Axially moving structures are of technological importance and present in a wide range of engineering problems. Typical examples are the chain and belt drives, aerial cable tramways, rolling systems, thread-lines in the textile industry, band saw blades and the like. The axially moving speed of a structure can significantly affect its dynamic characteristics, giving rise to variations in natural frequencies and vibration modes. Above a certain critical moving speed, the axially moving structure may experience severe vibration or structural instability resulting in structural failure. Thus, it is important to accurately predict the dynamic characteristics of such axially moving structures to achieve successful designs. Though conventional finite element (FE) method is versatile and widely used to model complex structures of arbitrary geometry for structural dynamics problem, it is highly unsuited for wave propagation analysis. Higher frequency content of the loading wave propagation problems requires very ?ne mesh with the element size comparable to the wavelengths, which are very small at higher frequencies. This results in large system size and huge computational cost. In addition to the ?ne mesh, to obtain system response, the time integration schemes have to be implemented after FE modeling. In these methods, analysis is performed over a small time step, which is a fraction of the total time for which the response histories are required. For some time integration schemes, however, a constraint is placed on the time step, and this coupled with large system sizes makes the FE solution of wave propagation problems computationally prohibitive. Thus, in general, alternative numerical techniques, like wavelet-based spectral finite element (WSFE) and conventional FFT based spectral finite element (SFE) models, are adopted for these problems. The WSFE is similar to SFE, except that, here Daubechies wavelet basis is used for approximation in time to reduce the governing partial differential equations (PDEs) to a set of ordinary differential equations (ODEs). The localized nature of the compactly supported Daubechies wavelet basis helps to circumvent several problems associated with SFE due to the required assumption of periodicity, particularly for time domain analysis. In this thesis, a WSFE model is developed for some 1D structures, especially for an axially moving Timoshenko beam subjected to axial tension. This study is done to use the formulated WSFE to perform both frequency and time domain analysis. The high accuracy of the WSFE model is then verified by comparing its results with the FE results and conventional SFE model results. The effects of the moving speed and axial tension on the vibration characteristics, wave characteristics (natural frequencies and wave number), and the static and dynamic stabilities of a moving beam are investigated. Keywords : Axially moving structures, Frequency and time domain analysis, Wavelet-based spectral finite element model, Daubechies wavelet basis function, and Dynamic stabilities