A novel vibration-based energy harvester which consists of a monostable Duffing oscillator connected to an electromagnetic generator with a mechanical motion rectifier (MMR-Duffing) is studied. The mechanical motion rectifier converts the bidirectional vibratory motion from ambient environments into unidirectional rotation to the generator and causes the harvester to periodically switch between a larger- and small-inertia system, resulting in nonlinearity in inertia. By means of the method of averaging, it is analytically shown that the proposed Duffing-MMR harvester outperforms traditional monostable Duffing oscillator energy harvesters in twofold. First of all, it increases the bandwidth of energy harvesting, given identical nonlinear stiffness. Second of all, it mitigates the jump phenomenon due to nonlinear stiffness and thus exploits more potential bandwidth of energy harvesting without inducing any jump phenomenon. Finally, the analytical analyses are verified via numerical simulations of a prototype of the proposed Duffing-MMR harvester.