Short Answer:

Steady-state vibration is the type of vibration that continues in a system when the effect of initial disturbances or transient vibrations has completely disappeared. It occurs when the system vibrates at a constant amplitude and frequency under a continuous external periodic force.

In simple words, steady-state vibration is the permanent or continuous vibration that remains after the system has settled into a regular pattern. It depends on the frequency and amplitude of the external force and remains constant as long as the force continues to act on the system.

Detailed Explanation :

Steady-State Vibration

Steady-state vibration is a very important concept in vibration analysis and mechanical engineering. When a system is subjected to a periodic external force, it initially undergoes transient vibration followed by steady-state vibration. Transient vibrations occur only for a short period and die out due to damping, while steady-state vibration remains as long as the external force acts.

In steady-state vibration, the system vibrates continuously at the same frequency as the external exciting force, not at its natural frequency. The amplitude of steady-state vibration depends on several factors such as the magnitude of the applied force, the stiffness of the system, and the damping present.

This steady-state condition represents the final and stable motion of the system after the transient effects have vanished. It is the condition in which the system reaches a constant response to the external excitation.

Explanation of Steady-State Vibration

When a mechanical system (such as a spring–mass–damper system) is subjected to a periodic external force like , it initially vibrates irregularly due to the combination of its own natural vibration and the applied force. This irregular motion is known as transient vibration and occurs due to the system’s initial energy.

As time passes, the transient effects are gradually absorbed or dissipated through damping, and only the vibration caused by the external force remains. This continuous, regular vibration with a constant amplitude and frequency is called steady-state vibration.

The displacement of the mass in steady-state vibration is given by:

Where:

= Amplitude of steady-state vibration
= Frequency of external excitation
= Phase angle between the displacement and external force

This equation represents a pure sinusoidal motion with constant amplitude and frequency, indicating a steady condition of vibration.

Derivation of Amplitude in Steady-State Vibration

For a damped system under forced vibration, the equation of motion is:

After solving this equation for steady-state conditions, the amplitude of vibration is obtained as:

Where:

= Frequency ratio (ratio of forcing frequency to natural frequency)
= Damping ratio
= Stiffness of the system

From this expression, we can understand the following:

When is far from , the amplitude is small.
When approaches (resonance), the amplitude increases sharply.
Higher damping () reduces the maximum amplitude at resonance and helps control vibration.

Thus, the steady-state amplitude depends on the relationship between the forcing frequency, natural frequency, and damping.

Phase Difference in Steady-State Vibration

In steady-state vibration, there is a phase difference (φ) between the external force and the system’s displacement. This phase difference depends on the frequency ratio and damping ratio.

The phase angle is given by:

When , φ ≈ 0°, meaning displacement is almost in phase with the force.
When , φ = 90°, meaning displacement lags the force by 90°.
When , φ ≈ 180°, meaning displacement and force are almost in opposite directions.

This phase relationship is very important in analyzing steady-state vibrations in machines and structures.

Characteristics of Steady-State Vibration

Constant Amplitude and Frequency:
The vibration continues with the same amplitude and frequency as long as the external force remains constant.
Depends on Forcing Frequency:
The system vibrates at the frequency of the external force, not at its natural frequency.
Independent of Initial Conditions:
It does not depend on how the motion started; it depends only on the continuous external excitation.
Controlled by Damping:
Damping reduces amplitude and prevents excessive vibrations at or near resonance.
Sinusoidal Response:
The steady-state motion is sinusoidal, meaning it repeats in a regular and predictable pattern.

Practical Examples of Steady-State Vibration

Engines and Rotating Machines:
The continuous vibration of a running motor or engine due to unbalanced rotating parts is a steady-state vibration.
Vehicle Suspension:
When a car moves on a rough road, the continuous motion of the suspension system due to road irregularities is steady-state vibration.
Building Vibrations:
Structures exposed to continuous wind or machine vibrations experience steady-state vibration.
Machinery Foundations:
Equipment like compressors and turbines produce steady-state vibrations that travel through their foundations.
Musical Instruments:
The steady vibration of a guitar string when continuously plucked or bowed is an example of steady-state vibration.

Importance of Studying Steady-State Vibration

Design of Machines and Structures: Helps in determining safe operating frequencies and avoiding resonance.
Noise and Vibration Control: Reduces unwanted continuous vibration in engines, vehicles, and buildings.
Durability Improvement: Ensures long life of components by minimizing fatigue due to continuous vibration.
Comfort Enhancement: Provides smoother operation and comfort in vehicles and mechanical systems.
Energy Efficiency: Prevents unnecessary energy losses due to continuous excessive vibrations.

Difference between Transient and Steady-State Vibration

Transient vibration occurs for a short period immediately after the system is disturbed and then dies out.
Steady-state vibration continues indefinitely as long as the external periodic force is applied.

In steady-state vibration, the system’s response becomes stable and predictable over time.

Conclusion

In conclusion, steady-state vibration is the constant, continuous vibration that remains in a system after transient effects disappear. It occurs when a periodic external force keeps acting on the system, making it vibrate with a fixed amplitude and frequency. The amplitude and phase of steady-state vibration depend on the damping ratio and frequency ratio of the system. Understanding steady-state vibration is very important in mechanical design to ensure stability, safety, and comfort in machines and structures operating under continuous dynamic forces.