PSI - Issue 73

Marek Kawulok et al. / Procedia Structural Integrity 73 (2025) 51–57 Marek Kawulok et al. / Structural Integrity Procedia 00 (20 2 5) 000 – 000

54 4

this information, the size of a single pixel along both axes is determined, and the resulting displacements are converted to metres. In this way, the trajectory of the centre is obtained over time. Further data processing allows the determination of the angle of displacement of the sphere relative to its equilibrium position. In the case of free vibration, this equilibrium position is defined as the position where the sphere comes to rest after the motion has been damped. To reduce measurement noise and smooth the results, a Savitzky-Golay filter (Schafer, 2011) is applied in the final step.

Fig. 2. Example of filter application on the frame, showing the ball in a steady position.

4. Analysis of the damping behaviour One of the key parameters influencing the response of a dynamic system is motion damping. It determines the rate at which the mechanical energy released during the oscillation is dissipated, and thus the rate at which the oscillations decay. In engineering practice, two basic types are most commonly considered. The first is viscous damping, which is proportional to the velocity of motion, and the second is frictional damping, which is based on Coulomb's model of shear friction and is characterised by a constant resistive force acting against the direction of motion. For numerical simulations, accurately identifying the type and magnitude of damping is essential for a realistic response of the system. Choosing the appropriate damping model and parameters is therefore crucial. Experimental measurements were made for two different rolling track surfaces. As mentioned in the description of the experimental setup, the application of rubber tubing slightly reduced the radius of curvature of the trajectory. The initial displacement was imparted manually by simply releasing the ball without giving it any initial velocity. Damping is analysed based on the angular displacement from the ball’s equilibrium position. As a first step, the value of the logarithmic decrement of damping δ for the entire recording was determined. It is given by the following relation: = 1 + , (1) where and + represent the amplitudes of the first and the last selected cycle, respectively, and n is the number of cycles between them. In both cases, the first amplitude was taken as the initial displacement at the beginning of the motion, and the last was defined as the final amplitude greater than 0.05 °. This threshold was chosen to avoid incorrect identification of additional oscillations in the low-amplitude region near the end of the motion. The logarithmic decrement can be used to calculate the damping ratio ξ . When assuming low damping, the following relation can be applied: ≈ 2 . (2) The input values for the calculations according to (1) and (2), as well as the resulting values, are presented in Table 1.