Realization of potential high-speed qualities of transport tracked vehicles is limited to a number of dynamic phenomena in linear motion. The analysis of the test results of investigating controlled movement dynamics and vehicle operating experience shows that linear motion process is accompanied by essential axial vibrations of the body. When moving along the roads with slightly deformable covering and low adhesion properties the grip of caterpillars with the ground is unilateral constraint. In such conditions a lateral track shift with essential amplitude of random nature is observed, it leads to deviation of a directional angle of vehicle movement within the range of frequencies, which the driver is not able to compensate. Consequently, in order to negotiate the limited corridor the driver reduces travelling speed.The research objective is to increase safe speed of linear motion of the tracked vehicle along the roads with slightly deformable covering and limited adhesion properties.The objective in view is achieved by investigation of parametrical oscillations in relation to the vertical axis of the tracked vehicle body with the asymmetric characteristic of the power train; the oscillations are induced by asynchronous wave processes in the caterpillar drive.This work puts forward and proves the hypothesis that the most plausible reason of fluctuations is high asymmetry of linear elasticity of caterpillars with the silent blocks of the left and right sides. When the vehicle enters the turn the system asymmetry increases, as far as the length of a driving track run of the lagging side as well as its elasticity increases in 6 - 7 times. Various element stiffness of the mechanical system may cause initiation of self-oscillations and with periodically changeable stiffness they will be parametrical ones as well. Initiation of vehicle body fluctuations occurs due to wave processes in an unloaded track run.