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The three damping regimes on explicit data: underdamped, critical and overdamped motions at $m = k = 1$ and $c = 1, 2, 5$ , with their decay bounds

Physical scope

A worked check of the damped oscillator at the declared data m=1 kg, k=1 N/m, datum (0, 1 m, 0 m/s) and the three drag coefficients c=1, 2, 5 kg/s: the three explicit motions of the classification, their initial data, the common initial energy 1/2 J, the strict energy dissipation identity E'=-c x-dot^2, the explicit decay bounds and the convergence to rest. The item asserts no measured value and no claim that a real damper realizes the model.

Example

Declare 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 $m = 1 k g, k = 1 N / m,$ and the common datum at $t_{0} = 0$ $x_{0} = 1 m, v_{0} = 0 m / s,$ for each of the three drag coefficients $c = 1, c = 2, c = 5 (k g / s) .$ In displayed numerical formulas, $t : = t_{p h y s} / (1 s)$ , $x : = x_{p h y s} / (1 m)$ , and $x^{'} (t)$ is the numerical velocity in metres per second; model parameters and rates are their SI numerical values. Energy values are numerical joules, so the numerical energy identity is $E^{'} (t) = - c x^{'} (t)^{2}$ with these conventions. The claims checked below are that the three displayed curves are the unique maximal motions with this datum in the three cases, that all three have initial energy $\frac{1}{2}$ J and satisfy $E^{'} (t) = - c x^{'} (t)^{2}$ , and that the decay bounds of Strictly damped oscillator: explicit decay bounds, convergence to rest and asymptotic stability of the origin hold with the displayed constants:

$\begin{aligned} c = 1 : & x (t) = e^{- t / 2} (\cos \frac{\sqrt{3} t}{2} + \frac{1}{\sqrt{3}} \sin \frac{\sqrt{3} t}{2}), \\ c = 2 : & x (t) = (1 + t) e^{- t}, \\ c = 5 : & x (t) = \frac{5 + \sqrt{21}}{2 \sqrt{21}} e^{\frac{- 5 + \sqrt{21}}{2} t} + \frac{\sqrt{21} - 5}{2 \sqrt{21}} e^{\frac{- 5 - \sqrt{21}}{2} t} . \end{aligned}$

Verification

Proof technique: cases; verify the initial data and the equations by direct computation from the classification theorem, and evaluate the energy identity and the decay bounds on the explicit data.

Given: The damped model with $m = k = 1$ , datum $(0, 1, 0)$ , the three values of $c$ , and the three displayed curves.

[L1] General theory (Classification of the motions of the damped oscillator by the discriminant, and nonincreasing energy $E^{'} = - c x^{' 2}$ clause 1, The homogeneous damped oscillator: solution classification by the discriminant and uniqueness for initial data, Strictly damped oscillator: explicit decay bounds, convergence to rest and asymptotic stability of the origin): with $m > 0, k > 0, c \geq 0$ the motions are the three families according to the sign of $Δ = c^{2} - 4 m k$ , with coefficients $A = (v_{0} - λ_{-} x_{0}) / (λ_{+} - λ_{-})$ , $B = (λ_{+} x_{0} - v_{0}) / (λ_{+} - λ_{-})$ for $Δ > 0$ , $A = x_{0}$ , $B = v_{0} - λ x_{0}$ for $Δ = 0$ , and $A = x_{0}$ , $B = (v_{0} + γ x_{0}) / ω$ for $Δ < 0$ ; the motions extend to $R$ ; the energy satisfies $E^{'} = - c x^{' 2}$ ; and for $c > 0$ the decay bounds of the three cases hold with $r_{0} = \sqrt{x_{0}^{2} + v_{0}^{2}}$ .

[L2] Derivatives and values: $(e^{μ t})^{'} = μ e^{μ t}$ (The exponential function is smooth and $(\exp)^{'} = \exp$ ), $(\sin)^{'} = \cos$ , $(\cos)^{'} = - \sin$ (The derivatives of sine and cosine are cosine and minus sine), sums and products are differentiated by the algebra of derivatives (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 $g (c) \neq 0$ ) and composites by the chain rule (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 $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ ); $\sin^{2} + \cos^{2} = 1$ (Parity and the Pythagorean identity for sine and cosine); $\sqrt{3} > 0$ , $\sqrt{21} > 5 / 2 > 0$ (Square roots exist: a unique $\sqrt{a} \geq 0$ with $(\sqrt{a})^{2} = a$ ; the positives are ${x^{2} : x \neq 0}$ ); $e^{s} \leq 1$ for $s \leq 0$ (The exponential function is strictly increasing, The real exponential function and the number $e$ by a power series); and $e^{s} \to 0$ as $s \to - \infty$ (Limits at $+ \infty$ and $- \infty$ , and infinite limits at a point combined with the exponential limits of The exponential tends to $+ \infty$ at $+ \infty$ and to $0$ at $- \infty$ ); $e^{s} > 0$ for every real $s$ by that same supplier, and $\sin u = 0$ exactly when $u \in π Z$ by The zero sets of sine and cosine and the least positive common period 2 pi.

[L3] Dimensions (Dimensions and SI unit conventions of the non-relativistic classical mechanics framework): $[m] = μ$ , $[k] = μ τ^{- 2}$ , $[c] = μ τ^{- 1}$ , so $c^{2}$ and $4 m k$ both have dimension $μ^{2} τ^{- 2}$ and the discriminant is dimensionally consistent; the roots $λ_{\pm}$ have dimension $τ^{- 1}$ , so the exponents are dimensionless. In the physical-time families both coefficients carry length for distinct real roots and for complex conjugate roots. In the critical family $(A + B t) e^{λ t}$ , $A$ carries length but $B = v_{0} - λ x_{0}$ carries length per time; here they are $1 m$ and $1 m / s$ . The displayed numerical-time formulas use their SI numerical values.

1.1assume-case 1assume-case 2assume-case 3givenL1L3algebra

Discriminants and coefficients. With $m = k = 1$ and $x_{0} = 1$ , $v_{0} = 0$ the three discriminants are $Δ = 1 - 4 = - 3$ , $Δ = 4 - 4 = 0$ and $Δ = 25 - 4 = 21$ . In the first case $γ = \frac{1}{2}$ , $ω = \sqrt{4 - 1} / 2 = \sqrt{3} / 2$ , $A = 1$ and $B = (0 + \frac{1}{2}) / (\sqrt{3} / 2) = 1 / \sqrt{3}$ . In the second $λ = - 2 / (2 \cdot 1) = - 1$ , $A = 1$ , $B = 0 - (- 1) \cdot 1 = 1$ . In the third $λ_{\pm} = (- 5 \pm \sqrt{21}) / 2$ , $λ_{+} - λ_{-} = \sqrt{21}$ , $A = (0 - λ_{-}) / \sqrt{21} = (5 + \sqrt{21}) / (2 \sqrt{21})$ and $B = (λ_{+} - 0) / \sqrt{21} = (\sqrt{21} - 5) / (2 \sqrt{21})$ . By [L1] the three displayed curves are the unique maximal motions with the datum in the three cases, and all are defined on $R$ .

2.1step 1.1L1L2algebra

Initial data. In the first case $x (0) = \cos 0 + (1 / \sqrt{3}) \sin 0 = 1$ and, by [L2], $x^{'} (t) = e^{- t / 2} [- \frac{1}{2} (\cos \frac{\sqrt{3} t}{2} + \frac{1}{\sqrt{3}} \sin \frac{\sqrt{3} t}{2}) - \frac{\sqrt{3}}{2} \sin \frac{\sqrt{3} t}{2} + \frac{1}{2} \cos \frac{\sqrt{3} t}{2}] = - \frac{2}{\sqrt{3}} e^{- t / 2} \sin \frac{\sqrt{3} t}{2},$ so $x^{'} (0) = 0$ . In the second case $x (0) = 1$ and $x^{'} (t) = e^{- t} - (1 + t) e^{- t} = - t e^{- t}$ by [L2], so $x^{'} (0) = 0$ . In the third case $x (0) = A + B = (5 + \sqrt{21} + \sqrt{21} - 5) / (2 \sqrt{21}) = 1$ and $x^{'} (0) = λ_{+} A + λ_{-} B = [- (5 - \sqrt{21}) (5 + \sqrt{21}) - (5 + \sqrt{21}) (\sqrt{21} - 5)] / (4 \sqrt{21}) = 0$ since $λ_{+} A = - 1 / \sqrt{21}$ and $λ_{-} B = 1 / \sqrt{21}$ . Hence $x^{'} (t) = (e^{λ_{-} t} - e^{λ_{+} t}) / \sqrt{21}$ ; strict monotonicity of the exponential and $λ_{-} < λ_{+}$ show that this vanishes exactly at $t = 0$ .

3.1step 1.1step 2.1L1L2algebra

Energy. With $m = k = 1$ the energy is $E = \frac{1}{2} v^{2} + \frac{1}{2} x^{2}$ , and $E (0) = \frac{1}{2} \cdot 0 + \frac{1}{2} \cdot 1 = \frac{1}{2}$ J in all three cases. By [L1] the identity $E^{'} (t) = - c x^{'} (t)^{2}$ holds along each motion, so $E^{'} (t) < 0$ exactly when $x^{'} (t) \neq 0$ . In the first case the exceptional times are $t = 2 π n / \sqrt{3}$ for every $n \in Z$ , by the sine zero set and the positive exponential factor in step 2.1. In the second and third cases the derivative vanishes only at $t = 0$ and $E^{'} (t) < 0$ for every $t \neq 0$ . In particular no motion here is a constant-energy motion other than the rest motion, consistent with clause 2 of Classification of the motions of the damped oscillator by the discriminant, and nonincreasing energy $E^{'} = - c x^{' 2}$ .

4.1cases-exhaustivestep 1.1step 2.1L1L2algebra∎

Decay bounds and limits. Here $r_{0} = 1$ . By clause 1 of Strictly damped oscillator: explicit decay bounds, convergence to rest and asymptotic stability of the origin the following bounds hold for $t \geq 0$ : in the first case $∣ x (t) ∣ \leq (1 + 1 / \sqrt{3}) e^{- t / 2}$ , in the second $∣ x (t) ∣ \leq (1 + t) e^{- t}$ and in the third $∣ x (t) ∣ \leq \frac{5}{\sqrt{21}} e^{(- 5 + \sqrt{21}) t / 2}$ , because $∣ A ∣ + ∣ B ∣ = (5 + \sqrt{21} + 5 - \sqrt{21}) / (2 \sqrt{21}) = 5 / \sqrt{21}$ as $\sqrt{21} < 5$ . Evaluated at $t = 4$ : the bounds are $(1 + 1 / \sqrt{3}) e^{- 2} \approx 0.2135$ , $5 e^{- 4} \approx 0.0916$ and $\frac{5}{\sqrt{21}} e^{- 10 + 2 \sqrt{21}} \approx 1.0911 \cdot 0.4339 \approx 0.4735$ , while the explicit curves give $∣ x (4) ∣ \approx 0.1531$ , $∣ x (4) ∣ = 5 e^{- 4} \approx 0.0916$ and $∣ x (4) ∣ \approx 0.4537$ . By clause 2 of Strictly damped oscillator: explicit decay bounds, convergence to rest and asymptotic stability of the origin, $x (t) \to 0$ and $x^{'} (t) \to 0$ in all three cases; by clause 3 the rest point $(0, 0)$ is asymptotically stable.

Remarks

The three regimes. The discriminant test $Δ < 0$ , $Δ = 0$ , $Δ > 0$ separates the decaying oscillation ( $ω = \sqrt{3} / 2$ , quasi-period $4 π / \sqrt{3}$ s), the critical case ( $x = (1 + t) e^{- t}$ , no oscillation) and the overdamped case (two decaying exponentials with rates $(- 5 \pm \sqrt{21}) / 2$ ). The numerical evaluations above are decimal approximations of the explicit expressions, quoted to display the order of magnitude; the exact inequalities are the displayed bounds.
Scope. The data $m = k = 1$ , $c = 1, 2, 5$ are declared model data of this check; no measured value is asserted and no real damper is claimed to obey the linear drag law in any regime.

The three damping regimes on explicit data: underdamped, critical and overdamped motions at $m = k = 1$ and $c = 1, 2, 5$ , with their decay bounds

Example

Verification

Remarks

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The three damping regimes on explicit data: underdamped, critical and overdamped motions at m=k=1m=k=1 and c=1,2,5c=1,2,5, with their decay bounds

Example

Verification

Remarks

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Dependency tree · 2 levels

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The three damping regimes on explicit data: underdamped, critical and overdamped motions at $m = k = 1$ and $c = 1, 2, 5$ , with their decay bounds