  We show that low temperature specific heat (C$_p$) of the incommensurate
chain-ladder system Sr$_{14}$Cu$_{24}$O$_{41}$ is enriched by the presence of a
rather large excess contribution of non-magnetic origin. Diluted Al doping at
the Cu site or annealing the crystal in an O$_2$ atmosphere suppresses this
feature considerably. Using the THz time-domain spectroscopy, we show that the
occurrence of excess specific heat is associated with the presence of very
low-energy ($\sim$ 1 meV) gapped phonon modes that originate due to the sliding
motion of oppositely charged mutually incommensurate chain and ladder layers.
