Core Engineering Mechanics and DC Motor Principles

Fundamental Principles of Engineering Mechanics

Principle of Transmissibility: A force acting on a rigid body can be moved to any other point along its line of action without changing its external effect on the body. The magnitude and direction of the force remain the same during this shift. This principle is applicable only to rigid bodies and not when internal stresses or deformations are considered.

Varignon’s Theorem: The moment of a force about any point is equal to the sum of the moments of its components about the same point. In other words, if a force is resolved into components, the algebraic sum of the moments of these components about a point is equal to the moment of the original force about that point.

Parallelogram Law of Forces: If two forces acting simultaneously at a point are represented in magnitude and direction by the two adjacent sides of a parallelogram, then their resultant is represented in magnitude and direction by the diagonal of the parallelogram passing through that point.

Lami’s Theorem: If a body is in equilibrium under the action of three coplanar, concurrent, and non-collinear forces, then each force is proportional to the sine of the angle between the other two forces; that is, the ratio of each force to the sine of the angle between the other two is constant.

Force System: A force system is a group of two or more forces acting simultaneously on a body. These forces may have different magnitudes, directions, and lines of action, and their combined effect determines the state of rest or motion of the body.

Moment of Force: The moment of a force is the turning effect produced by a force about a point or axis, and it is equal to the product of the force and the perpendicular distance of its line of action from the point.

Couple Law: A couple consists of two equal and opposite forces whose lines of action do not coincide, producing only rotation. Its moment is the same about any point and is given by M = F × d.

DC Machine Fundamentals and Operations

DC Machine: A DC machine is an electrical device that works as a generator to convert mechanical energy into DC electrical energy or as a motor to convert DC electrical energy into mechanical energy. It consists of a stator (field), rotor (armature), commutator, and brushes.

Principle of DC Machine

A DC machine works on the principle of electromagnetic induction, where a conductor moving in a magnetic field induces an EMF, and a current-carrying conductor in a magnetic field experiences a mechanical force.

Construction of DC Machine

A DC machine is made up of two main parts: the stator and the rotor. The stator consists of the yoke, poles, pole shoes, and field windings which produce the magnetic field, while the rotor includes the armature core, armature winding, commutator, and brushes that rotate and carry current.

Working of DC Machine

When working as a motor, electrical energy is supplied to the armature which interacts with the magnetic field to produce rotation and mechanical energy. In generator operation, mechanical energy rotates the armature in a magnetic field to induce EMF, and the commutator converts it into direct current.

Types of DC Machine

DC machines are classified into DC motors and DC generators. Based on excitation, they are either separately excited or self-excited. Self-excited machines are further divided into shunt, series, and compound types.

Applications of DC Machine

DC machines are used where variable speed and high starting torque are required, such as in electric traction, cranes, hoists, elevators, conveyors, and rolling mills. DC generators are used for battery charging, electroplating, welding, and as exciters.

EMF Equation of DC Generator (Derivation)

Torque Equation of DC Motor (Derivation)

Brushless DC (BLDC) Motor Technology

BLDC Motor (Brushless DC Motor): A BLDC motor is an electric motor that operates without brushes and uses electronic commutation instead of a mechanical commutator. It works on the principle of interaction between a rotating magnetic field and permanent magnets, providing high efficiency, low maintenance, and long life.

Principle of BLDC Motor

A BLDC motor works on the principle of electromagnetic interaction, where a current-carrying conductor placed in a magnetic field experiences a force. In this motor, a rotating magnetic field is produced electronically which interacts with the permanent magnets of the rotor to produce rotation.

Construction of BLDC Motor

A BLDC motor consists of a stator and a rotor. The stator is made up of laminated steel with windings placed in slots, while the rotor consists of permanent magnets. It also includes an electronic controller and position sensors (like Hall sensors) to control the switching of current in the stator windings.

Working of BLDC Motor

When power is supplied, the electronic controller energizes the stator windings in a sequence, creating a rotating magnetic field. This rotating field interacts with the permanent magnets on the rotor, causing it to rotate continuously. The position sensors provide feedback to ensure proper switching and smooth operation of the motor.

Applications of BLDC Motor

BLDC motors are widely used in applications such as electric vehicles, computer fans, drones, washing machines, air conditioners, robotics, and industrial automation due to their high efficiency, reliability, and precise speed control.

Classification of Force Systems

Nodal Analysis and KCL Problem Solutions

Apply KCL at Node 1:
(V1 - 3) / 1 + V1 / 2 + (V1 - V2) / 2 = 0
Multiply by 2:
2(V1 - 3) + V1 + (V1 - V2) = 0
4V1 - V2 = 6 (Equation 1)

Apply KCL at Node 2:
(V2 - V1) / 2 + V2 = 2
Multiply by 2:
(V2 - V1) + 2V2 = 4
-V1 + 3V2 = 4 (Equation 2)

From Equations 1 & 2:
V1 = 2V, V2 = 4V

Apply KCL at Node B:
(VB - VA) / 2 + (VB - VC) / 5 + VB / 100 = 0
Apply KCL at Node C:
(VC - VB) / 5 + VC / 20 = 9
On solving the two KCL equations:
VB = 10V, VC = 0V

Voltage across 5 Ohm resistor:
V = VB - VC = 10 - 0 = 10V
Current through 5 Ohm resistor:
I = (VB - VC) / 5 = 10 / 5 = 2A
Current through 12V source:
I = (12 - VB) / 2 = (12 - 10) / 2 = 1A

Apply KCL at Node 1:
V1 / 2 + (V1 - V2) / 3 + (V1 - V3) / 5 = 10
At Node 2:
(V2 - V1) / 3 + V2 / 5 + (V2 - V3) = 0
At Node 3:
V3 / 4 + (V3 - V2) + (V3 - V1) / 5 = 2
On Solving:
V1 = 4V, V2 = 2V, V3 = 1V

Current Calculations:
I (2 ohm) = 4 / 2 = 2A
I (3 ohm) = (4 - 2) / 3 = 0.67A
I (5 ohm) = (4 - 1) / 5 = 0.6A
I (4 ohm) = 1 / 4 = 0.25A

Mesh Analysis and KVL Applications

Apply KVL starting from node D clockwise:
20 - 6i1 - 5i1 - 9i1 = 0
20 - 20i1 = 0
i1 = 1A

Apply KVL starting from node G counter-clockwise:
40 - 8i2 - 5i2 - 7i2 = 0
40 - 20i2 = 0
i2 = 2A

Assume Node G as the reference (ground, 0V):

• VG = 0V
• VF = 40V
• VE = VF - (8 × i2) = 40 - 16 = 24V
• VH = VE - (5 × i2) = 24 - 10 = 14V
• VB = VH + 10 = 14 + 10 = 24V
• VA = VB + (6 × i1) = 24 + 6 = 30V
• VC = VB - (5 × i1) = 24 - 5 = 19V
• VCE = VC - VE = 19 - 24 = -5V
• VAG = VA - VG = 30 - 0 = 30V

Structural Engineering: Types of Beams

Simply Supported Beam

A beam that is supported at both ends (usually one hinge and one roller support) is called a simply supported beam.

Overhanging Beam

A beam in which one or both ends extend beyond the supports is called an overhanging beam.

Cantilever Beam

A beam that is fixed at one end and free at the other end is called a cantilever beam. At the fixed end, there are three reactions: Horizontal reaction (RH), Vertical reaction (RV), and Moment (M).

Propped Cantilever Beam

A beam that is fixed at one end and has an extra simple support at the other end (to reduce bending or deflection) is called a propped cantilever beam.

Continuous Beam

A beam that rests on more than two supports is called a continuous beam.

Advanced Nodal Analysis Problems

Step 1: Apply KCL at Node V1
(V1 - V2) / 1 + (V1 - V3) / 1 = 5
2V1 - V2 - V3 = 5 (Equation 1)

Step 2: Apply KCL at Node V2
(V2 - V1) / 1 + (V2 - V3) / 1 + V2 / 4 = 0
Multiplying by 4:
4(V2 - V1) + 4(V2 - V3) + V2 = 0
-4V1 + 9V2 - 4V3 = 0 (Equation 2)

Step 3: Apply KCL at Node V3
(V3 - V2) / 1 + (V3 - V1) / 1 = 10
-V1 - V2 + 2V3 = 10 (Equation 3)

Step 4: Solve the Equations
From Equations 1, 2, and 3:
V1 = 8.89V, V2 = 8.64V, V3 = 10.56V

Let: Left top node = VA, Middle top node = VB, Right top node = VC = 12V, Bottom node = 0V.

KCL at Node A:
VA / 5 + (VA - VB) / 2 = 7
Multiply by 10:
2VA + 5(VA - VB) = 70
7VA - 5VB = 70 (Equation 1)

KCL at Node B:
(VB - VA) / 2 + VB / 10 + (VB - 12) / 6 = 0
Multiply by 30:
15(VB - VA) + 3VB + 5(VB - 12) = 0
-15VA + 23VB = 60 (Equation 2)

Solving Equations 1 and 2:
VA = 22.21V, VB = 17.09V
Current through 2 ohm resistor:
I = (VA - VB) / 2 = (22.21 - 17.09) / 2 = 2.56A

Let the top center node be V1 and the bottom center node be V2.

Apply KCL at Node V1:
(V1 - 18) / 1 + (V1 - (V2 + 12)) / 3 + (V1 - V2) / 6 = 0
Multiply by 6:
6(V1 - 18) + 2(V1 - V2 - 12) + (V1 - V2) = 0
9V1 - 3V2 = 132 => 3V1 - V2 = 44

Apply KCL at Node V2:
V2 / 2 + (V2 - V1) / 6 + (V2 + 12 - V1) / 4 = 0
Multiply by 12:
6V2 + 2(V2 - V1) + 3(V2 + 12 - V1) = 0
-5V1 + 11V2 = -36

Solving for V1 and V2:
V1 = 16V, V2 = 4V
Current through the 6 ohm resistor:
I = (V1 - V2) / 6 = (16 - 4) / 6 = 2A