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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) 1 y1 22.4 0 1 x 4 —.-- \ Distributed parameter systems 363 tion (eigenvectors) with corresponding natural frequencies (eigenvalues).The mode shape describes the relative displacements of the various parts of the beam at any instant in time. A number of standard results are given in Tables 12.2 and 12.3, and more comprehensive lists are contained in References 4 and 5. Table 12.2 Natural frequencies of uniform flexural beams Simply-supported Encastré Cantilever Length = L (m) (typical units) Flexural rigidity = El (N m2 ) Mass/unit length = m (kg/m) kn fn = 2p EI ( Hz: values of kn given below 4 mL ) Mode Shape kn Nodal points at x/L = 1 9.87 0 1 2 39.5 0 0.5 1 3 88.8 0 0.333 0.667 1 n n2p21 n - 1 x n 0 ... 1 n n x sin p L = ( ) 2 61.7 0 0.5 1 3 121 0 0.359 0.641 1 1 3.52 0 2 22.0 0 0.774 3 61.7 0 0.5 0.868
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) 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
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