Natural frequencies of multiple pendulum systems under free condition

In this classical article, we study natural frequencies of the multiple pendulum systems (MPSs) in a plane under the free condition. The systems of governing differential equations for the MPSs such as triple pendulum (TP) and double pendulum (DP) are derived using the Euler–Lagrangian equation of second kind to validate the Braun’s generalized expressions (Arch Appl Mech 72:899–910, 2003) for natural frequencies of multiple pendulum systems. The governing equations of the TP and DP systems are also derived in terms of angular momentum and angular displacement to confirm the basic results obtained using the aforementioned approach. The eigenvalue analysis of the pendulum systems ranging from single pendulum to quintuple indicates that natural frequency increases with degree of freedom for equal mass and length of each pendulum in a MPS. The results show that the natural frequency of a distributed pendulum system is larger than the corresponding to the point mass pendulum system. Moreover, the natural frequency of the bottom pendulum is the most sensitive to change in length or mass of either pendulum of a MPS. However, unlike mass-dependent natural frequency, the natural frequency of all pendulums of a multiple pendulum always decreases with increasing length of a pendulum in MPS. These results are, in turn, validated with Braun’s formula for natural frequency of a MPS.