ESDU 08015:2009

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.