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Fault diagnosis under time-varying operational conditions is always the challenge in industrial systems. The chirplet transform (CT) provides an effective first-order approximation approach, but the slow operation speed and the drawback for high-order signal analysis limit its uses. Focusing on this problem, a fast nonlinear chirplet dictionary-based sparse decomposition (FNC-SD) method for nonlinear signal analysis is proposed. By replacing the time–frequency inclination parameter in the CT with a frequency bending parameter, the proposed method can track the nonstationary signals by arbitrary order polynomial law with time by an additional degree. Considering that too many parameters will slow down the operation speed, a frequency bending operator estimation algorithm is also proposed in this paper based on the calculated rotating frequency. Therefore, a laser vibrometer is needed in the experiment to collect the impulse signals, which can represent rotating speed signals. Then, the FNC-SD method is used to construct a redundant dictionary for signal sparse time–frequency representation. The high similarity between signals and atoms can make the method prefer to choose the right atoms, instead of being influenced by noise. Therefore, the proposed method also shows a strong noise robustness. Two simulation experiments are constructed to verify the performance of the proposed method. Finally, the FNC-SD method is used to diagnose a faulty motor with broken bar. The analyzed results and comparisons with respect to the state-of-the-art methods are illustrated in detail, as well as the convergence accuracy and speed, which highlight the superiority of the proposed method. [ABSTRACT FROM AUTHOR]