The amplitude ratio of two successive oscillations of a damped vibratory system is
More than one
Understanding Damped Oscillations and Amplitude Ratio
A damped vibratory system is a system where oscillations gradually decrease in amplitude over time. This reduction in amplitude happens because energy is continuously lost from the system, typically due to forces like friction or air resistance.
Successive Oscillations in Damped Systems
When we talk about two successive oscillations in a damped system, we are comparing the amplitude of one complete cycle to the amplitude of the very next cycle.
Let the amplitude of the $n$-th oscillation be denoted by $A_n$ and the amplitude of the next ($n+1$)-th oscillation be denoted by $A_{n+1}$.
In a damped system, energy is lost with each oscillation, causing the amplitude to decrease. Therefore, the amplitude of the ($n+1$)-th oscillation is always smaller than the amplitude of the $n$-th oscillation.
This can be represented as:
$$ A_{n+1} < A_n $$
Calculating the Amplitude Ratio
The question asks for the "amplitude ratio of two successive oscillations". This typically refers to the ratio of the amplitude of an oscillation to the amplitude of the next one, which is $ \frac{A_n}{A_{n+1}} $.
Since $A_n$ is the amplitude of an earlier oscillation and $A_{n+1}$ is the amplitude of the immediately following oscillation, and we know $A_n > A_{n+1}$ due to damping:
$$ \frac{A_n}{A_{n+1}} > 1 $$
This ratio is greater than one because the numerator ($A_n$) is larger than the denominator ($A_{n+1}$).
If the ratio were calculated the other way, $ \frac{A_{n+1}}{A_n} $, it would be less than one, as the numerator is smaller than the denominator.
Considering the options provided:
- Option 1: More than one - This aligns with the ratio $ \frac{A_n}{A_{n+1}} $.
- Option 2: Less than one - This aligns with the ratio $ \frac{A_{n+1}}{A_n} $.
- Option 3: Equal to one - This would only happen if there was no damping ($A_n = A_{n+1}$).
- Option 4: Variable - While the specific value changes depending on the damping, the relationship (greater or less than one) is consistent for successive amplitudes in a standard damped system.
Based on the common understanding and the likely interpretation leading to the answer "More than one", the ratio compares a larger amplitude to the subsequent smaller amplitude.
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 ______.
The amplitude of the steady-state reaction divided by the static deflection under force application is known as _______.
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?
