A little deeper look at epicyclic gear trains

Pop Quiz: Can you have an epicyclic gear train design with only one planet pinion? Yes, think of the final output drive of a KitchenAid mixer.

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As we know, epicyclic gear trains are very versatile and compact. They have multiple configurations, able to give you as many as seven ratios within one package. They have many attributes, which we can use to our advantage, including:

Power density (the transferred power divided by package volume) is lowest for most common mechanical power transmission gear systems.

The mechanical power put through the system is handled and manipulated by concentric members, which means that all off-axis forces are canceled out (theoretically).

All gears have centering tendencies, meaning that the internal off-axis forces not only cancel, but also motivate the individual gears to seek their own center position — or stated more correctly, seek their own off-axis equilibrium position.

Both spur and helical gears work nicely with an epicyclic gear train. Other than the requirement to address and negate the axial thrust of a helical gear, both tooth forms satisfy all operational and build requirements. (Don’t forget, a spur gear is the degenerate case of a helical gear; a helical gear with a helix angle of zero).

The seven ratios can be accessed by a simple change of configuration of the three main power path components (i.e. input, output, and fixed).

Fixed has traditionally been assigned a rotational speed value of zero (0 rpm); however, as we have learned from folks like Lepelletier and others, ‘fixed’ can mean any non-changing rotational speed value. Further, the duration of the use of the term ‘fixed’ can be momentary — meaning that the use of the basic operational ratios, etc., can be applied to any ‘fixed’ rotational speed.

Compounding of epicyclic gear trains can be either simple or complex.

> Simple compound epicyclic gear trains are generally just two epicyclic gear trains nested such that the output from the first stage is the drive (input) to the second stage; this also applies to more than two stages and the combination of more than one configuration (e.g. a planetary first stage and a solar second stage, as an example).

> Complex compounding is more like the Ravigneaux style wherein the planet pinion is still just one functional piece (i.e. one piece of metal) but has differing tooth counts on one end versus the other, which allows for more ratio variation within one stage.

The calculation of ratio as a function of configuration is widely documented and will not be repeated here. However, the ‘build requirement’ for equal spaced planet pinions will be reviewed in brief. The number of planet pinions (not tooth count) has to satisfy the relationship of the number of teeth in the sun gear minus the number of teeth in the internal (ring) gear (stated as a negative number; as an example, a ring gear has -65 teeth) divided by an assumed number of planet pinions that must be equal to an integer.

Where

k is an integer,

z is the number of teeth on the designated gear,

1, 2, 3 are the individual gears; 1 is the sun gear, 2 is the planet pinion, and 3 is the internal gear.

Essentially, once we know what overall ratio, we need from the epicyclic gear train, we select the tooth count for the sun and internal gears that provide that ratio (or close enough to satisfy the drive requirements) and then we determine the number of planet pinions required by the above equation.

Unequal planet pinion spacing, which does not provide many of the benefits listed above, must satisfy the following equation:

Where

θ is called the “tick angle,” and is the offset angle of one or more planet pinions that is required to assemble (‘build’) the gear train.

Generally, unequal spacing is used in four, or multiples of four, number of planet pinions and two of the four pinions are offset the same amount (degrees) and in the same direction, which is referred to as diametrically opposed spacing or the X-configuration.

Another concept and technique used to facilitate certain operational phenomenon and enhance certain running conditions is Mesh Phasing. Mesh Phasing addresses the cyclic behavior of the multipole gear meshes within the configuration and their interaction as a function of the various frequencies of vibration emanating from the epicyclic gear train. The energies emanating from the epicyclic gear train are the manifestation of transmission error associated with each individual gear and the cumulative effect of positioning error of both the tooth form and the bulk gear (primarily the planet pinions) relative to every other gear. It can also be driven by the excitation from the induced harmonics from other manufacturing errors and / or effect of tolerance stack-up.

As an aside, gear mesh excitation can be seen as a collection of sinusoidal signals, typically labeled as harmonics, which in turn each have a phase angle and an amplitude or signal strength (which is the measure of vibration energy). In terms of analysis, Fourier transforms are used to analyze these wave patterns by converting the measured signal in the time domain to the frequency domain for analysis. The details of this are beyond the scope of this article. However, if interested, research modal analysis.

The phasing of each gear mesh is the basis for the larger analysis of epicyclic phasing. Further, relative phasing within the various gear meshes are the only significant consideration in terms of phasing analysis. With that as a basis of our understanding, there are only two phasing relationships: Sun Gear to Planet Pinion, and Planet Pinion to Internal Gear.

Let’s start with the sun gear to planet pinion. We use the following relationship to determine the phase angle of the sun-planet spatial relationship. This relationship determines if we are designing a gear tooth mesh wherein the teeth are either equally spaced or diametrically opposed. The first step in the process is to calculate the transverse pitch angle in the internal gear, which is given by:

Where

ψt1 is the sun gear transverse pitch angle.

Now calculate the relationship (quotient) of the planet pinion angle as a function of the transverse pitch angle of the sun gear, given by

Where

i is either 1, 2, 3; the individual gears 1 is the sun gear, 2 is the planet pinion, and 3 is the internal (ring) gear.

q is just a quotient which represents the fraction of the mesh cycle.

Finally, the mesh phase angle is given by the remainder of the mesh cycle fraction quotient, as shown in

Where

ϕ1 i is the mesh phase angle.

Similarly, for the planet pinion to internal gear

Where

ψt3 is the internal (ring) gear transverse pitch angle.

Now, calculate the relationship (quotient) of the planet pinion angle as a function of the transverse pitch angle of the internal gear, given by

Where

1 is either 1, 2, 3; the individual gears, 1 is the sun gear, 2 is the planet pinion, and 3 is the internal (ring) gear

q is just a quotient that represents the fraction of the mesh cycle.

Finally, the mesh phase angle is given by the remainder of the mesh cycle fraction quotient, as shown in

Where

ϕ3 i is the mesh phase angle.

Summing the above into a usable set of relationships, the Phasing Types are defined by the first harmonic (and subsequent higher order harmonics, but generally these negligible) of the gear mesh excitation. The objective is to understand which of the gear mesh excitations either combine or cancel these energies. For equal-spaced planet pinions

Where

k is a factor that dictates the mesh harmonic of interest.

There are four types of phasing of interest:

In-Phase wherein the phase angle for each gear mesh reinforces the transmitted torques and axial thrust loads (due to either the helix angle and / or the misalignment consideration). The In-Phase mode also cancels the radial forces and excitations due to the gear meshes.

Sequential-Phase is defined as equally separated phase angles for subsequent gear meshes. Sequential phase epicyclic gear trains cancel the torques and thrusts forces, but reinforce radial forces.

In-Phase Counter-Phase is defined as opposite phase angles for opposing planet pinions, and have summed components of excitation canceled.

Mixed-Phase is the combination of various phasing types, which can only occur with unequally spaced planet pinions (e.g. diametrically opposed, etc.) wherein the phase angle(s) sum to determine which of the force components cancel or reinforce. 

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Dr. William Mark McVea, P.E., is President and Principal Engineer of KBE+, Inc. which develops complete powertrains for automotive and off-highway vehicles. He is the Principal Engineer with Kinatech, a joint venture with Gear Motions / Nixon Gear. He has published extensively and holds or is listed as co-inventor on numerous patents related to mechanical power transmissions. Mark, a licensed Professional Engineer, has a B.S. in Mechanical Engineering from the Rochester Institute of Technology, a Ph.D. in Design Engineering from Purdue University.