By Sanjay Sharma
Sharma S. utilized Nonlinear Programming (New Age foreign (P) Ltd, 2006)(ISBN 812241771X)
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Additional info for Applied nonlinear programming
Gears are used for transmission of the motion. Depending on the parameters such as speed and power to be transmitted, suitable material for the gear is selected. The objective function may include (i) minimize the transmission error, (ii) maximize the efficiency, and (iii) minimize the weight. F. In addition to the space constraint, the induced stresses such as crushing, bending and shear stress must be less than or equal to the respective allowable crushing, bending and shear stress of the selected material.
7) i=1 where wi = M*i f* Mi* = ωi . 6) can be written as, N ∑a * ki . Mi i =1 = 0, substituting the optimum values and GEOMETRIC PROGRAMMING 51 multiplying with xk* on both sides. 9) in above equation, N f* ∑a ki . wi =0 i=1 N ∑a ki . 10) are used to solve the problem. A unique solution is obtained for wi, i = 1, 2,.... N if N = n + 1, as will be observed later. This type of problem is having zero degree of difficulty as N – n – 1 = 0, where degree of difficulty = (N – n – 1). A comparatively complicated problem may occur if N > (n + 1) because degree of difficulty > 1.
A1i . x1a1i − 1 . x2a2i ... xnani = 1 x1 i =1 Similarly ∂f 1 = ∂x2 x2 N ∑a 2i i=1 . Mi i =1 =0 50 APPLIED NONLINEAR PROGRAMMING 1 ∂f = xn ∂xn N ∑a ni . Mi = 0 i=1 To generalize, ∂f 1 = ∂xk xk N ∑a ki . Mi = 0, k = 1, 2, ... 6) i =1 Let x1*, x2*, ... 5). F. , N f* = ∑M * i = M1* + M*2 + ... + M*N i =1 Dividing by f* on both sides, M1* M*2 M*N f* + + ... + = =1 f* f* f* f* or w1 + w2 + ... 7) i=1 where wi = M*i f* Mi* = ωi . 6) can be written as, N ∑a * ki . Mi i =1 = 0, substituting the optimum values and GEOMETRIC PROGRAMMING 51 multiplying with xk* on both sides.