The motion of an absolutely rigid biaxial trolley with an elastic body along a double rail curvilinear track

Authors: Gnezdilov V.A., Russkikh S.V. Published: 29.03.2016
Published in issue: #2(107)/2016  

DOI: 10.18698/0236-3941-2016-2-93-112

Category: Mechanics | Chapter: Dynamics and Strength of Machines, Instruments, and Equipment  
Keywords: elastic body, trolley, motion along a curvilinear track, nonlinear elastic rollers, a mathematical model, reaction, overloads

The research deals with the unsteady motion of an elastic body located on an absolutely rigid trolley moving on the elastic rollers along an arbitrary curvilinear track with two equidistant guide rails. The track is defined by the equations of bending and torsion of its midline. The contact between the trolley and the rails is carried out through the nonlinear elastic rollers of rear drive axle, rigidly connected with the trolley, and those of the front axle, coupled with it in the middle by the spherical hinge. We developed a new nonlinear mathematical model for calculating the relative and portable motions of a trolley with an elastic body along the surface and defined contact reactions and overloads. Initially, we conducted a geometric modeling of kinematic motion of a biaxial trolley, where the distance between the axes is considered to be small in comparison with the curvature radius of the trolley centerline. Next, we obtained the relations allowing us to further consider the curvature of the midline between the axes of the trolley. Then, we described dynamics of the relative motion of an elastic body on the trolley by finite deformations in the quadratic approximation. Finally, we worked out equations of system motion taking into account the nonlinear elasticity and damping of the rollers, on which the trolley slides along the surface at a predetermined speed or under a given traction.


[1] Gorshkov A.G., Morozov V.I., Ponomarev A.T., Shkliarchuk F.N. Aerogidrouprugost’ konstruktsiy [Construction aerohydroelasticity]. Moscow, Fizmatlit Publ., 2000. 592 p.

[2] Gnezdilov V.A. Design and fabrication of steel structures for complex mechanized attractions. Montazhnye i spetsial’nye raboty v stroitel’stve [Erecting and special works in construction], 2000, no. 6, pp. 20-24 (in Russ.).

[3] Wayne T. Roller Coaster Physics, Wayne, 1998. 154 p. Available at: http://vip.vast.org/BOOK/HOME.HTM (accessed 24 April 2015).

[4] Kulakov N.A. Vozdeistvie dinamicheskoi nagruzki na nazemnye transportnye sredstva. Izbrannye problemy prochnosti sovremennogo mashinostroeniia [The impact of the dynamic loads on ground vehicles. Selected problems of modern engineering strength]. Moscow, Fizmatlit Publ., 2008. 204 p.

[5] Selifonov V.V. Teoriya avtomobilya [The theory of car]. Moscow, Grinlait Publ., 2009. 206 p.

[6] Russkih S.V. Motion of a rigid body on two wheels on a plane curve. Izv. Vyssh. Uchebn. Zaved., Mashinostr. [Proc. of Higher Educational Institutions. Machine Building], 2014, no. 2 (647), pp. 52-58 (in Russ.).

[7] Shklyarchuk F.N. Non-linear and linearized equations of motion of the elastic space structures. Izv. RAN. MTT [Proceedings of the Russian Academy of Sciences. Mechanics of rigid body], 1996, no. 1, pp. 161-175 (in Russ.).

[8] Grishanina T.V., Shklyarchuk F.N. Dinamika uprugikh upravlyaemykh konstruktsii [Dynamics of elastic controlled constructions]. Moscow, MAI Publ., 2007. 328 p.

[9] Biushgens S.S. Differentsial’naia geometriia [Differential geometry]. Moscow, Komkniga Publ., 2006. 302 p.

[10] Rashevskiy P.K. Kurs differentsial’noi geometrii [A course of differential geometry]. Moscow, LKI Publ., 2013. 432 p.

[11] Banchoff T., Lovett S.T. Differential Geometry of Curves and Surfaces. AK Peters, Taylor & Francis, 2010. 352 p.

[12] Kosmodem’yanskiy A.A. Dinamika kosmicheskogo poleta [The dynamics of space flight]. Moscow, Librokom Publ., 2011. 248 p.

[13] Rabotnov Yu.N. Mekhanika deformiruemogo tverdogo tela [Mechanics of a deformable solid]. Moscow, Nauka Publ., 1988. 712 p.

[14] Rabinovich B.A. Bezopasnost’ cheloveka pri uskoreniyakh (Biomekhanicheskiy analiz) [Human safety when accelerating (Biomechanical analysis)]. Moscow, Kniga i biznes Publ., 2007. 208 p.