Short Answer:

The equation of torsional vibration represents the relationship between torque, angular displacement, moment of inertia, and stiffness of a rotating shaft. It is used to describe the twisting oscillations of a shaft or rotor system about its longitudinal axis. The basic equation of torsional vibration is similar to that of a simple harmonic motion but in rotational form.

The general form of the torsional vibration equation is:
I (d²θ/dt²) + C (dθ/dt) + Kθ = 0,
where I is the moment of inertia, C is the damping coefficient, K is the torsional stiffness, and θ is the angular displacement.

Detailed Explanation :

Equation of Torsional Vibration

Torsional vibration occurs when a rotating shaft or any mechanical system experiences twisting motion due to the action of fluctuating torque. The system resists this twisting motion because of its torsional stiffness, and the rotating parts oscillate back and forth around their mean position. The mathematical expression that describes this motion is called the equation of torsional vibration.

This equation helps engineers understand how the system behaves under different conditions, such as free vibration, damped vibration, and forced vibration. It is very important in designing mechanical components like shafts, crankshafts, and couplings to ensure smooth operation and to avoid resonance or failure.

1. Derivation of the Equation

Let us consider a simple torsional system having:

• A shaft of uniform circular cross-section, fixed at one end.
• A rotor or disk attached to the other end of the shaft.

When the disk is twisted through an angular displacement θ (theta) from its equilibrium position, a restoring torque is developed in the shaft due to its elasticity.

According to the torsion theory,
Restoring torque (T) = K × θ,
where,
K = torsional stiffness of the shaft (N·m/rad), and
θ = angular displacement (radians).

Now, the disk has a moment of inertia (I), and if it rotates with an angular acceleration (d²θ/dt²), the inertia torque acting on it is:
Inertia torque = I × (d²θ/dt²)

By applying Newton’s second law of motion for rotational systems,
Sum of torques on the system = 0

Hence,
Inertia torque + Restoring torque = 0

Substituting the values:
I (d²θ/dt²) + Kθ = 0

This is the basic differential equation of torsional vibration for an undamped system.

2. Equation for Damped Torsional Vibration

In practical systems, damping is always present due to material resistance and air friction. Therefore, a damping torque proportional to the angular velocity is introduced:
Damping torque = C (dθ/dt)
where C = damping coefficient.

Now, the complete equation becomes:
I (d²θ/dt²) + C (dθ/dt) + Kθ = 0

This is the differential equation of damped torsional vibration.

Depending on the damping, the system may exhibit three types of motion:

1. Overdamped motion – Vibration dies out slowly without oscillations.
2. Critically damped motion – System returns to equilibrium in the shortest time.
3. Underdamped motion – System oscillates with decreasing amplitude.

3. Natural Frequency of Torsional Vibration

If there is no damping (C = 0), the equation becomes:
I (d²θ/dt²) + Kθ = 0

This is similar to the simple harmonic motion (SHM) equation.
The natural frequency (ωₙ) of torsional vibration is given by:

Where,

• ωₙ = natural angular frequency (rad/s),
• K = torsional stiffness,
• I = moment of inertia.

The time period (T) of vibration is given by:

This means that the natural frequency of a torsional system depends directly on the stiffness of the shaft and inversely on the moment of inertia of the rotating body.

4. Physical Meaning of the Terms

• θ (Theta): Angular displacement or twist of the shaft in radians.
• dθ/dt: Angular velocity, rate at which twisting occurs.
• d²θ/dt²: Angular acceleration of the rotating mass.
• I: Moment of inertia of the rotating mass, representing its resistance to angular acceleration.
• K: Torsional stiffness, which shows how much torque is required to twist the shaft through one radian.
• C: Damping coefficient, representing the frictional or resistive torque that reduces oscillation.

5. Significance of the Equation

The torsional vibration equation is extremely important in mechanical design because:

• It helps determine the natural frequency to avoid resonance.
• It allows engineers to estimate vibration amplitude under dynamic loading.
• It is used to design torsional dampers and vibration absorbers.
• It assists in predicting fatigue life of shafts under repeated torque variations.

For example, in a crankshaft of an internal combustion engine, the periodic torque produced by combustion causes twisting of the shaft. If the frequency of these torque variations equals the natural frequency of the crankshaft, dangerous resonance may occur. The torsional vibration equation helps predict and prevent such failures.

Conclusion:

The equation of torsional vibration expresses the dynamic relationship between inertia, damping, stiffness, and angular motion of a rotating shaft. The general form is I (d²θ/dt²) + C (dθ/dt) + Kθ = 0. It represents a rotational version of the spring-mass-damper system used in linear vibration. This equation is essential in analyzing the natural frequency, amplitude, and damping behavior of rotating machinery, ensuring safe and efficient design of mechanical systems.