In this work we study the existence and uniqueness of solutions and decay rates to the total energy and the L2-norm of solution for a semilinear second order evolution equation with fractional damping term and under effects of a generalized rotational inertia term in the case of plate equation. This system also includes equations of Boussinesq type that model hydrodynamic problems. We show decay rates depend- ing on the fractional powers of the operators and using an asymptotic expansion of the solution to the linear problem, we prove for some cases depending on the exponents of the operators, the optimality of the decay rates.