Some light reading in terms of the collision between hammer and valve... "To precisely determine whether the hammer will open the valve when it hits the valve
stem you have to solve a complex open two-body impact problem. A collision problem is
“open” when external forces interfere with the momentum exchange between the colliding
bodies. You can solve the impact problem in a PCP airgun at several levels of complexity,
depending on the time scales involved. If the time scale of the collision is comparable to
the valve dwell time scale, you must solve the collision problem in full, accounting for
what happens during the collision. Hertz [13] laid out the elasticity framework for dealing
with colliding bodies when you need full resolution of the collision process. Analytical
solutions exist for relatively simple bodies, such as spheres, bars, and planes [14]. For
more complex colliding bodies, such as PCP hammers and valves, detailed resolution of
the collision process requires a numerical technique such as finite elements. Since the
focus here is on simplicity and computational economy, it is best to find a simpler alterna-
tive.
If the collision time scale is much shorter than the time the valve remains open, you can
simplify the problem by assuming the collision occurs instantaneously. In the case of a
PCP hammer-valve impact, the valve must overcome some deflection of its seating area,
and this means that the impact of the hammer with the valve stem will happen as an open
system, subject to external momentum transfer. Elementary collision theory, as is usually
presented in elementary Physics texts, assumes the colliding bodies form a closed system,
subject to no external forces.
I will assume the collision time is short enough that it can be neglected compared with the
dwell time of the valve. Without entering into the mathematical details, this assumption is
justified by comparing computed valve dwell times with characteristic collision times of
simple bodies (such as spheres of appropriate size and mass) using Hertz’ theory. If you
replace the hammer and valve with spheres of diameter and mass comparable to the diam-
eter and mass of the hammer and valve stem, Hertz theory gives you collision times of the
order of 50 microseconds, or 0.05 millisecond. Typical dwell times for the configurations
studied here are about one millisecond, or twenty times longer. To determine how good
this assumption really is would require empirical verification with sophisticated measure-
ment equipment. One of the consequences of the instantaneous collision assumption is the
possibility of multiple collisions between the hammer and the valve stem while the valve
is open.
The instantaneous collision assumption when there is an external source of impulse
amounts to a separation of the momentum exchange between hammer and valve during
collision, and the momentum exchange between the valve and the closing forces acting on
the valve (the valve spring and the reservoir pressure). In our case, the external impulse
comes from the resistance of the valve to be lifted against the reservoir pressure as the
valve separates from its deflected seat. In this approximation, the bouncing velocities of
the hammer and valve result from elementary collision theory, and whether the valve can
overcome the forces that keep is the outcome of a-posteriority verification.
If is the hammer velocity when it first makes contact with the valve stem, is the
hammer mass, and is the mass of the valve, the hammer and valve velocities after
impact are given by...
where is the restitution coefficient of the hammer and valve metal pair (typical values of
for steel are between 0.7 and 0.85.) The restitution coefficient represents the fraction of
momentum that is restored to the colliding bodies as they bounce away from each other.
In this simplified approach, in order for the valve to open, the energy of the valve after
impact with the hammer must be greater than the energy used in overcoming the forces
that keep the valve closed. Mathematically, this means the valve velocity calculated with
EQ 2 must be corrected by the energy consumed in restoring the valve seat to its unloaded
shape
where is the valve seating area, is a characteristic thickness of the valve seating area,
is the initial reservoir pressure, and is Young’s elasticity modulus of the valve seat-
ing material. The operator returns the positive value of its argument if the argument
is greater than zero, or zero otherwise. The thickness and surface are functions of the
valve and seat geometry, and it is best to treat these quantities as calibration parameters.
EQ 3 ignores the effect of the initial transfer plenum pressure on the valve poppet, which
is insignificant when compared with the initial reservoir pressure.
Once you know the minimum value of required to open the valve, you can use EQ 1 to
calculate the minimum hammer velocity upon valve contact, and with this it is trivial to
compute the minimum hammer spring strength required to open the valve. "
The rest can be read here...
