Free Body Diagram (FBD)

Sketching Free Body Diagrams (FBDs) is probably one of the most important skill you can acquire as an engineer.   Believe it or not, solving most engineering problems begin with a FBD.

To use the equation of equilibrium we must be aware of all the forces acting on the object.  If you're dealing with one or two forces then maybe it's easy to just visualize your problem but more often than not we are looking at problems with many many forces (well not that many) so it is best to put pen to paper.

These are the simple steps of producing an Free Body Diagram

1. The best way to start a FBD is to sketch the object we are analyzing in such a way that it is isolated from its surrounding.  
2. Once you have an outline you can start inserting all the known and unknown forces acting on the object.
3. Lastly, all the forces identified should be labeled.

• Known forces are labeled with its magnitude and direction
• As for unknown forces, its magnitudes are represented by alphabets while it's direction can be directed randomly (upon solving the problem, it's true direction will become known)

That's it for this post, stay tuned for my next post when I show you some Free Body Diagram sketching examples.

Condition for Equilibrium

An object is in equilibrium if it satisfies Newtons' first law:

"An object at rest or moving in a straight line with constant velocity will remain in this state unless it is subjected to an unbalanced force"

Quite simply put, an object is in equilibrium if the sum of forces acting on it equals zero:

This equation is called the equation of equilibrium.

Engineering Mechanics (Statics) - Equilibrium of a Particle

Before we discuss "equilibrium" lets go back a little and see what we understand about the whole subject of Mechanics.

Mechanics is a subject that is concerned with what happens to the motion of an object when forces are acting on it.  Mechanics can be divided into three branches¹:

1. Rigid-body mechanics (under the assumption that forces acting on the object cause no deformation upon the object)
2. Deformable-body mechanics
3. Fluid mechanics

At this point in time we are concerned with rigid-body mechanics which can be further divided into two areas:

1. Statics
2. Dynamics

When we talk of equilibrium we are in the Statics area of rigid-body mechanics.

An object is said to be in equilibrium when it is static or moving at constant velocity (not accelerating)

Dynamics on the other hand deals with object that are accelerating.

Addition of Coplanar Forces - Using Cartesian Vector Notation

Useful book to have

Schaum's Outline of Engineering Mechanics: Statics

For problems and solutions on addition of coplanar forces click HERE.

Given the coplanar forces F₁, F₂ and F₃ below, find the resultant force F_R = F₁ + F₂ + F₃

Step 1:

Represent each force as a Cartesian vector:

Step 2:

Finally, determine the resultant force: 

Addition of Coplanar Forces – Using Scalar Notation

For problems and solutions on addition of coplanar forces click HERE.

Given the coplanar forces F₁, F₂ and F₃ below, find the resultant force F_R = F₁ + F₂ + F₃

Step 1:

Resolve the forces into their rectangular components i.e.:





Step 2:

Determine the resultant force in the x-axis and y-axis independently i.e.:



Step 3:

Finally determine the magnitude of the resultant force F_R and its direction θ :

For problems and solutions on addition of coplanar forces click HERE.

Coplanar Forces and their Rectangular Components

Coplanar forces: a set of forces is coplanar if they all lie in the same geometric plane.  In the diagram below forces F₁, F₂ and F₃ are coplanar because the all lie in the x-y plane.  Refer to the image below.

Coplanar forces can be resolved into components along the x and y axes.  These component forces are called rectangular components.  Rectangular components can be represented in two ways:

1. Scalar notation


2. Cartesian vector notation

Scalar Notation: When using the scalar notation the rectangular components of force F is written F_x and F_y.

• F_x represents the component in the x-axis


• F_y represents the component in the y-axis

Refer to the image below.

Cartesian Vector Notation: When using the Cartesian vector notation, the rectangular components are represented in terms of unit vectors i and j.  Unit vectors are vectors that have a magnitude of one and they are used to represent direction.  In this case:

• i represents direction in x-axis


• j represents direction in y-axis

Refer to the image below.

Vector Subtraction and Resolution of a Vector

Vector Subtraction

Given vectors A and B as follows:



The parallelogram for vector subtraction takes the following form:





Resolution of a Vector

A vector may be resolved into to components having known lines of action using the parallelogram law:

Given the vector R below:



Extend parallel lines from the head of vector R to form components in a and b axis



The resolution of vector R into vector A and vector B is as follows:

Two Dimenstional Vector Addition - The Parallelogram Law

For problems and solutions on the parallelogram law, click HERE.

Two vectors of the same type (i.e. two force vectors or two velocity vectors) can be added together  to obtain a single resultant vector.

Addition of two vectors can be done using the parallelogram law.

Given two vectors A and B find the resultant vector R = A + B



Step 1:

Join vectors A and B at their tails:



Step 2:

Draw parallel lines from the head of each vector to form a parallelogram:



Step 3:

To obtain the resultant of the two vectors draw the diagonal of the parallelogram starting from the tail of the two vectors:



Step 4:

Sketch the triangle half of the parallelogram and  use “law of cosines” and “law of sines” to determine magnitude and direction of  R:




Law of cosines:

Law of sines:

For problems and solutions on the parallelogram law, click HERE.

Glossary:

Parallel lines

Two or more lines that will NEVER meet each other.

Parallelogram

A quadrilateral (a four sided polygon) whose opposite sides are both parallel and equal in length.

For problems and solutions on the parallelogram law, click HERE.

Multiplication and Division of a Vector by a Scalar

The diagram below shows that when a vector is multiplied by a scalar, its magnitude changes but not its direction.  For example if vector A is multiplied by 2, its magnitude is doubled (i.e. length of the arrow doubles).  The same concept applies to the division of vectors.

Force Vectors - Scalars and Vectors

 

Scalars and Vector Table of Comparison

Download Engineering Mechanics Lecture Notes

Please click the following for my blog notes on Engineering Mechanics.

Scalars and Vectors

Multiplication and Division of a Vector by a Scalar

Two Dimentional Vector Addision - Parallelogram law

Vector Subtraction and Resolution of a Vector

Coplanar Forces and their Rectangular Components

Addition of Coplanar Forces - Using Scalar Notation

Addition of Coplanar Forces - Using Cartesian Vector Notation

Equilibrium of a Particle

Condition for Equilibrium

Free Body Diagram (FBD)

Below is a list of engineering mechanics tutorials.  Click on the individual topic to download the notes and don’t forget to print a copy for the lectures.

Introduction

Force Vector Tutorial

Equilibrium of a Particle

Force System Resultant

As you probably notice the list is not complete, they will be in due time.  The lecture notes are based on your recommended textbooks.  You are encouraged to refer to them for a wider understanding of the subject.

Download Material Science Lecture Notes

In my list below are the lecture notes for materials science.  Unfortunately it is not a complete list as I can't find some of my files.  I'm hoping I would find the missing files soon as I dread having to do it all over again!  If that be the case it might take some time to get the missing notes online.

To download click on the individual titles below and remember to make copies for the lectures.

1. Introduction


2. Atomic Structures and Bonding Part 1


3. Atomic Structures and Bonding Part 2


4. Atomic Structures and Bonding Tutorial


5. Crystalline Structure


6. Defects in Solids


7. Diffusion


8. Mechanical Properties


9. Material Failure


10. Phase Diagrams


11. Engineering Alloys


12. Polymers


13. Corrosion


14. Non Destructive Testing

These notes are highly based on your recommended textbooks.  You are encouraged to refer to these books to gain higher understanding of the topics discussed.

Yajeta Jagadiswari

Download Heat Transfer Lecture Notes

Below is a list of heat transfer lecture notes.  Click on the individual topic to download the notes and don’t forget to print a copy for the lectures.

1. Introduction


2. Heat Transfer Summary Table


3. Introduction Tutorial


4. Conduction


5. Conduction Tutorial 1


6. Insulation


7. Convection Introduction and Dimensionless Numbers


8. External Forced Convection


9. Internal Forced Convection


10. Forced Convection Tutorial


11. External Free Convection


12. Free Convection Tutorial


13. Introduction and Types of Heat Exchangers


14. Analysis of Heat Exchangers


15. Log Mean Temperature Difference (LMTD) Method


16. Effectiveness NTU Method


17. Selection of Heat Exchangers


18. Radiation


19. View Factor Tables and Graphs

The lecture notes are based on your recommended textbooks.  You are encouraged to refer to them for a wider understanding of the subject.

Yajeta Jagadiswari

Download thermodynamics Lecture Notes

You lecture notes are given below.  Click on the individual topic to download the notes and don't forget to print a copy for the lecture.

1. Introduction


2. Pure Substances


3. First Law of Thermodynamics - Closed System


4. First Law of Thermodynamics - Closed System Tutorial


5. First Law of Thermodynamics Control Volume


6. Isentropic Efficiency


7. The Second Law of Thermodynamics


8. The Second Law of Thermodynamics Tutorial


9. Entropy


10. Entropy Tutorial


11. Carnot Cycle


12. Vapour Power Cycle


13. Gas Power Cycle


14. Refrigeration Cycle

The lecture notes are mostly based on your recommended text book, Thermodynamics An Engineering Approach by Y.A. Cengel and M.A. Boles.  I suggest you refer to the book to get a wider understanding of the subject.

Yajeta Jagadiswari

Recomended Textbooks

The following are recommended textbooks for subject I teach.  My notes are mostly based of these books.  You are encouraged to refer to these books as much as possible to have a greater understanding of the subject.

1) Thermodynamics

Main textbook:

Cengel Y.A., Boles M.A. (2007) Thermodynamics and Engineering Approach, McGraw Hill (6^th Edition)

Additional Textbook:

Moran M.J., Howard N.S., (2007), Fundamentals of Engineering Thermodynamics, Wiler (6th Edition)

 

2) Heat Transfer

Main textbook:

F.P. DeWitt, D.P., Bergman T. L.,Lavine A.S., (2006), Fundamentals to Heat Transfer, 5th Edition, John, Wiley & Sons.

Additional textbook:

Cengel Y.A (2006), Heat Transfer: A Practical Approach, McGraw-Hill Companies (2Edition)

 

3) Engineering Mechanics

Main textbook:

Russell C. Hibbeler, (2006), Engineering Mechanics – Statics, Prentice Hall (11th Edition)

Additional textbook:

J. L. Meriam, L.G. Kraige, (2006), Engineering Mechanics – Statics, Wiley (6 edition)

 

4) Materials Science

Main textbook:

Callister, W.D., Fundamentals of Materials Science and Engineering, John, Wiley & Sons

Additional textbook:

James F. Shackelford, (2009), Introduction to Materials Science for Engineers (7th Edition)

 

5) Industrial Management

Main textbook:

Mar, D.D., Operations and Industrial Management, McGraw Hill 1985.

Additional textbook:

Turner, W.C., Mize, J.H., Case, K.E., Introduction to Industrial and Systems Engineering, Prentice Hall, 3rd. edition, 1993.

 

Yajeta Jagadiswari