Brachistochrone with limited reaction of constraint in an arbitrary force field
Abstract
The paper considers brachistochronic motion of a particle along a curve y=y(x) in an arbitrary force field in the vertical plane of Cartesian coordinate system. The curve is treated as a bilateral or unilateral constraint that can be smooth or rough. The projection of the reaction force of the curve onto the normal to the curve is confined to the fixed limits. A control variable u is given as the second derivative of the function y(x) relative to the horizontal coordinate x of the particle, i.e., u=d (2) y/dx (2). Applying Pontryagin's maximum principle and singular optimal control theory, the problem is reduced to numerical solving of the corresponding two-point boundary value problem. The procedure based on the shooting method is used to solve the boundary value problem. Two examples with friction forces of the viscous friction and Coulomb friction type have been solved.
Keywords:
Viscous friction / Singular control / Pontryagin's maximum principle / Coulomb friction / BrachistochroneSource:
Nonlinear Dynamics, 2012, 69, 1-2, 211-222Publisher:
- Springer, Dordrecht
Funding / projects:
- Sustainability and improvement of mechanical systems in energetic, material handling and conveying by using forensic engineering, environmental and robust design (RS-MESTD-Technological Development (TD or TR)-35006)
DOI: 10.1007/s11071-011-0258-1
ISSN: 0924-090X
WoS: 000304651400016
Scopus: 2-s2.0-84861746884
Collections
Institution/Community
Mašinski fakultetTY - JOUR AU - Šalinić, Slaviša AU - Obradović, Aleksandar AU - Mitrović, Zoran AU - Rusov, Srđan PY - 2012 UR - https://machinery.mas.bg.ac.rs/handle/123456789/1501 AB - The paper considers brachistochronic motion of a particle along a curve y=y(x) in an arbitrary force field in the vertical plane of Cartesian coordinate system. The curve is treated as a bilateral or unilateral constraint that can be smooth or rough. The projection of the reaction force of the curve onto the normal to the curve is confined to the fixed limits. A control variable u is given as the second derivative of the function y(x) relative to the horizontal coordinate x of the particle, i.e., u=d (2) y/dx (2). Applying Pontryagin's maximum principle and singular optimal control theory, the problem is reduced to numerical solving of the corresponding two-point boundary value problem. The procedure based on the shooting method is used to solve the boundary value problem. Two examples with friction forces of the viscous friction and Coulomb friction type have been solved. PB - Springer, Dordrecht T2 - Nonlinear Dynamics T1 - Brachistochrone with limited reaction of constraint in an arbitrary force field EP - 222 IS - 1-2 SP - 211 VL - 69 DO - 10.1007/s11071-011-0258-1 ER -
@article{ author = "Šalinić, Slaviša and Obradović, Aleksandar and Mitrović, Zoran and Rusov, Srđan", year = "2012", abstract = "The paper considers brachistochronic motion of a particle along a curve y=y(x) in an arbitrary force field in the vertical plane of Cartesian coordinate system. The curve is treated as a bilateral or unilateral constraint that can be smooth or rough. The projection of the reaction force of the curve onto the normal to the curve is confined to the fixed limits. A control variable u is given as the second derivative of the function y(x) relative to the horizontal coordinate x of the particle, i.e., u=d (2) y/dx (2). Applying Pontryagin's maximum principle and singular optimal control theory, the problem is reduced to numerical solving of the corresponding two-point boundary value problem. The procedure based on the shooting method is used to solve the boundary value problem. Two examples with friction forces of the viscous friction and Coulomb friction type have been solved.", publisher = "Springer, Dordrecht", journal = "Nonlinear Dynamics", title = "Brachistochrone with limited reaction of constraint in an arbitrary force field", pages = "222-211", number = "1-2", volume = "69", doi = "10.1007/s11071-011-0258-1" }
Šalinić, S., Obradović, A., Mitrović, Z.,& Rusov, S.. (2012). Brachistochrone with limited reaction of constraint in an arbitrary force field. in Nonlinear Dynamics Springer, Dordrecht., 69(1-2), 211-222. https://doi.org/10.1007/s11071-011-0258-1
Šalinić S, Obradović A, Mitrović Z, Rusov S. Brachistochrone with limited reaction of constraint in an arbitrary force field. in Nonlinear Dynamics. 2012;69(1-2):211-222. doi:10.1007/s11071-011-0258-1 .
Šalinić, Slaviša, Obradović, Aleksandar, Mitrović, Zoran, Rusov, Srđan, "Brachistochrone with limited reaction of constraint in an arbitrary force field" in Nonlinear Dynamics, 69, no. 1-2 (2012):211-222, https://doi.org/10.1007/s11071-011-0258-1 . .