Short Answer:
Parametric resonance is a special type of resonance that occurs in a vibrating system when one or more system parameters, such as stiffness or mass, vary periodically with time. Unlike ordinary resonance, which happens due to external periodic forces, parametric resonance is caused by internal periodic variations within the system itself.
When these variations match certain critical conditions, the vibration amplitude increases rapidly, leading to instability and possible failure. This phenomenon is often seen in shafts, pendulums, or mechanical systems with time-varying stiffness or load conditions.
Detailed Explanation :
Parametric Resonance
Parametric resonance is a phenomenon in vibration systems where the amplitude of vibration grows continuously due to the periodic variation of system parameters like stiffness, mass, or length. It occurs not because of an external force but because the parameters that define the system’s dynamic behavior change periodically over time.
In simpler words, while normal resonance happens when the frequency of an external force matches the system’s natural frequency, parametric resonance occurs when the system’s internal characteristics change periodically in a way that excites the system itself. This can lead to rapidly increasing vibration amplitudes, instability, and even structural failure if not controlled.
Parametric resonance is common in rotating machinery, suspension bridges, ships, and aircraft components, where operating conditions or geometry vary with time. Engineers must analyze and design systems to avoid operating in such resonant conditions to ensure safety and reliability.
Principle of Parametric Resonance
The basic principle of parametric resonance is based on the time-varying stiffness or other parameters of a system.
Let us consider a simple system like a mass-spring system. If the stiffness of the spring changes periodically with time (for example, due to varying load or structural deformation), the natural frequency of the system also changes. When this variation occurs at twice the natural frequency of the system, the amplitude of vibration increases rapidly.
This condition is known as parametric resonance condition.
Mathematically, this phenomenon is described by the Mathieu’s differential equation:
Where,
- = displacement of the mass,
- = natural frequency of the system,
- = amplitude of the parametric excitation,
- = frequency of parameter variation.
Parametric resonance occurs when (the excitation frequency) satisfies certain critical conditions, such as:
where = 1, 2, 3, …
The most significant and dangerous condition occurs when .
At this condition, the system experiences primary parametric resonance, and the vibration amplitude grows exponentially with time.
Difference Between Ordinary and Parametric Resonance
| Aspect | Ordinary Resonance | Parametric Resonance |
| Cause | External periodic force | Periodic variation of system parameters |
| Example | Forced vibration by unbalanced rotor | Shaft with periodically varying stiffness |
| Frequency condition | Excitation frequency = natural frequency | Excitation frequency ≈ 2 × natural frequency |
| Effect | Amplitude increases gradually | Amplitude increases exponentially |
| Control | Reduce external forcing | Control parameter variation |
Thus, while ordinary resonance is caused by external forces acting on the system, parametric resonance originates from internal variations within the system’s own parameters.
Causes of Parametric Resonance
- Time-Varying Stiffness:
If stiffness changes periodically (due to load changes, shaft bending, or deformation), it can trigger resonance. - Variable Rotational Speed:
In rotating shafts, speed variation changes the centrifugal forces acting on the system, altering stiffness and mass distribution. - Structural Deformation:
Elastic structures like beams or plates can change their shape dynamically, resulting in fluctuating stiffness. - Periodic Load Variations:
Loads varying periodically (as in reciprocating engines or ships) can modify internal parameters, exciting parametric resonance. - Geometrical or Boundary Changes:
Changes in geometry or boundary conditions (like suspension bridges under wind forces) can cause time-dependent parameter variations.
Examples of Parametric Resonance in Engineering Systems
- Rotating Shafts:
Shafts with varying speed or eccentricity may experience time-dependent stiffness changes, leading to parametric resonance and whirling instability. - Pendulums:
When the length of a pendulum is varied periodically, its natural frequency changes, which can result in large oscillations if the variation frequency meets the resonance condition. - Ships and Offshore Structures:
The motion of ships due to waves can cause periodic variations in buoyancy and stiffness, leading to parametric rolling instability. - Bridges and Towers:
Wind or vehicle-induced load fluctuations can vary the stiffness of long-span bridges or towers, resulting in resonant vibrations. - Aircraft Wings:
Aerodynamic loads that vary periodically can induce resonance in wings and other structural components. - Suspension Systems:
Vehicle suspensions with variable spring stiffness or damping can enter a resonant condition under certain road conditions.
Effects of Parametric Resonance
- Rapid Amplitude Growth: Vibration amplitude increases exponentially, leading to large oscillations.
- System Instability: The system loses control and deviates from its normal operating condition.
- Fatigue Failure: Repeated high stress leads to material fatigue and cracks.
- Loss of Performance: Machine efficiency and precision decrease significantly.
- Safety Hazards: Severe instability can cause collapse or breakdown of the system.
Methods to Prevent Parametric Resonance
- Avoid Critical Speed or Frequency:
Ensure that operating conditions do not coincide with twice or any fractional multiple of the system’s natural frequency. - Increase Damping:
Use damping materials or devices to absorb excess vibration energy and limit amplitude growth. - Design Modification:
Change mass, stiffness, or geometry to shift the natural frequency away from excitation frequencies. - Control of Parameter Variation:
Avoid periodic variations in stiffness or mass by improving design and manufacturing precision. - Use of Feedback Control Systems:
In modern systems, active control devices detect instability and counteract it automatically to maintain stability.
Mathematical Analysis and Stability
The stability of a system under parametric excitation can be analyzed using Mathieu’s stability chart, which shows stable and unstable regions for different parameter values.
If the system operates within a stable region, vibration amplitude remains limited. However, if it falls into an unstable region, parametric resonance occurs, and oscillations grow without bound.
Engineers use this analysis during design to ensure the system’s operating range remains within stable zones.
Conclusion
Parametric resonance is a type of resonance that occurs when one or more system parameters, such as stiffness or mass, vary periodically with time, leading to excessive vibration amplitude. It differs from ordinary resonance because it arises from internal variations rather than external forces. The most critical condition occurs when the excitation frequency equals twice the natural frequency of the system. This phenomenon can cause serious damage in rotating machinery, bridges, ships, and aircraft if not properly controlled. Preventing parametric resonance requires proper design, adequate damping, and avoidance of critical frequency conditions to ensure stable and safe operation.