Coupled Resonators Have Normal Modes

Out-of-Phase Mode Uses the Coupling Spring

Opposite displacements include the coupling stiffness twice. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Out of phase stretches the coupling spring

The sign pattern is plus minus, so the coupling spring contributes twice in the checked ledger.

signs=+−\mathrm{signs}=+-
Out-of-phase modeThe checked sign pattern is plus minus.mode=out-of-phasem=2 kgk_wall=2 N/mk_couple=1 N/msigns=+,-omega2=2 1/s^2

Wall stiffness plus twice the coupling stiffness sets the value

The numerator is 2 plus two times 1, and dividing by mass 2 gives frequency squared 2.

ω−2=2+2⋅12=2\omega_-^{2}={2+2\cdot 1\over 2}=2
Out-of-phase frequency ledgerThe frequency-squared value is exact.mode=out-of-phasem=2 kgk_wall=2 N/mk_couple=1 N/msigns=+,-omega2=2 1/s^2