Establishing dynamic characteristics of structures is a challenging area of research. The dynamic characteristics of structures, such as natural frequencies, modeshapes, response levels and damping characteristics play an important role in identifying the condition of the structures. The assumed modes method is a particular analytical method used to estimate the dynamic characteristics of a structure. However, the eigenfunctions used in the assumed mode method often led to ill-conditioning due to the presence of hyperbolic functions. Furthermore, a change in the boundary conditions of the system usually necessitates a change in the choice of assumed mode. In this thesis, a set of Alternative Admissible Functions (AAF), along with penalty functions, are used to obtain closed form solutions for an Euler-Bernoulli beam with various boundary conditions. A key advantage of the proposed approach is that the choice of AAF does not depend on the boundary conditions since the boundary conditions are modelled via penalty functions. The mathematical formulation is validated with different boundary conditions, Clamped-Free (CF), Simply-Supported (SS), and Clamped-Clamped (CC). A specific relation between the penalty function and the system parameters are established for CF, SS and CC boundary conditions to obtain appropriate values of penalties. Validation of results with the reported literature indicates excellent agreement when compared with closed-form Euler-Bernoulli beam values. The AAF approach with penalties is extended to a beam with a shallow crack to estimate the dynamic characteristics. The crack is modelled as a penalty function via a massless rotational spring. This model has the advantage of simplifying parametric studies, because of its discrete nature, allowing easy modification in the crack position and depth of the crack. Therefore, once the model is established, various practical applications may be performed without reformulation of the problem. Validation of results with the reported literature on beams with shallow cracks indicates the suitability of the proposed approach.