Composite Trapezoidal Rule

Composite Simpson’s Rule

Trapezoidal Rule

Simpson’s Rule

Simpson’s Three-Eighths Rule

**DATA AND RESULT FINDING**

Real-life Application Question:

Real-life Application Question:

DATA AND RESULT FINDING

THANK YOU

Numerical methods are very useful in engineering field since analytical methods are not so essential as the accurate data with minor error is always loss than the data input.

Conclusion

Best method:

Runge-Kutta of Order Four

At 0.8 second, speed of motor equals to 303.0937 rad/s is slightly difference with actual solution equals to 303.0940156 rad/s

The smaller step size, the lower the absolute relative true error.

Table 4.5 Methods and Errors

Runge-Kutta Methods of Order Four

In this topic, a machine vision quality inspection system which inspects parts for defects. The parts are positioned in fixtures on a conveyor driven by a DC motor.

The conveyor is required to ramp up to a certain speed within a specific time, maintain a constant speed while the part is under the camera.

Many such applications which utilize DC motors have requirements associated with the ramp up time for the motor to reach this speed. Since, the speed of the motor can be controlled by changing the magnitude of this input voltage.

BACKGROUND OF STUDY IN CHAPTER 5

In this topic, an industrial engineer works as a quality control engineer for a company making toilet paper. The company advertises that every roll of toilet paper has at least 250 sheets.

To certify this claim by sampling rolls off of the assembly line and formulating the probability that the company can make the claim to prevent frivolous lawsuits related with false advertising.

BACKGROUND OF STUDY IN CHAPTER 4

Let Total Cost = Total Sales to find break-even point.

In this topic, a recent graduate with a bachelor’s degree in industrial engineering has been recently employed by a start-up computer assembly company called the “MOM AND POP COMPUTER SHOP” and he has been asked by the president, to determine the minimum number of computers that the shop have to sell to make a profit during the first year in business.

BACKGROUND OF STUDY IN CHAPTER 2

To solve real life application problem with using numerical methods in Industrial Engineering field.

OBJECTIVES OF THE STUDY

Industrial engineering is a division of engineering which deals with the optimization of complex processes or systems. In addition, industrial engineering is a traditional and longstanding engineering discipline subject to professional engineering licensure in most authorities, its underlying models overlap greatly with certain business-oriented disciplines such as operations management.

BACKGROUND OF THE STUDY

INDUSTRIAL ENGINEERING

GROUP 3

WONG ZHEN YUAN 219236

TAN JIT SENG 221042

IZZMIER IZZUDDIN BIN ZULKEPLI 221744

NURUL ATIKA BINTI NASRO 221975

SQQM 3054

NUMERICAL ANALYSIS

Best method:

Composite Simpson’s Rule

with n=20

Table 3.2 Methods and Errors

Trapezoidal Rule

Simpson’s Rule

Simpson’s Three-Eighths Rule

Best Method :

Lagrange and Divided Difference Order of 6 Polynomial

Table 2.3 Methods and Errors

1. Intake of air-fuel

mixture

2.Valves closed and piston moves upward

3.Ignited and combustion occurs

4.Exhaustion flow out

Euler’s Method

Modified Euler Method

Boole’s Rule

Composite Trapezoidal Rule

Composite Simpson’s Rule

A company advertises that every roll of toilet paper has at least 250 sheets. The probability that there are 250 or more sheets in the toilet paper is given by

Approximating the above integral as

Find the probability that there are between 250 and 270 sheets in each toilet roll.

Real-life Application Question:

Best Result

Lagrange Interpolation

Value = 0.92196 (Order 6)

Newton’s Divided Difference

Value = 0.92196 (Order 6)

Real Life Application Question

The geometry of a cam is given in Figure 1. A curve needs to be fit through the seven points given in Table 1 to fabricate the cam.

Figure 1 Schematic of cam profile.

Table 1 Geometry of the cam

Find the cam profile using all points in Table 1 to find the value of y at 1.10.

In this topic, a car company wants to fabricate a cam lobe. To design a cam lobe, we must by interpolate seven points given by the question.

Cam lobe is a part of camshaft which located in the engine,

Its main function is to control the opening and closing of valves which is depends on the lobes separation. The camshaft is shown as below.

BACKGROUND OF STUDY IN CHAPTER 3

Best method :

Newton’s Method

Least number of iteration

Easily to calculate

Minimum amount of computer (integer), thus error doesn’t influence much since tolerance is within 10^-5

Table 1.4 Comparisons of Methods and Errors

Estimated actual equation:

Source : Microsoft Excel

y(1.10) = 0.9235999

Figure 2.7 Best Fit of Graph for Cam Profile

INDUSTRIAL ENGINEERING

GROUP 3

WONG ZHEN YUAN 219236

TAN JIT SENG 221042

IZZMIER IZZUDDIN BIN ZULKEPLI 221744

NURUL ATIKA BINTI NASRO 221975

SQQM 3054

NUMERICAL ANALYSIS

SQQM 3054

NUMERICAL ANALYSIS

Industrial engineering is a division of engineering which deals with the optimization of complex processes or systems. In addition, industrial engineering is a traditional and longstanding engineering discipline subject to professional engineering licensure in most authorities, its underlying models overlap greatly with certain business-oriented disciplines such as operations management.

BACKGROUND OF THE STUDY

Problem Statement

Determine the minimum number of computers that the shop have to sell to make a profit.

Know how the carmakers design cam profile to developed schemes to vary the cam profile as the engine speed changes.

The quality engineer for a company making toilet paper.

Required to ramp up to a certain speed within a specific time, maintain a constant speed while the part is under the camera

OBJECTIVES OF THE STUDY

To solve real life application problem with using numerical methods in Industrial Engineering field.

METHODOLOGY

Numerical methods only show approximation solution as result. In most cases numerical solution will be close enough, which is adequate for most engineering problems. Normally mathematicians have more time and funding to find an analytical solution, but the demands of business usually demands an approximate solution for engineers working in industry.

In engineering field, no matter how exact and regulated the experiment was conducted, will involve some degree of error. It would be a waste of time to employ exact analytical techniques in such a situation because the answer can never be more accurate than the input data. Therefore, numerical methods work best in this situation.

BACKGROUND OF STUDY IN CHAPTER 2

In this topic, a recent graduate with a bachelor’s degree in industrial engineering has been recently employed by a start-up computer assembly company called the “MOM AND POP COMPUTER SHOP” and he has been asked by the president, to determine the minimum number of computers that the shop have to sell to make a profit during the first year in business.

Methods, Advantages and Disadvantages

Real-life Application Question:

You are working for a start-up computer assembly company and have been asked to determine the minimum number of computers that the shop will have to sell to make a profit. The equation that gives the minimum number of computers n to be sold after considering the total costs and the total sales is f(𝑛)=−40𝑛^1.5+875𝑛−35000.

Let Total Cost = Total Sales to find break-even point.

Result

Bisection >> Interval of [ 60, 90]

P_n=(a_1+b_1)/2

Value = 62.69169331

Newton >> 60 (initial guess)

𝑃_𝑛=𝑃_(𝑛−1)−(𝑓(𝑃_(𝑛−1)))/(𝑓′(𝑃_(𝑛−1)))

Value = 62.69169715

**Tolerance within 10^-5

Muller >> 60,75,90 (initial guess)

Value = 62.6916972

**Tolerance within 10^-5

Comparisons of Methods and Errors

Best method :

Newton’s Method

Least number of iteration

Easily to calculate

Minimum amount of computer (integer), thus error doesn’t influence much since tolerance is within 10^-5

BACKGROUND OF STUDY IN CHAPTER 3

In this topic, a car company wants to fabricate a cam lobe. To design a cam lobe, we must by interpolate seven points given by the question.

Cam lobe is a part of camshaft which located in the engine,

Its main function is to control the opening and closing of valves which is depends on the lobes separation. The camshaft is shown as below.

1. Intake of air-fuel mixture

2.Valves closed and piston moves upward

3.Ignited and combustion occurs

4.Exhaustion flow out

Real Life Application Question

The geometry of a cam is given in Figure 1. A curve needs to be fit through the seven points given in Table 1 to fabricate the cam.

Figure 1 Schematic of cam profile.

Table 1 Geometry of the cam

Find the cam profile using all points in Table 1 to find the value of y at 1.10.

Schematic of cam profile.

Best Result

Lagrange Interpolation

Value = 0.92196 (Order 6)

Newton’s Divided Difference

Value = 0.92196 (Order 6)

Source : Microsoft Excel

Figure 2.7 Best Fit of Graph for Cam Profile

Estimated actual equation

Best Method :

Lagrange and Divided Difference Order of 6 Polynomial

Methods and Errors

BACKGROUND OF STUDY IN CHAPTER 4

In this topic, an industrial engineer works as a quality control engineer for a company making toilet paper. The company advertises that every roll of toilet paper has at least 250 sheets.

To certify this claim by sampling rolls off of the assembly line and formulating the probability that the company can make the claim to prevent frivolous lawsuits related with false advertising.

Advantages and Disadvantages of Methods

Assume : Number of sheets in a roll of toilet paper, y, is ruled by the normal probability distribution, that is y~N(μ,σ^2) where N denotes to a normal random variable described by its mean, and standard deviation, .

Given that, a co-operative education student working for the industrial (quality) engineer samples 10 rolls of toilet paper on a monthly basis and defines the number of sheets in each roll and get the mean and standard deviation.

Samples of Toilet Roll

ȳ=252.2≈ μ

s=1.135 ≈σ

Real-life Application Question

A company advertises that every roll of toilet paper has at least 250 sheets. The probability that there are 250 or more sheets in the toilet paper is given by

Approximating the above integral as

Find the probability that there are between 250 and 270 sheets in each toilet roll.

Methods and Errors

Best method:

Composite Simpson’s Rule

with n=20

BACKGROUND OF STUDY IN CHAPTER 5

In this topic, a machine vision quality inspection system which inspects parts for defects. The parts are positioned in fixtures on a conveyor driven by a DC motor.

The conveyor is required to ramp up to a certain speed within a specific time, maintain a constant speed while the part is under the camera.

Many such applications which utilize DC motors have requirements associated with the ramp up time for the motor to reach this speed. Since, the speed of the motor can be controlled by changing the magnitude of this input voltage.

Methods, advantages and disadvantages

Real-life Application Question:

A machine vision quality inspection system which inspects parts for defects. The parts are positioned in fixtures on a conveyor driven by a DC motor. The conveyor is required to ramp up to a certain speed within a specific time, maintain a constant speed while the part is under the camera. If using the open loop response, the speed of the motor to a voltage input of 20 𝑉, assuming a system without damping is

If the motor is not run initially, what is the speed of motor at time 0.8 second?

Euler’s Method

Modified Euler Method

Runge-Kutta Methods of Order Four

Methods and Errors

Best method:

Runge-Kutta of Order Four

At 0.8 second, speed of motor equals to 303.0937 rad/s is slightly difference with actual solution equals to 303.0940156 rad/s

The smaller step size, the lower the absolute relative true error.

**Conclusion**

Numerical methods are very useful in engineering field since analytical methods are not so essential as the accurate data with minor error is always loss than the data input.

We let the equation (1) represent the linear relationship between the torque 𝑇 and the current 𝑖_𝑎 applied to the motor. The slope of the line is the torque constant 𝐾_𝑡 .

Equation (2) describes the linear relationship between the back 𝑒𝑚f 𝐸_𝐵 and the armature speed 𝑤(𝑡). The slope of the line 𝐾_𝐵 is the voltage constant.

Equation (3) describes the components of the armature voltage 𝑉 as sum of the back emf and the voltage drop across armature resistance Ri_𝑎.

Equation (4) describes the relationship between torque, acceleration and speed of the motor in a no-load system as the sum of the angular acceleration 𝑑𝑤/𝑑𝑡 multiplied by the inertia of the motor and the load, and the damping of the system 𝑓 multiplied by the armature speed.

Speed control of DC motors

Given:

Consider a DC motor which has the following specifications:

The open loop response of the motor to a voltage input using Equation (7) and assuming a system without damping. Therefore we get