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For rotary machines, the measured vibration signals contain multiple frequency modulation components with much machine health information. Decomposing vibration signals of rotary machines into a series of sub-signal components with physical meaning is of great significance for a deep understanding of the machine properties in solving fault diagnosis problems. However, accurately detecting and decomposing frequency modulation components is a challenging problem. This article proposed a multiple frequency modulation components detection and decomposition approach for rotary machine fault diagnosis. Specifically, the amplitude and concentration at the orders (defined as multiple of the rotating frequency) of each signal to be analyzed are calculated, and thresholds are set to distinguish noise and non-noise signal components. Then the cross correlation between the non-noise signal components and an order ruler (constructed with normalized amplitude only on the orders of characteristic components of the faulty signal) is calculated to obtain the instantaneous frequencies (IFs) of these signal components. Finally, a variational algorithm framework defined only by the characteristic components of the faulty signal is used to achieve signal decomposition. The proposed method fully uses the inherent geometric relationship of the frequency modulation signal in the angular domain and is robust to strong environmental noise. Both the simulation example and the actual application cases demonstrated the effectiveness of the proposed method, which shows the potential in solving real industrial problems. [ABSTRACT FROM AUTHOR]