Perturbation In Force Research Articles

Stability analysis in the perturbed CRR3BP finite straight segment model under the effect of viscosity

In this paper Robe-finite straight segment model is analyzed under the effects of viscosity and perturbations in the Coriolis and centrifugal forces. We have taken the first primary $P_{1}$ as a rigid spherical shell $m_{1}$ filled with viscous, homogeneous incompressible fluid of density $\rho _{1}$ , and the second primary $P_{2}$ as a finite straight segment of length $2l$ . A third body of mass $m_{3}$ , moving inside $m_{1}$ is a small solid sphere of density $\rho _{3}$ . We prove how the locations of equilibrium points $L_{i}, i=1,2,\ldots ,5$ are affected by the presence of perturbation in the centrifugal force. However, these remain unaffected by the viscosity and perturbation in the Coriolis force. The stability criteria for $L_{1,2}$ are investigated and it has been observed that their stability is affected by the viscosity and perturbation of the centrifugal force. It is prominently observed that the viscosity changes their nature of stability from being stable to asymptotically stable. The equilibrium points $L_{3,4,5}$ are unstable irrespective of the perturbation and viscosity.

Divulging the effect of small perturbations in the Coriolis and centrifugal forces in the photogravitational version of autonomous restricted four‐body problem with oblate primary

AbstractIn the present problem, we have studied the effect of small perturbations in the Coriolis and centrifugal forces on the locations and stability of libration points by taking the smaller primary as radiating oblate spheroid in the restricted four‐body problem. We have supposed that all the primaries are set in an equilateral triangle configuration, moving in circular orbit around their common center of mass. We have observed that the oblateness and radiation of the third primary, and the effect of the small perturbation in centrifugal force have substantial influences on the locations of libration points but the effect of the small perturbation in the Coriolis force has no impact on their locations. However, the stability of the libration points is highly influenced by the effect of the small perturbation in the Coriolis force. Furthermore, it is obtained that the effect of the small perturbation in the centrifugal force has a substantial influence on the regions of possible motion. In addition, we have also discussed how the oblateness and radiation pressure of the third primary, and the small perturbations in the Coriolis and centrifugal forces influence geometry of the basins of convergence by using multivariate version of the Newton–Raphson iterative scheme.

On the restricted four-body problem with the effect of small perturbations in the Coriolis and centrifugal forces

We have studied the restricted four-body problem (R4BP) with the effect of the small perturbation in the Coriolis and centrifugal forces on the libration points and zero velocity curves (ZVCs). Further, we have supposed that all the primaries are set in an equilateral triangle configuration, moving in the circular orbits around their common centre of mass. We have observed that the effect of the small perturbation in centrifugal force has a substantial effect on the location of libration points but a small perturbation in the Coriolis force has no impact on the location of libration points. But the stability of the libration points is highly influenced by the effect of the small perturbation in the Coriolis force. It is observed that as the Coriolis parameter increases, the libration points become stable. Further, it is found that the effect of the small perturbation in the centrifugal force has a substantial influence on the regions of possible motion. Also, when the effect of small perturbation in the centrifugal force increases the forbidden region decreases; here the motion is not possible for the infinitesimal mass. It is observed when the value of the Jacobian constant decreases, the regions of possible motion increase. In addition, we have also discussed how small perturbations in the Coriolis and centrifugal forces influence the Newton–Raphson basins of convergence.

On the stability of L4,5 in the perturbed relativistic R3BP with a triaxial bigger primary

In the present paper, we endeavor to study the stability of triangular points under the influence of small perturbations in the Coriolis and centrifugal forces, together with the triaxiality of the bigger primary in the framework of the relativistic R3BP. It is observed that the locations of these points are affected by the relativistic factor, triaxiality and a small perturbation in the centrifugal force, but are unaffected by that of the Coriolis force. It is also seen that for these points the range of stability region increases or decreases according as equation (14) without is greater or less than zero.

Existence and linear stability of triangular points in the perturbed relativistic R3BP when the bigger primary is an oblate spheroid

We study the effects of oblateness and small perturbations in the Coriolis and centrifugal forces on the locations and stability of the triangular points in the relativistic R3BP. It is observed that the positions are affected by the oblateness, relativistic, and a small perturbation in the centrifugal force, but are unaffected by that of Coriolis force. It is also seen that the relativistic terms, oblateness, small perturbations in the centrifugal and Coriolis forces influence the critical mass ratio. It is also noticed that all the former three and the latter one possess destabilizing and stabilizing behavior respectively. However, the range of stability increases or decreases according to as p >0 or p<0 where p depends upon the relativistic, oblateness and small perturbations in the Coriolis and centrifugal forces.

Effect of Perturbations in the Coriolis and Centrifugal Forces on the Stability of Equilibrium Points in the Restricted Four-Body Problem

This paper studies numerically the existence, location and stability of equilibrium points under the influence of small perturbations in the Coriolis and centrifugal forces in the restricted four-body problem, where three of the bodies are finite, moving in circles around their centre of mass fixed at the origin of the coordinate system, according to the solutions of Lagrange where they are always at the vertices of an equilateral triangle, while the fourth one is infinitesimal. The fourth body does not affect the motion of the three bodies (primaries). We consider that the two masses of the primaries m2 and m3 are equal to μ and the dominant mass m1 is 1 − 2μ. The allowed regions of motion as determined by the zero-velocity surfaces and the corresponding equipotential curves are given. The equations which determine equilibrium positions of the infinitesimal mass in the rotating coordinate system are found. This is observed that their positions are affected by a small perturbation in the centrifugal force, but are unaffected by that of the Coriolis force. For different values of the perturbation parameter $${\beta (\beta = 1 + \varepsilon^{\prime}, {\vert} \varepsilon^{\prime} {\vert} \ll 1)}$$ , we obtain two collinear and six non-collinear equilibrium points. We also examine the linear stability of these equilibrium points for a wide range of the small perturbations and all points found unstable except two (L7,8) which are stable.

The rotational feedback on linear-momentum balance in glacial isostatic adjustment

The influence of changes in surface ice-mass redistribution and associated viscoelastic response of the Earth, known as glacial isostatic adjustment (GIA), on the Earth's rotational dynamics has long been known. Equally important is the effect of the changes in the rotational dynamics on the viscoelastic deformation of the Earth. This signal, known as the rotational feedback, or more precisely, the rotational feedback on the sea level equation, has been mathematically described by the sea level equation extended for the term that is proportional to perturbation in the centrifugal potential and the second-degree tidal Love number. The perturbation in the centrifugal force due to changes in the Earth's rotational dynamics enters not only into the sea level equation, but also into the conservation law of linear momentum such that the internal viscoelastic force, the perturbation in the gravitational force and the perturbation in the centrifugal force are in balance. Adding the centrifugal-force perturbation to the linear-momentum balance creates an additional rotational feedback on the viscoelastic deformations of the Earth. We term this feedback mechanism, which is studied in this paper, as the rotational feedback on the linear-momentum balance. We extend both the time-domain method for modelling the GIA response of laterally heterogeneous earth models developed by Martinec and the traditional Laplace-domain method for modelling the GIA-induced rotational response to surface loading by considering the rotational feedback on linear-momentum balance. The correctness of the mathematical extensions of the methods is validated numerically by comparing the polar-motion response to the GIA process and the rotationally induced degree 2 and order 1 spherical harmonic component of the surface vertical displacement and gravity field. We present the difference between the case where the rotational feedback on linear-momentum balance is considered against that where it is not. Numerical simulations show that the resulting difference in radial displacement and sea level change between these situations since the Last Glacial Maximum reaches values of ±25 and ±1.8 m, respectively. Furthermore, the surface deformation pattern is modified by up to 10 per cent in areas of former or ongoing glaciation, but by up to 50 per cent at the bottom of the southern Indian ocean. This also results in the movement of coastlines during the last deglaciation to differ between the two cases due to the difference in the ocean loading, which is seen for instance in the area around Hudson Bay, Canada and along the Chinese, Australian or Argentinian coastlines.

Motion around L 4 in the perturbed relativistic R3BP

This paper studies the motion of a test particle (infinitesimal mass) in the neighborhood of the triangular point L4 in the framework of the perturbed relativistic restricted three-body problem (R3BP). The problem is perturbed in the sense that a small perturbation is given to the centrifugal force. It is found that the position and stability of the triangular point are affected by both the relativistic factor and a small perturbation in the centrifugal force.

Instability of triangular libration points in the perturbed photogravitational R3BP with Poynting-Robertson (P-R) drag

We consider the primaries of the circular restricted three-body problem (CR3BP) to be luminous and investigate the influences of small perturbations in the Coriolis and centrifugal forces together with Poynting-Robertson (P-R) drag from both primaries on the triangular points. It is seen, both analytically and numerically, that the positions of triangular points are affected by the radiation pressures, P-R drag and a small perturbation in the centrifugal force. This has been shown for the binary systems Luyten 726-8 and Kruger 60.1. These perturbing forces do not influence the nature of the stability of triangular points in the presence of P-R drag. They remain unstable in the linear sense.

Motion in a modified Chermnykh’s restricted three-body problem with oblateness

In this paper, the restricted problem of three bodies is generalized to include a case when the passively gravitating test particle is an oblate spheroid under effect of small perturbations in the Coriolis and centrifugal forces when the first primary is a source of radiation and the second one an oblate spheroid, coupled with the influence of the gravitational potential from the belt. The equilibrium points are found and it is seen that, in addition to the usual three collinear equilibrium points, there appear two new ones due to the potential from the belt and the mass ratio. Two triangular equilibrium points exist. These equilibria are affected by radiation of the first primary, small perturbation in the centrifugal force, oblateness of both the test particle and second primary and the effect arising from the mass of the belt. The linear stability of the equilibrium points is explored and the stability outcome of the collinear equilibrium points remains unstable. In the case of the triangular points, motion is stable with respect to some conditions which depend on the critical mass parameter; influenced by the small perturbations, radiating effect of the first primary, oblateness of the test body and second primary and the gravitational potential from the belt. The effects of each of the imposed free parameters are analyzed. The potential from the belt and small perturbation in the Coriolis force are stabilizing parameters while radiation, small perturbation in the centrifugal force and oblateness reduce the stable regions. The overall effect is that the region of stable motion increases under the combine action of these parameters. We have also found the frequencies of the long and short periodic motion around stable triangular points. Illustrative numerical exploration is rendered in the Sun–Jupiter and Sun–Earth systems where we show that in reality, for some values of the system parameters, the additional equilibrium points do not in general exist even when there is a belt to interact with.

Periodic Solution of the restricted three body problem

The effect of perturbation in centrifugal force on the periodic solution of the restricted three-body problem representing analytic continuation of Keplerian rectilinear periodic motion has been examined. However, we have taken the perturbation in the centrifugal force to be of the order of μ, the reduced mass of the smaller primary. We have calculated the first order perturbations also. BIBECHANA 10 (2014) 44-51 DOI: http://dx.doi.org/10.3126/bibechana.v10i0.9310

Equilibrium points and stability under effect of radiation and perturbing forces in the restricted problem of three oblate bodies

This paper presents a generalized problem of the restricted three body studied in Abdul Raheem and Singh with the inclusion that the third body is an oblate spheroidal test particle of infinitesimally mass. The positions and stability of the equilibrium point of this problem is studied for a model in which the primaries is the binary system Struve 2398 (Gliese 725) in the constellation Draco; which consist of a pair of radiating oblate stars. It is seen that additional equilibrium points exist on the line collinear with the primaries, for some combined parameters of the problem. Hence, there can be up to five collinear equilibrium points. Two triangular points exist and depends on the oblateness of the participating bodies, radiation pressure of the primaries and a small perturbation in the centrifugal force. The stability analysis ensures that, the collinear equilibrium points are unstable in the linear sense while the triangular points are stable under certain conditions. Illustrative numerical exploration is given to indicate significant improvement of the problem in Abdul Raheem and Singh.

The equilibrium points in the perturbed R3BP with triaxial and luminous primaries

This study explores the effects of small perturbations in the Coriolis and centrifugal forces, radiation pressures and triaxiality of the two stars (primaries) on the position and stability of an infinitesimal mass (third body) in the framework of the planar circular restricted three-body problem (R3BP). it is observed that the positions of the usual five (three collinear and two triangular) equilibrium points are affected by the radiation, triaxiality and a small perturbation in the centrifugal force, but are unaffected by that of the Coriolis force. The collinear points are found to remain unstable, while the triangular points are seen to be stable for 0<μ<μ c and unstable for \(\mu_{c} \le\mu\le\frac{1}{2}\), where μ c is the critical mass ratio influenced by the small perturbations in the Coriolis and centrifugal forces, radiation and triaxiality. It is also noticed that the former one and all the latter three posses stabilizing and destabilizing behavior respectively. Therefore, the overall effect is that the size of the region of stability decreases with increase in the values of the parameters involved.

Robe's Restricted Three-Body Problem with Variable Masses and Perturbing Forces

The linear stability of equilibrium points of a test particle of infinitesimal mass in the framework of Robe's circular restricted three-body problem, as in Hallan and Rana, together with effect of variation in masses of the primaries with time according to the combined Meshcherskii law, is investigated. It is seen that, due to a small perturbation in the centrifugal force and an arbitrary constant of a particular integral of the Gylden-Meshcherskii problem, every point on the line joining the centers of the primaries is an equilibrium point provided they lie within the shell. Further, a number of pairs of equilibrium points lying on the -plane and forming triangles with the centers of the shell and the second primary exist, for some values of . The points collinear with the center of the shell are found to be stable under some conditions and the range of stability depends on the small perturbations and , while the triangular points are unstable. Illustrative numerical exploration is given to indicate significant improvement of the problem in Hallan and Rana.

Motion around the out-of-plane equilibrium points of the perturbed restricted three-body problem

This paper studies the motion of an infinitesimal body near the out-of-plane equilibrium points, L6,7, in the perturbed restricted three-body problem. The problem is perturbed in the sense that the primaries of the system are oblate spheroids as well as sources of radiation and small perturbations are give to the Coriolis and centrifugal forces. It locates the positions and examines the stability of L6,7 with a particular application to the binary system Struve 2398. It is observed that their positions are affected by the radiation, oblateness and a small perturbation in the centrifugal force, but is unaffected by that of the Coriolis force. They are also found to be unstable.

An exploration of the function of the triceps surae during normal gait using functional electrical stimulation

Gastrocnemius and soleus have a common tendon and both are active during stance phase, where they are thought to arrest and control tibial advance. Soleus is associated with the production of an extending moment at the knee. The two-joint gastrocnemius, which crosses the knee joint, will have an additional contribution to the knee flexors. Recent work using induced acceleration analysis (IAA) has demonstrated distinct differences between the actions of gastrocnemius and soleus. This study aims to use gait analysis to provide in vivo examination of these theoretical predictions. Functional electrical stimulation (FES) was chosen to provide a perturbation in muscle force, a close physical analogue to the theoretical predictions of IAA. Five adult male subjects, with no gait problems, participated. Each had gastrocnemius and soleus stimulated at three different timings during normal gait, while 3D gait data were collected. The order of testing was randomised and unstimulated trials were randomly interspersed to act as a control. The results show very different actions for soleus (ankle plantarflexing/knee extending) and gastrocnemius (ankle dorsiflexing/knee flexing) in stance phase. The counterintuitive nature of the action of gastrocnemius suggests that further clinical and biomechanical investigation into this muscle's function is required. The actions of both muscles at the knee confirm published IAA predictions. In vivo evidence such as this gives greater confidence when using model predictions. The approach adopted in this study could eventually be extended to other muscles and patient populations.

Vectorsyn Force Generators for Three Axes Acceleration Measurement

This paper discusses the design and test results of a proposed electromagnetic component, to be named the Vectorsyn, which provides an improved means of mass element restraint and support for a three axes acceleration measurement system. The principle of operation of the Vectorsyn and a functional block diagram for a three axes acceleration measurement system are given. An expression is derived for the first-order perturbation in the output force of the Vectorsyn, which becomes zero under an ideal operating point. However, realizability of the desired operating point is determined by the relative magnitude of various physical parameters, such as the ratio of coil inductive reactance to effective resistance. The relationship between various parameters gives a bounded region. For physical realizability of the desired operating point, certain parameters must fall within this region. The experimental procedure is briefly explained and typical test results are given. The data indicate that the perturbation forces are less than the order of 1 part per 106 per microinch of displacement of the mass element. The test data support the conclusion that Vectorsyn force generators potentially are suitable for providing the means of mass element restraint and support for a three axes acceleration measurement system.