An analytical solution for a specific case of the forced Duffing oscillator is proposed. The excitation force contains two harmonics with significant difference frequencies. This case corresponds to a presence of a defect in the machinery and is in the art of the machinery vibration diagnostics. The results obtained show an amplitude modulation. Therefore, the presence of an amplitude modulation in the vibration signal may be used as an indicator for a malfunction. Analytical solution derived clarifies how the amplitude modulation occurs. Also, a numerical solution is realized and compared with the analytical one. For this, the Duffing equation is solved numerically and then, the spectrograms of vibrations are obtained through a Discrete-time Fourier Transform.
 Jia, S., I. Howard. Comparison of Localised Spalling and Crack Damage from Dynamic Modelling of Spur Gear Vibrations. Mechanical Systems and Signal Processing, 20 (2006), 332-349.
 Wu, S., M. J. Zuo, A. Parey. Simulation of Spur Gear Dynamics and Esti- mation of Fault Growth. Journal of Sound and Vibration, 317 (2008), 608-624.
 Chen, Z., Y. Shao. Dynamic Simulation of Spur Gear with Tooth Root Crack propagating Along Tooth Width and Crack Depth. Engineering Failure Analysis, 18 (2011), 2149-2164.
 Pedersen, N. L., M. F. Jorgensen. On Gear Tooth Stiffness Evaluation. Computers and Structures, 135 (2014), 109-117.
 Feldman, M. Non-linear System Vibration Analysis using Hilbert Transform-I. Free Vibration Analysis Method ’Freevib’, 8 (1994), No. 2, 119-127.
 Feldman, M. Non-linear System Vibration Analysis using Hilbert Transform-II. Forced vibration analysis method ’Forcevib’, 8 (1994), No. 3, 309-318.
 Feldman, M. Considering High Harmonics for Identification of Non-linear Sys- tems by Hilbert Transform.Mechanical Systems and Signal Processing, 21 (2007), 943-958.
 Feldman, M. Theoretical Analysis and Comparison of the Hilbert Transform Decomposition Methods. Mechanical Systems and Signal Processing, 22 (2008), 509-519.
 Feldman, M. Hilbert Transform in Vibration Analysis. Mechanical Systems and Signal Processing, 25 (2011), 735-802.
 Feldman, M. Hilbert Transform Applications in Mechanical Vibration, First Edition, Published 2011 by John Wiley & Sons Ltd., ISBN: 978-0-470-97827-6.
 Qu, L. L., L. X. Liao. Haar Wavelet for Machine Fault Diagnosis. Mechanical Systems and Signal Processing, 21 (2007), 1773-1786.
 Li, H., Y. Zhang, H. Zheng. Application of Hermitian Wavelet to Crack Fault Detection in Gearbox. Mechanical Systems and Signal Processing, 25 (2011), 1353-1363.
 Combet, F., L. Gelman, G. LaPayne. Novel Detection of Local Tooth Dam- age in Gears by the Wavelet Bicoherence. Mechanical Systems and Signal Pro- cessing, 26 (2012), 218-228.
 Rafiee, J., M. A. Rafiee, P. W. Tse. Application of Mother Wavelet Func- tions for Automatic Gear and Bearing Fault Diagnosis. Expert Systems with Ap- plications, 37 (2010), 4568-4579.
 Saravanan, N., K. I. Ramachandran. Fault Diagnosis of Spur Bevel Gear Box using Discrete Wavelet Features and Decision Tree Classification. Expert Systems with Applications, 36 (2009), 9564-9573.
 Tiwari, R. On-line Identification and Estimation of Non-linear Stiffness Param- eters of Bearings. Journal of Sound and Vibration, 234 (2000), No. 5, 906-910.
 Enns, R., G. C. McGuire. Nonlinear Physics with Mathematica for Scientist and Engineers, Birkhauser Boston, 2001, ISBN 0-8176-4223-4.
 Kovacic, I., M. J. Brennan. The Duffing Equation: Nonlinear Oscillators and their Phenomena, Edited by Ivana Kovacic, Michael J. Brennan, 2011, ISBN 978-0-470-71549-9.
 Patel, V. N., N. Tandon, R. K. Pandey. Defect Detection in Deep Groove Ball Bearing in Presence of External Vibration using Envelope Analysis and Duff- ing Oscillator. Measurement, 45 (2012), 960-970
 Cveticanin, L., G. Mester, I. Biro. Parameter Influence on the Harmonically Excited Duffing Oscillator. Acta Polytechnica Hungarica, 11 (2014), No. 5, 145-160.
 Li, C., L. Qu. Applications of Chaotic Oscillator in Machinery Fault Diagnosis. Mechanical Systems and Signal Processing, 21 (2007), 257-269.
 Zhao, Z., X. Wang, X. Zhang. Fault Diagnosis of Broken Rotor Bars in Squirrel-Cage Induction Motor of Hoister Based on Duffing Oscillator and Mul- tifractal Dimension, Hindawi Publishing Corporation, Advances in Mechanical Engineering, 2014, Article ID 849670.
 Shi, H., S. Fan, W. Xing, J. Sun. Study of Weak Vibrating Signal Detec- tion based on Chaotic Oscillator in MEMS Resonant Beam Sensor. Mechanical Systems and Signal Processing, 50-51 (2015), 535-547.
 Miller, B., S. Howard. Identifying Bearing Rotor-Dynamic Coefficients Using an Extended Kalman Filter. Tribology Transactions, 52 (2009), 671-679, Society of Tribologists and Lubrication Engineers ISSN: 1040-2004.
 Description and Use of LSODE, the Livermore Solver for Ordinary Differential Equations, NASA Reference Publication 1327, Lawrence Livermore Nationai Lab- oratory Report UCRL-ID-113855, 1993.