An Analytical Solution for Energy Harvesting Using a High-Order Shear Deformation Model in Functionally Graded Beams Subjected to Concentrated Moving Loads
Abstract
1. Introduction
2. Formulation for a Beam
2.1. Basic Equations FGM Beam
2.2. Governing Equations of Motion
3. Method of Solution
4. Piezoelectric Energy Harvesting
5. Numerical Examples
5.1. Verification Example
5.2. Investigation of Energy Harvesting from Vibrations of FGM Beams
- Metal phase:
- Ceramic phase:
6. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A. List of Parameters Used in the Analytical Model
Symbol | Description | Unit |
x, z | Longitudinal and thickness coordinates | m |
t | Time variable | s |
L | Beam length | m |
b | Beam width | m |
h | Beam thickness | m |
u(x,t) | Axial displacement at the neutral axis | m |
wb(x,t) | Transverse displacement due to bending | m |
ws(x,t) | Transverse displacement due to shear | m |
w(x,t) | Total transverse displacement | m |
ρ(z) | Density varying through thickness | kg/m³ |
ρm | Density of the metal phase | kg/m³ |
ρc | Density of the ceramic phase | kg/m³ |
E(z) | Young’s modulus varying through thickness | Pa |
Em | Young’s modulus of metal | Pa |
Ec | Young’s modulus of ceramic | Pa |
ν | Poisson’s ratio (assumed constant) | – |
G(z) | Shear modulus varying through thickness | Pa |
n | Material gradation index | – |
d31 | Piezoelectric strain constant | C/N |
e31 | Piezoelectric stress constant | C/m² |
Permittivity at constant stress in piezoelectric material | F/m | |
lP | Length of piezoelectric patch | m |
bp | Width of piezoelectric patch | m |
hp | Thickness of piezoelectric patch | m |
Cp | Capacitance of piezoelectric patch | F |
R | Load resistance (external circuit) | Ω |
V(t) | Output voltage across the electrodes | V |
P(t) | Harvested power | W |
P0 | Magnitude of moving concentrated load | N |
v | Velocity of moving load | m/s |
vr | Resonance velocity | m/s |
x(t) | Instantaneous position of the moving load | m |
δ(x − vt) | Dirac delta function representing load location | 1/m |
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Description | Parameter | Numerical Value |
---|---|---|
Permittivity component | ||
Plane stress piezoelectric stress constant | ||
High of piezoceramic patch | 0.0002 (m) | |
Width of piezoceramic patch | 0.05 (m) |
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Dao, S.-D.; Nguyen, D.-D.; Nguyen, T.-H.; Nguyen, N.-L. An Analytical Solution for Energy Harvesting Using a High-Order Shear Deformation Model in Functionally Graded Beams Subjected to Concentrated Moving Loads. Modelling 2025, 6, 55. https://doi.org/10.3390/modelling6030055
Dao S-D, Nguyen D-D, Nguyen T-H, Nguyen N-L. An Analytical Solution for Energy Harvesting Using a High-Order Shear Deformation Model in Functionally Graded Beams Subjected to Concentrated Moving Loads. Modelling. 2025; 6(3):55. https://doi.org/10.3390/modelling6030055
Chicago/Turabian StyleDao, Sy-Dan, Dang-Diem Nguyen, Trong-Hiep Nguyen, and Ngoc-Lam Nguyen. 2025. "An Analytical Solution for Energy Harvesting Using a High-Order Shear Deformation Model in Functionally Graded Beams Subjected to Concentrated Moving Loads" Modelling 6, no. 3: 55. https://doi.org/10.3390/modelling6030055
APA StyleDao, S.-D., Nguyen, D.-D., Nguyen, T.-H., & Nguyen, N.-L. (2025). An Analytical Solution for Energy Harvesting Using a High-Order Shear Deformation Model in Functionally Graded Beams Subjected to Concentrated Moving Loads. Modelling, 6(3), 55. https://doi.org/10.3390/modelling6030055