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Dimensional analysis of spring-wing systems reveals performance metrics for resonant flapping-wing flight


Gravish, Nicholas; Lynch, James; Gau, Jeff; Sponberg, Simon (2020), Dimensional analysis of spring-wing systems reveals performance metrics for resonant flapping-wing flight, Dryad, Dataset,


Flapping-wing insects, birds, and robots are thought to offset the high power cost of oscillatory wing motion by using elastic elements for energy storage and return. Insects possess highly resilient elastic regions in their flight anatomy that may enable high dynamic efficiency. However, recent experiments highlight losses due to damping in the insect thorax that could reduce the benefit of those elastic elements. We performed experiments on, and simulations of a dynamically-scaled robophysical flapping model with an elastic element and biologically-relevant structural damping to elucidate the roles of body mechanics, aerodynamics, and actuation in spring-wing energetics. We measured oscillatory flapping wing dynamics and energetics subject to a range of actuation parameters, system inertia, and spring elasticity. To generalize these results, we derive the non-dimensional spring-wing equation of motion and present variables that describe the resonance properties of flapping systems: the Weis-Fogh number, a measure of the relative influence of inertia and aerodynamics, and the reduced stiffness. We show that internal damping scales with Weis-Fogh number, revealing that dynamic efficiency monotonically decreases with increasing Weis-Fogh number. Based on these results, we introduce a general framework for understanding the roles of internal damping, aerodynamic and inertial forces, and elastic structures within all spring-wing systems.


This dataset consists of two elements:

1) A set of 9 experimental data folders from combined stiffness and inertia tests with dynamically scaled flapping robotic wing.

2) Simulation code to reproduce dynamic efficiency plots for series spring-wing systems. 

The data was processed in matlab, and simulation results were generated in matlab with all code provided to generate the datasets and plot.

Usage Notes

The data set includes a readme file process the data which we reproduce below.


In this directory we provide code to reproduce the experimental and simulation results for the manuscript:  "Dimensional analysis of spring-wing systems reveals performance metrics for resonant flapping-wing flight". James Lynch, Jeff Gau, Simon Sponberg, Nick Gravish

James Lynch (
Nick Gravish (

Folder subdirectories:
Functions -- This directory contains the functions used for analysis. Please add it to the Matlab path

Experiment_Data_analysis -- This directory contains the experimental data organized by spring stiffness K1 - K3, and inertia I1 - I3 totaling 9 datasets. The script Experiment_data_analysis.m will perform the experimental analysis.

Simulation_functions -- Contains the functions to perform simulations of spring-wing systems. Figure 10 simulation data is generated from series_parallel_comparison.m and then the plots are generated from sim_analysis_hysteretic.m


NSF, Award: 1554790

NSF, Award: 1205878