The Advanced Part of a Treatise on the Dynamics of a System of Rigid Bodies; Being Part II. of a Treatise on the Whole Subject

The Advanced Part of a Treatise on the Dynamics of a System of Rigid Bodies; Being Part II. of a Treatise on the Whole Subject

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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. 1905 edition. Excerpt: ...1842. Art. 278. 326. To find the Forced Vibration. To find a particular integral for any force Pe'Kt sin (Xt + a) we follow the methods already explained in Chap. VI. If A (8) be the determinant of the motion and /, (6"), 12 (&), &c. be the minors of the first, second, &c. terms in that row of A (8) which corresponds to the equation in which the force occurs, we have x = )Pe-Kt8m(Xt + a), y = Pe-'sin(i4-a), = &c. We shall now prove that these operators lead to two trigonometrical terms in each of the coordinates. These two terms constitute the forced vibration in that coordinate. 327. To perform the operations indicated by these functions of S, we use the following simple rule. To perform the operation F(S) = on PeKt 8ia (t + o) we A (5J cosv' write 6=-K + s/-I and reduce the operator to the form L + M J-1. The required result is then PeKt(L + H-) S1 (t + a). X/cos To prove this rule, we notice that, by Art. 265, F (J) emt= (L + If N/-1) emt where m=-Ac + X N/-1. If we now replace the imaginary part of the exponential by its trigonometrical value, and equate the real and imaginary parts on each side of the equation, the result follows at once. 328. If the force is permanent K = 0 and it immediately follows that the consequent forced vibration is also permanent. 329. If the determinant A(i) have a roots each equal to m, i.e.-k + Xz-i, the result assumes an infinite form. In this case the operator may be replaced by taI(S) + ata-1r(S) +...+la(S)la(S), where the coefficients follow the binomial law, and Aa(5), &c. have been written to express the ath differential coefficient of A (S), &c. Every one of these operations may now be performed by the rule given in the last article. To prove this, we replace the root m by m +...show more

Product details

  • Paperback | 212 pages
  • 189 x 246 x 11mm | 386g
  • Rarebooksclub.com
  • Miami Fl, United States
  • English
  • black & white illustrations
  • 1236494024
  • 9781236494023