The mathematical model with high accuracy for cantilevered piezoelectric vibration energy harvesters needs to be established to optimize the harvesters according to the characters of the environmental vibration and the electrical loads. The motion differential equation and the boundary conditions for a cantilevered piezoelectric vibration energy harvester were established

in which the position difference between the center of the proof mass and the free end of the cantilever is considered. A uniform coupled circuit equation

which is applicable to the cantilevered piezoelectric vibration energy harvester with one piezoelectric layer

two series connected piezoelectric layers and two parallel connected piezoelectric layers

was developed by introducing several constants. The expressions of the natural frequencies and corresponding mode shapes were obtained. The closed-form expressions of the output voltage were derived for harmonic base accelerometer excitations and arbitrary base accelerometer excitations. Experimental results show that the relative errors of the natural frequency expression and the output voltage expression are smaller than 10% and 20% for three harvester prototypes

which satisfy the requirements of the optimization of the harvesters.