This paper presents a new adaptive isogeometric method for structural vibration. Based on the newly introduced Geometry-Independent Field approximaTion (GIFT), generalized from Iso-Geometric Analysis (IGA), we exactly describe the geometry of the structure with NURBS (Non-Uniform Rational B-Splines), and independently employ PHT-splines (Polynomial splines over Hierarchical T-meshes) to achieve local refinement in the solution field. To deal with error estimation, we improve the MAC (Modal Assurance Criterion) method to locate unique, as well as multiple, modal correspondence between different meshes. Local adaptivity is carried out by sweeping modes from low to high frequency. Numerical examples show that a proper choice of the spline space in solution field (with GIFT) can deliver better accuracy than using NURBS solution field. In addition, for vibration of Reissner-Mindlin plates, our proposed method indicates that the adaptive local $ h-$refinement achieves a better solution accuracy than the uniform $h$-refinement.