Modelling the non-linear dynamics of the looping pendulum

Aneesh Agarwal; Avighna Daruka; Gyaneshwaran Gomathinayagam

REVIEW 3 major objections 4 minor 3 references

Modelling the non-linear dynamics of the looping pendulum

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper derives three coupled differential equations for the looping pendulum and reports that simulations built on them closely match video-tracked measurements of the heavy bob's fall.

desk verdict The paper's finite-string-mass generalization of the looping pendulum equations is a plausible extension, but the supplied MATLAB code does not solve those equations and the derivation has a real kinematic gap, so the reported validation does not hold up. read the letter →

arxiv 2506.21644 v1 pith:6LSN73EB submitted 2025-06-26 physics.class-ph nlin.CD

Avighna Daruka , Gyaneshwaran Gomathinayagam , Aneesh Agarwal This is my paper · ORCID

classification physics.class-phnlin.CD

keywords loopingpendulumIYPTrotationalmechanicstorquefrictioncapstanequationnonlineardynamicsdifferentialequations

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

The paper attempts to establish that the looping pendulum, in which a light bob sweeps around a cylindrical rod while supporting a heavier bob, is governed by a system of three ordinary differential equations. The derivation splits the motion into a rotating part and a non-rotating part, and treats the tension in the wrapped string through the capstan equation for a string with non-negligible mass. The paper solves these equations numerically and compares the predicted vertical fall of the heavy bob with video-tracked experiments across four mass ratios. It reports that the experimental fall distances follow the same nonlinear rising trend as the simulation, and takes this agreement as evidence that the derived theory is valid.

What carries the argument

The load-bearing object is the capstan-equation tension profile along the string wrapped around the rod. The paper derives, for an infinitesimal string element, a differential relation between tension and wrap angle, integrates it to find an exponential factor $e^{\mu_d(\pi+\theta)}$, and substitutes this into the heavy bob's vertical equation of motion. The other two equations are the angular equation obtained by writing torque as the time derivative of angular momentum with a time-varying moment of inertia, and the geometric relation $l = L + y - r_r(\pi+\theta)$ that couples the free length to the fall distance and the wrap angle.

What would settle it

Track the heavy bob's fall $y(t)$ and the free hanging length $l(t)$ separately with high-speed video; if the second time derivative of $l$ differs from the second derivative of $y$, the assumption used to eliminate the string tension is false and the two-equation system is not a faithful model.

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Extended reading notes

Core claim

The paper's central claim is that the looping pendulum is described by three equations: an angular equation for the light bob, a vertical equation for the heavy bob whose acceleration contains an exponential capstan factor, and a geometric constraint for the free string length. The equations extend earlier work by giving the string a non-negligible mass, so the light bob's mass does not need to be accounted for separately. The paper reports that numerical solutions of these equations reproduce the experimentally observed trend, in which the heavy bob's vertical fall increases nonlinearly with the mass ratio, and it takes this agreement as validation of the derived theory.

Load-bearing premise

The derivation assumes that the free hanging length of string changes exactly as fast as the heavy bob falls, which lets it equate the light bob's radial acceleration with the heavy bob's vertical acceleration minus the centripetal term.

Editorial extensions

If this is right

The heavy bob's fall distance rises nonlinearly with the mass ratio, with the increase flattening at larger ratios; this is the trend the paper's data and simulation both show.
A string with non-negligible mass can be modelled without separately tracking the mass of the light bob's string segment, because the capstan tension integration accounts for the string's linear density.
The same three-equation system yields both the spiral trajectory of the light bob and the vertical trajectory of the heavy bob, so a single simulation describes both components.
Because the tension scales as $e^{\mu_d(\pi+\theta)}$, small changes in the dynamic friction coefficient or the wrap angle translate into large changes in the force supporting the heavy bob.

Reading between the lines

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

The published numerical solver fixes $\omega$ at 1.5 rad/s instead of taking it from the state variable $d\theta/dt$; re-running with $\omega$ extracted from the solution would show whether the reported agreement comes from the equations themselves or from that fixed parameter.
The derivation sets the light bob's radial acceleration equal to the heavy bob's vertical acceleration minus a centripetal term, which treats the free length $l$ as if it changed at the same rate as $y$; a corrected treatment with a separate $d^2l/dt^2$ term could alter the predicted fall distances.
The model's structure matches tendon-pulley systems in robotics, where a thin string wraps around a pulley and friction governs the tension; testing the same equations on a robotic tendon with measured parameters would be a direct transfer.

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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 best thing about this paper is the attempt to generalize the capstan-based looping pendulum model to a string with mass. Equations (6), (21), and (22) are a genuine extension of the massless-string treatment in reference (4), and the mass-ratio versus fall-distance scan is a reasonable experimental question. The paper is clearly written, includes full code and data tables, and is honestly transparent about its limitations: only four mass ratios, a 30 fps camera, and the string-bending problem. Those are real assets for a school-level research report.

But the central claim—that the experimental findings fit the simulation and thereby accredit the theory—does not survive contact with the submitted code. In the odefun, the centripetal term in the d2y equation uses omega^2 with omega fixed as a scalar 1.5, not the state variable dtheta. The paper derives omega as dtheta/dt, so the simulation is not a solution of equation (21). At t=0 the initial angular velocity is zero, yet the code injects a nonzero centripetal acceleration. Later, the angular speed can be several rad/s, so a fixed 1.5 is not a good proxy. This alone invalidates the reported fit as evidence for the model.

The derivation also has a kinematic error. The paper sets the radial acceleration of the light bob to d2y/dt2 - l*omega^2. That ignores the time variation of the unwrapped length l. Since l = L + y - rr(pi+theta), the second derivative of l involves theta'', not just y''. The approximation might be defensible in a limit, but the paper does not state or justify that limit, and it is load-bearing because it enters the expression for tension and hence equation (21).

A third, smaller issue: the simulation time span is taken directly from the experimentally measured stopping time for each trial. So the final y(end) is not an independent prediction; it is partly a function of the data. The authors are upfront about this, but it further weakens the validation.

Who is this paper for? It could be a useful educational example of the looping pendulum, and the experimental data might be worth secondary use. But as a research contribution it does not work. The code-equation mismatch and the kinematic gap are visible from the text alone, and the 13% systematic discrepancy is blamed on string bending without a quantitative check. I would not send this to peer review in its current form; it needs a corrected derivation and a code that actually solves the stated equations before it deserves referee time.

Referee Report

3 major / 4 minor

Summary. The paper models the looping pendulum, in which a light bob wraps a massive string around a cylindrical rod and thereby supports a heavier bob that falls vertically. The author separates the dynamics into a rotating component and a non-rotating component, and derives three coupled equations: Eq. (6) for the angular acceleration of the light bob, Eq. (21) for the vertical acceleration of the heavy bob, and Eq. (22) for the time-varying free length of the string. A MATLAB ode45 simulation of these equations is compared with video-tracked experiments for four mass ratios (7.88, 12.92, 18.16, 29.03), and the paper claims close agreement between simulation and experiment, accrediting the derived theory.

Significance. If the derivation and validation were sound, the paper would offer a useful extension of existing massless-string looping-pendulum models to include string mass, packaged as a compact ODE system with a falsifiable prediction of fall distance versus mass ratio. The manuscript has strengths: the full MATLAB code is provided, the experimental procedure is described in detail, and the author acknowledges limitations such as low frame rate and a small number of trials. However, as detailed below, the central validation is undermined by a physically incorrect radial-acceleration assumption and by a simulation that does not solve the stated equations; the claimed agreement with experiment therefore does not support the paper's central claim.

major comments (3)

[§2.3.3.2, Eq. (17)] The radial acceleration of the light bob is set to a_r = d2y/dt2 - l*omega^2, with the justification that m and M are connected by an inextensible string and therefore have the same translational acceleration. This is not correct: the free length l = L + y - rr(pi+theta) changes at the rate dl/dt = dy/dt - rr*dtheta/dt, so the radial coordinate of m is l, and its radial acceleration is d2l/dt2 - l*omega^2 = d2y/dt2 - rr*d2theta/dt2 - l*omega^2. The term -rr*d2theta/dt2 is omitted. This omission propagates through Eq. (18) into Eq. (21), so the principal dynamical equation for the heavy bob does not follow from the stated assumptions.
[§2.6.1 and Appendix (odefun)] The MATLAB function odefun evaluates the centripetal contribution to d2y as Mu_d*m*l*omega^2*exp(Mu_d*(pi+theta)), where omega is a scalar parameter passed from the main script and set to omega=1.5. In the derivation, omega denotes dtheta/dt (Section 2.2.2), so the code should use the state variable dtheta = Y(2), not a fixed constant. At the initial condition dtheta0=0 the hard-coded omega injects a nonzero centripetal acceleration, and during the motion dtheta differs substantially from 1.5 rad/s. The simulation therefore does not solve Eqs. (6), (21), and (22); it solves a different dynamical system, so Figures 25 and 26 are not a test of the derived theory.
[§6.2 and main script (tspan)] For each trial, the simulation time span is set to the experimentally measured stopping time (0.3 s, 0.35 s, 0.4 s, 0.37 s), as stated in Section 6.2 and implemented in the main script. The reported predicted fall distance is y(end), evaluated at exactly the time when the experiment was observed to stop. Since the model has no stopping criterion, this is not a closed prediction of the total fall distance; it is an evaluation at a data-supplied time, so the comparison in Figure 25 is partly circular and cannot validate the model's predictive content.

minor comments (4)

[§2.2.2.1, Eq. (1)] The torque from the string's weight is divided by two with the comment that the string mass is 'averaged at the center of the string'; this should be justified explicitly, since the torque distribution along a finite-length string is not self-evidently equivalent to a point mass at its midpoint.
[§2.3.2.4, Eq. (14)] After dividing by e^{-mu_d phi}, the exponential in the constant term changes from e^{-mu_d phi} to e^{mu_d phi} without explanation; the algebra should be checked and the sign convention clarified.
[§5.2 and main script] The text states the rod radius is 0.003 m, but the MATLAB code sets rr = 0.0003. This factor-of-ten discrepancy affects Eq. (22) and the wrapped-length correction; it should be resolved.
[§6.3] The trajectory comparison for the heavy bob is described as having 'not much resemblance' between experiment and simulation, which is in tension with the abstract's claim of very close agreement; this discrepancy should be addressed.

Circularity Check

2 steps flagged · score 6.0 of 10

The simulation used to validate Eq. (21) does not solve Eq. (21): it injects a fixed omega=1.5 and is run to the experimentally measured stopping time, so the reported fit is partly forced by inputs.

fitted input called prediction [Section 2.2.2.3 and Section 2.6.1/Appendix code (d2y line; main script omega=1.5)]
"Since 𝜔 = 𝑑𝜃𝑑𝑡 ... d2y = (Mu_d*m*g*cos(theta)*exp(Mu_d*(pi+theta)) + Mu_d*m*l*omega^2*exp(Mu_d*(pi+theta)) - Mu_d*M*g) / ...; % Setting initial conditions ... omega=1.5;"

Equation (21) is the claimed prediction, and within it omega is the state variable dtheta/dt. The odefun evaluates d2y using the passed scalar omega^2, while the state vector Y(2) (dtheta) is extracted but never used in that term; the main script fixes omega=1.5. Hence the simulated heavy-bob acceleration, and therefore the final fall distance, is driven by an arbitrary constant rather than by the derived angular velocity. The 'simulation findings' that 'fit very closely' are not solutions of Eq. (21); the close fit cannot accredit that derivation. The constant is an unvalidated, data-independent input presented as part of the theoretical prediction.
fitted input called prediction [Section 6.2 Data Collection; code 'tspan = [0 0.37]; % Simulation time (changing as mentioned in the paper)']
"The start frame of each video and end frame (when the tape measure tool is used to measure length) was determined and the time taken for the pendulum bob to stop moving was used as the time for simulation running in MATLAB."

The paper's Research Question is the vertical distance travelled by M as a function of mass ratio. For each trial the simulation's time span is set to the experimentally measured time until the heavier bob stopped. The reported final value y_final = y(end) is therefore evaluated at a time coordinate supplied by the very measurement the model is supposed to predict. The model is not predicting when the bob stops; it is told when it stops. The Figure 25 comparison is thus partly retrodictive: the temporal endpoint is an experimental input, not an output of the derived ODE system.

full rationale

There is no self-citation load-bearing step and no uniqueness theorem imported from the authors; the ODE derivation itself is self-contained and not defined in terms of the measured fall distances. The circularity lies in the validation pathway. The paper claims 'the experimental findings fit very closely to the simulation findings, accrediting the validity and accuracy of the derived theory,' but the submitted MATLAB code does not implement Eq. (21): it uses a hard-coded omega=1.5 in the centripetal term where the derivation defines omega=dtheta/dt, so the simulated dynamics are not those of the derived system. In addition, each simulation is run for the experimentally observed stopping time of that trial, so y(end) is read at a time taken from the data. The agreement in Figure 25 therefore accredits a different, partly input-driven calculation rather than the derived equations. These are concrete reductions of the 'prediction' to inputs, not mere physical approximations. The paper's own discussion (Sections 7 and 9) candidly notes only four mass ratios and a 13% average difference, but that honesty does not remove the two input-substitution problems above. Overall the derivation has independent content; this is partial circularity, not a fully forced result.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The model rests on several unproved idealizations: a capstan friction law with a constant coefficient, a moving tangent-point origin treated as inertial, an averaging of string-mass torque, and an equality between the light bob's radial acceleration and the heavy bob's vertical acceleration that is not justified because the unwrapped length changes during winding. The MATLAB validation additionally introduces a fixed omega = 1.5 and per-trial integration times taken from the experiment.

free parameters (2)

omega (constant angular speed in centripetal term) = 1.5 rad/s
Used as a fixed number in the MATLAB implementation of equation (21) rather than the current dtheta from the ODE state. No derivation or measurement is provided for this value.
simulation time span tspan = 0.30, 0.35, 0.40, 0.37 s by trial
Chosen as the experimentally observed stop time for each mass ratio, so the model never predicts the duration and the final-y comparison is anchored to the measurement.

assumptions (5)

domain assumption The torque from the string's weight can be approximated by placing half the string mass at the midpoint of the unwrapped length.
Section 2.2.2.1, equation (1); this averaging is asserted without derivation.
ad hoc to paper The light bob's radial acceleration equals the heavy bob's vertical acceleration plus -l omega^2.
Section 2.3.3.2, equation (17); ignores the dl/dt term from wrapping, as shown by l = L + y - rr(pi+theta).
domain assumption The axis of rotation is the moving tangent point and can be used as an inertial frame for torque and angular momentum.
Sections 2.2.1.1 and 2.2.2; no transport theorem is applied for the moving origin.
standard math Capstan friction with a constant dynamic coefficient mu_d describes the string-rod contact.
Section 2.3.2, equation (13); mu_d = 0.257 is taken from literature reference (11).
standard math Small-angle Taylor approximations sin(deta) approx deta and cos(deta) approx 1 apply to capstan segments.
Section 2.3.2.2, equations (9) and (10).

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Cite this review

Pith. "Pith review of Modelling the non-linear dynamics of the looping pendulum." pith.science (2026). https://pith.science/paper/6LSN73EB

@misc{pith2026250621644,
  author       = {Pith},
  title        = {Pith review of: Modelling the non-linear dynamics of the looping pendulum},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6LSN73EB}},
  note         = {Machine review of arXiv:2506.21644}
}

read the original abstract

The Looping pendulum phenomenon was first introduced in 2019 at the 32nd edition of the IYPT, wherein a lighter bob sweeps around a cylindrical rod to support the weight of a heavier bob. In this paper, the phenomenon was divided based on rotating and non-rotating forces, and differential equations were derived for each. To verify the theoretical derivation, an experimental analysis was done, varying the mass ratio with the vertical distance travelled by the heavier bob. (Tracked using tracker) Experimental findings fit a logarithmic curve fit -- falling succinctly with a similar trend with the simulation run with MATLAB solving the derived differential equations. Furthermore, to verify the simulation, the trajectory of both the lighter and heavier mass was also compared for the simulation and experimental findings. The experimental findings fit very closely to the simulation findings, accrediting the validity and accuracy of the derived theory.

Figures

Figures reproduced from arXiv: 2506.21644 by the authors.

**Figure 2.** Classification of the Looping Pendulum Phenomenon into two components, the rotating [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** The Line of Action of Tension force intersects the Axis of Rotation, and hence does not [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

Figures from the paper (21 more)

**Figure 4.** Figure 4: The axis of rotation is the point at which the string is tangent to the metal rod, this is [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]

**Figure 5.** Figure 5: The angle theta is defined in the counterclockwise direction, and hence initially it is negative whereas as it sweeps around the angle with the -y axis it is positive [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]

**Figure 6.** Figure 6: The Angle progression as M sweeps around [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]

**Figure 7.** Figure 7: Angle between the radius and weight vectors has been defined as theta. …(3) [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]

**Figure 8.** Figure 8: There are two forces acting on bob m which represent the net force acting on the bob. The mg component has been resolved into its two components. …(7) [PITH_FULL_IMAGE:figures/full_fig_p012_8.png]

**Figure 9.** Figure 9: At position a, t=0, y=0. At position b, t>0, [PITH_FULL_IMAGE:figures/full_fig_p013_9.png]

**Figure 11.** Figure 11: The resultant force acts in the tangential [PITH_FULL_IMAGE:figures/full_fig_p013_11.png]

**Figure 12.** Figure 12: All the four forces acting upon the infinitesimal mass dm of the string have been represented in the free body diagram [PITH_FULL_IMAGE:figures/full_fig_p014_12.png]

**Figure 13.** Figure 13: The forces along the tangential direction have been resolved into their orthogonal components to help derive equations relating them. …(9) [PITH_FULL_IMAGE:figures/full_fig_p015_13.png]

**Figure 14.** Figure 14: Since I lies at the center of dm, a line drawn from the center bisects dø. [PITH_FULL_IMAGE:figures/full_fig_p016_14.png]

**Figure 15.** Figure 15: As J is pi/2, along with the point of intersection of NO, 𝑑η = !ø # …(12) [PITH_FULL_IMAGE:figures/full_fig_p017_15.png]

**Figure 16.** Figure 16: The angle of wrapping 𝜎 is defined from C to D. At C T = Tr and Ø = 0. While 𝜎 = ø at T = Tv , at D [PITH_FULL_IMAGE:figures/full_fig_p019_16.png]

**Figure 17.** Figure 17: illustrates 𝜃 and 𝜎 on the same diagram, so a clear distinction between the two can be established [PITH_FULL_IMAGE:figures/full_fig_p022_17.png]

**Figure 18.** Figure 18: The apparatus used for the preliminary investigation to help determine suitable [PITH_FULL_IMAGE:figures/full_fig_p027_18.png]

**Figure 20.** Figure 20: Apparatus Lay out [PITH_FULL_IMAGE:figures/full_fig_p029_20.png]

**Figure 21.** Figure 21: Point of view of the tracking software [PITH_FULL_IMAGE:figures/full_fig_p029_21.png]

**Figure 22.** Figure 22: The divot made in the rod to prevent the string from moving. [PITH_FULL_IMAGE:figures/full_fig_p030_22.png]

**Figure 24.** Figure 24: A plot comparison between the theoretical [PITH_FULL_IMAGE:figures/full_fig_p031_24.png]

**Figure 25.** Figure 25: After data processing, the experimental data [PITH_FULL_IMAGE:figures/full_fig_p032_25.png]

**Figure 26.** Figure 26: The simulation data and experimental data [PITH_FULL_IMAGE:figures/full_fig_p033_26.png]

**Figure 27.** Figure 27: The difference between the experimental and [PITH_FULL_IMAGE:figures/full_fig_p035_27.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

3 extracted references · 3 canonical work pages

[1]

Gyaneshwaran Gomathinayagam Mr

1 Modelling the non-linear dynamics of the looping pendulum Research Report Avighna Daruka* The Doon School, Dehradun India Mr. Gyaneshwaran Gomathinayagam Mr. Aneesh Agarwal Abstract The Looping pendulum phenomenon was first introduced in 2019 at the 32nd edition of the IYPT, wherein a lighter bob sweeps around a cylindrical rod to support the weight of ...

work page 2019
[3]

https://www.iypt.org/problems/ (2)

end 38 % Defining the parameters Ms = 0.0004; % Mass of string m = 0.0039; % Mass of moving part (lighter bob m) Mu_d = 0.257; % Friction coefficient (dynamic friction) g = 9.80665; % Gravity (constant) lambda = 0.00059; % Linear density of the string rr = 0.0003; % Radius of the cylindrical rod L = 0.500; % Length of the moving bob from the rod M = 0.021...

work page 1968
[2024]

Generate a MATLAB simulation for the following differential equations

end Verbatim 1: Defining the ODEFUN function in MATLAB. function [outputArg1,outputArg2] = untitled(inputArg1,inputArg2) outputArg1 = inputArg1; outputArg2 = inputArg2; end Verbatim 2: Defining a general function in MATLAB. 25 MATLAB ode45 solver. At the end of the script, the revised value of the dYdt function is written after the equations have been sol...

work page 2024

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

Gyaneshwaran Gomathinayagam Mr

1 Modelling the non-linear dynamics of the looping pendulum Research Report Avighna Daruka* The Doon School, Dehradun India Mr. Gyaneshwaran Gomathinayagam Mr. Aneesh Agarwal Abstract The Looping pendulum phenomenon was first introduced in 2019 at the 32nd edition of the IYPT, wherein a lighter bob sweeps around a cylindrical rod to support the weight of ...

work page 2019

[2] [3]

https://www.iypt.org/problems/ (2)

end 38 % Defining the parameters Ms = 0.0004; % Mass of string m = 0.0039; % Mass of moving part (lighter bob m) Mu_d = 0.257; % Friction coefficient (dynamic friction) g = 9.80665; % Gravity (constant) lambda = 0.00059; % Linear density of the string rr = 0.0003; % Radius of the cylindrical rod L = 0.500; % Length of the moving bob from the rod M = 0.021...

work page 1968

[3] [2024]

Generate a MATLAB simulation for the following differential equations

end Verbatim 1: Defining the ODEFUN function in MATLAB. function [outputArg1,outputArg2] = untitled(inputArg1,inputArg2) outputArg1 = inputArg1; outputArg2 = inputArg2; end Verbatim 2: Defining a general function in MATLAB. 25 MATLAB ode45 solver. At the end of the script, the revised value of the dYdt function is written after the equations have been sol...

work page 2024