodic motions established in these two Sections are then computed in Section 4 as power series in a small parameter. Finally, in Section 5 the motion of a symmetric rigid body moving in a viscous medium subject to a timedependent torque is studied. Its motion is compared with that in a vacuum. We show first that p,q, and r can be expressed in terms of certain integrals. For the special case where the selfexcitement is timeindependent and fixed in direction within the body, these integrals can be reduced to the generalized sine and cosine integrals. Their values can be computed from asymptotic and power series which are developed in the same Section. The asymptotic behavior of the spin vector is then discussed, yielding qualitative results which are summarized in three theorems.
Item Metadata
Title 
The motion of a selfexcited rigid body

Creator  
Publisher 
University of British Columbia

Date Issued 
1964

Description 
This thesis discusses the motion of a rigid body about a fixed point subject to a torque arising from internal reaction. Such a body is called selfexcited, and its motion is governed by Euler’s dynamical equations.
First, in Section 2 we consider the case of a torque vector which is fixed in direction along the largest or smallest principal axis of the body, and has a component in the chosen axis equal to a constant plus a perturbation term that is proportional to the square of the modulus of the spin vector [character omitted] (p,q,r). It is shown that Euler's equations can be integrated in terms of a variable φ, introduced by means of a differential relation. Further quadrature and inversion
yield p,q, and r as functions of the time t. Using the method of phaseplane analysis, we show that the spin vector can perform a variety of motions with respect to the bodyfixed trihedral. In particular,
when the perturbation is zero, we infer from the corresponding phaseplane trajectories that the spin vector can perform asymptotic motions of the first and second kinds and periodic motions about permanent axes lying in the principal plane perpendicular to the torque vector. Some of the results for this case were also obtained by Grammel, using different method. In the general case, when the perturbation is not zero, these motions are preserved. However, a second type of periodic motion exists; it occurs about the principal
axis containing the torque vector, the principal axis itself being a direction of stable permanent rotation.
In Section 3 we consider the same problem with the torque vector acting along the middle principal axis. Using the methods of the previous Section, we show that [character omitted] can assume periodic motions as well as asymptotic
motions of various kinds. The periodic motions established in these two Sections are then computed in Section 4 as power series in a small parameter.
Finally, in Section 5 the motion of a symmetric rigid body moving in a viscous medium subject to a timedependent torque is studied. Its motion is compared
with that in a vacuum. We show first that p,q, and r can be expressed in terms of certain integrals. For the special case where the selfexcitement is timeindependent and fixed in direction within the body, these integrals can be reduced to the generalized sine and cosine integrals. Their values can be computed from asymptotic and power series which are developed in the same Section. The asymptotic behavior of the spin vector is then discussed, yielding qualitative results which are summarized in three theorems.

Genre  
Type  
Language 
eng

Date Available 
20111017

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 
10.14288/1.0080543

URI  
Degree  
Program  
Affiliation  
Degree Grantor 
University of British Columbia

Campus  
Scholarly Level 
Graduate

Aggregated Source Repository 
DSpace

Item Media
Item Citations and Data
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.