A continuously scanning laser Doppler vibrometer (CSLDV) system is capable of rapidly obtaining spatially dense operating deflection shapes (ODSs) by continuously sweeping a laser spot from the system over a structure surface. This paper presents a new damage identification methodology for beams that uses their ODSs under sinusoidal excitation obtained by a CSLDV system, where baseline information of associated undamaged beams is not needed. A curvature damage index (CDI) is proposed to identify damage near a region with high values of the CDI at an excitation frequency. The CDI uses the difference between curvatures of ODSs (CODSs) associated with ODSs that are obtained by two different CSLDV measurement methods, i.e., demodulation and polynomial methods; the former provides rapid and spatially dense ODSs of beams, and the latter provides ODSs that can be considered as those of associated undamaged beams. Phase variables are introduced to the two methods for damage identification purposes. Effects of the order in the polynomial method on qualities of ODSs and CODSs are investigated. A convergence index and a criterion are proposed to determine a proper order in the polynomial method. Effects of scan and sampling frequencies of a CSLDV system on qualities of ODSs and CODSs from the two measurement methods are investigated. The proposed damage identification methodology was experimentally validated on a beam with damage in the form of machined thickness reduction. The damage and its region were successfully identified in neighborhoods of prominent peaks of CDIs at different excitation frequencies.