Short Answer:

Logarithmic decrement is a measure used to determine the rate at which the amplitude of a damped vibration decreases over successive cycles. It shows how quickly the vibration dies out in a damped system.

In simple terms, logarithmic decrement represents the natural reduction in vibration amplitude due to damping. It is defined as the natural logarithm of the ratio of two successive amplitudes of vibration. This parameter helps engineers to determine the damping characteristics of a mechanical system or structure.

Detailed Explanation :

Logarithmic Decrement

In mechanical vibration analysis, logarithmic decrement is an important parameter that measures how the amplitude of a damped vibration reduces with time. When a mechanical system vibrates, its amplitude gradually decreases because of damping — the resistance that dissipates energy in the form of heat or friction.

Logarithmic decrement helps to quantify the amount of damping present in a system by comparing the amplitudes of successive vibration cycles. It is particularly useful for lightly damped systems, where the amplitude decreases slowly and vibrations continue for several cycles before stopping completely.

Definition

Logarithmic decrement is defined as:

“The natural logarithm of the ratio of any two successive amplitudes on the same side of the mean position in a damped vibration is called logarithmic decrement.”

It provides a way to calculate damping from the observed amplitudes of vibration without directly measuring the damping force.

Mathematical Expression

Consider a damped free vibration system where the amplitude of vibration decreases exponentially with time. The displacement of the system can be represented as:

where,

•  = initial amplitude,
•  = damping ratio,
•  = natural angular frequency,
•  = damped natural frequency,
•  = time,
•  = phase angle.

After one complete cycle of vibration, the time period is:

Let  and  be the amplitudes of two successive cycles.

The ratio of amplitudes is:

Taking natural logarithm on both sides:

Hence,

Where,

•  = logarithmic decrement,
•  = damping ratio.

This relation shows that logarithmic decrement is directly proportional to the damping ratio.

Determination Using Multiple Cycles

For greater accuracy, logarithmic decrement can also be calculated using several cycles instead of just two.

If  and  are the amplitudes separated by n complete cycles, then:

This method reduces experimental error and gives more reliable results.

Interpretation of Logarithmic Decrement

• A large value of δ means the system is heavily damped, and the amplitude decreases quickly.
• A small value of δ means the system is lightly damped, and the amplitude decays slowly.

For example, in machinery, a small logarithmic decrement means that vibrations will persist longer, while a large value means the machine will settle quickly after disturbance.

Physical Significance

1. Measure of Damping Efficiency:
   Logarithmic decrement helps in evaluating how efficiently damping dissipates energy in a vibrating system.
2. System Stability Indicator:
   A high logarithmic decrement indicates a stable system that quickly returns to rest after being disturbed.
3. Used in Experimentation:
   Engineers use this method to determine damping ratio by observing amplitude decay in laboratory or field experiments.
4. Applicable to Real Systems:
   It is widely used in measuring damping in vehicle suspensions, rotating machinery, and structural components.

Experimental Determination

Step 1: Displace the system from its equilibrium position and allow it to vibrate freely.
Step 2: Record the amplitude of each successive vibration cycle (e.g., using a displacement sensor or graph).
Step 3: Measure the amplitudes  of consecutive peaks.
Step 4: Calculate logarithmic decrement using:

Step 5: Determine the damping ratio  using:

Example Calculation

Suppose the amplitude of a vibrating beam decreases from 10 mm to 7 mm in one cycle.
Then logarithmic decrement is:

If the damping ratio is to be found:

Hence, the system has a damping ratio of 0.0566, indicating light damping.

Applications of Logarithmic Decrement

1. Determining Damping Ratio:
   Used in laboratories to find damping ratios of mechanical and structural systems.
2. Machine Design:
   Helps in analyzing the performance of dampers and vibration isolators.
3. Vehicle Suspension Testing:
   Used to measure damping behavior of shock absorbers.
4. Structural Engineering:
   Used to evaluate the damping capacity of buildings and bridges under vibration or seismic loading.
5. Maintenance Diagnostics:
   Helps identify mechanical wear, cracks, or looseness through changes in damping behavior.

Advantages

1. Provides an accurate and simple method to measure damping.
2. Requires minimal equipment and data (amplitudes only).
3. Can be applied to both mechanical and structural systems.
4. Works well for light damping conditions common in real systems.

Limitations

1. Not suitable for systems with high damping (amplitude decays too fast).
2. Difficult to measure accurately in noisy environments.
3. Requires at least two clear vibration cycles for calculation.

Conclusion

Logarithmic decrement is a mathematical measure of how quickly the amplitude of a damped vibration decays over time. It is defined as the natural logarithm of the ratio of successive amplitudes and is directly related to the damping ratio. It helps engineers to analyze and quantify damping, predict vibration decay, and design systems that can safely control oscillations. In mechanical and structural applications, logarithmic decrement plays a key role in ensuring stability, safety, and long service life of vibrating components.