This paper presents the Fourier and Wavelet characterization of vibration problem. To determine the natural frequencies, modal damping and mass participation factors of bars, a rod element is established by means of isogeometric formulation. The non-uniform rational Bezier splines (NURBS) is presented to characterize the geometry and the deformation field in isogeometric Analysis (IGA). Non-proportional damping has been used to measure the real-state energy dissipation in vibration. Therefore, the stiffness, damping and mass matrices are derived by the NURBS basis functions. The efficiency and accuracy of the present isogeometric Analysis is demonstrated by using classical finite element method (FEM) models and closed-form analytical solutions. The frequency content, modal excitation energy and damping are measured as basis values. Results show that the present isogeometric formulation can determine the modal frequencies and inherent damping in an accurate way. Damping as an inherent characteristics of viscoelastic materials is treated in a realistic way in IGA method using non-proportional form. Based on results, k-refinement technique has enhanced the accuracy convergence with respect to other refinement methods. In addition, the half-power bandwidth method gives modal damping for the IGA solution with appropriate accuracy with respect to FEM. Accuracy difference between quadratic and cubic NURBS is significant in IGA h-refinement.