New Characterization and Classification of Synchronization of Multiple Metronomes on a Cart via Describing Function Method

Analysis of synchronization of multiple identical metronomes on a cart via describing function approach

“…At first, we here prove the exponential stability of W(s) in Equation ( 29) with its expression in Equation (32) for any n by showing the exponential stability of W − (s) in Equation (30) and W + (s) in Equation (31). Lemma 2.…”

“…Remark 2. From Equation (32), we obtain 30) and W + (s) in Equation ( 31) are low-pass filters, since their relative degrees are two and…”

“…Theorem 1. For the system depicted in Figure 5 with W(s) and N(a) (a > 0) being defined in Equation ( 29) (with its expression in Equation (32)) and ( 12), respectively, the following two types of synchronization are possible in the system shown in Figure 5: 1) The out-of-phase synchronization satisfies…”

“…By introducing a special matrix 𝚷 n captures the essential system structure of our target system, we can express W(s) in a simpler way in comparison with that given in our conference paper [30]. All these enable us to obtain a direct proof for the stability of W(s) in Equation (32), to clarify the possibility of existing two different types of synchronization, and to validate Assumption 2 theoretically and numerically.…”

“…Fifth, we investigate numerically the synchronization of non-identical multiple metronomes to assess the robustness regarding small parameter differences of the synchronization prediction, see Section 4.4. Finally, the content of the discussion in Section 4.5 has been completely rewritten including our discussion on our related conference papers [32] and [31].…”