Page 217 - Shigley's Mechanical Engineering Design

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192 Mechanical Engineering Design
Figure 4–27 W W

(a) A weight free to fall a
distance h to free end of a h h
EI, l
beam. (b) Equivalent spring
model.
k

(a) (b)

Figure 4–27b shows an abstract model of the system considering the beam as a sim-
3
ple spring. For beam 1 of Table A–9, we find the spring rate to be k = F/y = 3EI/l .
The beam mass and damping can be accounted for, but for this example will be con-
sidered negligible. If the beam is considered massless, there is no momentum transfer,
only energy. If the maximum deflection of the spring (beam) is considered to be δ, the
drop of the weight is h + δ, and the loss of potential energy is W(h + δ). The resulting
1
2
increase in potential (strain) energy of the spring is kδ . Thus, for energy conserva-
2
1
2
tion, kδ = W(h + δ). Rearranging this gives
2
W W
2
δ − 2 δ − 2 h = 0 (a)
k k
Solving for δ yields
1/2
W W 2hk

δ = ± 1 + (b)
k k W
The negative solution is possible only if the weight “sticks” to the beam and vibrates
between the limits of Eq. (b). Thus, the maximum deflection is
1/2

W W 2hk

δ = + 1 + (4–58)
k k W
The maximum force acting on the beam is now found to be
1/2

2hk
F = kδ = W + W 1 + (4–59)
W
Note, in this equation, that if h = 0, then F = 2W. This says that when the weight is
released while in contact with the spring but is not exerting any force on the spring, the
largest force is double the weight.
Most systems are not as ideal as those explored here, so be wary about using these
relations for nonideal systems.
PROBLEMS
Problems marked with an asterisk (*) are linked to problems in other chapters, as summarized in
Table 1–1 of Sec. 1–16, p. 24.
4–1 The figure shows a torsion bar OA fixed at O, simply supported at A, and connected to a can-
tilever AB. The spring rate of the torsion bar is k T , in newton-meters per radian, and that of the
cantilever is k l , in newtons per meter. What is the overall spring rate based on the deflection y at
point B?