The rolling unrestrained brachistochrone

Yang, Ju‐xing; Stork, David G.; Galloway, D. J.

doi:10.1119/1.15001

Cited by 6 publications

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“…The initial and final subarcs of the trajectory, as well as the transit time, are described in terms of Gaussian hypergeometric functions. These results correct and generalise those given in Yang and Stork (1986).…”

Section: Discussionsupporting

confidence: 88%

“…Equation (65) is equivalent to Equation (10) in Yang and Stork (1986). Equation (65) can be solved by separating the variables in combination with (46) to give…”

Section: Zero Normal Load Trajectorymentioning

confidence: 99%

“…This problem is still two-dimensional, but it contains rotational inertia effects and a rolling contact. We refer to this as the rolling unrestrained brachistochrone that was first studied in Yang and Stork (1986). We extend the problem formulation to allow for a wider set of terminal conditions as well as correct some technical errors in the original treatment.…”

Section: Introductionmentioning

confidence: 99%

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Time-optimal control of rolling bodies

Perantoni

Limebeer

2013

International Journal of Control

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The brachistochrone problem is usually solved in classical mechanics courses using the calculus of variations, although it is quintessentially an optimal control problem. In this paper, we address the classical brachistochrone problem and two vehicle-relevant generalisations from an optimal control perspective. We use optimal control arguments to derive closed-form solutions for both the optimal trajectory and the minimum achievable transit time for these generalisations. We then study optimal control problems involving a steerable disc rolling between prescribed points on the interior surface of a hemisphere. The effects of boundary and control constraints are examined. For three-dimensional problems of this type, which involve rolling bodies and nonholonomic constraints, numerical solutions are used.

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Section: Discussionsupporting

confidence: 88%

“…Equation (65) is equivalent to Equation (10) in Yang and Stork (1986). Equation (65) can be solved by separating the variables in combination with (46) to give…”

Section: Zero Normal Load Trajectorymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Time-optimal control of rolling bodies

Perantoni

Limebeer

2013

International Journal of Control

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“…Simple versions of the brachistochrone problem have been used for pedagogical purposes to introduce optimization problems to undergraduate physics students [3,5,6]. More complicated versions take into account friction, a linearly decreasing gravitational field, and a brachistochrone path for rolling beads [7,8].…”

Section: Introductionmentioning

confidence: 99%

From the discrete to the continuous brachistochrone: a tale of two proofs

Agmon

Yizhaq

2020

Eur. J. Phys.

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In a previous paper (2019 Eur. J. Phys. 40 035005) we showed how to design a discrete brachistochrone with an arbitrary number of segments. We have proved, numerically and graphically, that in the limit of a large number of segments, N ≫ 1, the discrete brachistochrone converges into the continuous brachistochrone, i.e. into a cycloid. Here we show this convergence analytically, in two different ways, based upon the results we obtained from investigating the characteristics of the discrete brachistochrone. We prove that at any arbitrary point, the sliding bead has the same velocity on both the continuous and discrete paths, and the radius of the curvature of both paths is the same at corresponding points. The proofs are based on the well-known fact that the curve of a cycloid is generated by a point attached to the circumference of a rolling wheel. We also show that the total acceleration magnitude of the bead along the cycloid is constant and equal to g, whereas the acceleration vector is directed toward the center of the wheel, and it rotates with a constant angular velocity.

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“…Various formulations of the brachistochrone problem for arbitrary rolling bodies were given in [22,19].…”

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confidence: 99%

Conditions for pure rolling of a heavy cylinder along a brachistochrone

Legeza

2010

Int Appl Mech

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Algebraic equations for the line of steepest descent of a cylinder are derived in parametric form. Conditions for rolling without slipping and separation of the cylinder along a brachistochrone are established based on the equations of motion with constraint reaction. The important conclusion is drawn that the center of mass of a cylinder moving along a brachistochrone describes a cycloid Introduction. Brief Review of Relevant Studies. In 1696, Bernoulli posed the following brachistochrone problem: find the shape of the curve down which a bead slipping from rest and accelerated by gravity will slip from one point to another in the least time. It is assumed that the bead (material point) moves in a vertical plane and in a uniform gravity field.Newton, Leibniz, l'Hopital, and two Bernoullis showed that the solution of this problem is a cycloid [14]. Analytic solutions of the brachistochrone problem obtained using geometrical optics and the calculus of variations can be found in [15,16]. Brachistochrone with Coulomb friction was studied in detail by Ashby et al. [10], Hayen [18], Heijden and Diepstraten [25]. The problem of a brachistochrone on a cylindrical surface with Coulomb friction was generalized by Covic and Veskovic in [12]. The problem of a brachistochrone in nonconservative force fields was addressed in [27]. The problem of a brachistochrone on a cylinder in uniform force fields was solved in [28]. Brachistochrones on cylinders and spheres were considered by Palmieri [20]. The same problem was generalized to nonuniform force fields by Aravind [9], Denman [13], and Venezian [26]. Venezian solved the problem of brachistochronein linear radial force fields. Denman [13], Parnovsky [21], and Tee [24] solved the same problem for radial force fields inversely proportional to the squared distance between the interacting points. The generalizations of the brachistochrone problem involving relativistic effects were outlined by Goldstein and Bender [17], Scarpello and Ritelli [23].The next natural step in the generalization of the brachistochrone problem is to derive the brachistochrone equation for rolling finite-size bodies (balls, cylinders, disks, wheels, etc.). Various formulations of the brachistochrone problem for arbitrary rolling bodies were given in [22,19].The brahistohrone problem for a heavy cylinder is of theoretical importance due to the following applied problem.There is a variety of antivibration devices for which the minimum time they take to reduce the amplitude of forced vibrations to a standard level is the performance criterion [6,7,11]. This criterion is applied to rolling antivibration devices [2,3]. In this connection, it is necessary to find the shape of the directrix of a cylindrical surface over which a heavy homogeneous cylinder rolls without slipping (pure rolling) in the least time. The requirement of no slipping is standard for vibration-protection problems.The present study continues the research reported in [4,5] where the following differential equation of brachistochrone along whi...

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The rolling unrestrained brachistochrone

Cited by 6 publications

References 0 publications

Time-optimal control of rolling bodies

Time-optimal control of rolling bodies

From the discrete to the continuous brachistochrone: a tale of two proofs

Conditions for pure rolling of a heavy cylinder along a brachistochrone

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