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Steel Designers' Manual - 6th Edition (2003)
364 Applicable dynamics
Table 12.3 Natural frequencies for uniform shear cantilevers
12.3.3 Calculation of responses
Length = L (m) (typical units)
Shear rigidity = GAs (N)
Mass/unit length = m (kg/m)
knGAs fn
= ( Hz: values of kn
given below
2 2p
mL )
Mode Shape kn Nodal points at x/L =
1 1.57 0
2 4.71 0 0.667
3 7.85 0 0.4 0.8
( )
n ( 2n-1) 0
2
p
px
y = sin ( 2n -1)
2L
2 4 6
etc.
( 2n -1) ( 2n -1) ( 2n -1)
For linear elastic behaviour, once the mode shapes and frequencies have been established,
dynamic responses can be calculated treating each mode as a single degree
of freedom system such as the one described in section 12.2.2. In theory, distributed
systems have an infinite number of modes of vibration, but in practice only a few
modes, usually those of lowest frequency, will contribute significantly to the overall
response.
It is convenient to describe a mode shape by a displacement parameter f defined
(or normalized) such that the maximum value at any point is 1.0. If the mode is
excited by dynamic forces resulting in a modal response amplitude of Ym, then the
displacement amplitude at a point i on the structure is
yi = fiYm where fi is the mode shape value at the point.
The magnitude of Ym can be calculated for simple dynamic loads treating the
mode as a single degree of freedom system having the following properties. These
are often referred to as modal or generalized properties:
2 modal mass M* =SMifi sum over whole structure
modal stiffness K* = w 2 nM*
modal force P* =SPifi sum for all loaded points