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.