Page 410 - Shigley's Mechanical Engineering Design

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Shafts and Shaft Components 385
which can be rewritten as
2
(m 1 δ 11 − 1/ω )y 1 + (m 2 δ 12 )y 2 + (m 3 δ 13 )y 3 = 0
2
(m 1 δ 21 )y 1 + (m 2 δ 22 − 1/ω )y 2 + (m 3 δ 23 )y 3 = 0 (a)
2
(m 1 δ 31 )y 1 + (m 2 δ 32 )y 2 + (m 3 δ 33 − 1/ω )y 3 = 0
Equation set (a) is three simultaneous equations in terms of y 1 , y 2 , and y 3 . To avoid the
trivial solution y 1 = y 2 = y 3 = 0, the determinant of the coefficients of y 1 , y 2 , and y 3
must be zero (eigenvalue problem). Thus,

# 2 #
# (m 1 δ 11 − 1/ω ) m 2 δ 12 m 3 δ 13 #
# #
2
#
# (m 2 δ 22 − 1/ω ) # = 0 (7–26)
m 1 δ 21 m 3 δ 23
#
# #
m 1 δ 31 m 2 δ 32
# (m 3 δ 33 − 1/ω )
2 #
which says that a deflection other than zero exists only at three distinct values of ω, the
critical speeds. Expanding the determinant, we obtain
3 2

1 1
− (m 1 δ 11 + m 2 δ 22 + m 3 δ 33 ) +· · · = 0 (7–27)
ω 2 ω 2
2
2
2
The three roots of Eq. (7–27) can be expressed as 1/ω , 1/ω , and 1/ω . Thus
1 2 3
Eq. (7–27) can be written in the form
1 1 1 1 1 1

− − − = 0
ω 2 ω 2 ω 2 ω 2 ω 2 ω 2
1 2 3
or
3 2
1 1 1 1 1

− + + + ··· = 0 (7–28)
ω 2 ω 2 ω 2 ω 2 ω 2
1 2 3
Comparing Eqs. (7–27) and (7–28) we see that
1 1 1
+ + = m 1 δ 11 + m 2 δ 22 + m 3 δ 33 (7–29)
ω 2 ω 2 ω 2
1 2 3
2
If we had only a single mass m 1 alone, the critical speed would be given by 1/ω =
m 1 δ 11 . Denote this critical speed as ω 11 (which considers only m 1 acting alone).
Likewise for m 2 or m 3 acting alone, we similarly define the terms 1/ω 2 22 = m 2 δ 22 or
1/ω 2 = m 3 δ 33 , respectively. Thus, Eq. (7–29) can be rewritten as
33
1 1 1 1 1 1
+ + = + + (7–30)
ω 2 ω 2 ω 2 ω 2 ω 2 ω 2
1 2 3 11 22 33
2
2
2
If we order the critical speeds such that ω 1 <ω 2 <ω 3 , then 1/ω 1/ω , and 1/ω .
1 2 3
So the first, or fundamental, critical speed ω 1 can be approximated by
1 . 1 1 1
= + + (7–31)
ω 2 ω 2 ω 2 ω 2
1 11 22 33
This idea can be extended to an n-body shaft:
n
1 . $ 1
= (7–32)
ω 2 ω 2
1 1=1 ii