Understanding Magnification Factor in Forced Vibrations

In the study of mechanical vibrations, we often encounter systems subjected to external forces. When a force is applied to a system, it causes a deflection or displacement. There are two main scenarios to consider regarding the force: static force and dynamic force.

Static Deflection vs. Dynamic Amplitude

• Static Deflection: This is the displacement of the system's mass when a constant, static force is applied. It can be calculated using Hooke's law for a simple spring-mass system as $\delta_{st} = \frac{F_0}{k}$, where $F_0$ is the static force and $k$ is the stiffness of the spring.
• Dynamic Amplitude: When the system is subjected to a time-varying force (like a harmonic force) that causes forced vibrations, the system oscillates. After some time, transient vibrations die out, and the system settles into steady-state vibrations. The maximum displacement from the equilibrium position during this steady-state vibration is called the amplitude of the steady-state response.

Defining the Magnification Factor

The question asks for the ratio of the amplitude of the steady-state response of forced vibrations to the static deflection under the action of a static force. This specific ratio is known as the Magnification factor. It represents how much larger the dynamic amplitude is compared to the static deflection caused by the same force magnitude applied statically.

Mathematically, the Magnification factor (MF) is defined as:

$\text{MF} = \frac{\text{Amplitude of steady-state response (Forced Vibration)}}{\text{Static deflection under same force}}$

For a simple viscously damped spring-mass system under a harmonic force $F(t) = F_0 \sin(\omega t)$, the steady-state amplitude $X$ is given by:

$X = \frac{F_0}{\sqrt{(k - m\omega^2)^2 + (c\omega)^2}}$

The static deflection $\delta_{st} = \frac{F_0}{k}$.

Therefore, the Magnification Factor is:

$\text{MF} = \frac{X}{\delta_{st}} = \frac{F_0 / \sqrt{(k - m\omega^2)^2 + (c\omega)^2}}{F_0 / k} = \frac{k}{\sqrt{(k - m\omega^2)^2 + (c\omega)^2}}$

This factor depends on the frequency ratio ($\frac{\omega}{\omega_n}$, where $\omega_n = \sqrt{\frac{k}{m}}$ is the natural frequency) and the damping ratio ($\zeta = \frac{c}{c_c}$, where $c_c$ is the critical damping coefficient).

Analyzing the Options

• Damping ratio: This is a dimensionless measure describing how oscillations decay after a disturbance. It is not the ratio described in the question.
• Damping factor: This term can sometimes refer to the damping coefficient (c) or be related to the damping ratio, but it does not represent the amplitude ratio defined.
• Transmissibility: This ratio relates the force transmitted to the support to the applied force (or displacement transmitted to the mass from base motion). It is different from the ratio of dynamic amplitude to static deflection.
• Magnification factor: This is the correct term for the ratio of the steady-state vibration amplitude to the static deflection caused by the same force magnitude.

Based on the definition in mechanical vibrations, the ratio of the amplitude of the steady-state response of forced vibrations to the static deflection under the action of a static force is indeed known as the Magnification factor.