Timothy A. Philpot - Mechanics of materials _ an integrated learning system-John Wiley (2017)

730
ENERgy METHOdS
Note that the negative root implies that the spring elongates when the block strikes it,
which clearly makes no sense for this system. Therefore, the positive root is the only meaningful
solution of equation (b).
If the block had been slowly and gradually placed on top of the spring, the deflection
corresponding to the static force W acting on the spring would be D st = W/k. Therefore,
Equation (b) can be restated in terms of the static spring deflection as
or
2
∆ max = D st + ( D st ) + 2D st h
⎡ 2h
⎤
D max = D st ⎢1+ 1+ ⎥ (17.21)
⎣ Dst ⎦
If Equation (17.21) is substituted into F max = k D max , the dynamic force acting on the
spring can be stated as
F
max
⎡ 2h
⎤
= kD st ⎢1+ 1+ ⎥
⎣ Dst ⎦
and since the weight of the block can be expressed in terms of the spring constant and the
static deflection as W = kD st , it follows that
F
max
⎡ 2h
⎤
= W⎢1+ 1+ ⎥ (17.22)
⎣ Dst ⎦
The force F max is the dynamic force that acts on the spring. This force, along with its
associated deflection D max , occurs only for an instant. Unless the block rebounds off of the
spring, the block will vibrate up and down until the motion dampens out and the block
comes to equilibrium in the static deflected position D st .
The expression in brackets in Equations (17.21) and (17.22) is termed an impact
factor, which will be denoted here by the symbol n:
2h
n = 1+ 1+ (17.23)
D
Thus, the maximum dynamic load F max can be replaced by an equivalent static load,
which is defined for the spring-and-block system as the product of the impact factor n and
the actual static load W of the block:
F
max =
The notion of an impact factor is useful in that both the dynamic force F max and the
dynamic deflection
nW
D = nD
max
can be readily expressed in terms of the static force and the static deflection for a particular
impact factor. Expressed another way, the impact factor is simply the ratio of the dynamic
effect to the static effect:
st
st
Fmax
max
n = = D W D st