  Floquet dynamical quantum phase transitions (FDQPTs) are signified by
recurrent nonanalytic behaviors of observables in time. In this work, we
introduce a quench-free and generic approach to engineer and control FDQPTs for
both pure and mixed Floquet states. By applying time-periodic modulations with
two commensurate driving frequencies to a general class of spin chain model, we
find multiple FDQPTs within each driving period. The nonanalytic cusps of
return probability form sublattice structures in time domain. Notably, the
number and time-locations of these cusps can be flexibly controlled by tuning
the Hamiltonian parameter and the higher frequency of the drive. We further
employ the dynamical topological order parameter (DTOP), which shows a
quantized jump whenever a DQPT happens, to identify the topological feature of
FDQPTs. Our findings reveal the advantage of engineering nonequilibrium phase
transitions with multi-frequency driving fields.
