The wooden trolley (in the sawing line using a vertical bandsaw) during work generally moves by iron wheels on two rails placed on the concrete floor. Due to the effect of noticeable cutting force applied by the saw blade and the vehicle's uneven weight on the wheels, it causes the vehicle to vibrate during the horizontal movement, affecting the stability of the wood sawing circuit. By modeling the problem of vibration of beams located on Winkler elastic foundations, subjected to mobile concentrated loads, the article has built a system of differential equations of vibration of two rails, giving an expression to calculate the vibration amplitude of the swan plane. The influence of the flexural rigidity EI of the rails and the elastic stiffness k of foundation on the vibration amplitude of the cutting plane is studied and analyzed. The use of the Dirac Delta function and the employed analytical solution allow the designers to evaluate the accuracy and reliability of the calculated results.