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
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