⚡ electrical engineering

Equivalent Mechanical Models for Sloshing

This paper presents a rigorous mathematical formulation of pendulum-based equivalent mechanical models for propellant sloshing in spacecraft, deriving both nonlinear and linearized equations of motion for single and multiple pendulums, demonstrating their equivalence to mass-spring-damper systems, and validating the approach through time-domain and frequency-domain analyses.

Original authors: Francesco Capolupo

Published 2026-07-07

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CC BY 4.0

Original authors: Francesco Capolupo

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are holding a bucket of water while riding a bicycle. If you ride smoothly, the water sits still. But if you hit a bump or turn sharply, the water sloshes around. That sloshing creates a force that pushes back against you, making the bike wobble or harder to control.

In space, rockets and satellites carry huge tanks of liquid fuel. When these vehicles accelerate, turn, or land, that fuel sloshes just like the water in your bucket. This "sloshing" is a major headache for engineers because it can make the spacecraft spin out of control or miss its target.

This paper by Francesco Capolupo is essentially a mathematical recipe book for predicting exactly how that fuel will move and how it will push back on the spacecraft. Instead of trying to simulate every single drop of liquid (which is incredibly complex), the author proposes replacing the messy liquid with simple mechanical toys: swinging pendulums and bouncing springs.

Here is a breakdown of the paper's main ideas using everyday analogies:

1. The "Swinging Ball" Trick (The Pendulum Model)

The paper starts by saying: "Instead of modeling the liquid as a fluid, let's pretend it's a heavy ball hanging from a string inside the rocket."

The Analogy: Imagine a chandelier hanging in a moving elevator. If the elevator jerks forward, the chandelier swings backward. The author developed a rigorous set of math equations to describe exactly how that chandelier swings, how it pulls on the ceiling, and how the elevator reacts to that pull.
The Innovation: The author didn't just look at one swing; they figured out how to handle multiple swings at once (representing different parts of the fuel tank or multiple tanks). They also added a "shock absorber" to the string to account for the fact that real fuel isn't perfectly frictionless—it slows down its own swinging over time.

2. The "Springy Box" Alternative (The Mass-Spring Model)

Engineers often use a different toy to model sloshing: a heavy block sitting on a spring.

The Analogy: Imagine a heavy box sitting on a mattress. If you push the mattress, the box bounces back and forth.
The Big Discovery: The paper proves mathematically that the swinging ball and the bouncing box are actually the same thing when the rocket is accelerating steadily. The author shows that you can translate the math of the pendulum directly into the math of the spring. This is huge because it means engineers can use whichever "toy" they prefer, knowing the results will be identical.

3. The "Push" Matters (High-G Conditions)

A crucial part of the paper focuses on what happens when the rocket is under a strong, constant push (like during launch or landing).

The Analogy: Think of a pendulum in a stationary room versus a pendulum in a rocket accelerating upward. In the rocket, the "gravity" feels stronger because of the acceleration. This changes how the pendulum swings.
The Finding: The author derived new equations that specifically account for this strong "push." They showed that if you ignore this push, your math is wrong. Their new formulas work perfectly whether the force is measured from the ground (inertial frame) or from the rocket's perspective (body frame).

4. Proving It Works (The Validation)

You can write all the math you want, but does it actually work?

The Test: The author built a virtual version of their math equations and compared them against a very famous, complex computer simulation software (called MATLAB Simscape) that engineers already trust.
The Result: The author's "simple" pendulum math matched the "complex" computer simulation almost perfectly. The differences were so tiny (less than 0.002%) that they were likely just due to how the computers counted numbers, not because the math was wrong.

Summary

In short, this paper gives engineers a precise, reliable, and simplified way to predict how fuel sloshes in a spacecraft.

It turns messy liquid physics into swinging pendulums.
It proves these pendulums are mathematically identical to bouncing springs.
It provides the exact math needed when the rocket is accelerating hard (high-g).
It proves this math is correct by comparing it to high-end computer simulations.

This allows engineers to design better control systems to keep their spacecraft steady, ensuring they don't spin out of control when their fuel starts sloshing around.

Technical Summary: Equivalent Mechanical Models for Sloshing

Problem Statement
Propellant sloshing remains a critical challenge for spacecraft stability and control, particularly in two distinct regimes: microgravity environments requiring high pointing stability (e.g., interplanetary missions) and high-acceleration ( $g$ ) conditions encountered during launch, re-entry, and landing. While pendulum-based and mass-spring-damper models are standard tools for control analysts to assess sloshing-induced perturbations, a rigorous mathematical formulation connecting these representations, specifically accounting for the effects of nominal longitudinal forces in high- $g$ regimes, is necessary for robust guidance, navigation, and control (GNC) design.

Methodology
The paper presents a rigorous derivation of equivalent mechanical models for sloshing, progressing from nonlinear dynamics to linearized representations suitable for control analysis.

Nonlinear Formulation:
- The study begins with a single spherical pendulum of mass $m_P$ and length $\ell$ attached to a rigid body $B$ of mass $m_B$ .
- Using a Newtonian approach, the authors derive the coupled equations of motion for the 8-degree-of-freedom system (rigid body translation/rotation + pendulum angles).
- The derivation explicitly handles the tension force $N$ and eliminates it to reduce the system variables, resulting in a compact matrix form $\mathcal{M}\ddot{\mathbf{x}} = \mathbf{u}$ .
- The formulation is extended to accommodate multiple pendulums, representing multiple tanks or sloshing modes, utilizing a block-structured mass matrix to exploit sparsity for efficient numerical solution.
- A viscous damping model is incorporated by adding terms proportional to angular rates, which affect the pendulum dynamics without altering the rigid-body equations directly.
Linearization and Modal Analysis:
- Recognizing that onboard control systems typically estimate the center of mass assuming a rigid propellant distribution, the authors reformulate the dynamics relative to a nominal global center of mass ( $\bar{G}$ ) rather than the rigid body's center of mass.
- The system is linearized about an equilibrium characterized by a constant attitude aligned with the inertial frame and a prescribed nominal longitudinal force $\bar{F}_z$ .
- Two cases are analyzed: forces defined in the inertial frame and forces defined in the body-fixed frame.
- Through coordinate transformations and algebraic manipulation, the linearized dynamics are cast into a "modal form." This involves defining new modal coordinates ( $\eta_\theta, \eta_\phi$ ) that decouple the pendulum dynamics into two independent planar oscillators.
- The resulting system is expressed as a second-order linear MIMO system ( $M\delta\ddot{\mathbf{x}} + K\delta\mathbf{x} = \delta\mathbf{u}_{ext}$ ), explicitly identifying the modal participation matrix and the stiffness terms induced by the nominal force.
Equivalence with Mass-Spring-Damper Models:
- The paper derives the linearized equations for a mass-spring-damper system where the sloshing mass is constrained to move laterally via springs of stiffness $k$ .
- By setting the spring stiffness $k = m_P \omega_0^2$ (where $\omega_0$ is the natural frequency derived from the pendulum model), the authors demonstrate that the linearized mass-spring-damper dynamics are mathematically identical to the linearized pendulum dynamics.
- The analysis highlights that the presence of a nominal longitudinal force $\bar{F}_z$ is essential for the pendulum to exhibit oscillatory behavior analogous to a spring; in the absence of this force ( $\bar{F}_z=0$ ), the pendulum model requires modification (e.g., adding a third modal degree of freedom) to represent 0-g sloshing.

Key Results

Mathematical Equivalence: The study formally establishes the exact equivalence between the linearized pendulum model and the classical mass-spring-damper model under high- $g$ conditions, clarifying how the nominal force acts as the restoring mechanism in the pendulum formulation.
Modal Representation: The derivation yields a clean modal representation where the sloshing modes are decoupled from the rigid-body dynamics in the stiffness matrix, with coupling appearing only in the mass matrix via the modal participation matrix.
Validation:
- Nonlinear Validation: The nonlinear model was validated against a MATLAB Simscape Multibody implementation (rigid hub with four pendulums). Time-domain simulations showed excellent agreement, with maximum errors below 0.002% of the response amplitude, with discrepancies attributed solely to differences in numerical integration schemes.
- Linear Validation: Bode plots comparing the analytical linearized model with the linearized Simscape plant demonstrated excellent agreement across the frequency range for both inertial and body-frame force inputs. Minor low-frequency discrepancies in the inertial frame case were attributed to numerical accuracy limits in the Simscape linearization process.

Significance and Claims
The paper claims to provide a rigorous mathematical foundation for modeling propellant sloshing in high-acceleration environments. Its primary contribution is the explicit derivation of the nonlinear equations of motion for a multi-pendulum system and the subsequent linearization that accounts for nominal longitudinal forces. By proving the mathematical equivalence between the pendulum and mass-spring-damper models, the work validates the use of the latter for control design while grounding it in the more fundamental pendulum dynamics. The authors emphasize that the proposed models are well-posed for realistic spacecraft applications where sloshing angles remain within reasonable limits, and the validation confirms the correctness of the derived formulations for both time-domain simulation and frequency-domain analysis.

1. The "Swinging Ball" Trick (The Pendulum Model)

2. The "Springy Box" Alternative (The Mass-Spring Model)

3. The "Push" Matters (High-G Conditions)

4. Proving It Works (The Validation)

Summary

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