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Resonant forcing at $Ω = ω_{0}$ and the two regimes of the amplitude maximum

Example

Consider the forced model $m x^{''} + c x^{'} + k x = f_{0} \cos (Ω t)$ on time domain $I = R$ with the values and conclusions below (The damped driven oscillator model: second-order equation, phase-plane system and characteristic data, Sinusoidal forcing: the steady response, its amplitude and phase, and resonance at $Ω = \sqrt{k / m}$ , The amplitude of the sinusoidally forced damped oscillator: location and value of its maximum).

(i) Resonance: $m = k = f_{0} = 1$ , $c = 0$ , $Ω = ω_{0} = 1$ . Then $x_{p} (t) = \frac{1}{2} t \sin t$ is a solution of $x^{''} + x = \cos t$ , it satisfies $x_{p} (0) = 0$ , $x_{p}^{'} (0) = 0$ , and $∣ x_{p} ∣$ is unbounded on $[0, \infty)$ .
(ii) Amplitude peak above zero frequency: $m = k = f_{0} = 1$ , $c = 1$ . Then $u_{*} = 1 - \frac{1}{2} = \frac{1}{2} > 0$ , the maximum of $A (Ω) = 1 / \sqrt{(1 - Ω^{2})^{2} + Ω^{2}}$ is attained at $Ω_{*} = 1 / \sqrt{2}$ , and $A (Ω_{*}) = 2 / \sqrt{3}$ , while $A (0) = A (1) = 1$ .
(iii) Amplitude peak at zero frequency: $m = k = f_{0} = 1$ , $c = 2$ . Then $u_{*} = 1 - 2 = - 1 \leq 0$ , and the maximum of $A (Ω) = 1 / \sqrt{(1 - Ω^{2})^{2} + 4 Ω^{2}}$ on $[0, \infty)$ is attained at $Ω = 0$ with value $1$ , while $A (1) = 1 / 2$ .

Facts & Assumptions

Given: The five parameters in each line and the steady-response formulae of Sinusoidal forcing: the steady response, its amplitude and phase, and resonance at $Ω = \sqrt{k / m}$ clause 1 and The amplitude of the sinusoidally forced damped oscillator: location and value of its maximum.

[L1]

In the resonant case $c = 0$ , $Ω = ω_{0} = \sqrt{k / m}$ the function $x_{p} (t) = f_{0} t \sin (ω_{0} t) / (2 m ω_{0})$ is a solution of the forced model on $R$ and is unbounded on $[0, \infty)$ when $f_{0} \neq 0$ (Sinusoidal forcing: the steady response, its amplitude and phase, and resonance at $Ω = \sqrt{k / m}$ clause 2). [given]

[L2]

If $c > 0$ the amplitude of the steady response is $A (Ω) = ∣ f_{0} ∣ / \sqrt{(k - m Ω^{2})^{2} + c^{2} Ω^{2}}$ ; its maximum is attained at $Ω_{*} = \sqrt{u_{*}}$ when $u_{*} = k / m - c^{2} / (2 m^{2}) > 0$ , with value $2 m ∣ f_{0} ∣ / (c \sqrt{4 m k - c^{2}})$ , and at $Ω = 0$ when $u_{*} \leq 0$ (The amplitude of the sinusoidally forced damped oscillator: location and value of its maximum; the connection to the steady response is clause 1 of Sinusoidal forcing: the steady response, its amplitude and phase, and resonance at $Ω = \sqrt{k / m}$ ). [given]

[L3]

$(\sin t)^{'} = \cos t$ , $(\cos t)^{'} = - \sin t$ ; sums and products are differentiated by the algebra of derivatives and the chain rule. [thm-sine-and-cosine-derivatives, thm-algebra-of-derivatives, thm-chain-rule]

Verification

technique · direct; substitute the resonant candidate, and evaluate the amplitude formula in the two regimes of the preceding proposition

1.1givenL1L3algebra

Line (i): with $m = ω_{0} = f_{0} = 1$ the candidate is $x_{p} (t) = \frac{1}{2} t \sin t$ ; by [L3], $x_{p}^{'} (t) = \frac{1}{2} (\sin t + t \cos t)$ and $x_{p}^{''} (t) = \frac{1}{2} (2 \cos t - t \sin t) = \cos t - \frac{1}{2} t \sin t$ , so $x_{p}^{''} + x_{p} = \cos t$ , as required, and $x_{p} (0) = 0 = x_{p}^{'} (0)$ . At $t_{k} = π / 2 + 2 π k$ one has $\sin t_{k} = 1$ and $∣ x_{p} (t_{k}) ∣ = t_{k} / 2 \to \infty$ , so $x_{p}$ is unbounded on $[0, \infty)$ , in agreement with [L1].

1.2givenL2algebra

Line (ii): here $u_{*} = 1 - 1 / 2 = 1 / 2 > 0$ and $4 m k - c^{2} = 3$ , so [L2] locates the maximum at $Ω_{*} = 1 / \sqrt{2}$ with value $2 / \sqrt{3}$ ; directly, $(1 - Ω_{*}^{2})^{2} + Ω_{*}^{2} = (1 / 2)^{2} + 1 / 2 = 3 / 4$ , so $A (Ω_{*}) = 2 / \sqrt{3}$ , while $A (0) = 1$ and $A (1) = 1 / \sqrt{0 + 1} = 1$ , both below $2 / \sqrt{3}$ .

2.1givenL2algebra∎

Line (iii): here $u_{*} = 1 - 2 = - 1 \leq 0$ , so [L2] locates the maximum at $0$ with value $A (0) = 1 / \sqrt{1} = 1$ ; directly, $A (1) = 1 / \sqrt{(0)^{2} + 4} = 1 / 2 < 1$ , and the same formula gives $A (Ω) \leq 1$ for all $Ω \geq 0$ because $(1 - Ω^{2})^{2} + 4 Ω^{2} = 1 + 2 Ω^{2} + Ω^{4} \geq 1$ .

Resonant forcing at $Ω = ω_{0}$ and the two regimes of the amplitude maximum

Example

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources

Resonant forcing at Ω=ω0Ω=ω0​ and the two regimes of the amplitude maximum

Example

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources

Resonant forcing at $Ω = ω_{0}$ and the two regimes of the amplitude maximum