Let $M$ be one of the projective spaces $\mathbb{C} \mathrm{P}^n$, $\mathbb{H} \mathrm{P}^n$ for $n\geq 2$ or the Cayley projective plane $\mathbb{O} \mathrm{P}^2$, and let $\Lambda M$ denote the free loop space on $M$. Using Morse theory methods, we prove that the suspension spectrum of $(\Lambda M )_+$ is homotopy equivalent to the suspension spectrum of $M_+$ wedge a family of Thom spaces of explicit vector bundles over the tangent sphere bundle of $M$.