Dynamics of n Coupled Double Pendula Suspended to the Moving Beam

Abstract

We consider the synchronization of n self-excited double pendula. For such pendula hanging on the same beam, different synchronous configurations can be obtained (in-phase and anti-phase states). An approximate analytical analysis allows to derive the synchronization condition and explain the observed types of synchronization for any number of coupled double pendula. The energy balance method is used to show how the energy between the pendula is transferred via the oscillating beam allowing their synchronization. We compute periodic solutions for n = 2, 3, 4, 5 coupled double pendula, based on analytical predictions. For all obtained periodic solutions, we investigate how the stability properties change with the varying natural frequency of the beam.

Synchronization and Stability of Two Unbalanced Rotors with Fast Antirotation considering Energy Balance

“…Substituting (16)∼ (19) into (15), the energy balance of the oscillating body in -direction can be written as follows:…”

“…By the Poincare method and the small parameter method, Jovanovic and Koshkin have studied synchronization and stability of Huygens' clocks [12]. Recently, Koluda and Perlikowski derived the synchronization conditions and explained the energy transmission between double pendula via an oscillating beam with energy balance method [13][14][15]. Kapitaniak et al explored synchronous states of slowly rotating pendula considering the oscillating beam moving in a single DOF [16,17].…”

Synchronization of Two Homodromy Rotors Installed on a Double Vibro-Body in a Coupling Vibration System

“…In the vibration system, the value of parameter is far larger than because the value of the damping ratio ( ξ ni < 0.05)is very small [ 14 ]. In the following calculation, we will ignore parameter to simplify H 0 , H 1 , H 2 and H 3 as …”

“…Jovanovic studied two models of connected pendulum clocks synchronizing their oscillations, a phenomenon originally observed by Huygens, with the Poincare´ method, and they found that the in-phase linear mode damps out faster than the anti-phase mode [ 13 ]. Koluda considered two and multiple self-excited double pendula hanging from a horizontal beam with the energy balanced method, on which they found how the energy is transferred between the pendula via the oscillating beam allowing the pendula’ synchronization[ 14 – 16 ]. For synchronization of multiple coupling rotors, Wen employed the average method to study synchronization and stability of multiple unbalanced rotors hung on a vibro-body in vibration systems, and applied such synchronization theory to invent many synchronization machines [ 17 ].…”

Synchronization and Stability of Elasticity Coupling Two Homodromy Rotors in a Vibration System

“…For the dynamics of coupled pendulums and rotors, Dutch scholar Huygens firstly reported the synchronization phenomenon in 1665, two pendulum clocks hanging the common base, and the clocks exhibit synchronized motion in a short while [4]. Subsequently, Koluda et al described the phenomenon of synchronization of clocks hanging on a common movable beam with the energy balance method and discovered how the energy is transferred between the pendula via the oscillating beam [5][6][7]. For the synchronization of mechanical rotors, Blekhman, the scholar of the former Soviet Union, proposed the Poincare method for the synchronization characteristics and synchronization theory of the vibrating machines [1].…”