Definition:Amplitude of Underdamped Oscillation

Definition

Consider a physical system $S$ whose behaviour can be described with the second order ODE in the form:

$(1): \quad \dfrac {\d^2 x} {\d t^2} + 2 b \dfrac {\d x} {\d t} + a^2 x = 0$

for $a, b \in \R_{>0}$.

Let $(1)$ be subject to the initial conditions:

$x = x_0$ at $t = 0$

$x' = 0$ at $t = 0$

Let $b < a$, so as to make $S$ underdamped.

The amplitude of the oscillation of $S$ is defined as:

$A = \dfrac {x_0 a} {\sqrt {a^2 - b^2} } e^{-b t}$

Examples

Arbitrary Example

Consider the equation describing underdamped oscillation:

$x = 5 e^{-2 t} \sin 3 t$

The amplitude of $x$ is $5 e^{-2 t}$.

Also see

• Definition:Period of Underdamped Oscillation

• Results about amplitudes of underdamped oscillations can be found here.

Sources

• 1972: George F. Simmons: Differential Equations ... (previous) ... (next): $\S 3.20$: Vibrations in Mechanical Systems
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): amplitude
• 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): amplitude