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AbstractWe develop a two-timing perturbation analysis to provide quantitative insights on the existence of temporal ratchets in an exemplary system of a particle moving in a tank of fluid in response to an external vibration of the tank. We consider two-mode vibrations with angular frequencies$$\omega $$ $\omega$and$$\alpha \omega $$ $\alpha \omega$, where$$\alpha $$ $\alpha$is a rational number. If$$\alpha $$ $\alpha$is a ratio of odd and even integers (e.g.,$$\tfrac{2}{1},\,\tfrac{3}{2},\,\tfrac{4}{3}$$ $\frac{2}{1},\frac{3}{2},\frac{4}{3}$), the system yields a net response: here, a nonzero time-average particle velocity. Our first-order perturbation solution predicts the existence of temporal ratchets for$$\alpha =2$$ $\alpha =2$. Furthermore, we demonstrate, for a reduced model, that the temporal ratcheting effect for$$\alpha =\tfrac{3}{2}$$ $\alpha =\frac{3}{2}$and$$\tfrac{4}{3}$$ $\frac{4}{3}$appears at the third-order perturbation solution. More importantly, we find closed-form formulas for the magnitude and direction of the induced net velocities for these$$\alpha $$ $\alpha$values. On a broader scale, our methodology offers a new mathematical approach to study the complicated nature of temporal ratchets in physical systems. Graphic abstract