Optimum Design in Mechanical Engineering Lecture 4

1. Optimum Designin Mechanical EngineeringLecture 4 โดย ผศ.ดร. มนต์ศักดิ์ พิมสาร

2. การวิเคราะห์ปัญหาออฟติไมเซชั่นด้วย Matlab ใน Matlab มี Optimization toolbox ซึ่งมีคำสั่งที่สำคัญดังนี้(บางส่วน)

3. ตัวอย่าง 1One variable minimization วิธีทำ ทำการวิเคราะห์โดยการใช้ Matlab

7. ตัวอย่าง 2Multi-variable unconstrained minimization วิธีทำ ทำการวิเคราะห์โดยการใช้ Matlab ค่าเดาเริ่มต้น (-1.2, 1.0) การวิเคราะห์จะใช้คำสั่งดังนี้ • fminsearch ซึ่งใช้วิธีของ Nelder-Mead • fminunc ซึ่งอาจจะใช้วิธี BFGS(default), Hessian matrix, Steepest descent methods คำสั่งนี้ต้องมีการกำหนดเกรเดียนท์ของฟังก์ชันเป้าหมาย แต่คำสั่งนี้จะมีสิทธิผลมากกว่า

11. ไฟล์ที่อธิบาย Objective function

12. ผลลัพธ์กรณีที่ 1

13. ผลลัพธ์กรณีที่ 2

14. ตัวอย่าง 3Multi-variable constrained minimization วิธีทำ ทำการวิเคราะห์โดยการใช้ Matlab ค่าเดาเริ่มต้น (20.1, 5.84) การวิเคราะห์จะใช้คำสั่ง fmincon

18. ไฟล์ที่อธิบาย Objective function ไฟล์ที่อธิบาย Constrained equations (Nonlinear equality and inequality)

19. ผลลัพธ์

20. Workshop 1 – Multivariable Optimization with equality and inequality constraints คำถาม จงทำการหาค่า x1 , x2ที่ทำให้ฟังก์ชั่นนี้มีค่าต่ำสุดกำหนดให้

21. Real life practical engineering design โดยทั่วไปแล้วปัญหาทางวิศวกรรมนั้น การทำการออกแบบเชิงออฟติมั่ม มีดังนี้ Step 1: การกำหนดเนื้อหาของปัญหา(Problem statement) Step 2: การรวบรวมข้อมูลที่เกี่ยวข้อง (Data and information collection) Step 3: การกำหนดตัวแปรออกแบบ (Identification/Definition of design variables) Step 4: การกำหนดเป้าหมายในการออกแบบ (Identification of a criterion to be optimized) Step 5: การกำหนดเงื่อนไขที่เกี่ยวข้องในการออกแบบ (Identification of constraints) Step 6: การคำนวณหาคำตอบ (Solving the problem)

22. Example 1Design of coil springs Step 1:Problem statement : Coil springs are used in numerous practical applications. Detailed methods for analyzing and designing such mechanical components have been developed over the years (e.g., Spotts, 1953; Wahl, 1963; Shigley, 1977; Haug and Arora, 1979). The purpose of this project is to design a minimum mass spring (shown in Figure) to carry a given axial load (called tension-compression spring) without material failure and while satisfying two performance requirements: the spring must deflect by at least D (in.) and the frequency of surge waves must not be less than w0 (Hertz, Hz).

23. Step 2:Data and information collection To formulate the problem of designing a coil spring, the following notation is defined: Deflection along the axis of the spring d, in. Mean coil diameter D, in. , Wire diameter d, in. Number of active coils,N, Gravitational constant g = 386 in./s2 Frequency of surge waves w, Hz Let the material properties be given as Weight density of spring material g = 0.285 lb/in.3 Shear modulus G = (1.15 x 107) lb/in.2 Mass density of material (r = g/g) r = (7.38342 x 10-4) lb-s2/in.4 Allowable shear stress ta = 80,000 lb/in.2

24. Step 2:Data and information collection (cont.) Other data for the problem are given as Number of inactive coils Q = 2 Applied load P = 10 lb Minimum spring deflection D = 0.5 in. Lower limit on surge wave frequency w0 = 100 Hz Limit on outer diameter of the coil Do= 1.5 in. The design equations for the spring are given as

25. Step 2:Data and information collection (cont.)

26. Step 3:Identification/Definition of design variables The three design variables for the problem are defined as d = wire diameter, in. D = mean coil diameter, in. N = number of active coils Step 4:Identification of a criterion to be optimized The problem is to minimize the mass of the spring, given as volume x mass density:

27. Step 5:Identification of constraints The constraints for the problem are defined as

28. FormulationWith design variables

29. Example 2Design of power screw Step 1:Problem statement : A power screw, having double square threads, is to be designed to lift a load with maximum efficiency. (See below figure). Formulate the optimization problem by treating the major diameter (d), the number of threads per inch (N), and the mean diameter of the collar (dc) as design variables.

30. Step 2:Data and information collection : Load, P = 1,500 lb, Coefficient of friction of threads in nut, m = 0.08 Coefficient of friction of thrust collar in bearing, mc = 0.08 Height of the screw, h = 6 in Material is steel and Young’s modulus, E = 30 x 106 psi Permissible shear stress, tmax = 12,000 psi Permissible compressive stress, smax = 20,000 psi

31. Step 2:Data and information collection : (cont.) The equations are as follows. (1) The efficiency of screw (e) is

32. Step 2:Data and information collection : (cont.) (2) The tensile area is

33. Step 3:Identification/Definition of design variables The three design variables for the problem are defined as d = major diameter, in. dc = mean diameter of the collar, in. N = number of threads per in Step 4:Identification of a criterion to be optimized The problem is to maximize the screw efficiency (e)

34. Step 5:Identification of constraints • The constraints for the problem are defined as • Direct compressive stress (s) must be less than smax. • Direct compressive stress (s) must be less than buckling stress sb. • Shearing of screw threads at minor diameter (dr) in nut must be avoided. • Shearing of nut at major diameter of the screw must be avoided. • Bearing stress in threads must be less than due to (smax). • Shear stress in screw due to applied torquetorque (t) must be less than tmax. • Design variable values must be positive.

35. Step 5:Identification of constraints (cont.) The constraint equations are

36. Step 5:Identification of constraints (cont.)

37. Workshop 2 – Multivariable Optimization with equality and inequality constraints Problem statement: Columns are used as structural members in many practical applications. Many times such members are subjected to eccentric loads such as a jib crane. The problem is to design a minimum mass tubular column that is subjected to an eccentric load, as shown in Figure. The cross section of the column is a hollow circular tube with R and t as the mean radius and wall thickness, respectively.

38. Eccentric column