[Paper Review] Pendulums, Drops and Rods: a physical analogy

Benoît Roman, Cyprien Gay+1 more|arXiv (Cornell University)|Jun 4, 2020

Advanced Materials and Mechanics4 references4 citations

TL;DR

This paper establishes a deep physical analogy between pendulums, liquid drops, and bending rods, showing they are all governed by the same non-dimensional equation of motion. Despite shared dynamics, the systems differ in physical constraints and boundary conditions, and the analogy extends to flexible membranes and pressure vessels, revealing universal shape behaviors across diverse systems.

ABSTRACT

A liquid meniscus, a bending rod (also called elastica) and a simple pendulum are all described by the same non-dimensional equation. The oscillatory regime of the pendulum corresponds to buckling rods and pendant drops, and the high-velocity regime corresponds to spherical drops, puddles and multiple rod loopings. We study this analogy in a didactic way and discuss how, despite this common governing equation, the three systems are not completely equivalent. We also consider the cylindrical deformations of an inextensible, flexible membrane containing a liquid, which in some sense interpolates between the meniscus and rod conformations.

Motivation & Objective

• To demonstrate that pendulums, liquid drops, and bending rods are governed by the same non-dimensional equation, highlighting a profound physical analogy.
• To explore the differences in behavior between these systems despite their shared governing equation, particularly in boundary conditions and physical constraints.
• To extend the analogy to cylindrical deformations of inextensible, flexible membranes containing liquid, which interpolate between meniscus and rod-like conformations.
• To show how real-world engineering systems like pressure storage tanks (Hortonspheroids) adopt shapes analogous to liquid drops due to force distribution principles.
• To clarify the limits of the analogy, especially in crossover regions where both bending rigidity and pressure effects are significant, and full equations deviate from pendulum-like solutions.

Proposed method

• Derive the non-dimensional equation of motion for a simple pendulum: $ \frac{d^2\theta}{dt^2} = -\sin\theta $, which governs oscillatory and high-velocity regimes.
• Apply the same equation to model the shape of 2D static drops (compressed, pendant, or meniscus), using curvature and surface tension balance.
• Use phase space analysis in $[\theta, \dot{\theta}]$ to distinguish between bounded (oscillatory) and unbounded (high-velocity) regimes, corresponding to different physical behaviors.
• Introduce a membrane model with variable curvature and pressure, governed by a modified equation that reduces to the pendulum equation under specific conditions.
• Analyze the crossover region where both membrane bending and liquid pressure are significant, requiring the full equation rather than pendulum approximation.
• Identify special initial conditions under which the full membrane equation admits pendulum-like solutions, derived from energy and force balance constraints.

Experimental results

Research questions

• RQ1 How can the same nonlinear differential equation describe the dynamics of pendulums, liquid drops, and bending rods?
• RQ2 What are the physical and geometric differences between these systems despite their shared governing equation?
• RQ3 In what way do cylindrical deformations of a flexible liquid-filled membrane interpolate between meniscus and rod-like shapes?
• RQ4 Why do pressure vessels like Hortonspheroids adopt shapes similar to liquid drops, and what physical principles underlie this similarity?
• RQ5 Under what conditions does the full membrane equation reduce to a pendulum-like solution, and what constraints must initial conditions satisfy?

Key findings

• Pendulums, drops, and rods are all described by the same non-dimensional equation $ \frac{d^2\theta}{dt^2} = -\sin\theta $, revealing a deep physical analogy.
• The oscillatory regime (energy $ E < 1 $) corresponds to buckling rods and pendant drops, while the high-velocity regime ( $ E > 1 $) corresponds to spherical drops, puddles, and multiple rod loopings.
• In the phase diagram $[\theta, \dot{\theta}]$, points with vanishing angular speed $ \dot{\theta} = 0 $ correspond to inflection points in rod and drop shapes.
• The special case $ E = 1 $ (separatrix) corresponds to soliton-like motion and marks the boundary between oscillatory and unbounded behavior.
• For flexible membranes with liquid, the shape transitions from a meniscus-like region (large length scale) to a rod-like region (small length scale), with distinct characteristic scales.
• Hortonspheroid storage tanks mimic liquid drop shapes because their wall forces are tangential and uniformly distributed, minimizing bending and fatigue—mirroring surface tension in drops.