1. Numerical Methods for Engineering MECN 3500
Department of Mechanical Engineering
Inter American University of Puerto Rico
Bayamon Campus
Dr. Omar E. Meza Castillo

2. Syllabus
Catalog Description: Study of errors in calculations. Analysis of the numerical methods used in engineering problem solving. Emphasis on the solution of linear and non linear equation systems, arrangement of curves, interpolation, integration and derivation by numerical approximation, numerical integration of differential equations and techniques of optimization. Application of computerized programs for problem solving.
Prerequisites: MATH 3400 – Differential Equation MECN 3110 – Fluid Mechanics and Application MECN 3135 – Solid Mechanics.
Course Text: Chapra , S and Canale, R, Numerical Methods For Engineering, 4th. Ed. McGraw-Hill 2002.

3. Syllabus
Absences: On those days when you will be absent, find a friend or an acquaintance to take notes for you or visit Blackboard. Do not call or send an the instructor and ask what went on in class, and what the homework assignment is.
Homework assignments: Homework problems will be assigned on a regular basis. Problems will be solved using the Problem-Solving Technique on any white paper with no more than one problem written on one sheet of paper. Homework will be collected when due, with your name written legibly on the from of the title page. It is graded on a 0 to 100 points scale. Late homework (any reason) will not be accepted.

4. Syllabus
Problem-Solving Technique:
Known
Find
Assumptions
Schematic
Analysis, and
Results
Quiz : There are four partial quizes during the semester.
Partial Exams and Final Exam: There are three partial exams during the semester, and a final exam at the end of the semester.

5. Syllabus
The total course grade is comprised of homework assignments, quiz, partial exams, final exam, and design project as follows:
Homework 15%
Quiz %
Partial Exam 25%
Final Exam %
Final Project 20% %
Cheating: You are allowed to cooperate on homework by sharing ideas and methods. Copying will not be tolerated. Submitted work copied from others will be considered academic misconduct and will get no points.

6. Syllabus
Most Course Material (Power Point Lectures and homework) will posted every week or two on Web Page of the course MECN 3500:
Office Hours: G235
Monday and Wednesday: 5:00-6:00 p.m.
Contact

7. Tentative Lectures Schedule
Topic
Lecture
Mathematical Modeling and Engineering Problem Solving 1
Introduction to Matlab
Numerical Error
Root Finding
System of Linear Equations
Least Square Curve Fitting
Polynomial Interpolation
Numerical Integration
Ordinary Differential Equations

8. Reference
Rao Singiresu S, Applied Numerical Methods for Engineers and Scientists, Prentice Hall, 2002.
Fausett, Laurene V, Numerical Methods: Algorithms and Applications, Prentice Hall, 2003.
John H. M, Kurtis K. F, Numerical Methods Using Matlab, 4th. Ed., Prentice Hall, 2004.
Mathews, J. H., Numerical Methods for Mathematics, Science, and Engineering, Prentice Hall, 2000.

9. Mathematical Modeling and Engineering Problem Solving
"Lo peor es educar por métodos basados en el temor, la fuerza, la autoridad, porque se destruye la sinceridad y la confianza, y sólo se consigue una falsa sumisión”
Einstein Albert
Mathematical Modeling and Engineering Problem Solving
Introduction

10. To understand the advantages and disadvantages the numerical methods.
Course Objectives
To introduce the mathematical modeling and its role in engineering problem solving.
To understand the advantages and disadvantages the numerical methods.
Thermal Systems Design Universidad del Turabo

11. Why Do We Need Numerical Methods?
Introduction
Why Do We Need Numerical Methods?
Solution of engineering and scientific problems can be done by theory or experiment.
An important third way is by computation
There are many problems which simply do not have analytical solutions, or those whose exact solution is beyond our current state of knowledge.
There are also many problems which are too long (or tedious) to solve by hand
When numerical answers are required one sometimes needs to rely on approximate methods to obtain useable answers 10

12. Analytical and Numerical Solutions
Numerical methods are techniques by which mathematical problems are formulated so that they can be solved with arithmetic operations
Numerical
Analytical
approximate
exact
more intuitive
less intuitive
easily coded
not so easy
easy to get 11

13. The Engineering Problem Solving Process
Requires understanding of engineering systems
By observation and experiment
Theoretical analysis and generalization
Computers are great tools, however, without fundamental understanding of engineering problems, they will be useless.
Simple mathematical models can be solved with pencil and paper. Realistic models usually require computer solution. 12

14. A Simple Mathematical Model
A mathematical model is represented as a functional relationship of the form
Dependent independent forcing
Variable = f variables, parameters, functions
Dependent variable: Characteristic that usually reflects the state of the system
Independent variables: Dimensions such as time and space along which the systems behavior is being determined
Parameters: reflect the system’s properties or composition
Forcing functions: external influences acting upon the system 13

15. Newton’s 2nd Law of Motion
States that “the time rate change of momentum of a body is equal to the resulting force acting on it.”
The model is formulated as
F = m a (1.2)
F=net force acting on the body (N)
m=mass of the object (kg)
a=its acceleration (m/s2) 14

16. Newton’s 2nd Law of Motion
Formulation of Newton’s 2nd Law has several characteristics that are typical of mathematical models of the physical world:
It describes a natural process or system in mathematical terms
It represents an idealization and simplification of reality
Finally, it yields reproducible results, consequently, can be used for predictive purposes.
Some mathematical models of physical phenomena may be much more complex. 15

17. Newton’s 2nd Law of Motion
Complex models may not be solved exactly or require more sophisticated mathematical techniques than simple algebra for their solution
Example, modeling of a falling parachutist:
Schematic diagram of the forces acting on a falling parachutist. FD is the downward force due to gravity. FU is the upward force due to air resistance 16

18. Newton’s 2nd Law of Motion17

19. Newton’s 2nd Law of Motion
(1.9)
This is a differential equation and is written in terms of the differential rate of change dv/dt of the variable that we are interested in predicting.
If the parachutist is initially at rest (v=0 at t=0), using calculus
(1.10)
Independent variable
Dependent variable
Parameters
Forcing function 18

20. Application Problems

21. Analytical Solution of the Falling Parachutist Problem
Example 1.1
Problem Statement: A parachutist of mass 68.1 kg jumps out of a stationary hot air balloon. Use Eq. (1.10) to compute velocity prior to opening the chute. The drag coefficient is equal to 12.5 kg/s.
Solution: Inserting the parameters into Eq. (1.10) yields

22. Analytical Solution of the Falling Parachutist Problem
We can use to compute
t,s
v,m/s
0.00 2
16.40 4
27.77 6
35.64 8
41.10 10
44.87 12
47.49
Infinite
53.39
Terminal Velocity
The analytical or exact solution to the falling parachutist problem shows that velocity increases with time and asymptotically approaches a terminal velocity.

23. Analytical vs. Numerical Solution
The previous solution is analytical, meaning that it is supplied by a single simple formula
We can solve this problem also numerically
Numerical solutions generalize
What if the drag is not linear in the velocity?
As mentioned previously, numerical methods are those in which the mathematical problem is reformulated so it can be solved by arithmetic operations.

24. Analytical vs. Numerical Solution`
True Slope
Approximate Slope
The use of a finite difference to approximate the first derivative of v with respect to t

25. Analytical vs. Numerical Solution
The Newton’s second law by realizing that the time rate of change of velocity can be approximated by:
(1.11)
The equation (1.11) is called a finite divided difference approximation of the derivative at time ti. It can be substituted into Eq. (1.9) to give

26. Analytical vs. Numerical Solution
This equation can then rearranged to yield
(1.12)
If you are given an initial value for velocity at some time ti, you can easily compute velocity at a later time ti+1. This new value of velocity at ti+1 can in turn be employed to extend the computation to velocity at ti+2 and so on. Thus, at any time along the way,
New value = old value + slope x step size
This approach is formally called Euler’s Method
slope

27. Application Problems

28. Numerical Solution of the Falling Parachutist Problem
Example 1.2
Problem Statement: Perform the same computation as in Example 1.1 but use Eq. (1.12) to compute the velocity. Employ a step size of 2 for the calculation.
Solution: At the start of computation (ti=0), the velocity of the parachutist is zero. Using this information and the parameter values from example 1.1 , Eq. (1.12) can be used to compute velocity at ti+1=2s.

29. Numerical Solution of the Falling Parachutist Problem
For the next interval (from t=2 to 4s), the computation is repeated, with the result
The calculation is continued in a similar fashion to obtain additional values:

30. Numerical Solution of the Falling Parachutist Problem
Terminal Velocity
t,s
v,m/s
0.00 2
19.60 4
32.00 6
39.85 8
44.82 10
47.97 12
49.96
Infinite
53.39

31. Homework1  www.bc.inter.edu/facultad/omeza
Omar E. Meza Castillo Ph.D.