  The observed behaviour of passive objects in simple flows can be surprisingly
intricate, and is complicated further by object activity. Inspired by the
motility of bacterial swimmers, in this two-part study we examine the
three-dimensional motion of rigid active particles in shear Stokes flow,
focusing on bodies that induce rapid rotation as part of their activity. Here,
in Part 1, we develop a multiscale framework to investigate these emergent
dynamics and apply it to simple spheroidal objects. In Part 2
(arXiv:2301.11032), we apply our framework to understand the emergent dynamics
of more complex shapes; helicoidal objects with chirality. Via a
multiple-scales asymptotic analysis for nonlinear systems, we systematically
derive emergent equations of motion for long-term trajectories that explicitly
account for the strong (leading-order) effects of fast spinning. Supported by
numerical examples, we constructively link these effective dynamics to the
well-known Jeffery's orbits for passive spheroids, deriving an explicit
closed-form expression for the effective shape of the active particle,
broadening the scope of Jeffery's seminal study to spinning spheroids.
