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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).
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) 366 Applicable dynamics — — (0) / ft t tt fit fit t I tilt t (b) UUUU (c) Fig. 12.6 Use of gravity deflections to estimate natural frequencies. In each case fn � 18/ √d. (a) For a vertical structure the gravity load is applied horizontally. (b) Continuous structure gravity load applied in opposite directions on alternate spans. (c) Continuous structure – symmetric mode has higher frequency. (3) The mode shape required must be carefully considered in multi-span structures. In the two-span beam of Fig. 12.6(b) the lowest frequency will correspond to an asymmetrical mode as illustrated; the corresponding d must be obtained by applying gravity in opposite directions on the two spans. The normal gravity deflection will correspond to the symmetrical mode with a higher natural frequency, Fig. 12.6(c). These concepts can be extended to estimating the natural frequencies of primary beam – secondary beam systems. If the static deflection of the primary beam is dp and the static deflection of a secondary beam is ds (relative to the primary) the combined natural frequency is approximately
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