Shaft Impact Problem Solution: Mechanics of Materials

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PROBLEM (12.102) A circular shaft of cross-sectional area A and length L with a helical gear
(of weight W and radius of gyration r) at left end, is supported by bearings in the pump housing (Fig.
P12.102). When turning at a speed (
ω
rad/s) the shaft is accidentally stopped because of jamming on a
bearing. Verify that the maximum angle of twist max
φ
and maximum shearing stress produced by the
impact in the shaft are
max max
22
k
LG
U
GJ AL
φτ ==
k
U
(P12.101)
where is the total kinetic energy (
k
U22 22rg ω )
SOLUTION
max
1
2k
T φ =U, where 4
max 2
GJ c
TJ
L
φ
π
==
Thus,
max max
1()
2k
GJ U
L
φ
φ =
Therefore
max 2k
LU
GJ
φ = Q.E.D. (1)
Similarly,
max max
max JL
TL L
GJ c GJ cG
τ
τ
φ
== =
Substitute this into Eq. (1):
22
max
22 2k
LLU
cG GJ
τ
=
or
22
2
max 4
22(2
kk
cG cG
UU
LJ c L
τπ ==
)
Solving
max 4k
GU
AL
τ = Q.E.D.
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