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:
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L. Blitzer, Equilibrium and stability of a pendulum in an orbiting spaceship, Am. J. Phys., 47 (1979), 241-246.  doi: 10.1119/1.11561.  Google Scholar

[3]

A. A. Burov, Oscillations of a vibrating dumbbell on an elliptic orbit, Dokl. Akad. Nauk, 437 (2011), 186-189.   Google Scholar

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A. A. BurovA. 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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

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A. BurovI. Kosenko and A. Guerman, Dynamics of a moon-anchored tether with variable lenght, Adv. Astronautical Sci., 142 (2012), 3495-3507.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[15]

A. K. MisraZ. Amier and V. J. Modi, Attitude dynamics of three-body tethered systems, Acta Astronautica, 17 (1988), 1059-1068.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

L. R. Valeriano, Parametric stability in Robe's problem, Regul. Chaotic Dyn., 21 (2016), 126-135.  doi: 10.1134/S156035471601007X.  Google Scholar

show all references

References:
[1]

V. S. Aslanov, Orbital oscillations of an elastic vertically-tethered satellite, Mech. Solids, 46 (2011), 657-668.  doi: 10.3103/S0025654411050013.  Google Scholar

[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.  Google Scholar

[3]

A. A. Burov, Oscillations of a vibrating dumbbell on an elliptic orbit, Dokl. Akad. Nauk, 437 (2011), 186-189.   Google Scholar

[4]

A. A. BurovA. 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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[7]

A. BurovI. Kosenko and A. Guerman, Dynamics of a moon-anchored tether with variable lenght, Adv. Astronautical Sci., 142 (2012), 3495-3507.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[15]

A. K. MisraZ. Amier and V. J. Modi, Attitude dynamics of three-body tethered systems, Acta Astronautica, 17 (1988), 1059-1068.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

L. R. Valeriano, Parametric stability in Robe's problem, Regul. Chaotic Dyn., 21 (2016), 126-135.  doi: 10.1134/S156035471601007X.  Google Scholar

Figure 1.  Double pendulum in an elliptical orbit. Orbital plane
Figure 2.  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 $
Figure 3.  Graphics of: (1) $ \beta^{(1)}(e) $ and (2) $ \beta^{(4)}(e) $ in the plane $ e\times \beta $
Figure 4.  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) $
Figure 5.  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) $
Figure 6.  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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