1. Dynamic Modeling of Contact-Mode Triboelectric Generators Using Lagrange’s Equation
Sean M Gauntt1, Prof. James Gibert2
1 Mechanical and Aeronautical Engineering Department, Clarkson University, Potsdam, NY
2 Mechanical Engineering Department, Purdue University, West Lafayette, IN
Introduction
Triboelectricity is derived from the Greek root tribos, meaning rubbing, and it means friction electricity.
Triboelectric Generators utilize static electricity and a variable
capacitive structure to convert mechanical energy into electrical
energy. Uses for these devices include harvesting ambient energy
in the environment and acting as sensors for a range of applications.
Typically the devices are made by rubbing two oppositely polarized
materials together; leaving a static charge on the two surfaces of the
materials. One material is a dielectric which is placed between two
metal electrodes. The materials used for charging and for the
composition of the two harvesters are determined by their polarity
based on the triboelectric series (Figure 1).
The literature is filled with a variety of designs for these devices,
constructed from paper and thin films, which render the generators
lightweight, flexible and inexpensive.
However, most designs of these devices are ad-hoc and not
based on the understanding of the underlying physics that govern
their behavior. Furthermore, the few models of the systems that are
present in the literature neglect the coupled electromechanical
behavior of the devices.
Objective: Investigate the effect of inertia and mass on
triboelectric generators by developing a coupled
electromechanical model.
Periodic Excitation
A sine wave was used to model periodic excitation
Figure 5: Plot of Power vs. Resistance and Frequency
Figure 4: Power vs. Frequency at Amplitude of 50 N (left). Power vs. Amplitude at Frequency of 1000 Hz (right)
Experimentation
Prototypes were made using paper and thin films to gather experimental data
Figure 9: Voltage vs. Time for Tapping (left). Power vs. Resistance for Tapping (right)
Figure 7: Steps for Construction of Prototype
Figure 8: Electret Charging Setup: Corona Treater, Foam Charging Platform, and Protective Equipment (left).
Experimental Setup: Oscilloscope, Resistor Bank and Finished Prototype (right).
Figure 1: Chart illustrating the different polarities of various materials in comparison to one another due to the triboelectric effect
Modeling
Relate charge to electric field by Gauss’s Law:
Derived governing equations to characterize generators with one and two dielectric layers
The resulting equations using Lagrange’s Equation in terms of mass (m), damping coefficient (ceq), stiffness (keq), displacement (x), induced charge (q) and forcing function (f):
Figure 2: Section of a Single Dielectric Generator
Figure 3: Section of a Double Dielectric Generator
Where: Lumped parameters in terms of total charges (Q), permittivity (ε) and surface area (A) are , Single: , Double:
Capacitance in terms of material thicknesses (d) and as a function of x is for single and double dielectric respectively:
Lagrangian Formulation
Where:
L is the Lagrangian in terms of kinetic (T) and potential (V) energies: L=T-V
U is the Rayleigh Dissipation Function
qr is a generalized coordinate
fr is a generalized forcing function
Human Inputs
Walking and running at various pacing frequencies was tested
Figure 6: Displacement of Upper Electrode vs. Time (bottom left). Voltage vs. Time (bottom right).
Power vs. Time (middle right). Power vs. Resistance (top right)
Conclusions
Lagrangian formulation yields complete set of coupled electromechanical equations of triboelectric generators not presently found in the existing scientific literature.
Currently, the devices are designed for applications to harvest from random vibrations and low frequency
applications. However, models indicate low frequency range inputs excite the harvester in a below optimal
regime.
Models allow for the prediction of novel energy scavenging applications; such as harvesting energy from
walking.
Future Work:
Refine model to include mechanical contact and contact resistance
Refine model to capture the forces caused by the volume change in an enclosed cavity
Examine applications that additional mass can be coupled to the generators lowering the natural frequency
Explore the fundamental nature of triboelectric charging
Improve the performance by increasing the charge density by increasing the apparent surface area.
Refine Experimentation for a one to one comparison of model and prototype
Acknowledgments
We would like to thank the SURF program for their funding and support, Amin Joodaky for his help in the experimentation and Dr. Gregory Batt for prototype discussion
References
[1] Hawley, M.S., and Romanow, F.F., “Electret Transducer Equation by Lagrange’s Equation,” J. Acoust. Soc. Am., 1978, 64, 2,
[2] Bachmann, H. and Ammann, W., 1987, Vibrations in Structures Induced by Man and Machines, International Association for Bridge and Structural Engineering, Zurich, Chap 2
[3] Karagozler, M.E., Poupyrev, I., Fedder, G.K., and Suzuki, Y., “Paper Generators: Harvesting Energy from Touching, Rubbing and Sliding,” Proc UIST ‘13
[4] Niu, S., Wang, S., Lin, L., Liu, Y., Zhou, Y.S., Hu, Y., and Wang, Z.L., “Theoretical Study of Contact-Mode Triboelecrtic Nanogenerators as an Effective Power Source,” Energy Envirn. Sci., 2013, 6,