The effect of inertia on the dynamics of a solid particle (a circular cylinder, an elliptical cylinder, and an ellipsoid) suspended in shear flow is studied by solving the discrete Boltzmann equation. At small Reynolds number, when inertia is negligible, the behaviour of the particle is in good agreement with the creeping flow solution showing periodic orbits. For an elliptical cylinder or an ellipsoid, the results show that by increasing the Reynolds number, the period of rotation increases, and eventually becomes infinitely large at a critical Reynolds number, Rec. At Reynolds numbers above Rec, the particle becomes stationary in a steady-state flow. It is found that the transition from a time-periodic to a steady state is through a saddle-node bifurcation, and, consequently, the period of oscillation near this transition is proportional to [mid ]p−pc[mid ]−1/2, where p is any parameter in the flow, such as the Reynolds number or the density ratio, which leads to this transition at p = pc. This universal scaling law is presented along with the physics of the transition and the effect of the inertia and the solid-to-fluid density ratio on the dynamics. It is conjectured that this transition and the scaling law are independent of the particle shape (excluding body of revolution) or the shear profile.