The amplitude of the steady-state reaction divided by the static deflection under force application is known as _______.
magnification factor
Understanding the Magnification Factor in Dynamics
The question asks for the term that describes the ratio of the amplitude of the steady-state reaction to the static deflection when a force is applied. This is a fundamental concept in the study of vibrations and dynamic systems.
What is Static Deflection?
Static deflection is the displacement of a structure or system under a constant applied force, when the system is at rest (not vibrating). For a simple spring-mass system, the static deflection ($\delta_{st}$) caused by a force $F$ is given by:
$\delta_{st} = \frac{F}{k}$
where $k$ is the stiffness of the system.
What is Steady-State Reaction Amplitude?
When a dynamic system is subjected to a continuous oscillating force (like $F_0 \sin(\omega t)$), it will eventually settle into a state of steady-state vibration. In this state, the system vibrates at the same frequency as the applied force. The amplitude of this vibration is the steady-state reaction amplitude.
The Ratio: Amplitude of Steady-State Reaction to Static Deflection
The ratio mentioned in the question compares how large the vibration amplitude is during steady-state vibration compared to the deflection that would occur if the same maximum force were applied statically. This ratio is a measure of how much the dynamic response is amplified relative to the static response.
Defining the Magnification Factor
This specific ratio is known as the magnification factor, often denoted by $M$ or $MF$. It quantifies the dynamic amplification. The formula is:
$\text{Magnification Factor} = \frac{\text{Amplitude of Steady-State Reaction}}{\text{Static Deflection}}$
The magnification factor depends on the forcing frequency, the natural frequency of the system, and the damping present in the system. It is a crucial parameter in understanding resonance.
Why Other Options are Incorrect
- Indicator factor: This term is not a standard concept in vibration analysis.
- Damping factor: Damping factor (or damping ratio) is a parameter that describes how oscillations in a system decay after a disturbance. While damping affects the magnification factor, the damping factor itself is not this specific ratio.
- Surcharge factor: This term is used in geotechnical engineering related to soil pressure, not in vibration analysis.
Based on the standard definitions in vibration theory, the ratio of the amplitude of the steady-state reaction divided by the static deflection under force application is precisely the definition of the magnification factor.
Important Questions from Forced Vibration
Which method is used to find the natural frequencies of a system with multi degrees of freedom?
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 known as
When the frequency of the forced vibration is equal to the frequency of the free vibration, this condition is called ______.
Consider a forced single degree-of-freedom system governed by \(\rm \ddot x(t) + 2 ζ ω_n \dot x (t) + ω_n^2 x(t) = ω_n^2 \cos (ω t)\), where ζ and ωn are the damping ratio and undamped natural frequency of the system, respectively, while ω is the forcing frequency. The amplitude of the forced steady state response of this system is given by [(1 − r2)2 + (2ζr)2]-1/2, where 𝑟 = ω/ω n. The peak amplitude of this response occurs at a frequency ω = ωp. If ωd denotes the damped natural frequency of this system, which one of the following options is true?
The amplitude ratio of two successive oscillations of a damped vibratory system is
