Brachistochrone with limited reaction of constraint in an arbitrary force field

Realizing brachistochronic planar motion of a variable mass nonholonomic mechanical system by an ideal holonomic constraint with restricted reaction

“…This is confirmed by a doctoral dissertation defended recently at the Lomonosov Moscow State University [1]. Throughout literature it is possible to encounter works related to the brachistochronic motion of a material point, of both constant and variable mass [2][3][4][5][6][7], as well as works related to the brachistochronic motion of mechanical systems [8][9][10][11][12][13][14][15]. Regarding variable mass nonholonomic mechanical systems, there is not a lot od works on that subject, whether it is the application of other types of equations in that field, such as Kane's [16] or Hamilton's equations [17], or the control of such systems [8][9][10][11][12][13][14]18].…”

Mass minimization of an Euler-Bernoulli beam with coupled bending and axial vibrations at prescribed fundamental frequency

“…Also, based on (61), ( 63) and ( 76), one has another initial condition: 0), and then to determine them from three final conditions (78). For a more detailed explanation of the procedure refer to [34][35][36]. 3 shows numerical solutions for the missing initial parameters for the cases α = π/6, α = π/4, and α = π/3 as well.…”

A new approach for the determination of the global minimum time for the Chaplygin sleigh brachistochrone problem

“…In Jeremić et al [1] and Šalinić et al [2,3], considerations of the brachistochronic motion of a particle involve the application of the shooting method [4] in solving a corresponding two-point boundary-value problem (TPBVP). The shooting method in these papers is reduced to solving a corresponding system of nonlinear algebraic equations, where the unknowns are values of some state and costate variables at characteristic instants.…”

“…Note that in solving the system of nonlinear algebraic equations that emerged in Jeremić et al [1] and Šalinić et al [2,3], in the application of the shooting method, it was necessary to estimate the values of the costate variables, which is often difficult because the costate variables usually have no physical interpretations. In this regard, the most convenient is the approach from Obradović et al [17], which does not require the estimation of the values of costate variables.…”