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Four-stage main helicopter transmission system occupies several strong points like good carrying capacity and is addressed by many scholars. However, the influence of variable parameters on natural characteristics in a three-engine helicopter, is not to be investigated, according to the limited published issues. Thus, in the study, a vibration model of the four-stage main helicopter transmission system is established through the lumped mass method. In the model, many factors, including time-varying meshing stiffness and torsional stiffness of gear shaft are considered. The differential equation of the system is solved via the Fourier method, 27-orders natural frequency and modal shape corresponded by each degree of freedom (DOF) of the system are obtained. The influence of the shaft torsional stiffness on the first five orders of the system natural frequency is studied. Besides, the impact of the matching relationship of key shafts’ torsional stiffness on the system stability is analyzed. These contributions would improve modification developments of four-stage deceleration helicopter in future.

The elastohydrodynamic lubrication (EHL) contact model is established, the oil film stiffness is calculated based on Dowson-Higginson empirical minimum thickness equation. The vibration model of single-rotor three-input helicopter main reducer is proposed by lumped mass method, and nonlinear factors like oil film stiffness, dynamic meshing force are taken into account. The influence of oil film stiffness on system natural frequency is analyzed as well. The results show that oil film stiffness of the internal and external meshing pairs in the planetary gear system have greater impact on the natural frequency, which tends to destabilize the system. Therefore, the planetary gear train is the most crucial branch regarding the system splash lubrication. When all the oil film stiffness in the system are greater than 4×109 N/m, the natural frequencies tend to be stable. This study can provide the theoretical reference for the lubrication characteristics in the single-rotor multi-input helicopter.

The vibration model of four-stage main transmission system in helicopter was established through lumped mass method. In the model, many factors, including time-varying meshing stiffness and torsional stiffness of gear shaft were considered. The differential equation of the system was solved via Fourier method, and the influence of the torsional stiffness of shafts on the first five orders of the system’s natural frequency was studied. Some theoretical results were summarized as guidelines for further research and design of four-stage deceleration helicopter at last.

Angiosperms, http://name.umdl.umich.edu/IC-HERB00IC-X-1715509%5DMICH-V-1715509, https://quod.lib.umich.edu/cgi/i/image/api/thumb/herb00ic/1715509/MICH-V-1715509/!250,250, The University of Michigan Library provides access to these materials for educational and research purposes. Some materials may be protected by copyright. If you decide to use any of these materials, you are responsible for making your own legal assessment and securing any necessary permission. If you have questions about the collection, please contact the Herbarium professional staff: herb-dlps-help@umich.edu. If you have concerns about the inclusion of an item in this collection, please contact Library Information Technology: libraryit-info@umich.edu., https://www.lib.umich.edu/about-us/policies/copyright-policy

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Angiosperms, http://name.umdl.umich.edu/IC-HERB00IC-X-1714088%5DMICH-V-1714088, https://quod.lib.umich.edu/cgi/i/image/api/thumb/herb00ic/1714088/MICH-V-1714088/!250,250, The University of Michigan Library provides access to these materials for educational and research purposes. Some materials may be protected by copyright. If you decide to use any of these materials, you are responsible for making your own legal assessment and securing any necessary permission. If you have questions about the collection, please contact the Herbarium professional staff: herb-dlps-help@umich.edu. If you have concerns about the inclusion of an item in this collection, please contact Library Information Technology: libraryit-info@umich.edu., https://www.lib.umich.edu/about-us/policies/copyright-policy

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Angiosperms, http://name.umdl.umich.edu/IC-HERB00IC-X-1714091%5DMICH-V-1714091, https://quod.lib.umich.edu/cgi/i/image/api/thumb/herb00ic/1714091/MICH-V-1714091/!250,250, The University of Michigan Library provides access to these materials for educational and research purposes. Some materials may be protected by copyright. If you decide to use any of these materials, you are responsible for making your own legal assessment and securing any necessary permission. If you have questions about the collection, please contact the Herbarium professional staff: herb-dlps-help@umich.edu. If you have concerns about the inclusion of an item in this collection, please contact Library Information Technology: libraryit-info@umich.edu., https://www.lib.umich.edu/about-us/policies/copyright-policy

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Angiosperms, http://name.umdl.umich.edu/IC-HERB00IC-X-1714102%5DMICH-V-1714102, https://quod.lib.umich.edu/cgi/i/image/api/thumb/herb00ic/1714102/MICH-V-1714102/!250,250, The University of Michigan Library provides access to these materials for educational and research purposes. Some materials may be protected by copyright. If you decide to use any of these materials, you are responsible for making your own legal assessment and securing any necessary permission. If you have questions about the collection, please contact the Herbarium professional staff: herb-dlps-help@umich.edu. If you have concerns about the inclusion of an item in this collection, please contact Library Information Technology: libraryit-info@umich.edu., https://www.lib.umich.edu/about-us/policies/copyright-policy

Angiosperms, http://name.umdl.umich.edu/IC-HERB00IC-X-1714100%5DMICH-V-1714100, https://quod.lib.umich.edu/cgi/i/image/api/thumb/herb00ic/1714100/MICH-V-1714100/!250,250, The University of Michigan Library provides access to these materials for educational and research purposes. Some materials may be protected by copyright. If you decide to use any of these materials, you are responsible for making your own legal assessment and securing any necessary permission. If you have questions about the collection, please contact the Herbarium professional staff: herb-dlps-help@umich.edu. If you have concerns about the inclusion of an item in this collection, please contact Library Information Technology: libraryit-info@umich.edu., https://www.lib.umich.edu/about-us/policies/copyright-policy

Angiosperms, http://name.umdl.umich.edu/IC-HERB00IC-X-1714101%5DMICH-V-1714101, https://quod.lib.umich.edu/cgi/i/image/api/thumb/herb00ic/1714101/MICH-V-1714101/!250,250, The University of Michigan Library provides access to these materials for educational and research purposes. Some materials may be protected by copyright. If you decide to use any of these materials, you are responsible for making your own legal assessment and securing any necessary permission. If you have questions about the collection, please contact the Herbarium professional staff: herb-dlps-help@umich.edu. If you have concerns about the inclusion of an item in this collection, please contact Library Information Technology: libraryit-info@umich.edu., https://www.lib.umich.edu/about-us/policies/copyright-policy

Angiosperms, http://name.umdl.umich.edu/IC-HERB00IC-X-1714103%5DMICH-V-1714103, https://quod.lib.umich.edu/cgi/i/image/api/thumb/herb00ic/1714103/MICH-V-1714103/!250,250, The University of Michigan Library provides access to these materials for educational and research purposes. Some materials may be protected by copyright. If you decide to use any of these materials, you are responsible for making your own legal assessment and securing any necessary permission. If you have questions about the collection, please contact the Herbarium professional staff: herb-dlps-help@umich.edu. If you have concerns about the inclusion of an item in this collection, please contact Library Information Technology: libraryit-info@umich.edu., https://www.lib.umich.edu/about-us/policies/copyright-policy

Angiosperms, http://name.umdl.umich.edu/IC-HERB00IC-X-1714082%5DMICH-V-1714082, https://quod.lib.umich.edu/cgi/i/image/api/thumb/herb00ic/1714082/MICH-V-1714082/!250,250, The University of Michigan Library provides access to these materials for educational and research purposes. Some materials may be protected by copyright. If you decide to use any of these materials, you are responsible for making your own legal assessment and securing any necessary permission. If you have questions about the collection, please contact the Herbarium professional staff: herb-dlps-help@umich.edu. If you have concerns about the inclusion of an item in this collection, please contact Library Information Technology: libraryit-info@umich.edu., https://www.lib.umich.edu/about-us/policies/copyright-policy

Angiosperms, http://name.umdl.umich.edu/IC-HERB00IC-X-1714098%5DMICH-V-1714098, https://quod.lib.umich.edu/cgi/i/image/api/thumb/herb00ic/1714098/MICH-V-1714098/!250,250, The University of Michigan Library provides access to these materials for educational and research purposes. Some materials may be protected by copyright. If you decide to use any of these materials, you are responsible for making your own legal assessment and securing any necessary permission. If you have questions about the collection, please contact the Herbarium professional staff: herb-dlps-help@umich.edu. If you have concerns about the inclusion of an item in this collection, please contact Library Information Technology: libraryit-info@umich.edu., https://www.lib.umich.edu/about-us/policies/copyright-policy

Angiosperms, http://name.umdl.umich.edu/IC-HERB00IC-X-1714094%5DMICH-V-1714094, https://quod.lib.umich.edu/cgi/i/image/api/thumb/herb00ic/1714094/MICH-V-1714094/!250,250, The University of Michigan Library provides access to these materials for educational and research purposes. Some materials may be protected by copyright. If you decide to use any of these materials, you are responsible for making your own legal assessment and securing any necessary permission. If you have questions about the collection, please contact the Herbarium professional staff: herb-dlps-help@umich.edu. If you have concerns about the inclusion of an item in this collection, please contact Library Information Technology: libraryit-info@umich.edu., https://www.lib.umich.edu/about-us/policies/copyright-policy

Angiosperms, http://name.umdl.umich.edu/IC-HERB00IC-X-1714097%5DMICH-V-1714097, https://quod.lib.umich.edu/cgi/i/image/api/thumb/herb00ic/1714097/MICH-V-1714097/!250,250, The University of Michigan Library provides access to these materials for educational and research purposes. Some materials may be protected by copyright. If you decide to use any of these materials, you are responsible for making your own legal assessment and securing any necessary permission. If you have questions about the collection, please contact the Herbarium professional staff: herb-dlps-help@umich.edu. If you have concerns about the inclusion of an item in this collection, please contact Library Information Technology: libraryit-info@umich.edu., https://www.lib.umich.edu/about-us/policies/copyright-policy

Angiosperms, http://name.umdl.umich.edu/IC-HERB00IC-X-1714104%5DMICH-V-1714104, https://quod.lib.umich.edu/cgi/i/image/api/thumb/herb00ic/1714104/MICH-V-1714104/!250,250, The University of Michigan Library provides access to these materials for educational and research purposes. Some materials may be protected by copyright. If you decide to use any of these materials, you are responsible for making your own legal assessment and securing any necessary permission. If you have questions about the collection, please contact the Herbarium professional staff: herb-dlps-help@umich.edu. If you have concerns about the inclusion of an item in this collection, please contact Library Information Technology: libraryit-info@umich.edu., https://www.lib.umich.edu/about-us/policies/copyright-policy