What is Engineering Optimization?

Optimization is the act of obtaining the best result under given circumstances. the word ‘optimum’ is taken to mean ‘maximum’ or ‘minimum’ depending on the circumstances. In design, construction, and maintenance of any engineering system, engineers have to take many technological and managerial decisions at several stages. The ultimate goal of all such decisions is either to minimize the effort required or to maximize the desired benefit. Since the effort required or the benefit desired in any practical situation can be expressed as a function of certain decision variables, so optimization can be defined as the process of finding the conditions that give the maximum or minimum value of a function

The optimum searching methods are also known as mathematical programming techniques and are generally studied as a part of operations research. Operations research is a branch of mathematics concerned with the application of scientific methods and techniques to decision making problems and with establishing the best or optimal solutions.

ENGINEERING APPLICATIONS OF OPTIMIZATION

Optimization, in its broadest sense, can be applied to solve any engineering problem e.g.

1. Running a business to maximize profit, minimize loss, maximize efficiency, or minimize risk.

2. It might mean designing a bridge to minimize weight or maximize strength. It might mean selecting a flight plan for an aircraft to minimize time or fuel use.

3. Design of water resources systems for maximum benefit

4. Planning the best strategy to obtain maximum profit in the presence of a competitor

5. Planning of maintenance and replacement of equipment to reduce operating costs

The power of optimization methods to determine the best case without actually testing all possible cases comes through the use of a modest level of mathematics and at the cost of performing iterative numerical calculations using clearly defined logical procedures or algorithms implemented on computing machines.

Design Vector

Any engineering system or component is defined by a set of quantities some of which are viewed as variables during the design process. In general, certain quantities are usually fixed at the outset and these are called pre-assigned parameters. All the other quantities are treated as variables in the design process and are called design or decision variables xi , i = 1, 2, . . . , n. The design variables are collectively represented as a design vector X = {x1, x2, . . . , xn}T.

Design Constraints

In many practical problems, the design variables cannot be chosen arbitrarily; rather, they have to satisfy certain specified functional and other requirements. The restrictions that must be satisfied to produce an acceptable design are collectively called design constraints.

Objective Function

The conventional design procedures aim at finding an acceptable or adequate design that merely satisfies the functional and other requirements of the problem. In general, there will be more than one acceptable design, and the purpose of optimization is to choose the best one of the many acceptable designs available. Thus a criterion has to be chosen for comparing the different alternative acceptable designs and for selecting the best one. The criterion with respect to which the design is optimized, when expressed as a function of the design variables, is known as the criterion or merit or objective function. The choice of objective function is governed by the nature of problem. The objective function for minimization is generally taken as weight in aircraft and aerospace structural design problems. In civil engineering structural designs, the objective is usually taken as the minimization of cost. The maximization of mechanical efficiency is the obvious choice of an objective in mechanical engineering systems

Constraints

Any optimization problem can be classified as constrained or unconstrained, depending on whether or not constraints exist in the problem. A problem that does not entail any equality or inequality constraints is said to be an unconstrained optimization problem.

MATLAB AN INTRODUCTION (PART 1)

MATLAB is a matrix-based computer system designed for scientific and engineering problem solving. The name MATLAB stands for Matrix Laboratory, because the system was designed to make matrix computations particularly easy. A matrix is an array of numbers organized in m rows and n columns. An example is the following m × n = 2 × 3 array:

Any one of the elements in a matrix can be taken out by using the row and column indices that identify its location. The elements in this example are taken out as follows: A(1, 1) = 1, A(1, 2) = 2, A(1, 3) = 3, A(2, 1) = 4, A(2, 2) = 5, A(2, 3) = 6. The first index identifies the row number counted from top to bottom; the second index is the column number counted from left to right. This is the convention used in MATLAB to locate information in an array.

You input commands to MATLAB in the Command Window. MATLAB returns output in two ways: typically, text or numerical output is returned in the same Command window, but graphical output appears in a separate figure window.

Arithmetic

MATLAB does arithmetic like a calculator. For simple arithmetic operations simply enter the command in the command line and press enter

2+3 Press Enter ans = 5

2-3 Press Enter ans = -1

2*3 Press Enter ans = 6

2/3 Press Enter ans = 0.6667

In above examples MATLAB prints the answer and assigns the value to a variable called ans. MATLAB uses double-precision floating-point arithmetic, which is accurate to approximately 15 digits; however, MATLAB displays only 5 digits by default.

Variables

Variable is a quantity that can assume any of a set of values. To assign a value of 3 to a variable ‘a’

a=3 Press Enter

The ‘a’ is a variable. This statement assigns the value of 3 to it.

Common commands

• The clear command clears all the values assigned to variables. Clear a will only clear the value assigned to variable a
• The command ‘whos’ is executed to determine the list of local variables or commands presently in the workspace.
• The semicolon (;) when placed in front of a variable, prevents its value from being displayed on the screen.
• clc stands for clear command window and clears the workspace.
• Pressing the up arrow key recalls all your previous commands without the need for you to retype.
• Enter quit or exit at the Command Window prompt will exit the software.

Vectors

When you assign values to variables such as a=2 and b =4, these are called scalars; because they are single-valued. MATLAB also handles vectors (generally referred to as arrays; array a collection of data items that are given a single name).The easiest way of defining a vector where the elements increase by the same amount is with a statement like

x=0:10

x = 0 1 2 3 4 5 6 7 8 9 10

To find its size give the following command

Size(x)

ans = 1 11 (means that it has single row and eleven columns)

MATLAB is case-sensitive, which means it distinguishes between upper and lowercase letters. Thus, logic, LOGIX and Logic are three different variables

Mathematical functions

MATLAB has all of the usual mathematical functions found on a scientific electronic calculator, like sin, cos and log.

>> sin(90)

ans = 0.8940 (You get the answer in radians)

Graph

a=1:5;

b=6:10;

plot(a,b),grid

Above command assigns values 1 to 5 to variable a and values 6 to 10 to variable b. Plot command plots a on x axis and b on y axis

Helical spring

A spring is defined as an elastic body, whose function is to distort when loaded and to recover its original shape when the load is removed.

Helical springs.

The helical springs are made up of a wire coiled in the form of a helix and are primarily intended for compressive or tensile loads. The cross-section of the wire from which the spring is made may be circular, square or rectangular. Helical compression springs have applications to resist applied compression forces or in the push mode, store energy to provide the "push". Different forms of compression springs are produced.

The helical springs are said to be closely coiled when the spring wire is coiled so close that the plane containing each turn is nearly at right angles to the axis of the helix and the wire is subjected to torsion. in other words, in a closely coiled helical spring, the helix angle is very small, it is usually less than 10 degree. The major stresses produced in helical springs are shear stresses due to twisting. The load applied is parallel to or along the axis of the spring. In open coiled helical springs, the spring wire is coiled in such a way that there is a gap between the two consecutive turns, as a result of which the helix angle is large. Terms used ¡n Compression Springs The following terms used in connection with compression springs Solid length (Ls). When the compression spring is compressed until the coils come in contact with each other, then the spring is said to be solid. The solid length of a spring is the product of total number of coils and the diameter of the wire. Free length (Lo). The free length of a compression spring is the length of the spring in the free or unloaded condition. Load(P) The force applied to a spring that causes a deflection. Deflection Motion of spring ends or legs under the application or removal of an external load (P). Wire Diameter (d) - The diameter of the wire that is wound into a helix. Spring Index (C) - The ratio of mean coil diameter to wire diameter. A low index indicates a tightly wound spring (a relatively large wire size wound around a relatively small diameter mandrel giving a high rate). Coil Diameter (D) - The mean diameter of the helix, i.e., (D outer + Dinner)/2. Active Coils (Na) - The number of coils which actually deform when the spring is loaded, as opposed to the inactive turns at each end which are in contact with the spring seat or base. Total Coils (Nt)- The number of coils or turns in the spring. Pitch (p) - The distance from center to center of the wire in adjacent active coils Pitch Angle (a) - The angle between the coils and the base of the spring. The pitch angle is calculated from the equation Compression Spring Rate: The change in load per unit of deflection, generally expressed in pounds per inch. Spring rate is determined by the amount of force, in pounds, required to constrict a spring by one inch.