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The
Dynamics of Reciprocating Compressor Valve Springs...continued
THEORY
Approximate Method
The approximate method for calculating spring dynamics
considers only torsion of the wire and ignores the effects of
closed end coils. It does take coil-to-coil contact into
consideration and can allow the end coils to leave their stops.
With the assumption that only torsional deflection of the wire
is important, the basic equation (Wahl 25-7) is:
where y = deflection of wire in direction of spring axis
t = time
a = speed of torsion wave in wire
s = distance along wire
b = damping factor (twice Wahl’s definition)
From Wahl (5-18)
where d = wire diameter
D = mean coil diameter
G = torsional modulus of wire
= density of wire material
The acceleration due to gravity is omitted on the assumption
that consistent units are used.
The basic equation can be written as:
Now:
(1) + a (2) gives:
That is:
where  = velocity of a point on the wire.
and  = angular deflection of a point on the wire.
This is directly related to the stress in the wire.
Similarly, subtracting a times equation 2 from equation 1
gives:
As the wave speed is constant, equations 3 and 4 can be solved
numerically using a rectangular grid. If the mesh dimensions are
such that the distance dimension ( s) is the wave speed (a) times
the time dimension (t), no interpolation will be required. The
values at position s and time t are obtained using equations 3 and
4 from those at time t - t and positions s -
s and s + s.
At the s = 0 boundary condition, u + a is known from equation
4 and at the other end u - a is known. The boundary conditions,
e.g. a known velocity u or a free end ( = 0), allow
and u to be calculated.
The stress can be calculated from the angular deflection using
Wahl 19-5 and 19-14. The maximum stress in the wire for a
statically deflected spring is:
where K = Wahl’s stress correction factor.
Several expressions resulting from different assumptions are given
by
Wahl. The results given here use the first terms of his equation
19-35 as given below.
= the static spring deflection
n = the number of coils
As used here:
where  = coil angle
For a dynamically loaded spring the stress 
must be
calculated from the local deflection as given by .
where S = total wire length
Using the above, the stress and velocity and hence the
deflection at every mesh point can be calculated. If two or more
coils contact each other, the position and velocity of all the
coils are set to the average position. Thus the effects of
coil-to-coil
contact are approximated.
Both boundary conditions in this application are velocity
inputs. The spring dynamics calculation is an integral part of the
valve dynamics calculation. The spring calculation gives the force
acting on the element at any instant, and the valve dynamics
calculation gives the resulting velocity. At the other end of the
spring, the input velocity is zero. At either end, the spring may
leave its stop. When this happens, the boundary condition is that
the stress is zero. The position of the end of the coil can then
be calculated from its velocity. Once the spring returns to its
stop, the boundary returns to a known velocity.
 
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