A 3-N degree of freedom system with two natural frequencies (Q1116724)

The study of linear N degree of freedom vibrating mechanical systems sometimes leaves one with the impression that there are usually N natural frequencies corresponding to the N freedoms, and that the system is linear only for small deformations. Some authors have pointed out that if the characteristic equation has multiple roots, then there are less than N distinct frequencies. The purpose of the present work is to investigate a system with 3-N degrees of freedom which has only two natural frequencies regardless of the value of N and which remains linear for very large deformations. Previously a vibrating system with N degrees of freedom was analyzed. With the appropriate choices of the masses and spring constants the system oscillated with a single natural frequency. Further a system with two degrees of freedom was considered. It was shown that the system had a single natural frequency even when the particle was connected to N arbitrary springs. The effective lengths of the springs were zero, and the deformations were arbitrarily large.