To improve the energy conversion efficiency of piezoelectric energy harvesters, a piezoelectric stack was coupled into a force amplification framework to construct a piezoelectric energy harvester with a force amplification function. Based on the constitutive equation of the piezoelectric effect, theoretical modeling, finite element simulations, and analyses were performed on a single piezoelectric stack. Based on the Euler Bernoulli beam principle, a theoretical model of the force amplification factor for the framework was established. The sensitivity analysis method of Sobol was used to determine the geometric parameters of the sensitive force amplification factor for the framework. A genetic algorithm and finite element modeling were applied to optimize the structure of the force amplification framework. Based on the optimal structure of the force amplification framework, a lumped parameter theoretical model of a piezoelectric energy harvester was established, and a numerical analysis was carried out. The results showed that under non-resonant excitation, the effective output voltage and maximum output power of the piezoelectric energy harvester were 3.022 V and 543.375 μW, respectively, which were 7.74 and 58.5 times the effective output voltage and maximum output power of the piezoelectric stack alone, respectively. Compared with the piezoelectric stack alone, the piezoelectric energy harvester could significantly improve the output voltage and output power under non-resonant excitation.