Shaft misalignment, rotor unbalance, rubbing, bearing faults and cracks are the commonly observed faults in the rotor-bearing systems. Vibration signature has been considered to be one of the best ways to detect and diagnosis subsequent condition monitoring of rotor-bearing faults. A flexible shaft with a rigid disc supported by flexible bearings at both the ends has been considered for the present critical observations. The bearing is considered as a spring with linear stiffness and damping. Further, Timoshenko beam theory is considered to describe mathematically the model of rotor element. We used finite element analysis to obtain the critical whirling speeds of rotor-bearing system for different modes at various system parameters with the illustration of Campbell diagram and Fourier spectrum. Vibration spectrums in the transverse direction are illustrated for different modes of vibration using successive approaches of Hamilton principle, Galerkin's method and Newmark's integral method. Condition monitoring of vibration spectrums considering rotor unbalance and shaft misalignment as well as their identification has been analysed with FFT, time responses, phase portrait and Poincare's map, numerically and experimentally.