Dynamics of multiple pendulum system under a translating and tilting pivot

In this article, we study the dynamics of multiple pendulum systems under translation and tilt. The main application considered for such systems is inertial sensing for high-precision instrumentation. To emulate the translating multiple pendulum system, we attach the pivot point of the pendulum to a cart that is free to move in the horizontal plane. Similarly, the pivot point of the tilting pendulum system is attached to a platform that rotates, enabling tilting motion for the system. First, we approach the problem from a Lagrangian dynamics perspective for a double-pendulum system under translation and tilt and then extend the solutions to a system of n pendulums, each hanging one below the other. Then, the natural frequencies of the systems are derived. The behavior of the systems under translation and tilt is studied and compared with that of fixed pivot point multiple pendulum systems, using eigenvalue analysis to understand how the natural frequency fluctuates with changes in degrees of freedom, mass, length and stiffness.