Reference Material

Background

Foundational concepts behind the research — what HASELs are, how they work, and how the electro-hydro-elastic (EHE) beam framework models submerged flexible propulsors.

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What is a HASEL?

Hydraulically Amplified Self-Healing Electrostatic actuators

HASEL actuators are soft electrostatic actuators inspired by biological muscle — inextensible polymer pouches of dielectric oil with conductive electrodes on the outside. Applied high voltage creates Maxwell stress between the electrodes, causing them to zip together, displacing fluid and contracting the pouch. Because the dielectric is a liquid, the device can self-heal after dielectric breakdown.

Operating Principle

Electrostatic Zipping

Maxwell stress (electrostatic pressure) between electrode-covered areas exceeds the hydraulic restoring pressure of the fluid → electrodes zip together → fluid displaced toward unelectroded region → pouch contracts laterally.

Geometry (Peano-HASEL)

Rectangular Pouch

Flat rectangular pouch; electrodes cover approximately half the length (L_e ≈ L_p/2). Unzipped region bulges into a circular segment. Kellaris (2019) analytic model: voltage → force/stroke.

Self-Healing

Dielectric Fluid Advantage

Unlike solid DEAs, HASELs use liquid dielectric oil. Pinhole breakdowns self-heal as oil fills the gap. Key advantage for longevity in underwater or harsh environments.

Variable Capacitor

Capacitance Change Mechanism

Zipped area acts as a parallel-plate capacitor: C = ε₀ε_r·w·l_e / 2t. As more electrode area zips, capacitance increases. Unzipped region contributes negligible capacitance (Kellaris 2019). C_max/C_min ≈ 11.7× for commercial HASELs.

Dynamic Regimes

Inertial vs. Viscous

Per Rothemund (2020): τ_v/τ_i = μwL^½/Mg^½. Inertial regime → t_n ∝ L^½, independent of viscosity. Artimus commercial HASELs operate in the inertial regime — response determined by inertia of the oil, not its viscosity.

Commercial Source

Artimus Robotics

Denver, CO. Commercial Peano-HASEL with thin-film unipolar design. 8-channel ARDI power supply used for Nebula v2. Self-sensing via AC impedance built into their newest multi-channel supplies.

HASEL Material Parameters

ε_rRelative permittivity, BOPP dielectric film2.2

ε_0Permittivity of free space8.854 × 10⁻¹² F/m

tFilm thickness12–18 µm

wPouch width49–53 mm

L_p / L_ePouch length / Electrode length17.0 / 9.37 mm

V_breakdownTypical dielectric breakdown voltage (BOPP film)~10 kV

Generation mode connection: The same variable capacitance that makes HASELs useful actuators also makes them viable generators. When a zipped (high-C) HASEL is mechanically unzipped at constant voltage, charge is ejected to an external storage element. This is the basis of the CCW constant-voltage generation cycle.

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EHE Modeling Background

Electro-hydro-elastic beam theory for submerged robotic propulsors

The electro-hydro-elastic (EHE) framework couples structural beam mechanics, fluid-structure interaction, and electromechanical actuation in a single governing equation. It is computationally efficient compared to coupled FEA-CFD and enables rapid parametric studies on stiffness, geometry, and actuation. The framework in this work follows Musgrave (2021).

Structural

Euler-Bernoulli Beam

Small-deflection assumption. Variable bending stiffness EI(x) and mass per unit length m(x). Augmented with nonlinear terms (Aureli 2012, Ramananarivo 2013) for large-amplitude cases.

Fluid — Quiescent

Lighthill + Nonlinear Damping

Lighthill (1971) elongated-body theory provides reactive (added mass) and resistive forces. Aureli (2012) nonlinear cubic damping term accounts for vortex shedding at moderate amplitudes and Keulegan-Carpenter numbers.

Fluid — Uniform Flow

Linearized Potential Flow

Garrick (1936) / Theodorsen unsteady lift on an oscillating flat plate. Valid at low reduced frequencies. Validated against LDV data from an aluminum propulsor in UF water tunnel.

Solution Method

Galerkin Projection

PDE projected onto N vacuum mode shapes as basis functions. Reduces to N coupled ODEs. Root-finding handles frequency-dependent nonlinear eigenvalue. N = 20 basis functions used for Nebula simulations.

Traveling Index

Complex Orthogonal Decomposition

Ti ∈ [0,1] quantifies the proportion of traveling vs. standing waves in the response waveform. Ti = 0 → pure standing wave; Ti = 1 → pure traveling wave. Method from Feeny (2008), used in Musgrave (2021) and Hess (2022).

Mode Shape Quality

MAC — Modal Assurance Criterion

MAC ∈ [0,1] measures correlation between a simulated mode shape and the corresponding experimentally measured mode shape. MAC = 1 → perfect agreement; MAC < 0.5 → poor agreement. Nebula Mode 1 MAC = 0.989.

Governing Equation Summary

∂²/∂x² [ EI(x) ∂²w/∂x² ] + [m_s(x) + m_p(x)] ∂²w/∂t² + F_fluid(w, ẇ, x, t) = M_HASEL(x, V(t))

F_fluid (quiescent) = m_f(x) Γ(ω) ∂²w/∂t² + m_f(x) Δ(β,ε) ∂²w/∂t² [Aureli 2012]

M_HASEL = ∂²/∂x² [ θ(x) χ_p(x) V(t) ] [linearized HASEL constitutive relation]

Validation strategy: Quiescent EHE model validated against Nebula DIC (natural frequencies, mode shapes, MAC). Uniform-flow model validated against a piezo-actuated thin aluminum beam in the UF recirculating water tunnel. Both models merged into one MATLAB package with a case structure switch.

Key Literature

Musgrave 2021EHE framework for structure-borne noise on flexible propulsors — basis for all Nebula simulations

Aureli 2012Nonlinear finite amplitude vibrations of a cantilever beam in quiescent fluid — nonlinear damping term

Lighthill 1971Large-amplitude elongated body theory — reactive and resistive fluid forces

Kellaris 2019Analytical HASEL actuator model — voltage, geometry, force, stroke relationship

Feeny 2008Complex orthogonal decomposition for traveling wave analysis — traveling index definition

Rothemund 2020HASEL actuator dynamics — inertial vs. viscous regime, response time scaling

Ramananarivo 2013Passive elastic mechanism for swimming locomotion — nonlinear EHE augmentation

DiSanto 2021Convergence in undulatory swimming — biological BCF quadratic midline model used for comparison