Power series expansions are useful in approximation theory and mathematical physics. The manuscript presents several types of fractional Taylor expansions of sufficiently smooth functions. This is achieved by employing an incremental regularization procedure to the computation of the derivative. The series are constructed in an algorithmic way, which allows for its implementation in computer algebra systems.

The paper presents the basic structural parts, a 3D model and the overall design of a laboratory machine, which was created to detect vibrations generated by the casing of a conveyor roller ro-tating at different speeds. The intention of the authors was to verify whether plastic brackets in-serted into the structurally modified trestles of a fixed conveyor idler can reduce the vibration values transmitted from the rotating conveyor roller to the trestle of a fixed idler. Experimental vibration measurements taken on the non-rotating parts of conveyor rollers, performed on a laboratory machine according to ISO 10816, are suitable for characterizing their operating condi-tions with regard to trouble-free operation. The aim of this paper is to detect the vibrations of a rotating conveyor roller on a laboratory machine in the defined places of a fixed conveyor idler and also on the steel frame of a laboratory machine that represents the supporting track of a belt conveyor. Vibrations detected by piezoelectric acceleration sensors were recorded by a measur-ing apparatus and displayed in the environment of Dewesoft X software. The measurements show that the vibration values grow with the increasing speed of the conveyor roller rotation. Experimental measurements have proven the correctness of the assumption that the vibrations transmitted to the trestle of a fixed conveyor idler are lower by up to 40% when using plastic brackets into which the axle of the conveyor roller is attached, compared to the solution where the axle of the conveyor roller is inserted into the notches of a steel trestle.