MATHEMATICAL MODELLING OF PIEZOELECTRIC HARVESTING FROM BRIDGE VIBRATION UNDER MOVING LOAD

Science and Techniques of Engineering and Innovation

This paper shows a possibility of harvesting electrical energy from mechanical vibration. In this way, a piezoelectric material is used to harvest energy from bridge vibration under a moving load. The response of bridge under moving load is studied first, and second the energy harvesting with a piezoelectric element, and third the combination of the two phenomena. The bridge is modelled as a simple beam with length l, supported at both the ends, and traversed by a force P moving with a constant velocity c. Its behaviour is described by the Euler-Bernoulli differential equation. Modal superposition is used to solve this equation, and a transformation is made from displacement coordinate to modal coordinate. A new differential equation is obtained and a resolution in the frequency domain by Fourier transform and Laplace transform is used to study the deflection of the bridge. A piezoelectric model in one dimension is also considered. The model is composed of a seismic mass that compresses all area of piezoelectric material. This means that the model is studied in longitudinal mode or mode33. The piezoelectric element is connected to a power harvesting circuit modelled as a single resistor R. A transformation in piezoelectric constitutive equation gives a differential equation of the electromechanical system. Laplace transform is used to solve the equation in frequency domain. The displacement and the generated voltage for the piezoelectric harvester are obtained. A mathematical study was done by combining the two models. The idea is to put a piezoelectric element directly under the bridge with glue. In this case, the deflection of the bridge is considered as a stress applied to the piezoelectric element. Numerical application is chosen for bridge and piezoelectric parameters. Results are obtained by varying the position of the harvester, the speed of the moving load and the length of the bridge. A maximum power between 99.9 µW and 438.20 µW is obtained with the bridge length between 25 m and 50 m.

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