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The driven oscillator: steady response and amplitude profile, bounded damped and off-resonant motions, and unbounded undamped resonance

Physical scope

The response of the driven oscillator model with m>0, k>0, c>=0, drive amplitude f_0 and angular frequency Omega>0: the explicit steady sinusoidal response and its amplitude |f_0|/sqrt((k-m Omega^2)^2+c^2 Omega^2) whenever the denominator does not vanish; the reduction of every motion to the steady response plus a homogeneous motion; boundedness of every motion in the strictly damped case, with exponential convergence of the transient to the steady response; boundedness for c=0 away from Omega=sqrt(k/m); unbounded motions at c=0 and Omega=sqrt(k/m) with nonzero amplitude, so boundedness fails exactly in the off-singular-case sense; and the positive-frequency amplitude peak when f_0 is nonzero and c^2<2mk. For c^2>=2mk and f_0!=0 the amplitude strictly decreases on Omega>0 and has no maximum on that domain; for f_0=0 it is constant. The result is a conditional deduction from the declared drive and damping models; it asserts no measured resonance curve, no value of any parameter and no behaviour of an apparatus.

Statement

Adopt the sinusoidally driven oscillator model of The sinusoidally driven oscillator model: a prescribed external force $f_{0} \cos (Ω t)$ added to the damped oscillator with data $m > 0$ , $k > 0$ , $c \geq 0$ , $f_{0} \in R$ and $Ω > 0$ , force law $F (t, x, v) = f_{0} \cos (Ω t) - k x - c v$ and equation $m x^{''} + c x^{'} + k x = f_{0} \cos (Ω t)$ . Put $ω_{0} : = \sqrt{\frac{k}{m}}, D : = k - m Ω^{2} + i c Ω \in C,$ and define the steady response $x_{p} (t) : = Re (\frac{f_{0} e^{i Ω t}}{D}) = \frac{f_{0} [(k - m Ω^{2}) \cos (Ω t) + c Ω \sin (Ω t)]}{(k - m Ω^{2})^{2} + c^{2} Ω^{2}} (D \neq 0),$ with amplitude $A (Ω) : = ∣ f_{0} ∣ / \sqrt{(k - m Ω^{2})^{2} + c^{2} Ω^{2}}$ , and in the exceptional case $c = 0$ , $Ω = ω_{0}$ define $x_{p} (t) : = f_{0} t \sin (ω_{0} t) / (2 m ω_{0})$ . All all-time boundedness, convergence and unboundedness claims below refer to maximal motions on $R$ ; local motions are their restrictions as in clause 2. Then:

Steady response. If $D \neq 0$ — that is $c > 0$ , or $c = 0$ and $Ω \neq ω_{0}$ — then $x_{p}$ is a motion of the model on $R$ , of period $2 π / Ω$ , with amplitude $A (Ω)$ in the sense $∣ x_{p} (t) ∣ \leq A (Ω)$ for all $t$ and equality at suitable times.
Reduction to the homogeneous model. Every datum has exactly one maximal motion, it is defined on all of $R$ , and every motion $x$ of the model on an interval with nonempty interior can be written $x = x_{p} + y$ , where $x_{p}$ is the particular motion of clause 1, or the resonant motion in the exceptional case, and $y$ is a motion of the damped homogeneous model with the same $m$ , $c$ , $k$ (and is a harmonic motion of The harmonic (linear restoring) model: a declared linear force law with parameter $k > 0$ when $c = 0$ ).
Strictly damped case $c > 0$ . For every $Ω > 0$ the following hold: every motion is bounded on $[0, \infty)$ ; and $x (t) - x_{p} (t) \to 0$ and $x^{'} (t) - x_{p}^{'} (t) \to 0$ as $t \to + \infty$ . Moreover $x_{p}$ is the unique motion with datum $(t_{0}, x_{p} (t_{0}), x_{p}^{'} (t_{0}))$ for any $t_{0} \in R$ . The amplitude function $Ω \mapsto A (Ω)$ has the following shape when $f_{0} \neq 0$ : if $c^{2} < 2 m k$ it is strictly increasing on $(0, Ω_{*})$ and strictly decreasing on $(Ω_{*}, \infty)$ , so has its unique maximum there, with $Ω_{*} = \sqrt{\frac{k}{m} - \frac{c^{2}}{2 m^{2}}}, A (Ω_{*}) = \frac{2 m ∣ f_{0} ∣}{c \sqrt{4 m k - c^{2}}};$ if $c^{2} \geq 2 m k$ it is strictly decreasing on $(0, \infty)$ and does not attain a maximum on the declared positive-frequency domain. Its continuous extension to $Ω = 0$ has its maximum there. For $f_{0} = 0$ the amplitude is identically zero on the domain, so every frequency has the same value.
Undamped case away from resonance. For $c = 0$ and $Ω \neq ω_{0}$ every motion is bounded on $R$ .
Undamped resonance. For $c = 0$ , $Ω = ω_{0}$ and $f_{0} \neq 0$ the motion $x_{p} (t) = f_{0} t \sin (ω_{0} t) / (2 m ω_{0})$ is unbounded on $[0, \infty)$ , and every motion of the model is unbounded on $[0, \infty)$ , although the drive $t \mapsto f_{0} \cos (ω_{0} t)$ is bounded.
Off the singular cases. For $c > 0$ the denominator never vanishes and every $Ω > 0$ is admissible; for $c = 0$ the exceptional frequency is exactly $Ω = ω_{0}$ . Consequently the boundedness assertions of clauses 3 and 4 hold for $c > 0$ , and for $c = 0$ away from $Ω = ω_{0}$ , and also at undamped resonance when $f_{0} = 0$ ; for a nonzero drive clause 5 gives unboundedness at undamped resonance.

The statement is read for the declared model data; no value of any parameter and no measured response is asserted.

Facts & Assumptions

Given: The driven model with data $(m, c, k, f_{0}, Ω)$ , its motions on intervals with nonempty interior, the constants $ω_{0}, D$ , the functions $x_{p}$ of clause 1, and the homogeneous damped model with the same $m, c, k$ .

[L1]

Model vocabulary (The sinusoidally driven oscillator model: a prescribed external force $f_{0} \cos (Ω t)$ added to the damped oscillator, The one-dimensional force models: data, motions, phase system, potential presentation, energy and equilibria clauses (b), (f), The damped driven oscillator model: second-order equation, phase-plane system and characteristic data): motions of the driven model are the solutions of $m x^{''} + c x^{'} + k x = f_{0} \cos (Ω t)$ , which is the forced oscillator model of The damped driven oscillator model: second-order equation, phase-plane system and characteristic data with the continuous function $f (t) = f_{0} \cos (Ω t)$ and $k > 0$ ; motions of the homogeneous damped model are the solutions with $f \equiv 0$ , and for $c = 0$ these are the motions of the harmonic model by clause 1 of Classification of the motions of the damped oscillator by the discriminant, and nonincreasing energy $E^{'} = - c x^{' 2}$ .

[L2]

Sinusoidal forcing: the steady response, its amplitude and phase, and resonance at $Ω = \sqrt{k / m}$ : if $D \neq 0$ then $x_{p} = Re (f_{0} e^{i Ω t} / D)$ is a solution of the forced equation on $R$ , with the displayed real form and amplitude $∣ f_{0} ∣ / \sqrt{(k - m Ω^{2})^{2} + c^{2} Ω^{2}}$ ; if $c = 0$ and $Ω = ω_{0}$ then $x_{p} (t) = f_{0} t \sin (ω_{0} t) / (2 m ω_{0})$ is a solution on $R$ and is unbounded on $[0, \infty)$ when $f_{0} \neq 0$ ; if $c > 0$ then every solution $x : [0, \infty) \to R$ equals $x_{p} + x_{h}$ with $x_{h}$ homogeneous, and $x (t) - x_{p} (t) \to 0$ , with $x$ bounded.

[L3]

The forced oscillator with $k \geq 0$ : unique global solution and the convolution (Green's function) formula: for $m > 0$ , $c \geq 0$ , $k > 0$ and continuous $f$ on an interval $I$ , the forced model has exactly one solution on $I$ for every datum; and The homogeneous damped oscillator: solution classification by the discriminant and uniqueness for initial data classifies the homogeneous solutions.

[L4]

Strictly damped oscillator: explicit decay bounds, convergence to rest and asymptotic stability of the origin clauses 2 and 3: a strictly damped homogeneous motion and its derivative tend to $0$ and are bounded; Classification of the motions of the damped oscillator by the discriminant, and nonincreasing energy $E^{'} = - c x^{' 2}$ clause 1: the maximal motion of the homogeneous model with a given datum is unique.

[L5]

The motions of the harmonic model are the sinusoids $A \cos (ω (t - t_{0})) + B \sin (ω (t - t_{0}))$ : amplitude and phase, the least period, the conserved energy $k R^{2} / 2$ and the elliptical phase curves clauses 2 and 3: a harmonic motion is bounded by its amplitude $R$ , and $∣ y^{'} (t) ∣ \leq ω_{0} R$ for the motion of amplitude $R$ because $y^{'} = - R ω_{0} \sin (θ - φ)$ .

[L6]

The amplitude of the sinusoidally forced damped oscillator: location and value of its maximum: for $c > 0$ and $f_{0} \neq 0$ the function $A (Ω) = ∣ f_{0} ∣ / \sqrt{(k - m Ω^{2})^{2} + c^{2} Ω^{2}}$ is strictly increasing then strictly decreasing with the maximum $Ω_{*} = \sqrt{k / m - c^{2} / (2 m^{2})}$ and value $2 m ∣ f_{0} ∣ / (c \sqrt{4 m k - c^{2}})$ when $c^{2} < 2 m k$ , and strictly decreasing on $(0, \infty)$ when $c^{2} \geq 2 m k$ .

[L7]

Derivatives (The derivatives of sine and cosine are cosine and minus sine, Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ ): the real form of $x_{p}$ is a sum of $\cos (Ω t)$ and $\sin (Ω t)$ with constant coefficients, so $x_{p}^{'} (t) = \frac{f_{0} Ω [- (k - m Ω^{2}) \sin (Ω t) + c Ω \cos (Ω t)]}{(k - m Ω^{2})^{2} + c^{2} Ω^{2}},$ which is bounded in absolute value by $Ω ∣ f_{0} ∣ (∣ k - m Ω^{2} ∣ + c Ω) / ((k - m Ω^{2})^{2} + c^{2} Ω^{2})$ .

Proof

technique · cases; verify the explicit particular motions, reduce an arbitrary motion to the particular one plus a homogeneous motion by linearity, and use the decay, boundedness and amplitude results of the suppliers

1.1assume-case 1givenL1L2L7

Clause 1. Assume $D \neq 0$ . By [L2] the function $x_{p}$ is a solution of the forced equation on $R$ , hence a motion of the model by [L1], with the displayed real form and amplitude $∣ x_{p} (t) ∣ \leq A (Ω)$ . Its periodicity: the real form is a constant linear combination of $\cos (Ω \cdot)$ and $\sin (Ω \cdot)$ , and for every real $u$ one has $\cos (u + 2 π) = \cos u$ and $\sin (u + 2 π) = \sin u$ by The zero sets of sine and cosine and the least positive common period 2 pi; taking $u = Ω t$ and $u = Ω (t + 2 π / Ω)$ gives $x_{p} (t + 2 π / Ω) = x_{p} (t)$ .

1.2givenL1L3algebra

Clause 2. The forced model has $k > 0 \geq 0$ and the continuous drive $f_{0} \cos (Ω t)$ on $I = R$ , so [L3] gives exactly one solution on $R$ for every datum; hence every maximal motion is defined on $R$ . Let $x$ be a motion on an interval $J$ with nonempty interior and let $x_{p}$ be the particular motion of clause 1 or the resonant motion of [L2]. The function $y : = x - x_{p}$ is of class $C^{2}$ on $J$ and $m y^{''} + c y^{'} + k y = (m x^{''} + c x^{'} + k x) - (m x_{p}^{''} + c x_{p}^{'} + k x_{p}) = f_{0} \cos (Ω t) - f_{0} \cos (Ω t) = 0$ by the linearity of the differential expression (Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ ; the two equations hold on $J$ ). Hence $y$ is a motion of the homogeneous damped model on $J$ , and for $c = 0$ it is a harmonic motion by [L1]. This is clause 2.

1.3givenL6

Clause 3, amplitude shape. For $f_{0} \neq 0$ use [L6] on the amplitude function $A$ of clause 1; for $f_{0} = 0$ , the strictly positive denominator for $c > 0$ gives $A (Ω) = 0$ for every $Ω$ directly.

2.1givenstep 1.2L2L4L7

Clause 3, transient and boundedness. Let $c > 0$ ; then $D \neq 0$ because $D = 0$ would force $c Ω = 0$ and hence $Ω = 0$ , contrary to $Ω > 0$ . Let $x$ be a motion; by clause 2, $y = x - x_{p}$ is a strictly damped homogeneous motion, so by [L4] $y (t) \to 0$ and $y^{'} (t) \to 0$ , and $x_{p}$ , $x_{p}^{'}$ are bounded by clause 1 and [L7]; hence $x (t) \to x_{p} (t)$ in the sense $x (t) - x_{p} (t) \to 0$ and $x^{'} (t) - x_{p}^{'} (t) \to 0$ , and $x$ is bounded on $[0, \infty)$ . If $x$ has datum $(t_{0}, x_{p} (t_{0}), x_{p}^{'} (t_{0}))$ , then $y$ is a homogeneous motion with $y (t_{0}) = y^{'} (t_{0}) = 0$ ; by the uniqueness of [L4] (with the datum $(t_{0}, 0, 0)$ ) $y \equiv 0$ , so $x = x_{p}$ .

2.2assume-case 2givenstep 1.1step 1.2L1L5

Clause 4. Let $c = 0$ and $Ω \neq ω_{0}$ . Then $D = k - m Ω^{2} = m (ω_{0}^{2} - Ω^{2}) \neq 0$ , so $x_{p}$ is a motion with $∣ x_{p} (t) ∣ \leq A (Ω)$ by clause 1. Let $x$ be a motion on an interval with nonempty interior; by clause 2, $y = x - x_{p}$ is a harmonic motion, so by [L5] there is $R_{y} \geq 0$ with $∣ y (t) ∣ \leq R_{y}$ for all $t$ ; hence $∣ x (t) ∣ \leq A (Ω) + R_{y}$ on the interval, and since clause 2 extends the motion to $R$ , the motion is bounded on $R$ .

2.3assume-case 3givenstep 1.2L2L5

Clause 5. Let $c = 0$ , $Ω = ω_{0}$ and $f_{0} \neq 0$ . By [L2] the function $x_{p} (t) = f_{0} t \sin (ω_{0} t) / (2 m ω_{0})$ is a motion on $R$ and is unbounded on $[0, \infty)$ . Let $x$ be any motion; by clause 2, $y = x - x_{p}$ is a harmonic motion, bounded by some $R_{y}$ by [L5]. Choose $t_{k} \to + \infty$ with $∣ x_{p} (t_{k}) ∣ \to + \infty$ ; then $∣ x (t_{k}) ∣ \geq ∣ x_{p} (t_{k}) ∣ - ∣ y (t_{k}) ∣ \geq ∣ x_{p} (t_{k}) ∣ - R_{y} \to + \infty$ , so $x$ is unbounded on $[0, \infty)$ as well, although $∣ f_{0} \cos (ω_{0} t) ∣ \leq ∣ f_{0} ∣$ is bounded by Parity and the Pythagorean identity for sine and cosine. This is clause 5.

3.1cases-exhaustivegivenstep 2.1step 2.2step 2.3algebra∎

Clause 6. For $c > 0$ the computation of step 2.1 gives $D \neq 0$ for every $Ω > 0$ , and clause 3 supplies the boundedness. For $c = 0$ , $D = 0 ⟺ k = m Ω^{2} ⟺ Ω = ω_{0}$ since $Ω > 0$ and $ω_{0} > 0$ ; clauses 4 and 5 give boundedness away from $Ω = ω_{0}$ and unboundedness at it when $f_{0} \neq 0$ . If $f_{0} = 0$ the equation is homogeneous, so the harmonic classification [L5] gives boundedness also at that frequency.

Remarks

What resonance is here. The exceptional case is a property of the declared model: with zero damping and a nonzero drive at the undamped frequency, the particular solution grows linearly and every maximal motion is unbounded. Clause 5 states no physical resonance phenomenon; an apparatus comparison would be a separate, empirically qualified step.
The quality factor. For $c > 0$ and $f_{0} \neq 0$ , use the continuous extension of $A$ to $0$ and define $Q : = A (ω_{0}) / A (0) = m ω_{0} / c = ω_{0} / (2 γ)$ , where $γ = c / (2 m)$ . Likharev §5.1, Eqs. (5.20)-(5.22), PDF page 91, concerns the normalized full width of the squared response, not a half width. In the source's low-damping, small-detuning approximation the squared amplitude is replaced by $f_{0}^{2} / [4 m^{2} ω_{0}^{2} ((Ω - ω_{0})^{2} + γ^{2})]$ . Its half-maximum points are $Ω = ω_{0} \pm γ$ , so their separation $Δ Ω = 2 γ$ satisfies $Δ Ω / ω_{0} = 1 / Q$ . This is an identity for that approximate profile; no exact width formula for the original amplitude curve or apparatus measurement is asserted.

The driven oscillator: steady response and amplitude profile, bounded damped and off-resonant motions, and unbounded undamped resonance

Statement

Facts & Assumptions

Proof

Remarks

Depends on

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