  Mechanics of Solids A Journal of Russian Academy of Sciences   Founded
in January 1966
Issued 6 times a year
Print ISSN 00256544 Online ISSN 19347936 
Archive of Issues
Total articles in the database:   9179 
In Russian (Èçâ. ÐÀÍ. ÌÒÒ):   6485

In English (Mech. Solids):   2694 

<< Previous article  Volume 53, Issue 4 / 2018  Next article >> 
T.V. Grishanina, S.V. Russkih, and F.N. Shklyarchuk, "Elimination of Nonstationary Oscillations of an Elastic System at the Stopping Time after Finite Rotation by the Given Law via the Tuning of Eigenfrequencies," Mech. Solids. 53 (4), 370380 (2018) 
Year 
2018 
Volume 
53 
Number 
4 
Pages 
370380 
DOI 
10.3103/S0025654418040027 
Title 
Elimination of Nonstationary Oscillations of an Elastic System at the Stopping Time after Finite Rotation by the Given Law via the Tuning of Eigenfrequencies 
Author(s) 
T.V. Grishanina (Moscow Aviation Institute (National Research University), Volokolamskoe sh. 4, Moscow, 125993 Russia, grishaninatat@list.ru)
S.V. Russkih (Moscow Aviation Institute (National Research University), Volokolamskoe sh. 4, Moscow, 125993 Russia; Institute of Applied Mechanics of the Russian Academy of Sciences, Leningradskii pr. 7, Moscow, 125040 Russia, sergey.russkih@rambler.ru)
F.N. Shklyarchuk (Moscow Aviation Institute (National Research University), Volokolamskoe sh. 4, Moscow, 125993 Russia; Institute of Applied Mechanics of the Russian Academy of Sciences, Leningradskii pr. 7, Moscow, 125040 Russia, shklyarchuk@list.ru) 
Abstract 
The article deals with an arbitrary elastic 3Dsystem (body) that performs a controlled finite rotation with respect to some fixed axis and small nonstationary oscillations.The system oscillations occur due to external load (power control) or inertial load of the rotational transportation of the carrying body (kinematic control).The linear equations of oscillations are used in normal coordinates, in which motion is represented by eigenmodes of vibrations for system that is free in the rotation angle (including system rotation as a solid body in the case of power control) and for system fixed in rotation angle in the case of kinematic control.It is assumed that the (power or inertial) load acting on the system is proportional to some controlling finite time function from a certain class. The purpose of this article is to solve the problem of system rotation for a certain time from one rest position to another at a given finite angle using the given control function and to eliminate the elastic oscillations on the several lowest eigenmodes at the stopping time.
The relations between the time of the system rotation under the action of a given control function and the eigenmodes frequencies for oscillations being eliminated are obtained on the basis of the exact solutions of the equations in normal coordinates. These relations satisfy the zero initial and final conditions. They are "tuned" by minimizing the positive definite quadratic form written for them by varying the system parameters to fulfill these relations simultaneously for several eigenfrequencies.
As an example, the calculations for a model of a symmetrical spacecraft with two identical elastic solar cell panels consisting of four planar nondeformable sections connected by elastic hinges are carried out for comparison and analysis of the results accuracy. The finite rolling motion of the system with the damping at the stopping time of rotation for several (from one to three) lowest eigenmodes of antisymmetric vibrations is considered. The comparisons of the initial equations of motion for the system in generalized coordinates using several simple control functions and the found parameters of the "tuned" system with numerical solutions are accomplished. 
Keywords 
control of oscillations, finite rotation of the system, nonstationary oscillations, damping of elastic vibrations, spacecraft turning 
References 
1.  F.L. Chernous'ko, N.N. Bolotnik, and V.G. Gradetsky,
Manipulation Robots: Dynamics, Control, Optimization (Nauka, Moscow, 1989) [in Russian]. 
2.  A.S. Kovaleva,
Control of Vibrational and VibroImpact Systems
(Nauka, Moscow, 1990) [in Russian]. 
3.  K.S. Kolesnikov, V.V. Kokushkin, S.V. Borzykh, and N.V. Pankova,
Calculation and Design of Rocket Separation Systems
(Izd.vo MGTU, Moscow, 2006) [in Russian]. 
4.  G.S. Nurre, R.S. Ryan, H.N. Scofield, and J.I. Sims,
"Dynamics and Control of Large Space Structures,"
J. Guid., Cont. Dyn.
7 (5), 514526 (1984). 
5.  G.L. Degtyarev and T.K. Sirazetdinov,
Theoretical Foundations of Optimal Control of Elastic Spacecraft
(Mashinostroenie, Moscow, 1986) [in Russian]. 
6.  S.K. Das, S. Utku, and B.K. Wada,
"Inverse Dynamics of Adaptive Space Cranes with Tip Point Adjustment,"
AIAA901166CP, 23672374 (1990). 
7.  P.M. Bainum and F. Li,
"Optimal Large Angle Maneuvers of a Flexible Spacecraft,"
Acta Astr.
25 (3), 141148 (1991). 
8.  J.K. Chan and V.J. Modi,
"A ClosedForm Dynamical Analysis of an Orbiting Flexible Manipulator,"
Acta Astr.
25 (2), 6776 (1991). 
9.  L. Meirovitch and M.K. Kwak,
"Control of Flexible Spacecraft with TimeVarying Configuration,"
J. Cont. Guid. Dynam.
15 (2), 314324 (1992). 
10.  D.W. Miller and E.F. Crawley,
"Theoretical and Experimental Investigation of SpaceRealizable Inertial Actuation for Passive and Active Structural Control,"
J. Cont. Guid. Dynam.
11 (5), 449458 (1988). 
11.  A.E. Zakrzhevskii,
"Optimalslewing of a Flexible Spacecraft,"
Prikl. Mekh.
39 (8), 106113 (2003)
[Int. Appl. Mech. (Engl. Transl.)
39 (10), 12081214 (2003)]. 
12.  J.N. Rotenberg,
Automatic Control
(Nauka, Moscow, 1971) [in Russian]. 
13.  A.A. Voronov,
Introduction to the Dynamics of Complex Controllable Systems
(Nauka, Moscow, 1985) [in Russian]. 
14.  A.P. Razygraev,
Fundamentals of Spacecraft Flight Control
(Mashinostroenie, Moscow, 1990) [in Russian]. 
15.  R.F. Ganiev and A.E. Zakrezhevskii,
Programmed Movements of Controlled Deformable Structures
(Nauka, Moscow, 1995) [in Russian]. 
16.  B.P. Masters and E.F. Crawley,
"Evolutionary Design of Controlled Structures,"
J. Air.
36 (1), 209217 (1999). 
17.  V.I. Matyukhin,
Controllability of Mechanical Systems
(Fizmatlit, Moscow, 2009) [in Russian]. 
18.  F.L. Chernousko, L.D. Akulenko, and B.N. Sokolov,
Control of Oscillations
(Nauka, Moscow, 1976) [in Russian]. 
19.  F.L. Chernousko, I.M. Anan'evskiy, and S.A. Reshmin,
Methods for Nonlinear Mechanical Systems Control
(Fizmatlit, Moscow, 2006) [in Russian]. 
20.  V.B. Berbyuk,
Dynamics and Optimization of Robotic Systems
(Nauova Dumka, Kiev, 1989) [in Russian]. 
21.  Ye.P. Kubyshkin,
"Optimum Control of Rotation of a Rigid Body with a Flexible Rod,"
Prikl. Mat. Mekh.,
56 (2), 240249 (1992)
[J. Appl. Math. Mech. (Engl. Transl.)
56 (2), 205214 (1992)]. 
22.  Ye.P. Kubyshkin,
"Optimal Control of the Rotation of a System of two Bodies Connected by an Elastic Rod,"
Prikl. Mat. Mekh.
78 (5), 656670 (2014)
[J. Appl. Math. Mech. (Engl. Transl.)
78 (5), 468479 (2014)]. 
23.  T.V. Grishanina,
"Controlled Turn of an Elastic Rod at the Finite Angle,"
Vest. MAI,
11 (1), 6468 (2004). 
24.  T.V. Grishanina,
"Elimination of Elastic System Vibrations after its Rapid Movement and Turn,"
Vest. MAI,
11 (2), 6875 (2004). 
25.  T.V. Grishanina.
"Dynamics of Controlled Motion of Elastic Systems Subject to Finite Displacements and Rotations,"
Izv. Akad. Nauk. Mekh. Tverd. Tela,
No. 6, 171186 (2004)
[Mech. Sol. (Engl. Transl.)
36 (6), 132144 (2004)]. 
26.  T.V. Grishanina and F.N. Shklyarchuk,
Dynamics of Elastic Controlled Structures
(Izd. MAI, Moscow, 2007) [in Russian]. 

Received 
10 April 2018 
Link to Fulltext 
https://link.springer.com/article/10.3103/S0025654418040027 
<< Previous article  Volume 53, Issue 4 / 2018  Next article >> 

If you find a misprint on a webpage, please help us correct it promptly  just highlight and press Ctrl+Enter

