This article is published in Journal of Sound and Vibration. Digital reference is http://dx.doi.org/10.1016/j.jsv.2009.03.002
Gears are used in almost all mechanical systems such as automobiles, airplanes, helicopters, and so on. In many such applications, gears rotate at high speeds, so they are prone to vibration. Excessive gear vibrations decrease the life and the durability of machinery and can cause premature failure. Gear vibration is a system level problem, that is, one cannot design “quiet” gears without considering the whole system, so each system must individually be designed to avoid resonances. This paper deals with the vibration of planetary gears, which are widely used in helicopters, automatic transmissions in cars, etc. Planetary gears have an added complexity compared with single gear pairs due to their configuration, so vibration issues of planetary gears do not lend themselves to direct approaches that simpler gear configurations do. The three-dimensional (3-D) layout of planetary gears further complicates the matters. The solution to vibration reduction lies in understanding the complex gear dynamic behavior. This work clarifies the 3-D vibration of planetary gears. The understanding of planetary gear dynamics yields design guidelines as to the action to be taken to eliminate vibration. The analysis in this work builds a foundation crucial to derive design changes that are very easy to implement, such as changing the tooth counts, to solve the vibration problem. The conclusions are widely applicable to real-world situations where 3-D effects are inevitably important. Introduction The structured modal properties of single-stage helical planetary gears with equally spaced planets are categorized and mathematically proved. Compared to prior two-dimensional analyses of spur gears, this study examines the three-dimensional motion of the helical gears and shafts. A lumped-parameter model (Figure 1) is formulated to obtain the equations of motion. The gear-shaft bodies are modeled as rigid bodies with compliant bearings at arbitrary axial locations on the shafts. A translational and a tilting stiffness account for the force and moment transmission at the gear mesh interface. The derived modal properties generalize those of two-dimensional spur planetary gears; there are twice as many degrees of freedom and natural frequencies due to the added tilting and axial motion. All vibration modes are categorized as rotational–axial, translational–tilting, and planet modes. Figures 1-5 show examples of each mode type. The modal properties are shown to hold even for configurations that are not symmetric about the gear plane, due to, for example, shaft bearings not being equidistant from the gear plane.   Figure 1. Lumped-parameter model of the planetary gear system.  Figure 2. A rotational-axial mode.
 Figure 3. A
translational-tilting mode  Figure 4. A translational-tilting mode  Figure 5. A planet mode. Gears are used in almost all mechanical systems such as automobiles, airplanes, helicopters, and so on. In many such applications, gears rotate at high speeds, so they are prone to vibration. Excessive gear vibrations decrease the life and the durability of machinery and can cause premature failure. Gear vibration is a system level problem, that is, one cannot design “quiet” gears without considering the whole system, so each system must individually be designed to avoid resonances. This paper deals with the vibration of planetary gears, which are widely used in helicopters, automatic transmissions in cars, etc. Planetary gears have an added complexity compared with single gear pairs due to their configuration, so vibration issues of planetary gears do not lend themselves to direct approaches that simpler gear configurations do. The three-dimensional (3-D) layout of planetary gears further complicates the matters. The solution to vibration reduction lies in understanding the complex gear dynamic behavior. This paper clarifies the 3-D vibration of planetary gears. The understanding of planetary gear dynamics yields design guidelines as to the action to be taken to eliminate vibration. The analysis in this paper builds a foundation crucial to derive design changes that are very easy to implement, such as changing the tooth counts, to solve the vibration problem. The conclusions are widely applicable to real-world situations where 3-D effects are inevitably important.
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