Short Answer:

Frequency response is the measure of how a system reacts or responds to different frequencies of an external force or input signal. It shows how the amplitude and phase of vibration change when the frequency of excitation varies.

In simple words, frequency response explains the behavior of a vibrating system when the forcing frequency changes. It helps to understand how a machine or structure behaves near resonance and how damping affects vibration amplitude at different frequencies.

Detailed Explanation :

Frequency Response

Frequency response is an important concept in vibration analysis and mechanical engineering. It describes how a mechanical system behaves when it is subjected to a range of excitation frequencies. In other words, it is the relationship between the input force frequency and the resulting vibration amplitude or output response of the system.

When an external periodic force acts on a system, the system’s vibration amplitude and phase depend on the frequency of the applied force. By studying the frequency response, we can understand how the system will behave across various frequencies, identify the resonance point, and determine the effect of damping on the amplitude of vibration.

The frequency response helps engineers design systems that can avoid resonance conditions and maintain stability during operation.

Mathematical Expression of Frequency Response

Consider a single degree of freedom (SDOF) system under a harmonic force:

Where:

•  = mass of the system
•  = damping coefficient
•  = stiffness of the system
•  = external harmonic force
•  = frequency of excitation

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

Here,  is the amplitude of steady-state vibration, and it can be expressed as:

Where:

•  = frequency ratio (ratio of forcing frequency to natural frequency)
•  = damping ratio

This expression defines the frequency response curve of the system. It shows how the amplitude  changes as the forcing frequency  changes from low to high values.

Interpretation of Frequency Response

1. At Low Frequency ():
   The system’s response is small because the restoring force dominates. The amplitude increases slightly as frequency increases.
2. At Resonance ():
   The amplitude of vibration becomes maximum because the frequency of the external force matches the natural frequency of the system.
   • Without damping, amplitude theoretically becomes infinite.
   • With damping, the amplitude remains finite and decreases as damping increases.
3. At High Frequency ():
   The amplitude decreases because the mass inertia force dominates. The system cannot follow the high-frequency motion of the external force.

Thus, the frequency response curve typically rises to a peak at resonance and then decreases at higher frequencies. The sharpness and height of the peak depend on the amount of damping present in the system.

Phase Response

Along with amplitude, frequency response also considers the phase difference (φ) between the applied force and the resulting vibration. The phase lag is given by:

• When  is very small (low frequency), the displacement is almost in phase with the applied force (φ ≈ 0°).
• At resonance (), the displacement lags the force by 90°.
• When  is very large (high frequency), the displacement lags the force by almost 180°.

Hence, the phase response shows how the system’s motion gradually shifts as the excitation frequency changes.

Frequency Response Curve

A frequency response curve is a graphical representation of how the amplitude or phase of a system varies with the frequency ratio .

• The x-axis represents the frequency ratio (r).
• The y-axis represents the amplitude ratio  or displacement amplitude .

The shape of this curve provides valuable information about:

• The resonant frequency of the system.
• The maximum amplitude at resonance.
• The effect of damping on reducing vibration amplitude.

With higher damping, the resonance peak becomes lower and broader, indicating a smoother and safer response. With very low damping, the resonance peak is sharp and high, showing the potential for excessive vibration.

Importance of Frequency Response

1. Resonance Prediction:
   Frequency response analysis helps identify the resonance point, which is essential to avoid mechanical failures.
2. Vibration Control:
   It allows engineers to design systems with suitable damping to minimize vibration amplitude.
3. Machine Design:
   Helps in selecting materials, stiffness, and damping elements to ensure smooth operation over a wide range of frequencies.
4. System Diagnosis:
   Used to detect unbalance, looseness, or defects in rotating machinery by analyzing vibration patterns.
5. Safety and Comfort:
   Ensures that structures, vehicles, and machines operate without excessive vibration or noise, improving comfort and reliability.

Applications of Frequency Response

1. Automobile Suspensions:
   Frequency response helps in designing suspension systems that absorb shocks and prevent resonance with road irregularities.
2. Rotating Machines:
   Used to study how unbalanced forces affect machine vibration at different speeds.
3. Building and Bridge Design:
   Helps engineers design structures that can resist dynamic forces such as wind or earthquakes.
4. Aerospace Engineering:
   Frequency response analysis ensures aircraft components can withstand vibrations due to aerodynamic forces.
5. Instrumentation:
   Measuring instruments like accelerometers and sensors are tested for accurate response over specific frequency ranges.

Factors Affecting Frequency Response

1. Damping Ratio (ξ):
   Higher damping reduces the peak amplitude and makes the system less sensitive to frequency changes.
2. Stiffness (k):
   Higher stiffness increases the natural frequency and affects how the system responds to excitation.
3. Mass (m):
   A higher mass decreases the natural frequency, shifting the resonance point.
4. Type of Excitation:
   The shape and nature of the external force also affect the system’s response.

Conclusion

In conclusion, frequency response describes how a mechanical system reacts to varying excitation frequencies by showing the relationship between vibration amplitude, phase, and frequency. It helps in understanding the behavior of systems near resonance and the role of damping in controlling vibration amplitude. By studying the frequency response curve, engineers can design stable and efficient machines and structures that operate safely under dynamic loads. Thus, frequency response analysis is a key part of vibration study and mechanical design.