Short Answer:
Amplitude ratio is the ratio of the steady-state amplitude of a forced vibrating system to the static deflection caused by the same force. It shows how much the vibration amplitude increases or decreases under dynamic conditions compared to static conditions. The amplitude ratio depends on the frequency of excitation, damping, and natural frequency of the system.
In simple words, amplitude ratio helps to understand how a system responds to external vibrations at different frequencies. If the excitation frequency is near the natural frequency, the amplitude ratio becomes very high, leading to resonance. Hence, it plays a key role in vibration analysis and control.
Detailed Explanation :
Amplitude Ratio
Amplitude ratio is an important term in the study of mechanical vibrations. It represents the relationship between the amplitude of forced vibration and the static deflection under a constant load. This ratio helps engineers and designers understand how a vibrating system behaves under external periodic forces.
When a system is subjected to an external force that varies with time, it vibrates with a certain amplitude depending on several factors such as the frequency of excitation, the natural frequency of the system, and the damping present. The amplitude ratio helps to express how much the vibration amplitude is magnified compared to the static condition.
Mathematically, the amplitude ratio (AR) is expressed as:
Where:
- = amplitude of forced vibration
- = static deflection of the system due to the same force
The amplitude ratio can also be derived from the dynamic equation of motion for a damped forced vibration system:
After solving this equation, the amplitude of steady-state vibration is given by:
Where,
- is the frequency ratio,
- is the damping ratio,
- is the excitation frequency, and
- is the natural frequency.
Therefore, the amplitude ratio becomes:
This equation shows how the amplitude ratio varies with the frequency ratio and damping.
When the damping ratio is small, the amplitude ratio increases rapidly near the resonance condition ( ). At resonance, the system vibrates with maximum amplitude, which may cause damage to mechanical components. However, when damping is increased, the peak of the amplitude ratio decreases, and the system becomes more stable.
The amplitude ratio curve helps to understand the vibration behavior of the system under different excitation frequencies. At very low frequencies ( ), the amplitude ratio is nearly 1, meaning the dynamic and static deflections are almost equal. As the excitation frequency increases and approaches the natural frequency, the amplitude ratio rises sharply. Beyond the resonance frequency ( ), the amplitude ratio decreases, and the system’s motion becomes stable again.
In practical applications, the concept of amplitude ratio is used to design vibration isolation systems and to prevent resonance in machines. For example, in rotating machinery or engines, designers use the amplitude ratio curve to determine safe operating speeds. In vehicle suspensions, damping is adjusted so that the amplitude ratio remains low even when road vibrations occur near the natural frequency of the suspension system.
Amplitude ratio is also important in structures like bridges, buildings, and towers to ensure they can safely withstand dynamic loads such as wind or earthquake forces without excessive vibration. By controlling damping and stiffness, engineers can limit the amplitude ratio to prevent damage and improve comfort and safety.
In short, amplitude ratio gives valuable information about how a system reacts to external vibrations, helps identify the resonance region, and assists in designing systems that can operate smoothly without excessive motion.
Conclusion:
Amplitude ratio is the measure of how much the vibration amplitude of a system under dynamic load differs from its static deflection. It depends on the damping ratio and the frequency ratio of the system. Understanding the amplitude ratio is very useful for avoiding resonance, designing vibration isolators, and improving the stability of mechanical and structural systems.