Define velocity in vibration.

Short Answer:

Velocity in vibration is defined as the rate of change of displacement of a vibrating body with respect to time. It shows how fast the body is moving while it vibrates about its mean or equilibrium position.

In simple words, velocity in vibration indicates the speed and direction of motion of the oscillating body at any instant. It is a vector quantity, meaning it has both magnitude and direction. The velocity keeps changing continuously — it is maximum at the mean position and zero at the extreme positions during vibration.

Detailed Explanation :

Velocity in Vibration

Velocity in vibration plays an important role in understanding the behavior of oscillating or vibrating systems. When a body vibrates, it moves back and forth around a fixed mean or equilibrium position. The displacement of the body changes continuously with time, and the rate at which this displacement changes is known as velocity.

In mechanical systems, the vibrating body does not move at a constant speed. Its velocity varies continuously — it reaches a maximum value when the body passes through the mean position and becomes zero at the extreme ends of motion. The direction of velocity also reverses as the body moves from one side of the mean position to the other.

Velocity helps engineers determine how fast the system vibrates, the energy involved in motion, and the forces acting on the body. It is one of the most important quantities in vibration analysis, along with displacement and acceleration.

Mathematical Expression of Velocity

For a body undergoing simple harmonic motion (SHM), the displacement at any instant is given by:

where,

  • x = instantaneous displacement
  • A = amplitude of vibration
  • ω = angular frequency (radians per second)
  • t = time
  • φ = phase angle

Velocity is obtained by differentiating displacement with respect to time:

Substituting the value of x, we get:

 

Hence, the equation for velocity becomes:

This expression shows that the velocity of a vibrating body depends on the angular frequency (ω), amplitude (A), and instantaneous displacement (x).

Nature of Velocity in Vibration

Velocity in a vibrating system varies continuously with time and position. The characteristics of velocity in simple harmonic motion are as follows:

  1. At Mean Position:
    The displacement (x) is zero, and velocity reaches its maximum value, which is given by:

At this point, the body moves fastest through the equilibrium position.

  1. At Extreme Positions:
    The displacement is maximum (x = ±A), and velocity becomes zero because the body momentarily stops before changing direction.
  2. Between Mean and Extreme Positions:
    The velocity decreases gradually from its maximum value to zero as the body moves away from the mean position.

Thus, velocity follows a cosine curve pattern when plotted against time, showing smooth periodic changes throughout the vibration cycle.

Graphical Representation

If the motion of a vibrating body is represented graphically:

  • The displacement-time curve follows a sine wave.
  • The velocity-time curve follows a cosine wave.

Both are sinusoidal in nature but shifted in phase by 90 degrees (π/2 radians). This means when displacement is zero (at mean position), velocity is maximum, and when displacement is maximum (at extremes), velocity is zero.

This phase relationship is an important concept in vibration theory as it helps in understanding the timing and motion characteristics of oscillating systems.

Physical Meaning of Velocity in Vibration

Velocity in vibration indicates how quickly and in which direction a vibrating body moves at any moment. It is directly related to the kinetic energy of the system, which is given by:

where m is the mass of the vibrating body and v is the instantaneous velocity.

Since velocity changes with time, the kinetic energy also varies during vibration. At the mean position, the body has maximum velocity and hence maximum kinetic energy. At extreme positions, velocity becomes zero and the kinetic energy is zero — all energy is then stored as potential energy.

This continuous exchange of kinetic and potential energy keeps the vibration going until external damping or resistance slows it down.

Practical Examples of Velocity in Vibration

  1. Spring-Mass System:
    In a vibrating spring-mass system, the mass moves fastest when passing through its equilibrium position and stops momentarily at the extreme points.
  2. Pendulum:
    The bob of a pendulum has maximum velocity at the lowest point (mean position) and zero velocity at the extreme ends of its swing.
  3. Machine Components:
    Rotating and reciprocating parts in machines such as engines, compressors, or turbines exhibit varying velocity during vibration, which must be analyzed for safe operation.
  4. Vehicle Suspension:
    The velocity of vibration of the vehicle’s wheels and body helps in studying ride comfort and shock absorption performance.
  5. Seismic Vibrations:
    In structures during earthquakes, the velocity of ground vibration determines the impact force acting on the building.

Importance of Velocity in Mechanical Engineering

Velocity in vibration is a critical parameter in the design and analysis of mechanical systems. Its importance includes:

  1. Vibration Measurement:
    Instruments like vibrometers and velocity transducers measure vibration velocity to assess machine health.
  2. Fault Diagnosis:
    Changes in vibration velocity help detect problems such as imbalance, misalignment, or bearing wear in rotating machines.
  3. Dynamic Balancing:
    Velocity data helps determine the dynamic forces generated in rotating components for balancing purposes.
  4. Resonance Control:
    By analyzing vibration velocity, engineers can avoid operating machines at dangerous resonance frequencies.
  5. Energy Estimation:
    Since velocity is directly related to kinetic energy, it helps calculate the energy transfer during vibration.
Conclusion

Velocity in vibration is the rate of change of displacement with respect to time. It indicates how fast and in which direction a vibrating body moves during its motion. The velocity is maximum at the mean position and zero at the extreme positions. It plays a vital role in vibration analysis as it determines the kinetic energy, speed, and dynamic behavior of mechanical systems. Understanding vibration velocity helps engineers design efficient, safe, and balanced machines and structures.