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.

Using the assumption that the load is evenly distributed in the horizontal direction, the article has given the cable deflection equation as a function that depends only on the horizontal coordinates, the length of the cable and horizontal distance between two supports. This result leads to the construction of a general system of equations to calculate the deflection, tension, and elongation of an elastic single cable resting on two supports with or without high difference, bearing uniformly distributed loads (or evenly distributed at intervals) and load is concentrated at many points. Calculations of examples to compare with results have been performed by other methods.

A fire fighting and rescue motorcycle fitted with a fire fighting and rescue vehicle cluster on the vehicle's suspension will change the vehicle's center of gravity, affecting the vehicle's stability during travel. By modeling the kinematics in the planar motion of a two-wheeled motorcycle with 4 degrees of freedom, the paper has built a system of differential equations for the planar motion of the vehicle. Investigate the influence of the installation position of the vehicle cluster, the road surface collision, and the vehicle's speed on the loading and unloading of the vehicle in the process of moving in a straight line. The test results evaluate the reliability of the theoretical model.