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Steel Designers' Manual - 6th Edition (2003)
362 Applicable dynamics
dx
small element Ox
g th d x
(a) (o)
M
(-*k) U+ dx
Ox
V
. x
Fig. 12.5 Dynamic equilibrium of vibrating beams. (a) Uniform beam mass/unit length = m,
(b) small element mass = mdx
For a flexural beam the deflection y is a function of the bending moment distribution,
with:
noting also that for moment equilibrium on the element:
M
V =-
x
V M
x x
Substituting these relationships into Equation (12.11) leads to the equation of
motion for a flexural beam:
4
2
∂ y m ∂ y
+ = 0
(12.12)
4
2
∂x
EI ∂t
It is sometimes convenient to represent an entire framed structure as a shear beam.
In this case the deflection y is a function of the shear force, with
∂
∂
∂
∂ =-∂
2 ∂ y M
= 2
∂x
EI
2
2
∂
∂
∂ =
y
x
V
GA s
The equation of motion for a shear beam is therefore:
2
2
∂ y m ∂ y
- = 0
2
2
∂x
GA ∂t
s
12.3.2 Modes of vibration
(12.13)
The partial differential equations derived above can be solved for given sets of
boundary conditions. The solution in each case identifies a family of modes of vibra-