Team Exercise  If you have enough money to buy a car, what kind of car do you like to buy?  If you are a car design engineer, identify design goal and design parameters from your team’s preference  Taken from - ETP 2005 – Dan Houston This material is based upon work supported by the National Science Foundation under Grant No Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the view of the National Science Foundation (NSF).

Team Exercise Well Posed Design Problem: Design a new car that can: 1. Go from mph in 6 seconds 2. Gets 50 miles/gal 3. Costs less than $10,000 to the consumer 4. Does not exceed government pollution standards 5. Appeals to aesthetic tastes

Team Exercise 1. Identify Problem e.g. we need to build a new car since we are losing market share 2. Synthesis (integrating parts to for a whole) e.g. we can combine an aerodynamic body with a fuel efficient engine to make a new car with very high fuel efficiency

Team Exercise 3. Analysis identify relationships, distinguish fact from opinion, detect logic information, make conclusions from evidence, select relevant information, TRANSLATE REAL-WORLD PROBLEM INTO MATHEMATICAL MODEL e.g. compare the drag of different body types and determine if engine can fit under the hood

Team Exercise 4. Application (identify the pertinent information) e.g. What force is required to allow the car to go 60 mph knowing the car has a 30ft 2 projected area and a 0.35 drag coefficient based on wind tunnel data?

Team Exercise 5. Comprehension (use the data and explicit theory to solve the problem) F = 1/2 C d  A V 2 F=force C d =drag coef.  =air density A=protected frontal area V=speed

Difficulties in Problem Solving  Most common difficulty: failure to use known information.  To avoid this problem:  Write the problem in primitive form and sketch an accurate picture of the setup (where applicable).  Transform the primitive statements to simpler language.  Translate verbal problems to more abstract mathematical statement(s) and figures, diagrams, charts, etc.

General Problem Solving Method Define and understand problem 1.Sketch the problem 2.Gather information 3.Generate and evaluate potential solutions  Use applicable theories and assumptions 4.Refine and implement solution 5.Verify and test solution

Define and Understand  Understand what is being asked  Describe input/output (I/O)  what are you given  knowns  what are you trying to find  unknowns  Sketch the problem

Gather Information  Collect necessary data  List relevant equations/theories  State all assumptions

Generate Solution Methods  Apply theories and assumptions.  Typically, there is more than one approach to solving a problem  Work problem by hand using the potential solution methods  Break problem into parts; scale it down; etc.  e.g., if the problem was to calculate the average of 1000 numbers, work the problem by hand using, say, 10 numbers, in order to establish a method

Refine and Implement  Evaluate solution methods.  accuracy  ease of implementation  etc.  Implement “best” solution.

Verify and Test  Compare solution to the problem statement  Is this what you were looking for?  Does your answer make sense?  Clearly identify the solution  Sketch if appropriate

CHECK YOUR WORK!!  Don’t stop at getting an answer!!  Think about whether the answer makes physical sense.  you are the instructor and you have to turn in final grades. In your haste, you calculate the average of Susie’s grades (100, 70, 90) to be 78 and give Susie a C...

Getting It Right The problem solving process may be an iterative process. If at first you don’t succeed (i.e., the algorithm test fails), try again… The more thorough you are at each step of the problem solving process, the more likely you are to get it right the first time!!

Team Exercise  Given: A student is in a stationary hot- air balloon that is momentarily fixed at 1325 ft above a piece of land. This pilot looks down 60 o (from horizontal) and turns laterally 360 o. Note: 1 acre = 43,560 ft 2

Team Exercise; cont’  Required:  a) Sketch the problem  b) How many acres of land are contained by the cone created by her line of site?  c) How high would the balloon be if, using the same procedure, an area four times greater is encompassed?

Creative Problem Solving  The nine dots shown are arranged in equally spaced rows and columns. Connect all nine points with four straight lines without lifting the pencil from the paper and without retracing any line. Individual Exercise (3 minutes)

Creative Problem Solving

If you enjoy solving puzzles, you will enjoy engineering Crick and Watson figured DNA when they were young Engineers create from nature what did not exist before In this creative process, the engineer marshals skills in mathematics, materials, and other engineering discipline and from these resources create a new solution for a human need

Creative Problem Solving Engineering is not dull or stifling; send people to moon, communication from battlefield, etc Creative artists spent many years perfecting their skills Engineers need patience, practice, and gaining problem-solving techniques by training

Self-Questions for Problem Solving How important is the answer to a given problem? Would a rough, preliminary estimate be satisfactory or high degree accuracy demanded? How much time do you have and what resources are at your disposal? Data available or should be collected, equipments and personnel, etc

Self-Questions for Problem Solving What about the theory you intend to use? Can you use it now or must learn to use it? Is it state of the art? Can you make assumptions that simplify without sacrificing needed accuracy? Are other assumptions valid and applicable? Optimize time and resources vs reliability

Engineering Method 1. Recognize and understand the problem (most difficult part) 2. Accumulate data and verify accuracy 3. Select the appropriate theory or principles 4. Make necessary assumptions 5. Solve the problem 6. Verify and check results

Engineering Method Perfect solutions to real problems do not exist. Simplify the problem to solve it; steady state, rigid body, adiabatic, isentropic, static etc To solve a problem, use mathematical model; direct methods, trial-and-error, graphic methods, etc.

Problem Presentation Problem statement Diagram Theory Assumptions Solution steps Identify results and verify accuracy

Standards of Problem Presentation Engineers should have ability to present information with great clarity in a neat, careful manner Poor engineering documents can be legal problems in courts Follow standard forms such as shown in the textbooks

Team Assignment Page 141 Problem 3.20

Algorithms Algorithm: “a step-by-step procedure for solving a problem or accomplishing an end” (Webster) Algorithms can be described by Pseudocode Flowcharts

Pseudocode English-like description of each step of algorithm Not computer code Example - take out trash barrels while there are more barrels take barrel to street return to garage end

Flowcharts Graphical description of algorithm Standard symbols used for specific operations Input/Output Start/Stop Branch Test Process Step Process Flow

Flowchart Example Define the problem Read input Solve the problem Can I solve this? Output results What do I need to know? Ask for more input Begin Can I solve this? End yes no yes no

Top Down Design State problem clearly Sketch problem Describe input/output (I/O) Work problem by hand Algorithm: pseudocode or flowchart Decomposition - break problem into steps Stepwise refinement - solve each step Test the algorithm/check your work!!

Example (Team exercise, 15 min) State problem clearly: Given ax 2 + bx + c = 0, find x. Describe I/O: Input: a, b, c Output: x

Example (cont.) Hand example: a=1, b=4, c=4 equation? (See Chapter 6, Mathematics Supplement) x=?

Example (cont.) Algorithm development write an algorithm in pseudocode to take any set of coefficients (i.e., a, b, c) and give the value of x for each set Test your algorithm a,b,c = 1,4,4 a,b,c = 1,1,-6 a,b,c = 1,0,1 other good test cases?