Phonon trapping has an immense impact in many areas of science and technology, from the antennas of interferometric gravitational wave detectors to chip-scale quantum micro- and nano-mechanical oscillators. It usually relies on the mechanical suspension--an approach, while isolating selected vibrational modes, leads to serious drawbacks for interrogation of the trapped phonons, including limited heat capacity and excess noises via measurements. To circumvent these constraints, we realize a new paradigm of phonon trapping using mechanical bound states in the continuum (BICs) with topological features and conducted an in-depth characterization of the mechanical losses both at room and cryogenic temperatures. Our findings of mechanical BICs combining the microwave frequency and macroscopic size unveil a unique platform for realizing mechanical oscillators in both classical and quantum regimes. The paradigm of mechanical BICs might lead to unprecedented sensing modalities for applications such as rare-event searches and the exploration of the foundations of quantum mechanics in unreached parameter spaces.
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