MIT2_017JF09_p25

# MIT2_017JF09_p25 - up with our explicit solution for radius...

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25 GEOMETRY OPTIMIZATION 84 25 Geometry Optimization Give the formulas for the radius and length of a closed, half-cylindrical tank (i.e., having a ”D”-like cross-section and ﬂat ends) that will hold a given volume V . Your objective is to make the tank with minimum surface area. What are the speciﬁc dimensions if the volume is 100 m 3 ? The volume and surface area of the tank are given respectively by: V = π r 2 l 2 A = πrl + πr 2 +2 rl With the volume ﬁxed, we seek the condition where the variation in A is zero, that is, ∂A/∂r = ∂A/∂l =0 , subject to the volume constraint. Following the example of Mid- dendorf and Engelmann (1998), we work this out using Lagrange multipliers, even though substitution as in the ﬁrst problem could also be used. The area itself is the cost function, and we augment it as A ± = A + λ ( V πr 2 l/ 2) A ± = πrl + πr 2 +2 rl + λ ( V πr 2 l/ 2) Now we have to set the derivatives of A ± with respect to r and l to zero: ∂A ± =0 = πl +2 πr +2 l λπrl ∂r ∂A ± =0 = πr +2 r λπr 2 / 2 ∂l These relations along with the constraint V πr 2 l/ 2=0 , form three equations in three unknowns. Use the second equation to write λ in terms of r , and then use the third to write l in terms of r , and then substitute both of these into the ﬁrst equation, to end

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Unformatted text preview: up with our explicit solution for radius: V (2 + π ) ± 1 / 3 r = π 2 This is exactly what you get if you use substitution to solve the problem, and in that case, we would put this r back into the constraint equation to solve for l (a little messy). But the Lagrange technique gives us more insight than this: if we use the ﬁrst and second of the three equations above, we can eliminate λ to ﬁnd that l = 2 πr/ (2 + π ) . This is a new property of the optimum solution, and we have simply 2 π V (2 + π ) ± 1 / 3 l = 2 + π π 2 If V = 100 m 3 , we ﬁnd r = 3 . 735 m and l = 4 . 564 m . MIT OpenCourseWare http://ocw.mit.edu 2.017J Design of Electromechanical Robotic Systems Fall 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms ....
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MIT2_017JF09_p25 - up with our explicit solution for radius...

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