Steel Designers Manual - TheBestFriend.org

This material is copyright - all rights reserved. Reproduced under licence from The Steel Construction Institute on 12/2/2007
To buy a hardcopy version of this document call 01344 872775 or go to http://shop.steelbiz.org/
Steel Designers' Manual - 6th Edition (2003)
Note that for mode n of a uniform simply-supported beam of span L and mass/unit
length m with f = sin(npx/L), the modal mass M* is:
2
M*= m sin ( npx/ L) dx
which is 0.5 mL (half the total mass of the beam).
The application of these concepts to the problems of floor vibration and windinduced
vibration is described in References 6 and 7.
12.3.4 Approximate methods to determine natural frequency
Approximate methods are useful for estimating the natural frequencies of structures
not conforming with one of the special cases for which standard solutions exist,
and for checking the predictions of computer analyses when these are used.
One of the most useful approximate methods relates the natural frequency of a
system to its static deflection under gravity load, d. With reference to section 12.2.2
the natural frequency of a single lumped mass system is:
1
fn
=
2p
This may be rewritten, replacing the mass term M by the corresponding weight Mg,
as:
g Ê K ˆ
fn
=
2p Ë Mg ¯
Since Mg/K is the static deflection under gravity load (d ):
f
n
Ú
L
0
[ ]
Ê K ˆ
Ë M ¯
g
=
2p d
= 15.76/ d when d is measured in mm
Distributed parameter systems 365
(12.14)
This formula is exact for any single lumped mass system.
For distributed parameter systems a similar correspondence is found, although
the numerical factor in Equation (12.14) varies from case to case, generally between
16 and 20. For practical purposes a value of 18 will give results of sufficient
accuracy.
When applying these formulae the following points should be noted.
(1) The static deflection should be calculated assuming a weight corresponding to
the loading for which the frequency is required. This is usually a dead load with
an allowance for expected imposed load.
(2) For horizontal modes of vibration (e.g. lateral vibration of an entire structure)
the gravity force must be applied laterally to obtain the appropriate lateral
deflection, as shown in Fig. 12.6(a).