  We study the dynamics of a three-dimensional generalization of Kitaev's
honeycomb lattice spin model (defined on the hyperhoneycomb lattice) subjected
to a harmonic driving of $J_z$, one of the three types of spin-couplings in the
Hamiltonian. Using numerical solutions supported by analytical calculations
based on a rotating wave approximation, we find that the system responds
nonmonotonically to variations in the frequency $\omega$ (while keeping the
driving amplitude $J$ fixed) and undergoes dynamical freezing, where at
specific values of $\omega$, it gets almost completely locked in the initial
state throughout the evolution. However, this freezing occurs only when a
constant bias is present in the driving, i.e., when $J_z = J'+ J\cos{\omega
t}$, with $J'\neq 0$. Consequently, the bias acts as a switch that triggers the
freezing phenomenon. Dynamical freezing has been previously observed in other
integrable systems, such as the one-dimensional transverse-field Ising model.
