To improve the energy-conversion efficiency of piezoelectric energy harvesters, a piezoelectric stack was intergrated with a force-amplification framework structure to construct a piezoelectric energy harvester featuring force amplification. Mechanical analysis and theoretical modeling of the force amplification coefficient were conducted, and Sobol sensitivity analysis and genetic algorithm optimization were applied to the structural parameters of the framework. Based on the optimal structure of the force-amplification framework, a lumped-parameter theoretical model of the piezoelectric energy harvester was established. The energy-harvesting performance of this energy harvester under non-resonant and multi-modal resonant excitations was compared and analyzed. The research results indicate that the open circuit output voltage peak and maximum output power of the piezoelectric energy harvester at the third and fourth natural frequency excitations are 55.07 V, 124.19 V and 1.63 W, 14.97 W, respectively. These values are 12.9, 29.1, 2 999.77, and 27 550 times greater than the open-circuit output voltage peak and maximum output power under 10 Hz non-resonant low-frequency excitation. Compared to non-resonant excitation, operating the harvester at the resonant excitation frequency significantly increases its output voltage and power. Moreover, the inclusion of mass blocks in energy harvesters effectively reduces their resonant frequency and increases output voltage, thereby enabling low-frequency extension applications for piezoelectric energy harvesters.