In the frame structure of stacker cranes during non-stationary phases of movement due to inertial forces undesirable mast vibrations may occur. This effect can reduce the stability and positioning accuracy of these machines. The aim of this paper is to introduce an accurate and quite simple dynamical model of single-mast stacker cranes, which is suitable for investigating the mast vibrations of these machines. The multi-body modelling approach is selected to generate the differential equations of motion for this model. The solution of these equations is performed by means of the so-called modal coordinate transformation or modal superposition method. In this model structural damping is taken into consideration by means of the so-called proportional damping (Rayleigh damping) approach. The main advantage of the presented multi-body model is that with this model the mast-vibrations can be investigated in various positions of the mast. Dynamic models with varying lifted load positions can also be generated in simple way by using the introduced modelling technique. The main properties, i.e., the state space representation of our model as well as time domain simulation results, are also introduced.