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Strictly damped oscillator: explicit decay bounds, convergence to rest and asymptotic stability of the origin

Physical scope

The strictly damped oscillator with m>0, k>0, c>0: every motion has explicit position and velocity bounds in normalized SI coordinates, tends to zero together with its velocity, and the origin of the phase plane is a globally asymptotically stable rest point in the phase norm defined by the numerical SI coordinates. The result is a conditional deduction from the declared model data; it asserts no value of the parameters, no measured decay rate and no behaviour of any real damper, and its constants are the explicit ones produced by the classification.

Statement

Adopt the damped oscillator model of The damped oscillator model: a harmonic restoring force with a phenomenological linear drag $- c \dot{x}$ , $c \geq 0$ with $m > 0$ , $k > 0$ and $c > 0$ (strict damping), force law $F (x, v) = - k x - c v$ and energy $E (x, v) = \frac{1}{2} m v^{2} + \frac{1}{2} k x^{2}$ . Position and velocity coordinates in every phase norm below are their normalized numerical SI values, as defined in clause (a) of The phase plane of a one-dimensional model: phase curves, the energy function and its level sets, and turning points. For the datum $(0, x_{0}, v_{0})$ , put $r_{0} : = ∥ (x_{0}, v_{0}) ∥_{2}$ in that norm. In displayed time-dependent formulas, $t$ is read numerically in seconds, and rates in exponents are read in inverse seconds. Write $Δ = c^{2} - 4 m k$ , $γ = c / (2 m) > 0$ , $λ_{\pm} = (- c \pm \sqrt{Δ}) / (2 m)$ for $Δ \geq 0$ , $λ = - c / (2 m)$ for $Δ = 0$ and $ω = \sqrt{4 m k - c^{2}} / (2 m)$ for $Δ < 0$ . Then:

Motion bounds. For every motion $x : R \to R$ $x : R \to R$ of the model with datum $(0, x_{0}, v_{0})$ $(0, x_{0}, v_{0})$ and phase norm $r_{0}$ $r_{0}$ the following bounds hold for all $t \geq 0$ $t \geq 0$ , with the coefficients $A, B$ $A, B$ of the classification of Classification of the motions of the damped oscillator by the discriminant, and nonincreasing energy $E^{'} = - c x^{' 2}$ :
- $Δ > 0$ : $∣ x (t) ∣ \leq (∣ A ∣ + ∣ B ∣) e^{λ_{+} t}$ and $∣ x^{'} (t) ∣ \leq (∣ λ_{+} ∣ ∣ A ∣ + ∣ λ_{-} ∣ ∣ B ∣) e^{λ_{+} t}$ ;
- $Δ = 0$ : $∣ x (t) ∣ \leq (∣ A ∣ + ∣ B ∣ t) e^{λ t}$ and $∣ x^{'} (t) ∣ \leq (∣ λ A + B ∣ + ∣ λ B ∣ t) e^{λ t}$ ;
- $Δ < 0$ : $∣ x (t) ∣ \leq (∣ A ∣ + ∣ B ∣) e^{- γ t}$ and $∣ x^{'} (t) ∣ \leq (∣ ω B - γ A ∣ + ∣ γ B + ω A ∣) e^{- γ t}$ ; and there are constants $C_{1}, C_{2} > 0$ , depending only on $(m, k, c)$ , with $∣ x (t) ∣ \leq C_{1} r_{0}, ∣ x^{'} (t) ∣ \leq C_{2} r_{0} (t \geq 0),$ for the numerical SI coordinate values; equivalently the corresponding physical position and velocity are bounded by $(1 m) C_{1} r_{0}$ and $(1 m / s) C_{2} r_{0}$ . Here $C_{1}, C_{2} > 0$ are dimensionless constants in the normalized SI coordinates and depend only on the numerical values of $(m, k, c)$ in the units declared by Dimensions and SI unit conventions of the non-relativistic classical mechanics framework.
Convergence. For every maximal motion, $x (t) \to 0$ and $x^{'} (t) \to 0$ as $t \to + \infty$ (Limits at $+ \infty$ and $- \infty$ , and infinite limits at a point).
Asymptotic stability. The rest point $(0, 0)$ is Lyapunov stable and globally asymptotically stable in the sense of Equilibria, Lyapunov stability and asymptotic stability of an autonomous first-order system for the model's phase system: for every $ε > 0$ there is $δ > 0$ such that every maximal motion with $∥ (x (t_{0}), x^{'} (t_{0})) ∥_{2} < δ$ for some $t_{0}$ is defined for all $t \geq t_{0}$ and satisfies $∥ (x (t), x^{'} (t)) ∥_{2} < ε$ for all $t \geq t_{0}$ ; and every maximal motion satisfies $∥ (x (t), x^{'} (t)) ∥_{2} \to 0$ as $t \to + \infty$ . The norm is the Euclidean norm on the normalized numerical SI coordinates of The phase plane of a one-dimensional model: phase curves, the energy function and its level sets, and turning points.

The constants are explicit in the proof and no value of $m, k, c$ is asserted.

Facts & Assumptions

Given: The strictly damped model with $m > 0$ , $k > 0$ , $c > 0$ , a motion $x : R \to R$ with datum $(0, x_{0}, v_{0})$ , the phase norm $r_{0} = ∥ (x_{0}, v_{0}) ∥_{2}$ of clause (a) of The phase plane of a one-dimensional model: phase curves, the energy function and its level sets, and turning points, and the dimensionless parameters $M : = m / (1 k g)$ , $K : = k / (1 N / m)$ and $C : = c / (1 k g / s)$ . Numerical rate values are denoted by $Λ_{\pm} : = (1 s) λ_{\pm}$ , $Λ : = (1 s) λ$ , $Γ : = (1 s) γ$ and $W : = (1 s) ω$ where applicable.

[L1]

Model vocabulary (The damped oscillator model: a harmonic restoring force with a phenomenological linear drag $- c \dot{x}$ , $c \geq 0$ , The one-dimensional force models: data, motions, phase system, potential presentation, energy and equilibria, Classification of the motions of the damped oscillator by the discriminant, and nonincreasing energy $E^{'} = - c x^{' 2}$ ): all motions extend to $R$ and are classified by the sign of $Δ$ into the three families, with the coefficients $A, B$ of clause 4 of The homogeneous damped oscillator: solution classification by the discriminant and uniqueness for initial data applied to the datum; the energy satisfies $E (x (t), x^{'} (t)) \leq E (x (0), x^{'} (0))$ for every $t \geq 0$ because $E$ is nonincreasing, and $E (0) = E (x_{0}, v_{0}) = \frac{1}{2} m v_{0}^{2} + \frac{1}{2} k x_{0}^{2}$ ; writing $X : = x$ and $V : = x^{'}$ for the numerical SI coordinates, its numerical energy in joules is $E : = E / (1 J) = \frac{1}{2} M V^{2} + \frac{1}{2} K X^{2}$ ; the equation is $m x^{''} + c x^{'} + k x = 0$ ; the phase system is the one of The damped driven oscillator model: second-order equation, phase-plane system and characteristic data.

[L2]

Strictly damped homogeneous oscillator with $k > 0$ : every solution tends to zero: in the three cases, with the coefficients $A, B$ of [L1], $∣ x (t) ∣ \leq (∣ A ∣ + ∣ B ∣) e^{λ_{+} t}$ for $Δ > 0$ , $∣ x (t) ∣ \leq (∣ A ∣ + ∣ B ∣ t) e^{λ t}$ for $Δ = 0$ , and $∣ x (t) ∣ \leq (∣ A ∣ + ∣ B ∣) e^{- γ t}$ for $Δ < 0$ , for $t \geq 0$ ; in each case the decay exponent is negative: $λ_{+} < 0$ for $Δ > 0$ (because $\sqrt{Δ} < c$ when $k > 0$ ), $λ = - c / (2 m) < 0$ for $Δ = 0$ , and $γ > 0$ for $Δ < 0$ .

[L3]

Exponential calculus and its values (The exponential function is smooth and $(\exp)^{'} = \exp$ , 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 derivatives of sine and cosine are cosine and minus sine, 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 exponential function is strictly increasing, The exponential tends to $+ \infty$ at $+ \infty$ and to $0$ at $- \infty$ , The real exponential function and the number $e$ by a power series): $(e^{μ t})^{'} = μ e^{μ t}$ for real $μ$ ; products and sums are differentiated by the algebra of derivatives and composites by the chain rule; $e^{0} = 1$ ; $e^{s} \leq e^{0} = 1$ for $s \leq 0$ ; and $\exp (s) \to 0$ as $s \to - \infty$ , so for every $η > 0$ there is $M$ with $e^{s} < η$ for every $s < - M$ .

[L4]

A series of nonnegative terms converges iff its partial sums are bounded, and then the sum is their supremum with The real exponential function and the number $e$ by a power series: for $u \geq 0$ the exponential series has nonnegative terms, and the partial sum $1 + u$ is at most the sum, so $u \leq e^{u} (u \geq 0) .$

[L5]

Square roots exist: a unique $\sqrt{a} \geq 0$ with $(\sqrt{a})^{2} = a$ ; the positives are ${x^{2} : x \neq 0}$ , Squaring is monotone on the nonnegatives: square roots of nonnegative reals exist and are unique, and the square root is strictly increasing on $[0, \infty)$ ; in particular $∣ x^{'} (t) ∣ \leq \sqrt{2 E (x (t), x^{'} (t)) / m}$ for every motion.

Proof

technique · cases on the sign of $\Delta$; use the classification and the decay corollary for the position in normalized SI coordinates, the energy bound for the normalized velocity, and then assemble the uniform bound and stability statement

1.1givenL1L5algebra

Preliminary. Because the phase system is autonomous, all statements are invariant under translating time, and it suffices to consider the datum at $t_{0} = 0$ ; the functions $x$ and $x^{'}$ of the statement are then defined for all $t \geq 0$ by [L1]. Let $X_{0} : = x_{0}$ and $V_{0} : = v_{0}$ be the normalized numerical SI coordinates of the datum, and let $M, K$ be the dimensionless numerical values of $m, k$ defined in Given; then $r_{0}^{2} = X_{0}^{2} + V_{0}^{2}$ . With $E : = E / (1 J)$ , the normalized initial energy is $E (0) = \frac{1}{2} M V_{0}^{2} + \frac{1}{2} K X_{0}^{2} \leq \frac{1}{2} \max (M, K) r_{0}^{2},$ since $X_{0}^{2}, V_{0}^{2} \leq r_{0}^{2}$ . By [L1] the energy is nonincreasing and $E (t) \geq \frac{1}{2} M V (t)^{2}$ , where $V (t) : = x^{'} (t)$ is the normalized numerical SI velocity coordinate; hence $∣ V (t) ∣ \leq \sqrt{\frac{2 E (t)}{M}} \leq \sqrt{\frac{\max (M, K)}{M}} r_{0} (t \geq 0) .$ This is the uniform normalized velocity bound with $C_{2}^{E} : = \sqrt{\max (M, K) / M}$ ; and every maximal motion is defined on all of $R$ , while arbitrary local motions are its restrictions by [L1].

2.1assume-case 1givenstep 1.1L1L2L3algebra

Case $Δ > 0$ . Let $Λ_{\pm} : = (1 s) λ_{\pm}$ and $d : = Λ_{+} - Λ_{-} > 0$ be the numerical SI rates and their difference. Write $\overset{ˉ}{A}, \overset{ˉ}{B}$ for the numerical SI values of the position coefficients $A, B$ in [L1]. Their formulas give $\overset{ˉ}{A} = \frac{V_{0} - Λ_{-} X_{0}}{d}, \overset{ˉ}{B} = \frac{Λ_{+} X_{0} - V_{0}}{d} .$ Hence $∣ \overset{ˉ}{A} ∣ \leq (1 + ∣ Λ_{-} ∣) r_{0} / d$ and $∣ \overset{ˉ}{B} ∣ \leq (1 + ∣ Λ_{+} ∣) r_{0} / d$ , since $∣ X_{0} ∣, ∣ V_{0} ∣ \leq r_{0}$ . By [L2], $∣ x (t) ∣ \leq (∣ \overset{ˉ}{A} ∣ + ∣ \overset{ˉ}{B} ∣) e^{Λ_{+} t}$ , so $∣ x (t) ∣ \leq C_{1} r_{0}$ with $C_{1} : = (2 + ∣ Λ_{-} ∣ + ∣ Λ_{+} ∣) / d$ , using $e^{Λ_{+} t} \leq 1$ for $t \geq 0$ by [L3]. Differentiating $x (t) = \overset{ˉ}{A} e^{Λ_{+} t} + \overset{ˉ}{B} e^{Λ_{-} t}$ gives the numerical velocity $x^{'} (t) = Λ_{+} \overset{ˉ}{A} e^{Λ_{+} t} + Λ_{-} \overset{ˉ}{B} e^{Λ_{-} t}$ , so $∣ x^{'} (t) ∣ \leq (∣ Λ_{+} ∣ ∣ \overset{ˉ}{A} ∣ + ∣ Λ_{-} ∣ ∣ \overset{ˉ}{B} ∣) e^{Λ_{+} t}$ , the normalized form of the displayed case bound; it tends to $0$ because $e^{Λ_{+} t} \to 0$ by [L3].

2.2assume-case 2givenstep 1.1L1L2L3L4algebra

Case $Δ = 0$ . Write $Λ : = (1 s) λ < 0$ , and let $\overset{ˉ}{A}, \overset{ˉ}{B}$ be the numerical SI values of the critical-case coefficients in [L1]. The coefficient formulas give $\overset{ˉ}{A} = X_{0}$ and $\overset{ˉ}{B} = V_{0} - Λ X_{0}$ , so $∣ \overset{ˉ}{A} ∣ \leq r_{0}$ and $∣ \overset{ˉ}{B} ∣ \leq (1 + ∣ Λ ∣) r_{0}$ . For $t \geq 0$ put $u = ∣ Λ ∣ t \geq 0$ ; by [L4], $u \leq e^{u}$ , so $t e^{Λ t} \leq 1 / ∣ Λ ∣$ . By [L2], the normalized position satisfies $∣ x (t) ∣ \leq (∣ \overset{ˉ}{A} ∣ + ∣ \overset{ˉ}{B} ∣ t) e^{Λ t} \leq ∣ \overset{ˉ}{A} ∣ + ∣ \overset{ˉ}{B} ∣ / ∣ Λ ∣ \leq (2 + 1 / ∣ Λ ∣) r_{0}$ ; take $C_{1} : = 2 + 1 / ∣ Λ ∣$ . For convergence use $t e^{- ∣ Λ ∣ t} \leq (2 / ∣ Λ ∣) e^{- ∣ Λ ∣ t / 2}$ , obtained from [L4] with $u = ∣ Λ ∣ t / 2$ . Thus $∣ x (t) ∣ \leq ∣ \overset{ˉ}{A} ∣ e^{- ∣ Λ ∣ t} + (2 ∣ \overset{ˉ}{B} ∣ / ∣ Λ ∣) e^{- ∣ Λ ∣ t / 2} \to 0$ by [L3]. Differentiating $x (t) = (\overset{ˉ}{A} + \overset{ˉ}{B} t) e^{Λ t}$ gives the numerical velocity $x^{'} (t) = (Λ \overset{ˉ}{A} + \overset{ˉ}{B} + Λ \overset{ˉ}{B} t) e^{Λ t}$ ; its $t$ -term tends to $0$ by the same inequality and its other term tends to $0$ by [L3]. This proves the displayed case bounds and $x^{'} (t) \to 0$ .

2.3assume-case 3givenstep 1.1L1L2L3algebra

Case $Δ < 0$ . Write $Γ : = (1 s) γ > 0$ and $W : = (1 s) ω > 0$ , and let $\overset{ˉ}{A}, \overset{ˉ}{B}$ be the numerical SI values of the oscillatory-case coefficients in [L1]. Then $\overset{ˉ}{A} = X_{0}$ and $\overset{ˉ}{B} = (V_{0} + Γ X_{0}) / W$ , so $∣ \overset{ˉ}{A} ∣ \leq r_{0}$ and $∣ \overset{ˉ}{B} ∣ \leq (1 + Γ) r_{0} / W$ . By [L2], the numerical position satisfies $∣ x (t) ∣ \leq (∣ \overset{ˉ}{A} ∣ + ∣ \overset{ˉ}{B} ∣) e^{- Γ t}$ , which tends to $0$ by [L3]; take $C_{1} : = 1 + (1 + Γ) / W$ . Differentiating $x (t) = e^{- Γ t} (\overset{ˉ}{A} \cos (W t) + \overset{ˉ}{B} \sin (W t))$ gives the numerical velocity $x^{'} (t) = e^{- Γ t} [(W \overset{ˉ}{B} - Γ \overset{ˉ}{A}) \cos (W t) - (Γ \overset{ˉ}{B} + W \overset{ˉ}{A}) \sin (W t)],$ so its absolute value is bounded by the normalized form of the displayed case bound and tends to $0$ by [L3].

3.1cases-exhaustivegivenstep 1.1step 2.1step 2.2step 2.3algebra∎

Uniform bound and stability. By the three cases of steps 2.1-2.3 the numerical position bound $∣ x (t) ∣ \leq C_{1} r_{0}$ holds for all $t \geq 0$ with the constant $C_{1}$ of the case at hand, and by step 1.1 the numerical velocity bound $∣ x^{'} (t) ∣ \leq C_{2} r_{0}$ holds with the universal constant $C_{2} : = C_{2}^{E}$ from energy monotonicity; both dimensionless constants depend only on the numerical SI values of $(m, k, c)$ . Let $ε > 0$ and set $δ : = ε / (C_{1} + C_{2}) > 0$ . If a maximal motion satisfies $r_{0} = ∥ (x (t_{0}), x^{'} (t_{0})) ∥_{2} < δ$ for some $t_{0}$ , then the shifted curve $s \mapsto x (t_{0} + s)$ is again a motion of the autonomous model, with datum $(0, x (t_{0}), x^{'} (t_{0}))$ and the same initial phase norm $r_{0}$ . Applying the uniform bounds of steps 1.1 and 2.1-2.3 gives $∥ (x (t), x^{'} (t)) ∥_{2} = \sqrt{x (t)^{2} + x^{'} (t)^{2}} \leq ∣ x (t) ∣ + ∣ x^{'} (t) ∣ \leq (C_{1} + C_{2}) r_{0} < ε (t \geq t_{0}),$ where the norm is on the normalized SI phase coordinates of The phase plane of a one-dimensional model: phase curves, the energy function and its level sets, and turning points. Thus $(0, 0)$ is Lyapunov stable in the sense of Equilibria, Lyapunov stability and asymptotic stability of an autonomous first-order system; and clauses 1 and 2 give $x (t) \to 0$ , $x^{'} (t) \to 0$ for every maximal motion, so every maximal motion tends to the rest point in the phase-plane norm and $(0, 0)$ is globally asymptotically stable.

Remarks

The three regimes at a glance. For $Δ > 0$ the motion is a combination of two decaying exponentials (overdamped), for $Δ = 0$ of $e^{λ t}$ and $t e^{λ t}$ (critically damped) and for $Δ < 0$ of a decaying oscillation (underdamped); the overdamped case has rate $∣ λ_{+} ∣$ , the underdamped case rate $γ$ , and the critical case has a polynomial factor multiplying $e^{λ t}$ . In that critical case the proof bounds this factor by an exponential with rate $∣ λ ∣ / 2$ ; these rates are positive and the bounds are uniform for data of bounded phase norm.
Stability is a statement about the declared model. The asymptotic stability of $(0, 0)$ follows from the explicit families of the linear model; it is not a claim that any real oscillator inevitably comes to rest, nor does it carry any empirical qualification beyond the model's own.

Strictly damped oscillator: explicit decay bounds, convergence to rest and asymptotic stability of the origin

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources