Dynamic characteristics of the curved track are very complex due to the in-plane and out-of-plane vibrations. Modelling the curved track in theoretical method is important to understand its physical properties. The curved track with a certain length can be regarded as one part of a circular structure with the cyclic length of fastener spacing. Dynamic response of the curved track can be solved within one basic cell of track in the frequency domain according to the properties of the periodic structure. A periodic discretely supported curved Euler beam is used to simulate the curved track. Mathematical modes of rail in the frequency domain are proposed and dynamic response of the curved track in 3 dimensions are expressed by series superposition of the mathematical modes. It is shown that the accurate frequency response of the curved track can be obtained by the curved track model proposed in this paper. Several conclusions can be obtained according to the model. The influence range of a single moving axle on the vertical, lateral and torsional vibrations of the curved track is about 5m on both sides of the load point. The velocity has a significant effect on lateral and torsional vibrations of the curved track. The directions of the lateral and torsional vibrations will change when the velocity increases. As the speed increases, the curved track transfers from the surplus superelevation state to the inadequate superelevation state gradually and the response amplitude decreases first and then increases. The radius, elevation of curve and velocity have great influence on the vertical, lateral and torsional vibrations of the curved track. As the radius increases, the effect of velocity on dynamic response is reduced. The dynamic response of the vertical, lateral and torsional vibrations of the curved track under a sequence of moving axles in the frequency domain is mainly within 0~40Hz. The spectrum distribution of horizontal, torsional vibrations is wider than vertical vibrations.