# American Institute of Mathematical Sciences

doi: 10.3934/jgm.2021031
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## Parametric stability of a double pendulum with variable length and with its center of mass in an elliptic orbit

 1 Departamento de Ciências Exatas, Centro de Ciências Aplicadas e Educação, Universidade Federal da Paraíba, Rio Tinto, Brazil 2 Departamento de Matemática, Centro de Ciências Exatas e Tecnologia, Universidade Federal de Sergipe, São Cristovão, Brazil 3 Departamento de Matemática, Facultad de Ciencias, Universidad del Bío-Bío, Concepción, Chile

* Corresponding author: José Laudelino de Menezes Neto

Received  September 2021 Early access January 2022

We consider the planar double pendulum where its center of mass is attached in an elliptic orbit. We consider the case where the rods of the pendulum have variable length, varying according to the radius vector of the elliptic orbit. We make an Hamiltonian view of the problem, find four linearly stable equilibrium positions and construct the boundary curves of the stability/instability regions in the space of the parameters associated with the pendulum length and the eccentricity of the orbit.

Citation: José Laudelino de Menezes Neto, Gerson Cruz Araujo, Yocelyn Pérez Rothen, Claudio Vidal. Parametric stability of a double pendulum with variable length and with its center of mass in an elliptic orbit. Journal of Geometric Mechanics, doi: 10.3934/jgm.2021031
##### References:
 [1] V. S. Aslanov, Orbital oscillations of an elastic vertically-tethered satellite, Mech. Solids, 46 (2011), 657-668.  doi: 10.3103/S0025654411050013. [2] L. Blitzer, Equilibrium and stability of a pendulum in an orbiting spaceship, Am. J. Phys., 47 (1979), 241-246.  doi: 10.1119/1.11561. [3] A. A. Burov, Oscillations of a vibrating dumbbell on an elliptic orbit, Dokl. Akad. Nauk, 437 (2011), 186-189. [4] A. A. Burov, A. D. Guerman and I. I. Kosenko, On plane oscillations of a pendulum with variable length suspended on the surface of a planet's satellite, Cosmic Research, 52 (2014), 289-294.  doi: 10.1134/S0010952514040029. [5] A. A. Burov and I. I. Kosenko, Planar vibrations of a solid with variable mass distribution in an elliptic orbit, Dokl. Phys., 56 (2011), 760-764. [6] A. A. Burov and I. I. Kosenko, Planar oscillations of a dumb-bell of a variable length in a central field of newtonian attraction, exact approach, Int. J. of Non-Linear Mech., 72 (2015), 1-5.  doi: 10.1016/j.ijnonlinmec.2015.01.011. [7] A. Burov, I. Kosenko and A. Guerman, Dynamics of a moon-anchored tether with variable lenght, Adv. Astronautical Sci., 142 (2012), 3495-3507. [8] G. Cruz Araujo and H. E. Cabral, Parametric stability in a $P +2$-body problem, J. Dyn. Diff. Equat., 30 (2018), 719-742.  doi: 10.1007/s10884-017-9570-x. [9] J. L. de Menezes Neto and H. E. Cabral, Parametric stability of a pendulum with variable length in an elliptic orbit, Regul. Chaotic Dyn., 25 (2020), 323-329.  doi: 10.1134/S1560354720040012. [10] A. D. Guerman, Spatial equilibria of multibody chain in a circular orbit, Acta Astronautica, 58 (2006), 1-14.  doi: 10.1016/j.actaastro.2005.05.002. [11] O. V. Kholostova, On the motions of a double pendulum with vibrating suspension point, Mechanics of Solids, 44 (2009), 184-197.  doi: 10.3103/S0025654409020034. [12] M. Lavagna and A. E. Finzi, Large multi-hinged space systems: A parametric stability analysis, Acta Astronautica, 54 (2004), 295-305.  doi: 10.1016/S0094-5765(02)00304-1. [13] A. P. Markeev, Linear Hamiltonian systems and some applications to the problem of stability of motion of satellites relative to the center of mass, $\mathcal{R}$ & $\mathcal{C}$ $\mathcal{D}$ynamics, Izhevsk-Moscow, 2009. [14] A. P. Markeev, On one special case of parametric resonance in problems of celestial mechanics, Astronomy Letters, 31 (2005), 350-356.  doi: 10.1134/1.1922534. [15] A. K. Misra, Z. Amier and V. J. Modi, Attitude dynamics of three-body tethered systems, Acta Astronautica, 17 (1988), 1059-1068. [16] A. V. Sarychev, Equilibria of a double pendulum in a circular orbit, Acta Astronautica, 44 (1999), 63-65.  doi: 10.1016/S0094-5765(99)00015-6. [17] J. L. Synge, On the behavior, according to Newtonian theory of a plumb line or pendulum attached to an artificial satellite, Proc. Roy. Irish Acad. Sect. A, 60 (1959), 6 pp. [18] L. R. Valeriano, Parametric stability in Robe's problem, Regul. Chaotic Dyn., 21 (2016), 126-135.  doi: 10.1134/S156035471601007X.

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##### References:
 [1] V. S. Aslanov, Orbital oscillations of an elastic vertically-tethered satellite, Mech. Solids, 46 (2011), 657-668.  doi: 10.3103/S0025654411050013. [2] L. Blitzer, Equilibrium and stability of a pendulum in an orbiting spaceship, Am. J. Phys., 47 (1979), 241-246.  doi: 10.1119/1.11561. [3] A. A. Burov, Oscillations of a vibrating dumbbell on an elliptic orbit, Dokl. Akad. Nauk, 437 (2011), 186-189. [4] A. A. Burov, A. D. Guerman and I. I. Kosenko, On plane oscillations of a pendulum with variable length suspended on the surface of a planet's satellite, Cosmic Research, 52 (2014), 289-294.  doi: 10.1134/S0010952514040029. [5] A. A. Burov and I. I. Kosenko, Planar vibrations of a solid with variable mass distribution in an elliptic orbit, Dokl. Phys., 56 (2011), 760-764. [6] A. A. Burov and I. I. Kosenko, Planar oscillations of a dumb-bell of a variable length in a central field of newtonian attraction, exact approach, Int. J. of Non-Linear Mech., 72 (2015), 1-5.  doi: 10.1016/j.ijnonlinmec.2015.01.011. [7] A. Burov, I. Kosenko and A. Guerman, Dynamics of a moon-anchored tether with variable lenght, Adv. Astronautical Sci., 142 (2012), 3495-3507. [8] G. Cruz Araujo and H. E. Cabral, Parametric stability in a $P +2$-body problem, J. Dyn. Diff. Equat., 30 (2018), 719-742.  doi: 10.1007/s10884-017-9570-x. [9] J. L. de Menezes Neto and H. E. Cabral, Parametric stability of a pendulum with variable length in an elliptic orbit, Regul. Chaotic Dyn., 25 (2020), 323-329.  doi: 10.1134/S1560354720040012. [10] A. D. Guerman, Spatial equilibria of multibody chain in a circular orbit, Acta Astronautica, 58 (2006), 1-14.  doi: 10.1016/j.actaastro.2005.05.002. [11] O. V. Kholostova, On the motions of a double pendulum with vibrating suspension point, Mechanics of Solids, 44 (2009), 184-197.  doi: 10.3103/S0025654409020034. [12] M. Lavagna and A. E. Finzi, Large multi-hinged space systems: A parametric stability analysis, Acta Astronautica, 54 (2004), 295-305.  doi: 10.1016/S0094-5765(02)00304-1. [13] A. P. Markeev, Linear Hamiltonian systems and some applications to the problem of stability of motion of satellites relative to the center of mass, $\mathcal{R}$ & $\mathcal{C}$ $\mathcal{D}$ynamics, Izhevsk-Moscow, 2009. [14] A. P. Markeev, On one special case of parametric resonance in problems of celestial mechanics, Astronomy Letters, 31 (2005), 350-356.  doi: 10.1134/1.1922534. [15] A. K. Misra, Z. Amier and V. J. Modi, Attitude dynamics of three-body tethered systems, Acta Astronautica, 17 (1988), 1059-1068. [16] A. V. Sarychev, Equilibria of a double pendulum in a circular orbit, Acta Astronautica, 44 (1999), 63-65.  doi: 10.1016/S0094-5765(99)00015-6. [17] J. L. Synge, On the behavior, according to Newtonian theory of a plumb line or pendulum attached to an artificial satellite, Proc. Roy. Irish Acad. Sect. A, 60 (1959), 6 pp. [18] L. R. Valeriano, Parametric stability in Robe's problem, Regul. Chaotic Dyn., 21 (2016), 126-135.  doi: 10.1134/S156035471601007X.
Double pendulum in an elliptical orbit. Orbital plane
Graphics of: (1) $\beta^{(2)}(e)$, (2) $\beta^{(3)}(e)$, (3) $\beta^{(5}(e)$, (4) $\beta^{(6)}(e)$, (5) $\beta^{(7)}(e)$, (6) $\beta^{(8)}(e)$ in the plane $e\times \beta$
Graphics of: (1) $\beta^{(1)}(e)$ and (2) $\beta^{(4)}(e)$ in the plane $e\times \beta$
Regions of stability and instability for the equilibrium point $E_2$. Graphics of (1): $\beta^{(1)}(e)$, (2): $\beta^{(2)}(e)$, (3): $\beta^{(3)}(e)$, (4): $\beta^{(4)}(e)$, (5): $\beta^{(5)}(e)$, (6): $\beta^{(6)}(e)$, (7): $\beta^{(7)}(e)$, (8): $\beta^{(8)}(e)$
Regions of stability and instability for the equilibrium point $E_3$. Graphics of (1): $\beta^{(1)}(e)$, (2): $\beta^{(2)}(e)$, (3): $\beta^{(3)}(e)$, (4): $\beta^{(4)}(e)$, (5): $\beta^{(5)}(e)$, (6): $\beta^{(6)}(e)$, (7): $\beta^{(7)}_+(e)$, (8): $\beta^{(7}_-(e)$, (9): $\beta^{(8)}(e)$, (10): $\beta^{(9)}(e)$
Regions of stability and instability for the equilibrium point $E_3$. Graphics of $\beta^{(7)}_+(e)$ and $\beta^{(7)}_-(e)$. The shaded region is where the equilibrium point is unstable
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