Modal localisation in a beam modelled as a continuous system: A discussion on the use of auxiliary oscillators

In this work, we show that perfectly localised modal vibrations can be obtained in continuous linear elastic structures, provided that, in addition to the regular boundary and continuity/jump conditions, extra (admissibility) conditions are met. They are derived with the aid of: (i) new parameters and/or DOFs of auxiliary systems, and/or (ii) admissible relationships amongst the original structural parameters. In both cases, however, the solution resorts to a fine-tuning problem. To illustrate the main ideas, we discuss three examples: two simply-supported beams with cantilever extensions and attached lumped oscillators, and a “tuning-fork-like” structure. Finally, to show that there are also systems in which fine-tuning is not possible, we discuss the case of two lumped oscillators attached to the free end of a cantilever beam.