Modelling and experimental validation of 1-degree-of-freedom impacting oscillator

Abstract

In this article, a mechanical 1-degree-of-freedom oscillator with harmonic forcing and impacts was analysed both numerically and experimentally. A special attention was paid to the mathematical modelling and realistic numerical simulations of a real system. The developed experimental rig consists of a cart mounted on a guide, connected with springs to the support, and equipped with one-sided stiff limiter of motion. The cart was excited by an unbalanced disc mounted on the shaft of a stepper motor. Modelling was focused on the mathematical description of the impact process, where soft obstacle with Hertzian stiffness was assumed and different forms of non-linear damping were tested, including original modifications of the already used models. Model parameters are identified based on two experimental solutions corresponding to free vibrations – with and without impacts. Then, the model was validated by means of experimental and numerical analysis of bifurcation dynamics of the forced system.

Properties of impact events in the model of forced impacting oscillator: Experimental and numerical investigations

“…For studies let us chose the case at ω = 40.86 rad/s (period 0.154). Recall [20] that the trajectory after some transition time leads to the chaotic attractor, the phase portrait of which is depicted in Fig. 6a.…”

“…The base of our studies is the dynamical regimes observed at the experimental rig comprising an impacting oscillator. Since the detailed discussion of this rig can be found in the paper [20], we give the short description only. The experimental stand corresponding to the physical model presented in Fig.…”

“…It is connected with the support by the use of linear spring of total stiffness coefficient k. The position x = 0 corresponds to the resultant force in the springs equal to zero and a gap x I between the two bumpers on the right side of the cart. Angular position of the disk mounted on the cart Figure 1: The schematic representation of the experimental stand [20].…”

“…In the work [20] the following set of parameters was estimated or assumed a priori leading to good agreement between numerical simulations and experimental investigations: m = 8.735 kg, m 0 e = 0.01805 kg·m, k = 1418.9 N/m, c = 6.6511 N·s/m, T = 0.63133 N, ε = 10 −6 m/s, k I = 2.3983 · 10 8 N/ m 3/2 , b = 0.8485 m −n 3 s n 3 , n 1 = n 2 = 3/2, n 3 = 0.18667. In the paper [20], the model (1) specified by this set of parameters is called the model BC and provides the optimal one from a group of models tested in the work [20]. The comparison of experimental and numerical studies is presented in Fig.…”

“…Here it should be mentioned that the paper [20] contains the description for experimental investigations of the oscillating system with impacts, construction and validation of different mathematical models, and bifurcation diagrams exhibiting the general properties of observed oscillating regimes.…”