Locally Resonant Metamaterial: Vibration Band Gap

Drive

Pass band — propagating

Drive frequency 0.50 Hz

Unit cell

Resonator mass ratio (m_r/m) 0.30

Resonator frequency f₀ 1.20 Hz

Chain stiffness k 50

Live readout

Gap: f_low–f_high

1.20–1.42 Hz

Transmission

100%

Effective mass m_eff

+1.00

Sim time

0.0 s

How it works

Each unit cell is a "mass-in-mass": a host mass m coupled to its neighbours by springs of stiffness k, with a small resonator mass m_r tucked inside it on its own spring k_r. This is the canonical model of a locally-resonant elastic/acoustic metamaterial (Liu et al. 2000; Huang & Sun 2009) — the same trick used in real vibration-damping panels and seismic metamaterials.

m ü1_n = k(u1_{n+1} − 2u1_n + u1_{n−1}) + k_r(u2_n − u1_n)
m_r ü2_n = −k_r(u2_n − u1_n)

resonator frequency:  ω0 = √(k_r / m_r)
effective mass:        m_eff(ω) = m + m_r·ω0² / (ω0² − ω²)

Near ω0 the resonator moves out of phase with the host, and the unit cell's effective mass m_eff(ω) turns negative. A negative effective mass cannot satisfy Newton's second law for a propagating wave, so no real wavenumber solves the dispersion relation — the wave becomes evanescent and decays exponentially along the chain instead of propagating. This forbidden window is the band gap, bounded by:

f_low  = f0
f_high = f0 · √(1 + m_r/m)

Drive frequency — sets the frequency of the sinusoidal displacement fed into the left end of the chain.
Resonator mass ratio / frequency — reshape the band gap: a heavier or softer resonator opens a wider, lower gap.
Chain stiffness — sets the host-to-host coupling, which controls how fast waves travel in the pass bands.
Transmission — the ratio of vibration amplitude near the far (undriven) end to the near (driven) end, after the response settles. Inside the gap it collapses toward zero; in a pass band it stays near 100%.

Host mass Internal resonator

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