14.26 For the optimal control problem of minimization of error in the state variable formulated and solved in Section 14.8.2 study the effect of including a 1 percent critical damping in the formulation. 14.27 For the minimum control effort problem formulated and solved in Section 14.8.3, study the effect of including a 1 percent critical damping in the formulation. 14.28 For the minimum time control problem formulated and solved in Section 14.8.4, study the effect of including a 1 percent critical damping in the formulation. 14.29 For the spring-mass-damper system shown in Fig. E14-29, formulate and solve the problem of determining the spring constant and damping coefficient to minimize the maximum acceleration of the system over a period of 10s when it is subjected to an initial velocity of 5m/s. The mass is specified as 5kg. The displacement of the mass should not exceed 5cm for the entire time interval of 10s. The spring constant and the damping coefficient must also remain within the 510 INTRODUCTION TO OPTIMUM DESIGN x L Base Motion y (x, t ) q (t ) E, I, m – m FIGURE E14-20 Cantilever structure with mass at the tip. c k m x (t ) FIGURE E14-29 Damped single-degree-of-freedom system. limits 1000 £ k £ 3000N/m; 0 £ c £ 300N·S/m. (Hint: The objective of minimizing the maximum acceleration is a min–max problem, which can be converted to a nonlinear programming problem by introducing an artificial design variable. Let a(t) be the acceleration and A be the artificial variable. Then the objective can be to minimize A subject to an additional constraint |a(t)| £ A for 0 £ t £ 10). 14.30 Formulate the problem of optimum design of steel transmission poles described in Kocer and Arora (1996b). Solve the problem as a continuous variable optimization problem. Design Optimization Applications with Implicit Functions 511 [...]... Ix Sx rx Iy Sy ry W36 ¥ 300 W36 ¥ 280 W36 ¥ 260 W36 ¥ 245 W36 ¥ 230 W36 ¥ 210 W36 ¥ 194 88 .30 82 .40 76.50 72.10 67.60 61 .80 57.00 36.74 36.52 36.26 36. 08 35.90 36.69 36.49 0.945 0 .88 5 0 .84 0 0 .80 0 0.760 0 .83 0 0.765 16.655 16.595 16.550 16.510 16.470 12. 180 12.115 1. 680 1.570 1.440 1.350 1.260 1.360 1.260 20300 189 00 17300 16100 15000 13200 12100 1110 1030 953 89 5 83 7 719 664 15.20 15.10 15.00 15.00... solutions 530 INTRODUCTION TO OPTIMUM DESIGN 16 Genetic Algorithms for Optimum Design Upon completion of this chapter, you will be able to: • Explain basic concepts and terminology associated with genetic algorithms • Explain basic steps of a genetic algorithm • Use a software based on genetic algorithm to solve your optimum design problem Genetic algorithms (GA) belong to the class of stochastic search... (refer to Section 15 .8 for more discussion of the problem) (Huang and Arora, 1997) Obtain a solution for the problem 542 INTRODUCTION TO OPTIMUM DESIGN 17 Multiobjective Optimum Design Concepts and Methods Upon completion of this chapter, you will be able to: • Explain basic terminology and concepts related to multiobjective optimization problems • Explain the concepts of Pareto optimality and Pareto optimal... = 81 .5 x = (2, 4) f = 80 x2 £ 6 Subproblem 3 x2 ≥ 7 Subproblem 4 x = (1, 6) x = (0.5, 7) f = 80 Stop—Discrete feasible solution f = 80 Stop—Discrete feasible solution Stop—Discrete solution will have cost higher than 80 Figure 15-2 Branch and bound method with solution of continuous subproblems Since the foregoing problem has only two design variables, it is fairly straightforward to decide how to. .. represents a design point where the seventh, sixth, and fourth allowable discrete values are assigned to x1, x2, and x3, respectively Initial Generation/Starting Design Set With a method to represent a design point defined, the first population consisting of Np designs needs to be created This means that Np D-strings need to be created In some cases, the designer already knows some good usable designs for... continued until a stopping criterion is satisfied or the number of iterations exceeds a specified limit Three genetic operators are used to accomplish this task: reproduction, crossover, and mutation Reproduction is an operator where an old design (D-string) is copied into the new population according to the design s fitness There are many different strategies to implement this reproduction operator This is... chromosome represents a design of the system, whether feasible or infeasible It contains values for all the design variables of the system Gene This term is used for a scalar component of the design vector; i.e., it represents the value of a particular design variable Design Representation A method is needed to represent design variable values in the allowable sets and to represent design points so that... string For example, a chosen member with a string of “345216” and two randomly selected sites of “4” and “1”, is changed to “312546” 1 2 10" 8 7 5 4" 10" 6 9 10 4 3 8" 4" 8" FIGURE 16-4 Bolt insertion sequence determination at 10 locations (Huang et al., 1997) 5 38 INTRODUCTION TO OPTIMUM DESIGN Permutation Type 2 Let n2 be a fraction for selection of the mating pool members for carrying out the Type 2 permutation... is set to 150 and Ig is set to 10 No seed designs are used for the problem The optimum bolting sequence is not unique for the problem With hole 1 as the starting point, the optimum sequence is determined as (1, 5, 4, 10, 7, 8, 9, 3, 6, 2) and the cost function value is 74.63 in The number of function evaluations is 1445, which is much smaller than the total number of possibilities (10! = 3,6 28, 800) Two... to Step 4 Otherwise, continue Step 7 Stopping criterion: If after the mutation fraction Pm is doubled, the best value of the fitness is not updated for the past Ig consecutive generations, then stop Otherwise, go to Step 2 Genetic Algorithms for Optimum Design 537 Immigration It may be useful to introduce completely new designs into the population in an effort to increase diversity This is called immigration, . 1300 156 3 .83 0 W36 ¥ 280 82 .40 36.52 0 .88 5 16.595 1.570 189 00 1030 15.10 1200 144 3 .81 0 W36 ¥ 260 76.50 36.26 0 .84 0 16.550 1.440 17300 953 15.00 1090 132 3. 780 W36 ¥ 245 72.10 36. 08 0 .80 0 16.510. f=–70 g=(5,–10,–10) Node 11 STOP—Since no other feasible points with smaller cost STOP—Since no other feasible points with smaller cost STOP–Since cost is larger than 80 STOP—Since cost is larger than -80 STOP—Feasible. other variations of the 520 INTRODUCTION TO OPTIMUM DESIGN problem 4 does not lead to a discrete solution with f = -80 . Since further branching from this node cannot lead to a discrete solution with