An oscillation discovery of the forced vibrating system predicted by the multi-time differential equation

In this article, an experiment of the forced oscillating pendulum system was set up. Here, a pendulum is driven sinusoidally by the loudspeaker. From this experiment, the spectrum of output signal of a Hall-effect sensor (UGN3503) which is used to transform the oscillatory motion of the pendulum into an electrical signal, exhibits four sharp frequency peaks, and it is also found that two of these frequency peaks cannot be described by the solution of ordinary differential equation from conventional ordinary differential equation text books. However, it can be solved by multi-time variable technique, a mathematical tool. The latter solution consists of the sum of four terms: Natural response, forced response and the two new terms being the result of multiplying between natural and forced responses. This analytical solution reveals the frequency components and their behaviors more precisely and corresponds well to the spectrum of the experimental result.   Key words: Vibration, pendulum, model, multi-time.