Evolution of Rotations of a Rigid Body Under the Action of Restoring and Control Moments

Abstract

Perturbed rotations of a rigid body close to the regular precession in the Lagrangian case under the action of a restoring moment depending on slow time and nutation angle, as well as a perturbing moment slowly varying with time, are studied. The body is assumed to spin rapidly, and the restoring and perturbing moments are assumed to be small with a certain hierarchy of smallness of the components. A first approximation averaged system of equations of motion for an essentially nonlinear two-frequency system is obtained in the nonresonance case. Examples of motion of a body under the action of particular restoring, perturbing, and control moments of force are considered.

On the motion of a gyro in the presence of a Newtonian force field and applied moments

“…The same previous discussion about both Figures 1 and 2 is satisfied for the plotted curves in Figures 3 and 4 for the solution q, respectively. For the second application and according to the calculations depicted in Figures 5 and 7, we found that when 3 grows from 100 kg.m 2 .s −1 to 120 kg.m 2 .s −1 through the value 110 kg.m 2 .s −1 and 2 takes a constant value, the amplitude of the fluctuates decreases owing to the fact that both the solutions p and q are inversely proportional to 3 as indicated by equations (20). Moreover, the number of fluctuations increases, which in turn increases the spindle vibration about the x-axis.…”

“…It is obvious that, the previous system (12) contains slow variables represented by a , b , r , ψ , θ , τ and fast variables denoted by γ , α . On the other hand, it is considered as a general system of the corresponding ones in the work by Akulenko et al [20] when R → ∞ and ( ℓ j = 0 ; 0.25em j = 1 , 2 , 3 ) and as in the work by Akulenko et al [23] at R → ∞ , ℓ j = 0 , K = normalconst . ; besides, M j is independent of τ .…”

“…From several vantage points, some limited special cases can be obtained from this case as in the work by Akulenko et al [20] if R → ∞ , ℓ j = 0 , as in the work by Akulenko et al [23] when R → ∞ , ℓ j = 0 besides M 1 , M 2 and M 3 are linear functions of P , Q and r respectively, and as in the work by Ismail et al [17] when ℓ j = 0 and M j depends on one component only of the angular velocity.…”

“…By virtue of equations (16) we observe that, the first two projections of the angular velocity p and q are achieved in terms of the constants p 0 , q 0 , θ 0 and k 0 , and they have an exponential decay behavior; bearing in mind that p depends on 1 and 3 , while q depends on 2 and 3 . From several vantage points, some limited special cases can be obtained from this case as in the work by Akulenko et al [20] if R → ∞, j = 0, as in the work by Akulenko et al [23] when R → ∞, j = 0 besides M 1 , M 2 and M 3 are linear functions of P, Q and r respectively, and as in the work by Ismail et al [17] when j = 0 and M j depends on one component only of the angular velocity.…”

“…The author evaluated the nutation and precession angles up to the first and second approximations. In the work by Akulenko et al [20], the authors investigated a new class for the rotations of a gyro about one of its fixed points in the presence of a non-stationary perturbing moment, which varied with time and was dependent upon the nutation angle. Recently, the perturbed rotatory motion of a Lagrangian top was studied by Akulenko et al [21] in the presence of perturbed torques, which vary slowly with respect to time.…”