Quickest-descent curve in the problem of rolling of a homogeneous cylinder

Abstract

The motion of a heavy homogeneous cylinder is considered as rolling without slipping along an unknown curve. A functional in the form of the total time of rolling is found and minimized by solving a variational problem. The algebraic equation of the quickest-descent directrix is derived in parametric form Keywords: heavy homogeneous cylinder, rolling without slipping, variational problem, functional, quickest-descent directrixIntroduction. The main performance criterion for many vibration-protection devices, shock-absorbers, dampers, and stabilizers is minimum displacements of some points of the bearing object or minimum forces (moments) in the most critical sections during forced vibrations [6,9,13,15,19,20]. Stability of buildings, mobile robots [1,5,8], wheeled vehicles, and carrying and carried bodies is analyzed in [9,12,14,16,18] taking into account the possible bifurcations of their dynamic equilibrium. However, there are a number of vibration-protection devices for which the main performance criterion is the minimum time it takes to reduce the amplitudes of forced vibrations of load-bearing structures to an admissible level [4,7,10,14].Here we use such a criterion for roller-type vibration-protection devices [2,3,9,11,17]. In this connection, it is necessary to identify the directrix of a cylindrical surface along which a heavy cylinder descends the quickest. It is assumed that the motion of the cylinder, which is considered homogeneous, along the directrix is rolling without slipping.Thus, we deal with a variational problem for a heavy homogeneous cylinder that rolls without slipping in a cylindrical hollow with a quickest-descent directrix. This study is a generalization of Bernoulli's study where he derived the equation of a brachistochrone along which a material point descends the quickest. This curve is known to be a cycloid [1,5,8].1. Problem Formulation and Objective Functional. Consider a homogeneous cylinder of mass m and radius r rolling without slipping from a point A without initial velocity over a cylindrical valley with a directrix AKL (Fig. 1). The curve AKL lies in a vertical plane. Let us determine the time it takes the cylinder to move from the point A to the point K. We choose the origin of coordinates at the point O and direct the OZ-axis vertically downwards and the OX-axis horizontally to the right (the point A is on the OZ-axis).The inward unit normal vector to the unknown curve at its arbitrary point K is expressed as

Conditions for pure rolling of a heavy cylinder along a brachistochrone

“…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 which a cylinder rolls without slipping (Fig. 1) was for the first time derived by setting up and minimizing a functional T z x [ ( )]: ( )[ ( ) ] ( ) z C z r z C + + ¢ -+ ¢ = 1 1 2 2 1 , z zx z L L ( ) , ( ) 0 0 = = .…”

“…The present study continues the research reported in [4,5] where the following differential equation of brachistochrone along which a cylinder rolls without slipping (Fig. 1) was for the first time derived by setting up and minimizing a functional T z x [ ( )]: ( )[ ( ) ] ( ) z C z r z C + + ¢ -+ ¢ = 1 1 2 2 1 , z zx z L L ( ) , ( ) 0 0 = = .…”

“…We have established that a special parametrization ( ¢ = z x ( ) cot( ) j / 2 ) allows quadratures to be used to solve the differential equation (1) derived in [4,5]. We have derived a system of algebraic parametric equations describing the brachistochrone for a cylinder.…”

“…(1), we will use a special parametrization of the curve z z x = ( ) different from that in [5]. The parametrization is done as follows ( / ) a j = 2 : …”

Dynamics of vibration isolation system with a quasi-isochronous roller shock absorber

“…This is because over time, these structures wear out and may require urgent restoration for (i) reduction of the dynamic loads on these structures and (ii) strengthening of their load-carrying components. Here we address only case (i) and propose a new engineering solution based on a roller shock absorber with working surface in the form of a brachistochrone [10,16,32].The dynamic analysis of high-rise structures for wind forces deals with forced vibrations excited by pulsating wind pressure and self-excited vibrations such as wind resonance [1,3,4]. The principal natural frequencies of such structures are low, and the displacements of the upper parts of high-rise structures are great (1 to 3 m).…”