Welcome ME 482/582 OPTIMAL DESIGN Rudy J. Eggert, Professor Emeritus Mechanical & Biomedical Engineering http://coen.boisestate.edu/reggert http://highpeakpress.com/eggert/

Today’s lecture Optimization Design Analysis versus design Phases of design Parametric design Mathematics review

OPTIMAL DESIGN Definition: The development and use of analytical and computer methods to provide an optimal design of a product or process with minimal computational effort. That’s right… The thing we design will be optimal AND the methods we use will be optimal. .

Product Realization Process Realized Product Product Development Product Develpoment Disposal Production Design Service Engineering Design Distribution Industrial Design Manufacturing (Production) Sales / Marketing Customer Need

Design Set of decision making processes and activities to determine:     Design Set of decision making processes and activities to determine: the form of an object, given the customer’s desired function. Design control hold move protect store  decision making processes shape configuration size materials manufacturing processes Function Form  

Analysis is not Design Which of the following is design and which is analysis? A. Given that the customer wishes to fasten together two steel plates, select appropriate sizes for the bolt, nut and washer. B. Given the cross-section geometry of a new airplane wing we determine the lift it produces by conducting wind tunnel experiments. Problem Type Solution Design Form (size, shape, matls,cnfg, mfg ) Analysis Predicted behavior (performance)

System Evolution (Arora) Figure 1.1 System evolution model.

Design Phases Formulation Concept Preliminary Design Configuration Embodiment Design Parametric Detail

From Customer Needs thru Concept Design ? Formulation Customer Needs Customer requirements Importance weights Eng. characteristics House of Quality Eng. Design Spec’s Concept Design Abstract embodiment Physical principles Material Geometry

Configuration Design Architecture Abstract embodiment Special Purpose Parts: Features Arrangements Relative dimensions Attribute list (variables) Standard Parts: Type Abstract embodiment Physical principles Material Geometry Architecture

Design Phases Cont’d Special Purpose Parts: Features Arrangements Relative dimensions Variable list Standard Parts: Type Design variable values e.g. Sizes, dimensions Materials Mfg. processes Performance predictions Overall satisfaction Prototype test results Parametric Design Detail Design Product specifications Production drawings Performance Tests Bills of materials Mfg. specifications

Design Optimal Design Figure 1.2 Comparison of (a) conventional design method and (b) optimum design method.

Systematic Parametric Design Engineering Design, Eggert, 2010

Tools used in Optimal Design Algebra Calculus Vector and matrix aritmetic Excel (computation & graphing) Graphing (hand) Computer Programming (any language) Engineering principles

Systematic Parametric Design

Mathematical Notation Recall from Calculus, a function of many variables: We shall use vectors for multiple variables: All vectors are columns The transpose is used to show a row

Handwritten vectors The book shows vectors as lower case bolded, for example: For handwritten homework and tests… we will use lower case hand-printed with an underscore, for example:

Points P, x(1) and x(2) Superscripts (1),(2) Figure 1.3 Vector representation of a point P that is in 3-dimensional space.

Vector or point? Is a “vector” a “point” in n-dimensional space denoted as R(n) ?

Set of Points, S S = {x|(x1 – 4)2 + (x2 – 4)2  9}. Figure 1.4 Image of a geometrical representation for the set S = {x|(x1 – 4)2 + (x2 – 4)2  9}.

Dot Product From Engineering Statics: In optimal Design: How do we know if two vectors are orthogonal (normal) ? In optimal Design:

Vector or Scalar? Is a dot product of two vectors a vector or scalar quantity?

Norm of a vector The length or magnitude of a vector is called the NORM.

Product of vector and matrix Is the product a scalar or vector?

Triple Product Rusty? …. Review appendix A, pgs 785-822

Function continuity Figure 1.5 Continuous and discontinuous functions: (a) and (b) continuous functions; (c) not a function; (d) discontinuous function.

First Partial Derivatives of a function Gradient vector We’ll se a lot of these in chapter 4.

Second Partial Deriivatives of a function… Hessian Matrix What does the x* mean?

Summary Design Optimal design Design Phases Systematic Parametric Design Vector, matrix review