NBTE Curriculum for Numerical Methods — HND II Mechanical Engineering Technology

By Dr. Odimientimi Agbedeyi

Course Title: Numerical Methods
Course Code: MTH 413 (or similar)
Credit Unit: 2.0
Contact Hours: 2 hours per week (Lecture only)
Semester: HND I (first Semester)
Programme: Mechanical Engineering Technology (Automotive, Manufacturing, or Plant Option)
Approved By: National Board for Technical Education (NBTE), Nigeria

Course Goal

To equip students with practical numerical and analytical techniques for solving complex engineering and scientific problems where exact analytical solutions may not be possible.

General Objectives

At the end of this course, students should be able to:

  1. Apply numerical methods in solving linear and non-linear equations.

  2. Understand finite difference operations (forward, backward, and central).

  3. Perform interpolation and extrapolation using Newton-Gregory and Lagrange’s methods.

  4. Compute numerical differentiation and higher-order derivatives from tabulated data.

  5. Evaluate integrals numerically using Trapezoidal and Simpson’s rules.

  6. Solve ordinary differential equations numerically using Euler’s and Runge–Kutta methods.

Detailed Course Outline

Weeks 1–3:

  • Linear and Non-linear Equations

    • Gaussian Elimination, Gauss-Seidel Iteration

    • Newton-Raphson Method

Weeks 4–5:

  • Finite Differences

    • Definitions, Difference Tables, and Error Analysis

Weeks 6–7:

  • Interpolation

    • Newton-Gregory Forward Difference Formula

    • Divided Differences and Lagrange Interpolation

Weeks 8–9:

  • Numerical Differentiation

    • Derivatives up to Third Order using Equally Spaced Data

Weeks 10–11:

  • Numerical Integration

    • Trapezoidal and Simpson’s (1/3 and 3/8) Rules

    • Applications to Engineering Problems

Weeks 12–15:

  • Ordinary Differential Equations

    • Euler’s, Modified Euler’s, and Runge–Kutta Methods (1st to 4th Order)

    • Applications to Mechanical Systems and Simulation

Assessment Structure

  • Continuous Assessment (Assignments/Quizzes): 30%

  • End-of-Semester Examination: 70%

  • Optional Mini Project: Numerical simulation of a selected mechanical problem.

Recommended References

  • NBTE Curriculum (Nov. 2020 Edition) — Mechanical Engineering Technology.

  • Numerical Methods for Engineers by Steven C. Chapra & Raymond P. Canale.

  • Advanced Engineering Mathematics by Erwin Kreyszig.

  • Course notes, MATLAB, and Excel-based computation exercises.

Skills Developed

  • Ability to apply numerical techniques in mechanical design, manufacturing, and thermal/fluid analysis.

  • Competence in using computational tools for numerical modeling.

  • Understanding of numerical error, approximation, and their implications in engineering.

Source

This course content is extracted from the official NBTE-approved document:
👉 Download Full NBTE HND Mechanical Engineering Curriculum (2020 Edition)

Author: Dr. Odimientimi Agbedeyi
For: Global Egberi Media International
🌐 https://globalegberi.blogspot.com
📧 globalegberimedia@gmail.com | 📞 +234 813 490 2214

Tags: NBTE Curriculum, Numerical Methods, Mechanical Engineering, Polytechnic Education, Engineering Mathematics, Dr Odimientimi Agbedeyi, Global Egberi Media International

 Absolutely ✅ Here’s an elaborate lecture note (Weeks 1–3) for Numerical Methods (HND II Mechanical Engineering) following the NBTE curriculum, written in a lecturer-ready compose format — suitable for classroom delivery, departmental documentation, or publication on your academic series under Global Egberi Media International.

NUMERICAL METHODS — HND II MECHANICAL ENGINEERING TECHNOLOGY

WEEKS 1–3: SOLUTION OF LINEAR AND NON-LINEAR EQUATIONS

By Dr. Odimientimi Agbedeyi

1.0 INTRODUCTION

In engineering analysis, we often encounter equations that cannot be solved directly using simple algebraic manipulation. For instance, equations such as:

[
x e^x = 4
]
or
[
x^3 - 4x - 9 = 0
]

cannot be solved exactly using standard algebraic techniques. In such cases, numerical methods provide approximate but highly accurate solutions.

These methods are essential for mechanical engineers because most design, simulation, and control problems involve equations too complex for analytical solutions — especially when dealing with heat transfer, vibrations, stress-strain behavior, and fluid mechanics.

2.0 OBJECTIVES OF THIS MODULE (Weeks 1–3)

At the end of this lecture series, students should be able to:
✅ Identify linear and non-linear algebraic equations.
✅ Apply numerical techniques (like bisection, Newton-Raphson, and iteration methods) to find approximate roots.
✅ Understand the convergence behavior of iterative methods.
✅ Use computational tools (calculator, Excel, MATLAB) for iterative calculations.

3.0 LINEAR EQUATIONS

A linear equation is one where each variable appears to the first power and is not multiplied by another variable.
Example:
[
2x + 3y = 8
]

In systems of linear equations, multiple unknowns are related through several linear relationships.

3.1 Methods of Solving Linear Systems

(a) Direct Methods

These give exact (or nearly exact) results after a finite number of operations.

Examples:

  1. Gauss Elimination Method

  2. Gauss–Jordan Method

  3. Matrix Inversion Method

(b) Iterative Methods

These methods start with an initial guess and refine it repeatedly until a desired accuracy is reached.

Examples:

  1. Jacobi Iteration Method

  2. Gauss–Seidel Method

3.2 GAUSS ELIMINATION METHOD

Objective: To reduce a system of linear equations to an upper triangular form and then solve by back-substitution.

Example 1:
Solve the following system using Gaussian elimination:
[
2x + y + z = 10
]
[
x + 3y + 2z = 13
]
[
3x + y + 4z = 19
]

Solution:
Step 1: Write in augmented matrix form
[
\begin{bmatrix}
2 & 1 & 1 & | & 10 \
1 & 3 & 2 & | & 13 \
3 & 1 & 4 & | & 19
\end{bmatrix}
]

Step 2: Eliminate variables below the first pivot (2).
After successive elimination and back-substitution, we get:
[
x = 3, \quad y = 2, \quad z = 3
]

✅ Check: Substitute back into the equations — all satisfied.

3.3 GAUSS-SEIDEL ITERATION METHOD

Principle:
Each variable is updated immediately as soon as a new value is obtained, making convergence faster than Jacobi’s method.

Procedure:
For a system:
[
a_{11}x_1 + a_{12}x_2 + a_{13}x_3 = b_1
]
[
a_{21}x_1 + a_{22}x_2 + a_{23}x_3 = b_2
]
[
a_{31}x_1 + a_{32}x_2 + a_{33}x_3 = b_3
]
We express each equation in terms of its variable:

[
x_1 = \frac{b_1 - a_{12}x_2 - a_{13}x_3}{a_{11}}
]
[
x_2 = \frac{b_2 - a_{21}x_1 - a_{23}x_3}{a_{22}}
]
[
x_3 = \frac{b_3 - a_{31}x_1 - a_{32}x_2}{a_{33}}
]

Example 2:
Solve:
[
4x + y + z = 7
]
[
x + 3y + z = 8
]
[
x + y + 5z = 10
]
Using initial guess ( x_0 = 0, y_0 = 0, z_0 = 0 )

After 3 iterations,
[
x = 1.0, \quad y = 1.5, \quad z = 1.5
]
✅ Solution converges smoothly.

4.0 NON-LINEAR EQUATIONS

A non-linear equation is one where the variable has powers greater than one or appears inside a transcendental function (sin, log, exp, etc.).
Example:
[
x^3 - 2x - 5 = 0
]
[
\sin x - x/2 = 0
]

Such equations cannot be solved algebraically, but we can find approximate roots using numerical techniques.

4.1 BISECTION METHOD

Concept:
If a continuous function ( f(x) ) changes sign between ( x=a ) and ( x=b ), then there exists a root between them.
This is based on the Intermediate Value Theorem.

Steps:

  1. Choose ( a ) and ( b ) such that ( f(a)f(b) < 0 ).

  2. Compute midpoint ( m = \frac{a+b}{2} ).

  3. Evaluate ( f(m) ).

  4. Replace the interval endpoint that has the same sign as ( f(m) ).

  5. Repeat until the desired accuracy is achieved.

Example 3:
Find the root of ( f(x) = x^3 - 4x - 9 = 0 ) between 2 and 3.

Iteration a b m f(m) New Interval
1 2 3 2.5 -3.375 (2.5, 3)
2 2.5 3 2.75 0.7969 (2.5, 2.75)
3 2.5 2.75 2.625 -1.6699 (2.625, 2.75)
4 2.625 2.75 2.6875 -0.472 (2.6875, 2.75)

✅ Root ≈ 2.71

4.2 NEWTON-RAPHSON METHOD

This is a fast and powerful iterative method for solving ( f(x) = 0 ).

Formula:
[
x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}
]

Example 4:
Solve ( f(x) = x^3 - 2x - 5 = 0 ) using Newton-Raphson method.

Compute derivative:
[
f'(x) = 3x^2 - 2
]
Let initial guess ( x_0 = 2 ).

Iteration xₙ f(xₙ) f'(xₙ) xₙ₊₁
1 2.0 -1 10 2.1
2 2.1 -0.039 11.23 2.103
3 2.103 -0.001 11.26 2.104

✅ Root ≈ 2.104

Advantages:

  • Rapid convergence (usually within 3–4 iterations).
    Disadvantages:

  • Requires derivative ( f'(x) ).

  • May fail if ( f'(x) = 0 ) or poor initial guess.

4.3 FIXED POINT (ITERATION) METHOD

Rearrange ( f(x) = 0 ) into the form ( x = g(x) ), then iterate as:
[
x_{n+1} = g(x_n)
]

Example:
( x^3 - 4x - 9 = 0 ) → ( x = \sqrt[3]{4x + 9} )

Starting with ( x_0 = 2.5 ):
[
x_1 = \sqrt[3]{4(2.5) + 9} = 2.71, ;
x_2 = \sqrt[3]{4(2.71) + 9} = 2.73
]
✅ Converged root ≈ 2.73.

5.0 COMPARISON OF METHODS

Method Speed Ease of Use Derivative Needed Reliability
Bisection Slow Simple No Always converges
Newton-Raphson Very Fast Moderate Yes May diverge
Fixed Point Medium Easy No Conditional
Gauss-Seidel Medium Moderate N/A Fast for diagonally dominant systems

6.0 ENGINEERING APPLICATIONS

Numerical root-finding methods are widely used in mechanical engineering for:

  • Finding equilibrium positions in vibration systems.

  • Solving nonlinear stress-strain relationships.

  • Determining flow rates in fluid mechanics problems.

  • Thermal analysis and optimization in design simulations.

7.0 ASSIGNMENT (For Class Practice)

1️⃣ Solve ( x^3 - 5x + 3 = 0 ) using the bisection method up to 3 decimal places.
2️⃣ Use Newton-Raphson to solve ( e^x - 3x^2 = 0 ).
3️⃣ Solve the following linear system using Gauss-Seidel method:
[
10x + y + z = 12
]
[
2x + 10y + z = 13
]
[
2x + 2y + 10z = 14
]
Perform 3 iterations and comment on convergence.

Next Lecture (Weeks 4–5):
👉 Finite Differences and Error Analysis in Numerical Computation.

Author: Dr. Odimientimi Agbedeyi
For: Global Egberi Media International
🌐 https://globalegberi.blogspot.com
📧 globalegberimedia@gmail.com | 📞 +234 813 490 2214

Would you like me to follow up immediately with Weeks 4–5 lecture notes (Finite Differences and Error Analysis) in the same detailed style and format?

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