Addendum: Modeling the amplitude and energy decay of a weakly damped harmonic oscillator using the energy dissipation rate and a simple trick (2025 Eur. J. Phys. 46(1) 015004)

Karlo Lelas; Robert Pezer

REVIEW 2 major objections 4 minor 14 references

Addendum: Modeling the amplitude and energy decay of a weakly damped harmonic oscillator using the energy dissipation rate and a simple trick (2025 Eur. J. Phys. 46(1) 015004)

T0 review · 2 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read This addendum extends the energy-dissipation-rate trick from viscous damping to sliding friction and velocity-squared air resistance, producing closed-form amplitude envelopes that need no equation-of-motion solution.

desk verdict A clear pedagogical addendum whose central derivation step is an implicit RMS averaging, not the derivation it claims to be; still worth a referee for classroom use. read the letter →

arxiv 2411.15588 v2 pith:M5W2ICTK submitted 2024-11-23 physics.class-ph

Karlo Lelas , Robert Pezer This is my paper · ORCID

classification physics.class-ph

keywords dampedharmonicoscillatorenergydissipationrateslidingfrictionairresistanceamplitudedecayweakdampingundergraduatephysicspedagogicalderivation

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This addendum aims to show that the amplitude envelope of a weakly damped harmonic oscillator can be derived for two nonlinear damping forces without solving the equation of motion or averaging over a period. Starting from a cosine solution and a sine solution that share one envelope function $f(t)$, the paper computes the energy dissipation rate for each and combines the two rates by adding their squares (sliding friction) or their $2/3$ powers (air resistance). The results are $f(t) = 1 - \mu g t/(\sqrt{2}\,\omega_0 x_0)$ for a constant friction force and $f(t) = \left[1 + C\omega_0 x_0 t/(2^{3/2}m)\right]^{-1}$ for drag quadratic in velocity. If correct, the approach gives first-year undergraduates a low-math route to amplitude-decay behavior that textbooks usually omit.

What carries the argument

The central object is the common amplitude envelope $f(t)$ appearing in both approximate solutions, and the 'simple trick' of combining the energy-dissipation rates from two equal-energy, quarter-cycle-shifted initial conditions. For each damping law the trick turns two phase-dependent equations into one equation for $f$ by exploiting $|\sin(\omega_0 t)|^2 + |\cos(\omega_0 t)|^2 = 1$ after an appropriate power (squaring for sliding friction, $2/3$ power for air resistance). This is what lets the derivation bypass both the equation of motion and the period-averaging used in the comparison method.

What would settle it

Measure the displacement maxima of a spring-mass oscillator with velocity-squared drag over at least ten periods and fit them to $x_0\left(1 + C\omega_0 x_0 t/(2^{3/2}m)\right)^{-1}$; if the value of $C$ required to fit the decay disagrees with an independent measurement beyond the weak-damping tolerance, the envelope formula fails. For sliding friction, compare the predicted stop time $\tau = \sqrt{2}\omega_0 x_0/(\mu g)$ with the exact piecewise solution of the Coulomb-damped equation of motion, which stops at nonzero displacement.

Watch

Extended reading notes

Core claim

The central claim is that the 'simple trick' previously used for viscous damping can be adapted to sliding friction and air resistance by writing the two weakly damped solutions as $x_0 f(t)\cos(\omega_0 t)$ and $x_0 f(t)\sin(\omega_0 t)$ with a common envelope. Equating the time derivative of the shared energy $E = m\omega_0^2 x_0^2 f^2/2$ to the instantaneous damping power for each initial condition gives two equations for $df/dt$ containing $|\sin(\omega_0 t)|$ and $|\cos(\omega_0 t)|$ (or their cubes). The paper squares these equations for sliding friction and raises them to the $2/3$ power for air resistance, then adds them; the identity $|\sin|^2 + |\cos|^2 = 1$ removes the oscillatory factors and leaves a single first-order ODE whose solutions are the two envelopes above. The paper's comparisons show the air-resistance envelope agrees well with the period-averaged and numerical solutions, while the sliding-friction envelope is adequate but stops somewhat earlier than the exact piecewise solution.

Load-bearing premise

The load-bearing premise is that the two motions share a single envelope $f(t)$ and that adding the squared (or appropriately powered) dissipation rates gives the true average decay rate; this averaging step is not derived from the dynamics and must be checked case by case.

Editorial extensions

If this is right

First-year students can derive the sliding-friction and air-resistance envelopes without solving equations of motion or performing period averages.
For sliding friction the derivation yields a simple stopping-time estimate $\tau = \sqrt{2}\omega_0 x_0/(\mu g)$ and a weak-damping condition $\mu \ll \sqrt{2}\, k x_0/(mg)$.
For air resistance it yields a reciprocal-in-time envelope and a weak-damping condition $C \ll 2^{3/2}m/x_0$.
Because the air-resistance derivation needs only one integral, it offers a shorter path to the known result than the period-averaging treatment.
The paper shows the trick works for equal-energy, $\pi/2$-shifted initial conditions for $n=0$ and $n=2$ damping, while for viscous damping the energy amplitudes need not be equal.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A natural extension not explored in the paper is to damping forces proportional to $|v|^n$: the same construction would use the exponent $2/(n+1)$ to make $|\sin(\omega_0 t)|^{n+1}$ and $|\cos(\omega_0 t)|^{n+1}$ combine through the identity $|\sin|^2+|\cos|^2=1$, and one could test numerically whether the resulting envelope remains accurate for fractional $n$.
Because the weak-damping conditions for sliding friction and air resistance depend on the initial displacement $x_0$, the method implies that initial energy sets the validity regime; measuring the same system at several starting amplitudes could separate the damping law's nonlinearity from ordinary viscous behavior.
One can test whether the agreement with numerics persists as $C$ or $\mu$ grows toward the weak-damping bound, since the paper only checks one or two parameter values for each damping law.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The new thing here is the algebraic trick: for a single envelope f(t), write the energy balance for two quadrature solutions, get two first-order equations for df/dt that contain |sin| and |cos| respectively, then square (or raise to 2/3) and add to eliminate time. That is a genuine twist, and the resulting coefficients 1/sqrt(2) and 2^{-3/2} are not in the prior work. The paper is clearly written, the comparisons with exact and numerical solutions are honest, and the discussion in Section VII about which initial-condition pairs work is useful.

The soft spot is load-bearing. Equations (17) and (18) are mutually inconsistent for a common f(t) unless |sin|=|cos|, and the same holds for (29)-(30) with cubes. Squaring and adding two equations that cannot both be true is not a derivation; it is an algebraic prescription that implicitly averages over the oscillation. The paper's claim that the approach needs 'no time averaging' is therefore overstated. The coefficients 1/sqrt(2) and 2^{-3/2} are not derived from the dynamics; they are consequences of the exponent chosen to make sin^2+cos^2=1. The cycle-averaged values 2/pi and 4/(3pi) differ by about 11% and 20%. For sliding friction the paper itself admits its result is less accurate than the standard period-averaged one; for air resistance the agreement is fine but the derivation still hides an average.

That said, the paper is not a fraud. It tests its results, reports the comparisons, and the approximation is reasonable in the weak-damping regime. The flaw is that a pedagogical trick is presented as a derivation when it is actually a heuristic. A referee should require the authors to state explicitly that (17)-(18) and (29)-(30) cannot hold simultaneously for one f, and that the squaring-and-adding step is a case-by-case averaging whose accuracy must be checked numerically or against known results. With that disclosure, the paper is a serviceable classroom resource.

For a first-year course, the simpler math is a real benefit. But the central step should be labeled honestly. I would not cite this in my own research, but I would send it to a serious referee for a teaching journal. It deserves peer review because the flaw is fixable by reframing and the intended audience is real.

Referee Report

2 major / 4 minor

Summary. This addendum extends a previously published 'simple trick' for deriving the amplitude decay of a weakly damped harmonic oscillator from energy dissipation rates. For sliding friction and quadratic air resistance, the authors posit two approximate solutions sharing a common envelope f(t), write the energy balance dE/dt = F_d v for each of two initial-condition pairs, and then combine the two resulting equations by squaring (friction) or raising to the 2/3 power (air resistance) and adding, exploiting sin^2 + cos^2 = 1. This yields a linear decay f(t) = 1 - μg t/(√2 ω0 x0) for sliding friction and f(t) = [1 + Cω0 x0 t/(2^{3/2} m)]^{-1} for air resistance. The results are compared with the period-averaged method of Wang et al. and with exact/numerical solutions, showing reasonable agreement in the weak-damping regime.

Significance. The paper addresses a genuine pedagogical gap: amplitude decay for non-viscous damping is usually omitted from introductory textbooks. The final formulas are correct approximations, and the comparisons with exact and numerical solutions are a strength, as are the explicit weak-damping conditions (24) and (37). However, the novelty is modest, since the same final expressions are obtained by standard cycle-averaging; the contribution is the claim that the 'trick' avoids time averaging and is suitable for first-year students. The central derivation step is logically problematic, as detailed below, so the significance currently rests on a heuristic that is presented as a derivation.

major comments (2)

[§III, Eqs. (17)-(18) and §IV, Eqs. (29)-(30)] For a single envelope f(t), Eqs. (17) and (18) cannot both hold except at instants where |sin ω0t| = |cos ω0t|. The step of squaring and adding them (Eqs. (19)-(21)) is therefore not a logical consequence of the energy balance; it is an algebraic averaging prescription. The same issue occurs in Section IV, where Eqs. (29) and (30) are mutually inconsistent unless |sin ω0t|^3 = |cos ω0t|^3, and the 2/3 power is chosen only to make the trigonometric terms add to unity. The paper should explicitly acknowledge this inconsistency and present the squaring-and-adding step as a heuristic that effectively averages over phase, rather than as a derivation. The advertised advantage 'no need for time averaging' is misleading: the operation is an implicit RMS-type average. The authors should also note that the resulting coefficients 1/√2 and 2^(-3/2) differ from the period-averaged values 2/π and 4/(3π) by about 11% and 20%, respectively, so the agreement with known results is not exact.
[Abstract and §VII] The abstract and conclusion state that the approach 'derives' the amplitude decay and avoids time averaging. Because of the inconsistency in Eqs. (17)-(18) and (29)-(30), the derivation is not logically forced; it is a plausible extension of the trick in [1] that happens to yield useful approximations. The authors should either prove that the combined equation is a valid approximation in some quantified sense or clearly label it as an approximate construction validated by comparison with exact/numerical solutions. This is load-bearing, because the paper's central claim is the availability of a low-math derivation, not just the final formulas.

minor comments (4)

[§VII, Eqs. (45)-(46)] The notation sin^{n+1}(ω0t + φ0) should be |sin|^{n+1} (or the absolute value should be explicitly indicated), since the power of the damping force is -C|v|^{n+1}.
[§V, Fig. 1] The caption says 'See text for details' but the text does not explicitly identify which curve corresponds to which solution; adding labels or a legend to the figure would improve clarity.
[§V, text near Eq. (40)] The phrase 'at displacement +0.02x0' is ambiguous; it should read 'at displacement 0.02 x0' or 'at +0.02x0 relative to the equilibrium position'.
[References] Reference [4] lists the author as 'Anastasios Adamopoulosa'; the final 'a' appears to be a typo and should be 'Adamopoulos'.

Circularity Check

1 steps flagged · score 2.0 of 10

Self-contained and externally benchmarked; the square-and-add 'trick' is an implicit averaging of two mutually inconsistent envelope equations, so the 'no time averaging' claim is overstated, but there is no parameter fit or load-bearing self-citation chain.

other [Section III, Eqs. (17)-(22) (sliding friction); Section IV, Eqs. (29)-(34) (air resistance); 'no time averaging' claims in Introduction, Section VI, and Conclusion.]
"By adding (19) and (20) we get (df/dt)^2 = 1/2 (µg/(ω0x0))^2, since identity |sin(ω0t)|^2 + |cos(ω0t)|^2 = 1 is valid... We raise the relations (29) and (30) to the power of 2/3... We add (31) and (32) and get (−df/dt)^(2/3) = 1/2 (Cω0x0/m)^(2/3) f^(4/3)(t)... In this approach, there is no need for the time averaging used in [2, 3]."

For a single envelope f(t), Eq. (17) gives df/dt = −(µg/(ω0x0))|sin ω0t| and Eq. (18) gives df/dt = −(µg/(ω0x0))|cos ω0t|; these are incompatible wherever |sin|≠|cos|, and the air-resistance pair Eqs. (29)-(30) requires |sin|^3=|cos|^3. Squaring and adding two mutually contradictory equations is not a consequence of the dynamics: the exponent 2 (respectively 2/3) is chosen so that sin^2+cos^2=1 cancels the time dependence, so the output coefficients 1/√2 and 2^(−3/2) are fixed by algebraic construction (a hidden RMS-style average), not by the physics.

full rationale

This addendum extends the energy-dissipation 'trick' of the authors' prior paper [1] to Coulomb (n=0) and quadratic (n=2) damping. The load-bearing derivation is fully restated in this paper (Eqs. 7-34): two equal-energy initial conditions are assumed to share one envelope f(t), the energy balance dE/dt = P_d gives two first-order equations for df/dt, and the combination is performed via sin^2+cos^2=1 (exponent 2 for friction, 2/3 for air resistance). No parameter is fitted to data, and no unverified result is imported: the citation of [1] supplies the strategy, not the n=0/n=2 results, and the paper openly states in the Introduction that [1] had found the trick not to work for these damping types. The final formulas are benchmarked against the exact Coulomb solution (Eq. 40) and against numerical ode45 solutions (Fig. 2), so the central claim has independent content. The reviewer's legitimate concern is mathematical validity rather than circularity: Eqs. (17) and (18) assign two different values to df/dt of the same f and are inconsistent except at instants where |sin|=|cos| (similarly Eqs. (29)-(30) require |sin|^3=|cos|^3). Squaring (or raising to 2/3) and adding imposes an RMS-style average whose coefficient (1/√2, 2^(−3/2)) is fixed by the trigonometric identity rather than by the dynamics, and which differs from the cycle-averaged coefficients (2/π, 4/(3π)) by 11-20%. Thus the claim 'there is no need for the time averaging' is overstated, but the step is a heuristic averaging and a transparency issue, not a self-referential derivation: the final answer is not presupposed in the inputs. The paper also discloses in Section VII that for n≠1 the approach works only for equal-energy, phase-shifted initial-condition pairs. Overall circularity is minimal: one construction-based averaging step with external benchmarks, and a non-load-bearing self-citation, giving a score of 2.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central derivation rests on the standard energy-dissipation relation dE/dt=Fv plus an ad hoc common-envelope assumption and a squaring-and-adding step that is not derived from the individual equations. No parameters are fitted to data; all constants are physical inputs or derived coefficients.

assumptions (4)

ad hoc to paper The damped solutions for both pairs of initial conditions are x1=x0 f(t) cos(ω0t) and x2=x0 f(t) sin(ω0t) with one common envelope f(t).
For linear damping the envelope is phase-independent, but for Coulomb friction and quadratic drag the dissipation rate depends on phase, so a single common envelope is an imposed simplification. It enters in Section II, Eqs. (7)-(10).
standard math The energy dissipation rate equals the power of the damping force: dE/dt = F_d v.
Standard one-dimensional energy theorem; used to convert force power into amplitude decay in Eq. (6).
domain assumption Weak damping justifies neglecting \dot f compared to ω0 f in the velocities, giving v1=-ω0 x0 f sin and v2=ω0 x0 f cos.
The weak-damping condition |\dot f| << ω0 is stated after Eq. (10) and is needed for the simple energy expressions in Eq. (11).
ad hoc to paper The squared and added equations (19)+(20) and (31)+(32) determine a single ODE for the common envelope.
This is the 'simple trick'; it is not a logical consequence of the individual equations, which are mutually inconsistent for a common f unless |sin|=|cos|. It acts as an implicit phase average. See Section III, Eqs. (17)-(22).

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Pith. "Pith review of Addendum: Modeling the amplitude and energy decay of a weakly damped harmonic oscillator using the energy dissipation rate and a simple trick (2025 Eur. J. Phys. 46(1) 015004)." pith.science (2026). https://pith.science/paper/M5W2ICTK

@misc{pith2026241115588,
  author       = {Pith},
  title        = {Pith review of: Addendum: Modeling the amplitude and energy decay of a weakly damped harmonic oscillator using the energy dissipation rate and a simple trick (2025 Eur. J. Phys. 46(1) 015004)},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/M5W2ICTK}},
  note         = {Machine review of arXiv:2411.15588}
}

read the original abstract

We show how to adapt the approach introduced for viscous damping in [1] to derive the approximate amplitude decay in the case of damping by a force of constant magnitude (sliding friction) and in the case of damping by a force proportional to the square of velocity (air resistance). We obtain two first-order differential equations from which we obtain the approximate time-dependent amplitudes corresponding to the considered damping forces. Our approach is suitable for first-year undergraduates, as it relies on the physical concepts and mathematical techniques they are familiar with.

Figures

Figures reproduced from arXiv: 2411.15588 by the authors.

**Figure 2.** FIG. 2: Solutions (42) (solid red curve) and (43) (solid blue curve), for (a) [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

14 extracted references · 3 canonical work pages

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K. Lelas and R. Pezer. Modeling the amplitude and energy decay of a weakly damped harmonic oscillator using the energy dissipation rate and a simple trick. European Journal of Physics , 46(1):015004, dec 2024

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Lelas and R

K. Lelas and R. Pezer. Modeling the amplitude and energy decay of a weakly damped harmonic oscillator using the energy dissipation rate and a simple trick. European Journal of Physics , 46(1):015004, dec 2024

2024

[2] [2]

X. Wang, C. Schmitt, and M. Payne. Oscillations with three damping effects. European Journal of Physics, 23(2):155–164, jan 2002

2002

[3] [3]

An elementary proof of the amplitude’s exponential decrease in damped oscillations

Stylianos-Vasileios Kontomaris and Anna Malamou. An elementary proof of the amplitude’s exponential decrease in damped oscillations. Physics Education, 59(2):025030, feb 2024

work page 2024

[4] [4]

Anastasios Adamopoulosa and N

A. Anastasios Adamopoulosa and N. Adamopoulos. Constant and quadratic damping of free oscillations: easy solutions. International Journal of Mathematical Education in Science and Technology , 53(11):3151–3161, jan 2022

2022

[5] [5]

I. R. Lapidus. Motion of a harmonic oscillator with sliding friction. American Journal of Physics , 38(11):1360–1361, nov 1970

work page 1970

[6] [6]

Marchewka, David

A. Marchewka, David. S. Abbott, and R. J. BeichnerKamela. Oscillator damped by a constant-magnitude friction force. American Journal of Physics , 72(4):477–483, apr 2004

2004

[7] [7]

Young and Roger A

Hugh D. Young and Roger A. Freedman. University Physics with Modern Physics . Pearson, 2020

2020

[8] [8]

Fundamentals of Physics

David Halliday, Robert Resnick, and Jearl Walker. Fundamentals of Physics . John Wiley & Sons, 2013

2013