The present paper is the first to classify the phenomenon of trapped mode initiation in an elastic system with mass-spring inclusions. Necessary and sufficient conditions for low and high frequency trapped modes were found. It is shown then that in elastic systems which are bounded by a continuous spectrum shifted from 0 and have mass inclusions, a discrete spectrum of natural frequencies occurs before the first boundary frequency. Trapped modes of oscillation corresponding to those frequencies possess a particularly pronounced local character in the region of inclusion. The discrete spectrum shift to the continuous spectrum axis and initiation of high frequency trapping modes are accompanied by a phenomenon of travelling waves damping (antiresonance).

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