The evolution of the motions of a rigid body close to the Lagrange case under the action of an unsteady torque

Акуленко, Л. Д.; Zinkevich, Ya. S.; Kozachenko, T. A.; Leshchenko, Dmytro

doi:10.1016/j.jappmathmech.2017.08.001

Cited by 19 publications

(14 citation statements)

References 7 publications

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“…By comparing the results obtained here with the conclusions of works 5 and 11 where μ and M i are independent of the slow time τ , paper 12 where M i changes slowly in time, and article 13, where μ and M i are slowly varying in time, we can note their distinct mechanical content. The relation of the restoring torque on τ and λ = ε θ and the perturbation torque on slow time reduces to the rise in the system (12) for slow variables of functions a ( τ ) , b ( τ ) and μ ( τ , λ ) , which smooth out the behavior of u i ( i = 1,2,3 ) , G z , …”

Section: Discussionsupporting

confidence: 56%

“…As an example of the developed procedure, we investigate a perturbed Lagrange motion with the regard to the torques acting on a rigid body from the surrounding medium. We express these torques M i ( i = 1,2,3 ) as 5,12,13 , 43 where a ( τ ) , b ( τ ) are positive integrable functions, depending on the surrounding medium and the body’s shape. Torques (13) comply with terms (4) of the possibility of averaging with respect to phase of θ only.…”

Section: The Test Examplementioning

confidence: 99%

“…Paper 12, investigates the evolution of the rigid body’s motion under the effect of an unsteady perturbation torque, the rigid body being close to the Lagrange gyroscope. The concept of the given problem is analyzed in article 13, when restoring and perturbation torques vary slowly in time.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Evolution of motion of a rigid body similar to Lagrange top under the influence of slowly time varying torques

Leshchenko

Ershkov

Kozachenko

2022

Proceedings of the Institution of Mechanical Engineers, Part C:

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Perturbed rotational rigid body motions similar to the case of Lagrange top subjected to restoring and perturbation slowly time-varying torques of forces are studied in this paper. The restoring torque also depends on the small angle of nutation. The following problem is formulated in the current research: analyzing the solution’s behavior of system of motion equations for the values of small parameter different from zero on a sufficiently large interval of time. An averaging method is used for solving the problem. We have established the terms for feasibility to average (with respect to phase of nutation angle) the equations of rigid body motion related to Lagrange case. The averaged system of equations is received in the first approximation. Asymptotic approach permits to obtain some qualitative results and to describe evolution of motion using simplified averaged equations. In the case of rotational motion of a body immersed in the linear-dissipative medium, the numerical integration of the averaged system of equations is conducted. A new class of rotational motions of the Lagrange top is studied for an unsteady perturbation torque, as well as restoring torque that slowly varies with time and depends on the small nutation angle. These results are an extension of our former works (in the absence of slow time τ in the expressions for μ and M i or μ, where M i are dependent on τ only).

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Section: Discussionsupporting

confidence: 56%

Section: The Test Examplementioning

confidence: 99%

See 1 more Smart Citation

Evolution of motion of a rigid body similar to Lagrange top under the influence of slowly time varying torques

Leshchenko

Ershkov

Kozachenko

2022

Proceedings of the Institution of Mechanical Engineers, Part C:

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“…The periodic solutions for this problem are constructed applying the periodicity conditions and assuming a large parameter [20] proportional to 1/r 0 . We used here the large parameter technique instead of the small one well-known in [21][22][23][24][25][26]. The advantage of this technique comes from the saving of the high initial energy which is given for the body to start the motion, and the solving of the problem in a new domain of the motion F(t, μ ⟶ ∞, r o ⟶ 0) and under new considerations.…”

Section: Discussionmentioning

confidence: 99%

Applying the Large Parameter Technique for Solving a Slow Rotary Motion of a Disc about a Fixed Point

Ismail

2020

International Journal of Aerospace Engineering

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In this paper, the motion of a disk about a fixed point under the influence of a Newtonian force field and gravity one is considered. We modify the large parameter technique which is achieved by giving the body a sufficiently small angular velocity component r0 about the fixed z-axis of the disk. The periodic solutions of motion are obtained in the neighborhood r0 tends to 0. This case of study is excluded from the previous works because of the appearance of a singular point in the denominator of the obtained solutions. Euler-Poison equations of motion are obtained with their first integrals. These equations are reduced to a quasilinear autonomous system of two degrees of freedom and one first integral. The periodic solutions for this system are obtained under the new initial conditions. Computerizing the obtained periodic solutions through a numerical technique for validation of results is done. Two types of analytical and numerical solutions in the new domain of the angular velocity are obtained. Geometric interpretations of motion are presented to show the orientation of the body at any instant of time t.

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“…In this section, we describe the body motion using Euler's angles ξ, ζ, and η which come from the obtained solutions (Figure 2). Replacing the time t by t + t 0 where t 0 is an arbitrary interval, the periodic solutions remain periodic since the initial system is autonomous [9]. For this case, we obtain from (32),…”

Section: Geometric Interpretation Of Motionmentioning

confidence: 99%

New Treatment of the Rotary Motion of a Rigid Body with Estimated Natural Frequency

Ismail

2020

Advances in Astronomy

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In this paper, the problem of the motion of a rigid body about a fixed point under the action of a Newtonian force field is studied when the natural frequency ω = 0.5 . This case of singularity appears in the previous works and deals with different bodies which are classified according to the moments of inertia. Using the large parameter method, the periodic solutions for the equations of motion of this problem are obtained in terms of a large parameter, which will be defined later. The geometric interpretation of the considered motion will be given in terms of Euler’s angles. The numerical solutions for the system of equations of motion are obtained by one of the well-known numerical methods. The comparison between the obtained numerical solutions and analytical ones is carried out to show the errors between them and to prove the accuracy of both used techniques. In the end, we obtain the case of the regular precession type as a special case. The stability of the motion is considered by the phase diagram procedures.

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The evolution of the motions of a rigid body close to the Lagrange case under the action of an unsteady torque

Cited by 19 publications

References 7 publications

Evolution of motion of a rigid body similar to Lagrange top under the influence of slowly time varying torques

Evolution of motion of a rigid body similar to Lagrange top under the influence of slowly time varying torques

Applying the Large Parameter Technique for Solving a Slow Rotary Motion of a Disc about a Fixed Point

New Treatment of the Rotary Motion of a Rigid Body with Estimated Natural Frequency

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