This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1894 edition. Excerpt: ...general theory of approximately simple systems. It will be sufficient here to repeat the result. If the density be p0 + BPt tne period rr of the rth component vibration is given by If the irregularity take the form of a small load of mass m at the point x = b, the formula may be written These values of T2 are ...
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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1894 edition. Excerpt: ...general theory of approximately simple systems. It will be sufficient here to repeat the result. If the density be p0 + BPt tne period rr of the rth component vibration is given by If the irregularity take the form of a small load of mass m at the point x = b, the formula may be written These values of T2 are correct as far as the first power of the small quantities 8p and m, and give the means of calculating a correction for such slight departures from uniformity as must always occur in practice. As might be expected, the effect of a small load vanishes at nodes, and rises to a maximum at the points midway between consecutive nodes. When it is desired merely to make a rough estimate of the effective density of a nearly uniform string, the formula indicates that attention is to be given to the neighbourhood of loops rather than to that of nodes. The effect of a small variation of density upon the period is the same whether it occur at a distance x from one end of the string, or at an equal distance from the other end. The mean variation at poiuts equidistant from the centre is all that we need regard, and thus no generality will be lost if we suppose that the density remains symmetricallydistributed with respect to the centre. Thus we may write V-O+Or) (3) where or = y--I1-cos. j dx (4). In this equation 8p may be expanded from 0 to l in the series k-A.l A-TM 2ttx _l. L A TM 2TTrx P0 Accordingly, ar = A0-Ar (8). This equation, as it stands, gives the changes in period in terms of the changes of density supposed to be known. And it shews conversely that a variation of density may always be found which will give prescribed arbitrary displacements to all the periods. This is a point of some interest. In order to secure a reasonable continuity...
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