Do coupled nested pendula have the same eigenfrequencies as pendula in cascade?

Over the last few years, several vibration isolation projects for gravitational wave detectors have been proposed. Some of them are related to N-stages cascaded systems which can achieve great factors of attenuation. However, these systems are usually very tall, which makes it difficult to build an efficient vaccum chamber for them. For this reason, this paper makes a comparison between a theoretical N-stage cascaded pendula with an experimental N-stage nested one. Preliminary results show that their resonant modes are very similar. The ratio between the results of experiment and theory is between 0.94 and 1.01 for the pendular modes and systematically close to 1.05 for the rotational modes. This result implies that an N-stage nested pendula set of about 1.4 m height can have resonant modes similar to a cascaded pendula set of about 6.6 m, which may suggests that a nested system can be theoretically treated as a cascaded one. So, the development of nested pendula can be an effective alternative of vibration isolation systems for future generations of gravitational wave detectors.