The correct answer is option 'D': magnification factor. Let's understand why.

Magnification Factor:
The magnification factor is the ratio of the maximum displacement of the forced vibration to the deflection due to the static force. It represents the amplification or multiplication of the static deflection due to the application of an external force.

In forced vibration, an external force is applied to a system at a specific frequency. This force causes the system to vibrate with an amplitude and phase that depends on the frequency of the force and the characteristics of the system.

The magnification factor is a measure of how much the system amplifies the force at the resonant frequency. It helps us understand the response of the system to the applied force and provides insight into the dynamic behavior of the system.

The magnification factor can be calculated using the equation:

Magnification factor = Maximum displacement of forced vibration / Static deflection due to static force

Damping Factor and Damping Coefficient:
Damping factor and damping coefficient are not directly related to the ratio of maximum displacement to static deflection.

Damping factor is a measure of the rate at which the amplitude of the vibrations decreases over time due to the dissipation of energy. It is typically represented by the symbol ξ (xi) and is a dimensionless quantity. Damping factor is related to the damping ratio, which is the ratio of actual damping to critical damping.

Damping coefficient, on the other hand, is a parameter that represents the resistance to motion or the dissipative properties of a system. It is denoted by the symbol c and has units of force per unit velocity. Damping coefficient is used in the equation of motion to model the damping effect in a system.

Logarithmic Decrement:
Logarithmic decrement is a measure of the rate at which the amplitude of the vibrations decreases over time due to damping. It is defined as the natural logarithm of the ratio of two successive maximum amplitudes.

Conclusion:
In conclusion, the correct answer is option 'D': magnification factor. The magnification factor represents the ratio of the maximum displacement of the forced vibration to the deflection due to the static force. It helps us understand the amplification of the static deflection by the applied force in a dynamic system. Damping factor, damping coefficient, and logarithmic decrement are different concepts related to damping in a vibrating system.