Short Answer:

The inertia torque is the torque required to overcome the inertia of a rotating or reciprocating mass when its angular velocity changes. It is the opposing torque developed due to the mass moment of inertia of the moving parts. In simple words, inertia torque resists any change in rotational speed and must be balanced by the applied or driving torque to maintain motion.

It is an important concept in dynamics of machines because it affects the smoothness of motion in mechanisms like engines and flywheels. The inertia torque acts in a direction opposite to the angular acceleration of the rotating part and plays a key role in power transmission and vibration control.

Detailed Explanation :

Inertia Torque

The concept of inertia torque arises from Newton’s second law of motion applied to rotational systems. When a rotating body or system is subjected to a torque, it starts to rotate with some angular acceleration. However, every mass resists a change in its state of motion due to its inertia. This resistance in rotational motion is expressed as a torque and is called inertia torque.

Inertia torque is the torque developed by the mass moment of inertia of a rotating body when it experiences angular acceleration or deceleration. It always acts in the opposite direction to the applied torque that causes the angular acceleration. In an engine, when the piston and crank rotate, the speed is not constant throughout the stroke. The change in angular velocity gives rise to the inertia torque, which must be considered for balancing and smooth running of the engine.

Derivation of Inertia Torque

Let a rotating body have:

• Mass moment of inertia =  (kg·m²)
• Angular acceleration =  (rad/s²)

According to Newton’s second law for rotational motion,

Here,  is the torque required to produce an angular acceleration . This torque is called inertia torque.

The direction of inertia torque is always opposite to the direction of the applied torque or angular acceleration. This means if the body is accelerating in the clockwise direction, the inertia torque acts anticlockwise, and vice versa.

Hence,

and

for linear motion.

Physical Meaning

Inertia torque represents the tendency of a rotating body to resist any change in its angular motion. For example, when a flywheel is rotating and an external torque is applied to increase its speed, some part of that torque is spent in overcoming the inertia torque. Similarly, when the speed decreases, the inertia torque assists in maintaining motion.

It can be said that the inertia torque stores or releases energy depending on whether the system is accelerating or decelerating. This property is utilized in mechanical systems like flywheels, which help maintain a uniform speed by balancing out the effect of varying inertia torques.

Inertia Torque in Engines

In reciprocating engines, various parts like the crankshaft, connecting rod, and piston undergo periodic acceleration and deceleration during each stroke. Due to this, they produce inertia forces and inertia torques. These torques cause fluctuations in the engine speed and vibration in the mechanism.

To maintain uniform motion, the engine must develop sufficient turning moment (driving torque) to overcome these varying inertia torques. The difference between the driving torque and inertia torque at any instant gives the net torque acting on the crankshaft.

The magnitude of the inertia torque depends on the following factors:

1. Mass Moment of Inertia (I): Heavier parts produce larger inertia torque.
2. Angular Acceleration (α): Higher acceleration results in a greater inertia torque.
3. Speed of Rotation: Fluctuations in speed affect the variation of inertia torque.

Example

Consider a rotating wheel having a moment of inertia of 2 kg·m² and an angular acceleration of 4 rad/s². The inertia torque will be:

This means that a torque of 8 N·m acts opposite to the direction of angular acceleration to resist the change in motion.

Importance of Inertia Torque

• It helps in the analysis of dynamic balancing of engines and machines.
• It is essential for the design of flywheels which regulate energy fluctuations.
• It affects vibration and stability in rotating and reciprocating systems.
• It determines the power required to accelerate or decelerate machine parts.

Practical Applications

1. Flywheels: Used to absorb and release energy due to inertia torque variation, maintaining constant engine speed.
2. Automobiles: Inertia torque affects acceleration and deceleration of vehicles, influencing fuel efficiency.
3. Machine Tools: Helps in controlling sudden torque variations in rotating shafts or cutting tools.
4. Engines: Used in designing crankshaft and balancing systems to reduce vibrations caused by inertia torque.

In engineering practice, the knowledge of inertia torque is used to predict and minimize unwanted oscillations in rotating machinery and to ensure steady performance of mechanisms.

Conclusion:

The inertia torque is the opposing torque produced by the mass moment of inertia of rotating parts when their speed changes. It resists acceleration or deceleration and plays a vital role in analyzing the motion and balance of engines, machines, and mechanical systems. Understanding and compensating for inertia torque ensures smoother operation, less vibration, and efficient performance of mechanical devices.