Read e-book online Analytical Elements of Mechanics PDF

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By Thomas R. Kane

ISBN-10: 1483231283

ISBN-13: 9781483231280

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Number the elements as shown in Fig. 5b. FIG. 5b FIG. 5C (b) Selection of a point in each element: use the left-most point of each element, calling these points Pi, P 2 , . . , P n , as shown in Fig. 5c. P t is a typical point of this set of points. (c) Strengths of the points P ; (i = 1, . . , n ) : let Ni be the strength of Pi. The length of each element is wR/n, as noted in (a). Hence, all of the Ni(i = 1, . . , n) must be taken equal to each other. Take N4: = 1, i = 1, 2, . . , n (1) (d) Location of the centroid of the set of points P t (i = 1, .

N, with respect to 0 is equal Pi (Hi) / P Ρ*(Σ Νί) " FIG. 4 0 Σ η i=l NiPi. 5 The cartesian coordinates of the centroid P* of a set of points P», i = 1, 2, . . , n, of strengths Niy i = 1, . . , n, are given by three expressions of the form £#<** Σ* x* = ^ 1 r= l Proof (see Fig. 2) Substitute these into p * ΣΝίΡί = LULL Σ* and write the three scalar equations corresponding to the vector equation thus obtained. CENTROIDS AND MASS CENTERS; SECTION 2 . 6 If the points of a set are arranged in such a way that corresponding to every point on one side of a certain plane there exists a point of equal strength on the other side, the two points being equidistant from the plane, but not necessarily lying on the same normal to it, then the centroid of the set lies in this plane.

Nn = 1 + 1 + . . + 1 = n Substitute into Eqs. (3): x = - Σ cos (iir/n), y = - ]T) sin (ζττ/η) (4) i=l t=l These results can be simplified as follows: For any angle Θ not Σ η t= 1 cos (ιθ) can be written (see W. E. Byerly, "Fourier Series and Spherical Har­ monics," p. , Boston) as [(2yi + 1W2] t cos m « -21 ^+2 *sin sin (0/2) Z^ C0S W 2+ 2 sin Ϊ0/2Ϊ i=l and, similarly, n Σ sin (*) sin (nfl/2) sin Γ(η + 1)0/2] sin (0/2) 56 CENTROIDS AND MASS CENTERS; SECTION 2 . 5 Hence, for Θ = x/n, V^ /-Λ V^ f / \ 1 , l s i n ( x + ir/2n) ζ cos (τθ) = £ cos (tx/n) = - ^ + § s i n ( r / 2 n ) I-I 2 2 in sin (x/2) sin y~] sin (ίθ) = ^ sin (ίτ/η) = t=l i (1+L·) sin (π/2η) = cotan (π/2η) Substitute into Eqs.

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Analytical Elements of Mechanics by Thomas R. Kane

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