What is phase angle at resonance?

Short Answer:

The phase angle at resonance is the phase difference between the applied external force and the resulting displacement of a vibrating system when resonance occurs. At this condition, the forcing frequency becomes equal to the natural frequency of the system.

In simple words, when a system vibrates at resonance, the phase angle between the external force and the displacement is 90 degrees (or π/2 radians). This means the displacement lags behind the applied force by 90°, indicating that the energy transfer between the force and the system is at its maximum.

Detailed Explanation :

Phase Angle at Resonance

When a mechanical system is subjected to a harmonic external force, the applied force and the resulting vibration (displacement) are generally not in phase. The difference in timing between the applied force and the system’s response is known as the phase angle (φ).

The phase angle at resonance refers specifically to the condition when the excitation frequency (ω) equals the natural frequency (ωₙ) of the system. At this condition, the amplitude of vibration becomes maximum, and the phase difference between the force and displacement becomes exactly 90°.

This means the displacement lags the applied force by one-quarter of a complete vibration cycle. Understanding this relationship is essential for analyzing vibration behavior, especially near resonance conditions.

Equation of Motion for a Forced Damped Vibration

Consider a single-degree-of-freedom (SDOF) damped vibration system subjected to a harmonic external force:

Where:

  •  = mass of the system
  •  = damping coefficient
  •  = stiffness of the system
  •  = externally applied periodic force
  •  = frequency of excitation
  •  = displacement

The steady-state solution of this equation gives the displacement response as:

Here,  is the amplitude of vibration, and  is the phase angle between the applied force and displacement.

Expression for Phase Angle

The phase angle  is given by the following equation:

Where:

  •  = damping ratio
  •  = frequency ratio
  •  = natural frequency of the system

This equation shows that the phase angle depends on both the frequency ratio (r) and the damping ratio (ξ) of the system.

Phase Relationship at Different Frequencies

The phase angle changes as the excitation frequency varies:

  1. At Low Frequency (ω << ωₙ):
    • The frequency ratio  is much less than 1.
    •  becomes very small, so .
    • The displacement and force are almost in phase.
  2. At Resonance (ω = ωₙ):
    • The frequency ratio .
    • The denominator .
    • Therefore,  becomes infinite.
    • Hence, the phase angle  or  radians.
    • This means that at resonance, the displacement lags behind the applied force by 90°.
  3. At High Frequency (ω >> ωₙ):
    • The frequency ratio  becomes much greater than 1.
    • The denominator  becomes large and negative.
    •  approaches zero in the negative direction, giving .
    • The displacement is almost opposite in phase to the applied force.

Hence, as frequency increases from zero to very high values, the phase angle shifts gradually from 0° to 180°, passing through 90° at resonance.

Physical Meaning of Phase Angle at Resonance

At resonance, the system’s natural frequency matches the forcing frequency, and energy transfer from the external force to the system is at its maximum.

However, at this condition:

  • The displacement does not occur at the same time as the applied force.
  • The maximum displacement occurs when the applied force crosses zero, which means a phase difference of 90° exists.
  • The velocity of the system, however, is in phase with the applied force.

This phase difference is significant because it explains the condition of maximum energy transfer and maximum amplitude during resonance.

Effect of Damping on Phase Angle at Resonance

The phase angle at resonance is always 90°, regardless of the amount of damping present in the system.
However, damping affects how quickly the phase angle approaches 90° as the frequency increases:

  • In a lightly damped system, the transition from 0° to 180° is sharp, and the phase angle becomes 90° very close to resonance.
  • In a heavily damped system, the transition is gradual, and the resonance curve becomes smoother.

Thus, while damping reduces amplitude, it does not change the fact that at resonance, the phase angle remains 90°.

Graphical Explanation (Concept)

If we plot the phase angle (φ) on the vertical axis and the frequency ratio (r) on the horizontal axis, the graph will show:

  • At , φ = 0°
  • At , φ = 90°
  • At , φ → 180°

This graph helps visualize how the phase relationship changes as the frequency of excitation increases and clearly shows the special condition of 90° at resonance.

Practical Importance of Phase Angle at Resonance

  1. Machine Design:
    Helps engineers identify and control resonance conditions to avoid excessive vibrations.
  2. Balancing and Alignment:
    Phase information is used to find the direction of unbalance in rotating systems.
  3. Vibration Measurement:
    Phase difference is used in vibration analysis to determine the relationship between input and output responses.
  4. Damping Design:
    Understanding phase behavior assists in designing damping systems to limit amplitude at resonance.
  5. Dynamic Testing:
    Phase data helps determine system properties like damping ratio and stiffness experimentally.
Conclusion

In conclusion, the phase angle at resonance is the phase difference between the external force and the resulting displacement when the excitation frequency equals the system’s natural frequency. At this condition, the phase angle becomes 90 degrees, meaning the displacement lags the applied force by a quarter cycle. This phase shift signifies maximum energy transfer and maximum vibration amplitude. The phase angle at resonance remains constant regardless of damping, making it an important parameter in vibration analysis and machine design.