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.

The first Data Item in this series on springs, No. 06024, defines 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 provides 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 is discussed and formulae given for ensuring stability.

The second Data Item, No. 08015, extends the scope of the earlier Item, presenting the vibration characteristics of cylindrical helical springs. The Item 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 also discussed.

ESDU 08015 further considers free and forced vibrations of a spring mass system, dealing with the two cases when the system mass is large compared to the mass of the spring and when it is of comparable size. The influence of various kinds of damping, Coulomb and viscous friction, material hysteresis, etc. is 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.

The present Item extends the scope of the earlier Items to impact loading.

In the majority of machines, particularly those which execute alternating motion, impacts occur during their normal operation. Often displacement impacts are small, such as clearances in bearings or joints, in other cases the impacts are an integral part of the normal functioning of the machines or mechanisms, for example in vehicle suspensions systems, valves of internal combustion engines, forging hammers, firearms, etc. In these latter cases springs are normally employed to absorb or store the energy of the impact. When designing machine components for impact loading, the stresses and deformations in the components must be considered. It is also necessary to find out what effects these stresses and deformations have on the materials involved.

The classical theory of impact regards the bodies involved as rigid and the impact as being instantaneous and so it is suitable only for determining the kinetic consequences of the impact. When the impact process itself is to be investigated, i.e. its duration and the deformations, forces and stresses involved, much more advanced theories must be used. However, when the dynamic behaviour of only a spring mass system after impact is the subject of the investigation, and not that of the spring itself, the classical theory of impact proves to be very useful.

As far as the impact process is concerned, there are two extreme idealised cases, the perfectly non elastic impact and the perfectly elastic impact. The reality lies somewhere in between these two extremes. Accordingly, in practical calculations a so-called "impact coefficient", designated k, is introduced which has the values in the range 0 lessthan or equal to k lessthan or equal to 1. The value of must be determined by experiment. In the case of a perfectly non elastic impact its value is zero, i.e. k = 0 , and in the case of a perfectly elastic impact, k = 1.

In the first part of this Item, the impact on spring mass systems is investigated using the classical theory of impact. In the latter part of the Item, the problem of an impact on an elastic rod is considered. This latter problem has a special practical significance, since an elastic rod can be used as an "equivalent rod" representing a helical spring.