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In the last two decades, the emphasis in aircraft maintenance has been on developing online structural health monitoring systems to replace conventional non destructive inspection techniques which require considerable down-time, human effort and cost. Vibration based damage detection is one of the most promising techniques for implementation in Structural Health Monitoring (SHM). In vibration based methods, the presence of damage is detected by monitoring changes in one of the dynamic parameters of the structure, resonant frequencies, modeshapes or damping characteristics. Compared to modeshape based methods, frequency based methods have the advantage that measurements need to be taken only at a single location. Previous developments on frequency based techniques have relied on Finite Element Model updating; analytical techniques have hitherto been restricted to beams due to the complexity in developing equations for cracked two dimensional structures. In this thesis the analytical approach using an energy formulation is extended to plates with through-thickness cracks, where modeshapes from either numerical modelling or experimental measurements can be employed to determine the energy of vibration. It is demonstrated that by using a hybrid approach, incorporating experimentally measured modeshapes along with measured changes in frequencies, the damage parameters can be estimated without resorting to theoretical modelling or numerical analysis. The inverse problem of finding the crack location, size and orientation from measured changes in frequencies is addressed using minimisation techniques. The forward problem and the inverse algorithm is first validated using numerical simulation and experimental testing of beams with edge cracks and centre cracks. The application of the methodology to the two dimensional case is then validated by numerical simulation and experimental modal analysis of plates with through thickness cracks. A statistical procedure is developed for determination of the 90/95 probability of crack detection and the minimum detectable crack size in both cases. It is demonstrated that the measurement of frequency changes can be successfully employed to detect and assess the location and size of cracks in beams and plates, using modeshapes from theory, Finite Element Analysis.