Identify $\mathbb R^2$ with $\mathbb C$ and consider the $S^1$ action on $\mathbb R^{2n} \simeq \mathbb C^n$ induced by cordinatewise complex multiplication. These of course lead to the trivial examples in on $G_{n,1}$. G_{2n,1}$. For$n$even and$r$odd the very same examples do the trick. One has just to observe that these$S^1$actions have no invariant odd dimensional subspaces, and therefore induce$r$-dimensional subspace.S^1$-actions without fixed points on $G_{n,r}$.
Identify $\mathbb R^2$ with $\mathbb C$ and consider the $S^1$ action on $\mathbb R^{2n} \simeq \mathbb C^n$ induced by cordinatewise complex multiplication. These of course lead to the trivial examples in $G_{n,1}$. For $n$ even and $r$ odd the very same examples do the trick. One has just to observe that these $S^1$ actions have no invariant $r$-dimensional subspace.