Condition for Resonance in Forced Vibration of Underdamped System

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Theorem

Consider a physical system $S$ whose behaviour is defined by the second order ODE:

$(1): \quad \dfrac {\d^2 y} {\d x^2} + 2 b \dfrac {\d y} {\d x} + a^2 x = K \cos \omega x$

where:

$K \in \R: k > 0$
$a, b \in \R_{>0}: b < a$

Then $S$ is in resonance when:

$\omega = \sqrt {a^2 - 2 b^2}$

and thus the resonance frequency is:

$\nu_R = \dfrac {\sqrt {a^2 - 2 b^2} } {2 \pi}$


This resonance frequency exists only when $a^2 - 2 b^2 > 0$.


Proof

From Linear Second Order ODE: $y + 2 b y' + a^2 y = K \cos \omega x$ where $b < a$ the general solution of $(1)$ is:

$(2): \quad y = e^{-b x} \paren {C_1 \cos \alpha x + C_2 \sin \alpha x} + \dfrac K {\sqrt {4 b^2 \omega^2 + \paren {a^2 - \omega^2}^2} } \map \cos {\omega x - \phi}$

where:

$\alpha = \sqrt {a^2 - b^2}$
$\phi = \map \arctan {\dfrac {2 b \omega} {a^2 - \omega^2} }$


Consider the steady-state component of $(2)$:

$y_s = \dfrac K {\sqrt {4 b^2 \omega^2 + \paren {a^2 - \omega^2}^2} } \map \cos {\omega x - \phi}$

By definition, the amplitude of $y_s$ is:

$A = \dfrac K {\sqrt {4 b^2 \omega^2 + \paren {a^2 - \omega^2}^2} }$

We have that both $a$ and $b$ are greater than $0$.

It is also assumed that $\omega$ is also greater than $0$.

Thus:

$4 b^2 \omega^2 > 0$
$a^2 > 0$
$\omega^2 > 0$

Then $y_s$ is at a maximum when:

$\map f \omega = 4 b^2 \omega^2 + \paren {a^2 - \omega^2}^2$

is at a minimum.

Differentiating $\map f \omega$ with respect to $\omega$:

\(\ds \map {f'} \omega\) \(=\) \(\ds 8 b^2 \omega + 2 \paren {a^2 - \omega^2} \paren {-2 \omega}\)
\(\ds \) \(=\) \(\ds 8 b^2 \omega - 4 a^2 \omega + 4 \omega^3\)
\(\ds \) \(=\) \(\ds 4 \omega \paren {2 b^2 - a^2 + \omega^2}\)


Setting $\map {f'} \omega = 0$:

\(\ds 4 \omega \paren {2 b^2 - a^2 + \omega^2}\) \(=\) \(\ds 0\)
\(\ds \leadsto \ \ \) \(\ds \omega\) \(=\) \(\ds 0\)
\(\, \ds \lor \, \) \(\ds \omega^2\) \(=\) \(\ds a^2 - 2 b^2\)

As $\omega > 0$ the first of these cannot apply.

If $a^2 < 2 b^2$ then $a^2 - 2 b^2 < 0$ and so $\omega^2$ has no real roots.

Hence the resonance frequency is given by:

$\omega = \sqrt {a^2 - 2 b^2}$

if and only if $a^2 > 2 b^2$.

$\blacksquare$


Sources

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