A Treatise on Dynamics of a System of Rigid Bodies; With Numerous Examples Volume 2

A Treatise on Dynamics of a System of Rigid Bodies; With Numerous Examples Volume 2

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This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1905 edition. Excerpt: ...resisting forces. 322. The series for any coordinate now takes the form (see Art. 317) x = Xtert cospt + X en sin pt+... where p is the same as before and, by Art. 319, r is negative. With the same given initial values of x, y, &c, dxjdt, dy/dt, &c. the coefficients Xlt &c. are changed only by terms which contain the factor K, and being themselves small, these changes may be neglected. The value of r may be deduced from the expressions given at the end of Art. 318. If the forces of resistance were zero, the real exponentials would be absent and the ratios Aj/A2, YJY2 would all be equal (Arts. 50 and 115). With small forces of resistance these ratios differ from each other by quantities which contain the small factor K. It follows that the ratios B(X1X1)IA (X) and B (X2X2)//1 (X2X2) are also equal when we reject the square of the small quantity. The expression for r therefore reduces to the simple form BjXM B11A7 + 2B12Xirl+... 3 A (A, A, ) 4 AnX + 2A,2X1Yl +...' Translating this formula into English we see by Art. 73 that the numerical value of r, for any one principal oscillation, is one-half the ratio of the mean value of the dissipation function to the mean value of the kinetic energy for that oscillation. Forced Oscillations. 323. We may suppose a system to be moving in a given state of motion denned, as explained in Art. 257, by the coordinates 0 = #0, f = /0, &c. where 0O, f0, &c. are known functions of the time. This motion we shall call sometimes the undisturbed motion and sometimes the steady motion. If the system be now disturbed in any manner, we write 0 = 0o + x, j) = j)0 + y, &c. where x, y, &c. are so small that we may reject their squares. This disturbance may have been made by some small...show more

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  • Paperback
  • 189 x 246 x 11mm | 390g
  • Rarebooksclub.com
  • United States
  • English
  • black & white illustrations
  • 1236895711
  • 9781236895714