What is frequency ratio?

Short Answer:

The frequency ratio is the ratio of the excitation frequency (the frequency of an external periodic force) to the natural frequency of a vibrating system. It is a dimensionless quantity used to compare how close the operating frequency is to the system’s natural frequency.

In simple words, the frequency ratio helps determine the vibration behavior of a system. When the frequency ratio equals one, resonance occurs. If it is less than or greater than one, the vibration amplitude changes accordingly, helping engineers analyze and control vibration performance.

Detailed Explanation :

Frequency Ratio

In mechanical vibration analysis, every system has its own natural frequency, which is the frequency at which it tends to vibrate freely without any external force. When an external periodic force acts on the system, it vibrates at the forcing frequency or excitation frequency.

The frequency ratio is a dimensionless term that represents the ratio between these two frequencies — the excitation frequency and the natural frequency. It is expressed as:

Where:

  •  = frequency ratio (dimensionless)
  •  = excitation or forcing frequency (radians per second)
  •  = natural frequency of the system (radians per second)

This ratio plays a very important role in understanding the vibration response of a system. It helps determine how the amplitude, phase angle, and energy response of the system will vary with changes in the excitation frequency.

Meaning of Frequency Ratio

The frequency ratio indicates how the system behaves when subjected to an external periodic force. Based on its value, different vibration conditions occur:

  1. When :
    • The excitation frequency is less than the natural frequency.
    • The system moves almost in phase with the applied force.
    • The amplitude of vibration is small, and the motion is stable.
  2. When :
    • The excitation frequency equals the natural frequency.
    • The system reaches resonance condition, and the amplitude becomes maximum.
    • At this point, the system absorbs maximum energy from the external force.
  3. When :
    • The excitation frequency is greater than the natural frequency.
    • The system displacement becomes out of phase with the applied force (almost opposite).
    • The amplitude of vibration decreases as the frequency ratio increases further.

Therefore, the frequency ratio helps engineers predict whether a system is operating below, at, or above resonance and what its response will be in each region.

Relationship Between Frequency Ratio and Amplitude

The amplitude of forced vibration depends directly on the frequency ratio and damping. The equation for amplitude in a forced damped system is:

Where:

  •  = steady-state amplitude
  •  = amplitude of external force
  •  = stiffness of the system
  •  = damping ratio

From this equation, it can be seen that the amplitude  changes with the frequency ratio :

  • For small values of r, the amplitude is small.
  • As r approaches 1, the amplitude increases sharply (resonance).
  • For r greater than 1, the amplitude decreases again.

Thus, the frequency ratio is one of the most important parameters to study the vibration behavior of systems under forced excitation.

Effect of Frequency Ratio on Phase Angle

The phase angle (φ) between the applied force and the system displacement also depends on the frequency ratio. The relationship is given by:

As  increases from 0 to very large values:

  • At ,  → force and displacement are in phase.
  • At ,  → displacement lags the force by 90° (resonance).
  • At ,  → displacement is opposite in phase to the applied force.

This shows that the frequency ratio not only affects amplitude but also the phase relationship between force and motion.

Graphical Representation (Concept Explanation)

When we plot the amplitude ratio () against the frequency ratio :

  • The curve starts with a small amplitude at low .
  • It rises sharply near  (resonance).
  • After resonance, the amplitude decreases rapidly as  increases.

This resonance curve is very important in vibration analysis, as it visually shows the effect of frequency ratio on system response.

Practical Significance of Frequency Ratio

  1. In Machine Design:
    Engineers use the frequency ratio to ensure that the operating frequency of a machine is not close to its natural frequency, preventing resonance and damage.
  2. In Structural Engineering:
    Buildings, bridges, and towers are designed with frequency ratios that avoid resonance caused by wind, earthquakes, or machinery.
  3. In Automotive Engineering:
    Suspension systems are designed with proper frequency ratios so that the vehicle body does not resonate with road vibrations, improving ride comfort.
  4. In Testing and Analysis:
    The frequency ratio is used in laboratory vibration tests to identify the natural frequency and damping characteristics of materials or components.
  5. In Rotating Machinery:
    It helps determine critical speeds, which are the speeds at which the rotating shaft’s frequency matches its natural frequency (r = 1).

Thus, the concept of frequency ratio is crucial for predicting vibration performance and preventing mechanical failure due to resonance.

Example of Frequency Ratio

Suppose a machine has a natural frequency () of 20 rad/s, and it is subjected to a periodic force with frequency () of 10 rad/s.
Then the frequency ratio is:

This means the machine is operating at half of its natural frequency, and no resonance will occur. If the excitation frequency increases to 20 rad/s, , and resonance will occur, causing maximum vibration amplitude.

Importance of Controlling Frequency Ratio

To prevent resonance and ensure smooth operation, engineers maintain the frequency ratio either much less than 1 (for isolation) or much greater than 1 (for safety in rotating systems).
Controlling the frequency ratio helps reduce vibration, noise, and mechanical wear.

Conclusion

In conclusion, the frequency ratio is the ratio of the excitation frequency to the natural frequency of a vibrating system. It is a key factor in determining the vibration behavior, amplitude, and phase response of a system. When the frequency ratio equals one, resonance occurs, leading to large vibrations. By controlling the frequency ratio through proper design and damping, engineers can ensure stable and safe operation of machines and structures, avoiding resonance-related failures.