In isoparametric foliations in the unit sphere, the OT-FKM type plays an important role. The focal submanifold M+ of OT-FKM type is a sphere bundle over sphere, including the Stiefel manifolds as special cases. We construct the characteristic maps corresponding to vector bundle of M+ for m=4 and 8, generalizing the results of Steenrod and James. As an application, we establish new harmonic representatives of elements in certain homotopy groups of spheres. Moreover, inspired by the OT-FKM construction, we give partial answers to two questions of James. This talk is based on joint work with Professor Zizhou Tang and Professor Chao Qian.