AAE550_HW1_2011

AAE550_HW1_2011 - AAE 550 MULTIDISCIPLINARY DESIGN...

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AAE 550 MULTIDISCIPLINARY DESIGN OPTIMIZATION FALL 2011 HOMEWORK ASSIGNMENT #1 DUE OCTOBER 14, 2011 I. ENGINEERING PROBLEM IN N VARIABLES For the three-bar non-symmetric truss presented below, minimizing the total potential energy as a function of the displacement can determine the equilibrium position of the “free” node (i.e. the node not attached to a support) under the applied load P . For this, neglect the self-weight of the truss elements. This potential energy function is: Π u ( ) = 1 2 u T Ku p T u Here, u is the displacement vector; this will be the design variable vector – u 1 is the displacement in the horizontal ( x -axis) direction and u 2 is the displacement in the vertical ( y -axis) direction. K is the stiffness matrix, and p is the applied load vector. The global stiffness matrix is the sum of the stiffness matrices for the individual elements. The following expressions describe the element stiffness matrices in terms of the x - and y - coordinates. K 1 = cos 0 ° ( ) sin 0 ° ( ) EA 1 L 1 cos 0 ° ( ) sin 0 ° ( ) { } , K 2 = cos 29.05 ° ( ) sin 29.05 ° ( ) EA 2 L 2 cos 29.05 ° ( ) sin 29.05 ° ( ) { } , and x,u 1 y,u 2 P = 15000 lb 4.5 ft 2.5 ft 3.25 ft = 45° 1 2 3
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AAE 550, FALL 2011 HOMEWORK #1, PAGE 2 K 3 = cos 51.95 ° ( ) sin 51.95 ° ( ) EA 3 L 3 cos 51.95 ° ( ) sin 51.95 ° ( ) { } . E is Young’s modulus; use a value of 16 × 10 6 psi. A 1 is the cross-sectional area of the lowest truss member; this element has a solid circular cross-section with a diameter of 0.8 in. Similarly, A 2 and A 3 are the cross-sectional areas of the middle and top truss members, respectively; these elements have a solid circular cross-section with a diameter of 0.65 in. L 1 , L 2 , and L 3 are the lengths of the two elements; use the dimensions on the sketch above to compute these values. With the element stiffness matrices computed, the stiffness matrix for the truss is K = K 1 + K 2 + K 3 . Minimizing Π ( u ) will provide the displacements u under the applied load vector, p . Here, this load vector is: p = P cos 45 ° ( ) P sin 45 ° ( ) When you submit your work, include a copy of your Matlab function files and scripts used to call fminunc . These should be included, even if you have simply modified the provided examples. For the Excel problems, you can cut and paste your spreadsheet at the x 0 values , the answer report and the sensitivity report as objects or tables in your submittal. You can use the example submittal from HW 0 as a template for this assignment, too. 1) Develop analytic expressions for the gradient vector components and the Hessian matrix components. You may do this either in matrix form, or you may wish to expand Π ( u ) into a polynomial and then find derivatives of the polynomial.
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AAE550_HW1_2011 - AAE 550 MULTIDISCIPLINARY DESIGN...

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