INTRODUCTION The dynamic behaviour of cylindrical helical springs@ comprising both tension/compression and torsion springs@ is extremely difficult to calculate since its geometrical shape is a curve in three-dimensional space. To make the calculations manageable@ simple but representative mathematical models are required. The simplest of such models is the straight elastic rod@ the so called ??equivalent rod' which clearly must have the same elastic properties as the helical spring it represents. It is rather surprising@ but fortunate@ that the use of this very simple mathematical model should yield such reasonable results@ certainly accurate enough for most practical purposes An earlier Data Item No. 06024[2] defined the assumptions and limitations that apply to the calculation procedure for estimating the dynamic characteristics of springs@ together with the prescribed loading conditions assumed to apply to the spring. The Item also provided derivation of the deformation@ stresses and transverse loading on the spring and the form design of the spring ends which will affect the loading characteristics. The elastic stability of compression and torsion springs are discussed and formulae given for ensuring stability. The present Item extends the scope of the earlier Item@ presenting the vibration characteristics of cylindrical helical springs. Section 3 discusses the axial vibration of compression/tension helical springs on the basis of the ??equivalent rod' approximation@ dealing with both free and forced axial vibration. For free vibration@ cases when both ends of the rod are free@ one end of the rod is clamped and the other end is free and both ends of the rod are clamped@ are considered. For forced vibration@ the case when one end of the spring is forced to follow a cyclic motion and the stresses induced by the cyclic motion is discussed . Section 4@ considers the free and forced vibrations of a spring-mass system in reasonable detail@ dealing with the cases when the system mass is large compared to the mass of the spring and of comparable size. The influence of various kinds of damping@ Coulomb and viscous friction@ material hysteresis@ etc. are also discussed. In conjunction with forced vibration@ the resonance phenomenon is dealt with in a number of sections. Although it is an important design principle to avoid resonance whenever possible@ in high speed applications it is sometimes inevitable that the elastic system during its normal operation must pass through the resonance domain. In such cases the only practical possibility is to try to avoid sustained resonance. Recognising the engineering importance of this problem a separate section is devoted to the discussion of the transition through resonance. A further Data Item in this series on springs@ No. 09003[3]@ considers the dynamic characteristics of cylindrical helical springs due to impact loading@ which is an integral part of the normal operation of the majority of machines that execute rapid alternating motion. The Item also provides worked examples that estimate spring dynamic performance.