  The variational static formulation contributed in [International Journal of
Engineering Science 143, 73-91 (2019)] is generalized in the present paper to
model axial and flexural dynamic behaviors of elastic nano-beams by nonlocal
strain and stress gradient approaches. Appropriate forms of non-standard
boundary conditions are detected and properly prescribed. Equivalence between
differential laws and integral convolutions is elucidated and invoked to
analytically evaluate size-dependent axial and flexural fundamental frequencies
of cantilever and fully-clamped beams which significantly characterize
new-generation nano-actuators. The proposed methodology and ensuing results are
tested by pertinent outcomes in literature. Advantageously, in comparison with
available nonlocal gradient models, the developed formulation of elasticity
leads to well-posed dynamic structural problems of nano-mechanics. Outcomes
obtained by the well-established strain-driven and stress-driven nonlocal and
gradient theories of Engineering Science are recovered under special ad hoc
assumptions. The present study offers a simple and effective strategy to
predict peculiar stiffening and softening dynamic responses of nano-scaled
components of advanced technological devices, such as
Nano-Electro-Mechanical-Systems (NEMS), and modern composite nano-structures.
