Vertical vibrations of a freight car as a uniaxial carriage from a wave of unevenness of the path

Abstract

The paper gives the results of mathematical modeling of movement of a loaded wagon with elastic suspension under viscous friction force equivalent to the dry friction force. Parameters of vertical relative forced vibrations have been found, which are stipulated by the wave of road irregularity.

Simulation of movements of cargo with a flat base on a four-axle platform

“…Friction force projections 𝐹𝐹 ̄𝑓𝑓𝑓𝑓 on the x and y axis -𝐹𝐹 𝑓𝑓𝑓𝑓х and 𝐹𝐹 𝑓𝑓𝑓𝑓у can be represented in the form 𝐹𝐹 𝑓𝑓𝑓𝑓х = −𝑓𝑓𝑓𝑓 𝑐𝑐𝑐𝑐𝑐𝑐 𝜆𝜆 (13) 𝐹𝐹 𝑓𝑓𝑓𝑓𝑖𝑖 = −𝑓𝑓𝑓𝑓 𝑐𝑐𝑠𝑠𝑠𝑠 𝜆𝜆 (14) From ( 7),…, (9) one can find the generalized forces corresponding to the coordinates С…”

“…1 In equations ( 18) and ( 19) reactions 1 11 , R R flexible elastic elements 1 and 11, necessary for calculating their projections on the x and y axes, can be determined according to Hooke's law if the stiffness of each elastic element is known i c and the magnitude of load shifts both Let the cargo placed in the car supported on wooden supports be held from translational movements by wire flexible elastic fastening elements with rigidities ci, fixed at i points by one end by mounting loops, for example, at point M (see Fig. 2), and by the other -behind platform rack brackets or gondola car tie-down devices, for example, at point A [8,10,12].…”

“…First, we project in a vertical plane onto the direction of projection of the flexible fastening element onto a vertical plane, and then we project directly onto the elastic element in a plane perpendicular to this vertical plane (i.e., we first lower the perpendicular to line A1М -point B, and then lower the perpendicular to line AМ -point В1) [7,9,11,14,16,18].…”

Mathematical modeling of the rolling speed of a car on the first profile site of the slide

“…Let us consider the case of a flat system of forces when the trailer rolls down a hill along a straight track profile, taking into account that the resulting aerodynamic drag force is directed at a certain angle to the direction of the car rolling down the hill, i.e.𝐹𝐹 ̄𝑟𝑟𝑟𝑟 ∈ 𝐹𝐹 ̄𝑟𝑟𝑟𝑟𝑟𝑟 ′ , 𝐹𝐹 ̄𝑟𝑟𝑟𝑟𝑦𝑦 ′ [5,15]. Introducing the concepts of "shearing" and "holding" forces, taking into account all the active and reactive forces that were found in [5,7,11,13], we obtain: -in a headwind 𝐹𝐹 sh.х = 𝐺𝐺 𝑟𝑟 = 𝐺𝐺 𝑠𝑠𝑠𝑠𝑠𝑠 𝜓𝜓 0 ; -with a fair wind 𝐹𝐹 sh.х = 𝐺𝐺 𝑠𝑠𝑠𝑠𝑠𝑠 𝜓𝜓 0 + 𝐹𝐹 𝑟𝑟𝑟𝑟𝑟𝑟 ′ 𝑐𝑐𝑐𝑐𝑠𝑠 𝜓𝜓 0 ; -in a headwind 𝐹𝐹 h.х = 𝐹𝐹 𝜏𝜏 + 𝐹𝐹 𝑟𝑟𝑟𝑟𝑟𝑟 ′ 𝑐𝑐𝑐𝑐𝑠𝑠 𝜓𝜓 0 ; -with a fair wind 𝐹𝐹 h.х = 𝐹𝐹 𝜏𝜏 . The carriage will roll down the hill only if the following conditions are met:…”

“…where Mmass of the car with cargo, kg; tftime of passage of the car on any section of the hump, s; vfacquired final speed of the car (the value being sought), mps. Substituting (12) into (13) and dividing the result by the mass of the cut M, and given that 𝐺𝐺 = 𝑀𝑀𝑀𝑀, will have:…”