Solving Mechanical Problem Using

Numerical Integration on Speed of Car: Simpon’s 1/3

Rule Method

Fauzan

Fauzi Bin Mohamad Nora’eni (AD150132) [email protected]

Mohd

Ruzaine Bin Kushairi, (AD150076) [email protected]

Muhammad Taufiq Bin Rohaizad (AD150185) [email protected]

Najmi Haziq bin Hanis Muzafery (AD150146) [email protected]

Universiti Tun Hussein Onn

Malaysia, Batu Pahat, Johor

Abstract: The

field of engineering study in solid mechanics of a rigid body which applied

force are frequently known as force associated with these motion called

kinematics motion. The

main point is on defining quantities like position, velocity, and acceleration.

It need to identify a reference frame and a coordinate system in it to get the

vector expression. The aim of this project is to relate the concept used in

Dynamics analysis using the mathematical analysis. Mathematical analysis used

in Chapter 6: Numerical Integration by using

Simpson’s 1/3 Rule. The results of the project show that the Solid

Mechanics 1 analysis is related with the mathematical analysis that have been

used. The conclusion can be drawn that the relative error between these two

methods were also had been calculated. Thus, it can determine which method was

accurate and precise

1.0

INTRODUCTION

1.1

OBJECTIVES

The

aim of this project is to:

1. To Determine the distance travelled by the

particle by using the actual method and computational method.

2. To

investigate the distance travelled by using the mathematical method which is

Simpson’s Rule.

3. Understanding

the concept of Numerical Integration and their application in mechanical

problems.

1.2 BACKGROUND OF STUDY

A study field of the solid mechanics with the

state of rest or motion of bodies subjected to the action of forces is called

mechanics. Engineering mechanics have two areas of study, where first is statics

and next is dynamics. Statics is related with the equilibrium of a body that is

either at rest or moves with constant velocity. Here we will consider dynamics,

which negotiated with the accelerated motion of a body. The subject of dynamics

will be presented in two parts: kinematics, which treats only the geometric

aspects of the motion, and kinematics.

Integration is the process of calculating the

area under a certain function plotted on a graph. Among the most common

examples are finding the velocity of a body from an acceleration function, and

displacement of a body from a velocity function. There are too many complex

problem and difficult equation in engineering study. For this reason, a variety

of numerical methods has been developed to simplify the integral.

Here, we

will discuss the Simpson’s 1/3 rule of approximating integrals of the form, , where f(x)

is the integrand, a is the lower

limit of the integration and b is the

upper limit of the integration. The trapezoidal rule was based on approximating

the integrand by a first order polynomial, and then integrating the polynomial

over interval of integration. Simpson’s

1/3 rule is an extension of Trapezoidal rule where a second order

polynomial approximates the integrand.

1.3 METHODS OF INVESTIGATION

The objective of this project is to

introduce and develop computational skills to student doing mathematical

calculation with calculating manual or using calculator. It also expose student

an easily computed method for solving 1/3 Simpson Method based on models using

Microsoft Excel spreadsheet. The approach comprises entering the key parameter

values into the spreadsheet and leading the model by answering a set of

equations based on these parameter values. For an example, in this project we

used distance travel by a particle moves along a horizontal path with a given

equation of velocity.

So in this project we do

calculations based on the Simpson 1/3 mathematics that we learn before this in

Engineering Mathematic 4 and used to find the value of numerical integral

equation. The methodology in this present study should help student to create

simple simulations in excel without they need to learn a programming language

or purchase expensive software.

Computational Method

The computational method was used to

conduct the simulation involved writing out the equations of velocity that was

given in the equation, keying the parameter values and equations into the

spreadsheet and leading the model by solving the equations.

The purpose in modelling the

velocity is to use experimentally obtained data to produce an accurate model of

a system. Besides, we use excel to make sure that our calculation are accurate

and it make us easy to know the correct graph. Excel has a beneficial feature

where in cell formulas are colour coded such as each of the cell denoted to in

a formula is emphasized with same colour as expression in the formula making

identification of cells referred to within formula easy.

Calculation Method

For calculation method in this project

we use Simpson 1/3 rule is to develop appropriate formulas for approximating

the integral of the form

(1) Most of the formulas

given for integration are based on a simple idea of approximating a given

function by

a simpler function (usually a polynomial function),

where represents the order of the polynomial

function. Simpsons 1/3 rule for integration was derived by approximating the

integrand with

a 2nd order (quadratic) polynomial function.

For this method, the number of

division or segments for N must be multiplication of 2.

This is an example for our

calculation method which is Simpson 1/3 rule:

0

1

0.7071

1

1.25

0.8333

2

1.5

0.9487

3

1.75

1.0553

4

2

1.1547

5

2.25

1.2481

6

2.5

1.3363

7

2.75

1.4201

8

3

1.5000

9

3.25

1.5765

10

3.5

1.6499

11

3.75

1.7206

12

4

1.7889

After

that, substitute this result to equation form

The

final result:

=

2.0 RESULT

Kinematics of particle

A car is moving

along a straight road for a short time. Its velocity is defined by, v = (3-5t)

m/s, where t is in seconds. Determine the distance travelled by the car from 1s

to 13s.

Solution:

Step 1: Write the

equation of the problem which is (3- 5t).

Step 2: Write the

formula of Simpson’s 1/3 Rule.

Where h = 1

Calculate manually,

n = 13

N = n – 1 = 12

h = = = 1

Table of result 1: Calculate manually.

t

y(t) = 3- 5t

1

-2

2

2

3

12

4

28

5

50

6

78

7

112

8

152

9

198

10

250

11

308

12

372

13

442

?

440

882

680

= (1) (440 + 4(882) +2(680)

=1776

a.

Calculation of Simpson’s ? rule

Step 1: in cell C4,

type = t for the value of time that used and drag the pointer until 13.

Step 2: in cell D4,

type = y(t) for the equation.

Step 3: in cell D5,

type = “=3*C5^2-5*C5”

Step 4: in cell E4,

type = multiplier

Step 5: in cell E5,

type = 1, in cell E6, type = 4 and in cell E7 type = 2. These value act as

multiplier.

Step 6: in cell F4,

type = product of the sum and in cell F5, type = “=D5*E5”.

Step 7: in cell F19,

type = “=(1/3)*SUM(F5:F17)*1″. This is the equation for Simpson’s ? rule.

Table of result 2: Calculation of Simpson’s ? rule.

Graph 1: Calculation of Simpson’s ? rule.

Calculation by integrate of the function from

graph

Step 1: in cell B3,

type = t and in cell B4, type = 1 and drag until 13.

Step 2: in cell C3,

type = the function and in cell C4 =” =B4^3-2.5*(B4^2)”.

Step 3: in cell C18,

type = total value of the equation and in cell D18, type = “=C16-C4”.

Table of result 3: Calculation by integrate of the function from the graph.

Graph 2: Calculation by integrate of the function from the graph.

Calculation of error

Step 1: in cell B2,

type = t, while in cell B3, type = 1 and drag until 13.

Step 2: in cell C2,

type = velocity, v (exact)

Step 3: in cell D2,

type = velocity, v (approximate).

Step 4: in cell E2,

type = error and in cell E3, type = “=ABS ((C3-D3)/C3)”

Step 5: in cell D19,

type = “= (C16-D16)/C16”.

Table of result 4: Calculation of error

b.

Another way to calculate the Simpson’s ? rule using excel

Step 1: in cell B4,

type = h and in cell C4, type = 1.

Step 2: in cell C6,

type = t and in cell C7, type = 1 and drag until 13.

Step 3: in cell D6,

type = Then, in cell D7 type = “=3*(C7^2)-5*C7” and

drag until 13.

Step 4: in cell E6,

type = Simpson’s and in cell E9 type = “= ($C$4/3)*(D7+4*D8+D9)” and drag until

13.

Table of result 5: Another way to calculate the Simpson’s ? rule

using excel

3.0 DISCUSSION

For this mathematics engineering 4

group project, we has choose the problem which is a moving object where the subtopic is

kinematics of a particle. The main purpose of this project problem is to find out

the distance travel from 1 to 13 seconds by using. For the solution we

use chapter 6 in our syllables which is Numerical Integration by using

Simpson’s 1/3 rules. The calculation is easy to find and can get it accurately

by using Excel spreadsheet. In Excel spreadsheet, we apply relative row,

relative column, and fixed column concepts to solve. We just type the value of

h which is 1 and then we list the value of t from 1 until13 which is to put in

the equation of y(t). For y(t), we just have to type an equation of it and

Excel will give the value of y(t). Based on the final report, the actual result

and theoretical for the equation is 1776. So, the error between the actual

result and theoretical is 0 due to the same answer.

4.0 CONCLUSION

For

the conclusion, the result showed from

Simpsons Rule was always accurate. We see that

some of the situations were hardly to know the function governing some

phenomenon exactly and it is still possible to derive a reasonable estimate for

the integral of the function based on data points. The idea is to choose a

model function going through the data points and integrate the model function.

We also seen that there

are many theoretical factors that affect the numerical integration works such

as the number of data point located is affect the result. Simpsons way more accurate

compared to the other methods such as trapezoidal method and rectangular

method.

In this group project

assignment, the value of N that we used which is 13 and that makes the value of

. The increment is 1 starting from 1 until

reading 13. From the result, we clearly see that it is accurate. The final reading

is 1776 which it can be take out from calculation. Besides that, we have

compared our calculation result with excel spreadsheet and the excel give the

value also 1776 which is 100% accurate and same. We inserted a graph for the

theoretical calculation and actual calculation to show the related between

calculate manually and by computerisation method.

The

excel method proved that it is the fast and accurate way compared to manual

calculation. For an example, by simply entering the value in the Row below it can

calculate the increment in the fast and accurate way which is simple procedure.

Once we put in the precise formula into the spreadsheet, an Autofill which is

the process of copying will ensure that the value in-cell formula is correct

and error free.

5.0 REFERRENCE

1. Retrieved

from http://www2.math.umd.edu/~dlevy/classes/amsc466/lecture-notes/integration-chap.pdf

2. Retrieved

from https://www.math.ust.hk/~mamu/courses/231/Slides/CH04_3A.pdf

3. R.C,

Hibbeler (2010) Dynamics 12th

edition, United State of America: Pearson.

4. BDA34003 Engineering

Mathematics iv module (2017) 1st

edition, UTHM Publisher, Ong Pauline, Waluyo Adi Siswanto, Saifulnizan

Jamian.

5. Retrieved from www.damtp.cam.ac.uk/lab/people/sd/lectures/nummeth98/integration.htm

6. Retrieved form https://en.wikipedia.org/wiki/Numerical_integration