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The HelicalGearSet element automatically creates a pair of two helical gears based on user input and calculated values. However, if the HelicalGear element is used, then some parameters cannot be auto-calculated, such as center distance, addendum lowering, and so forth. You must consider this when creating gear sets using individual HelicalGear elements.

Backlash ratio

(j_{i})

$(j_{i})$ is defined as the parameter that if multiplied by the half circular pitch of the gear tooth, the circumferential backlash for each gear is calculated.

Backlash (circumferential) or tooth thickness allowance is the slight reduction of the theoretical tooth thickness on the pitch circle (calculated as backlash – free state) to prevent gear jamming. The center distance is not changed by this; the ratio only affects the gear tooth thickness.

Another way to introduce extra backlash to the gear tooth is to increase the center distance. Profile shift parameters are also adjusted.

The total backlash (for both gears) is calculated as:

b a c k l a s h = 0.5 \cdot (j_{1} + j_{2}) \cdot c p

$b a c k l a s h = 0.5 \cdot (j_{1} + j_{2}) \cdot c p$

where,

c p

$c p$ is the circular pitch and calculated as:

c p = \frac{(m_{n} \cdot π)}{c o s (β)}

$c p = \frac{(m_{n} \cdot π)}{c o s (β)}$

Addendum lowering is calculated from the summation of the profile shifts to ensure that tip clearance does not change.

It is calculated as:

k m = a_{w} - a_{i} - m_{n} \cdot (x_{1}^{*} + x_{2}^{*})

$k m = a_{w} - a_{i} - m_{n} \cdot (x_{1}^{*} + x_{2}^{*})$

When using a single gear, this parameter cannot be calculated and the default value is zero. For the HelicalGearSet, this parameter is calculated internally and provided to the created HelicalGear elements.

The default value is calculated internally with respect to the units of the system. S45C (AISI 1045) steel has been chosen as the default material.

Two of the num_of_teeth_1, num_of_teeth_2 and gear_ratio must be specified as inputs. The third is calculated consistently.

The profile shift coefficient must always be higher than a critical value to avoid the undercut phenomenon. An error message is displayed if this value is less than this limit.

When the number of teeth of the gear element parameter is negative, then an internal helical gear is created.

Use the negative gear ratio to create an external/internal gear set with the HelicalGearSet element.

Do not use a negative number of teeth on both gears.

If the hub diameter is not provided as an input, then it is calculated internally as a function of the addendum and the dedendum of the gear. The hub diameter is the inner diameter when creating external gears and the outer diameter when using internal gears.

In the HelicalGear element, the following parameters are defined:

The pitch diameter is defined as:

d = \frac{m_{n}}{c o s (β)} \cdot z

$d = \frac{m_{n}}{c o s (β)} \cdot z$

The tip diameter is defined as:

d_{a}^{*} = d + 2 \cdot m_{n} (x^{*} + h_{a}^{*})

$d_{a}^{*} = d + 2 \cdot m_{n} (x^{*} + h_{a}^{*})$

d_{a c} = d_{a}^{*} + 2 \cdot k m

$d_{a c} = d_{a}^{*} + 2 \cdot k m$

d_{a} = max (| d_{a c} |, | d_{a}^{*} ∣ ∣)

$d_{a} = \max (| d_{a c} |,| d_{a}^{*} |)$

The root diameter is defined as:

d_{f} = d - 2 \cdot m_{n} (h_{f}^{*} - x^{*})

$d_{f} = d - 2 \cdot m_{n} (h_{f}^{*} - x^{*})$

i f z < 0 : d f = | d f |

$i f z < 0 : d f = | d f |$

In the HelicalGearSet element, the operating center distance

(a_{w})

$(a_{w})$ is calculated internally from the definition of the two reference markers of the element. This center distance might differ from the initial one calculated as

a_{i} = 0.5 \cdot (d_{1} + d_{2})

$a_{i} = 0.5 \cdot (d_{1} + d_{2})$ . Ensure that the working pressure angle and the working pitch diameters

(d_{w 1}, d_{w 2})

$(d_{w 1}, d_{w 2})$ comply with the calculated center distance.

The center distance of the planetary gear set must have specific limits to avoid planet interference. The planet's distance

l

$l$ , as can be seen in the figure of the PlanetaryGearSet, must follow the following equation:

l > d_{a_{p l a n e t}}

$l > d_{a_{p l a n e t}}$

w h e r e :

$w h e r e :$

l = 2 \cdot a_{w} \cdot s i n (\frac{π}{N})

$l = 2 \cdot a_{w} \cdot s i n (\frac{π}{N})$

d_{a_{p l a n e t}} : t h e t i p d i a m e t e r o f p l a n e t s

$d_{a_{p l a n e t}} : t h e t i p d i a m e t e r o f p l a n e t s$

a_{w} : w o r k i n g c e n t e r d i s t a n c e

$a_{w} : w o r k i n g c e n t e r d i s t a n c e$

N : N u m b e r o f p l a n e t s

$N : N u m b e r o f p l a n e t s$

The number of teeth of the planetary gear set must follow specific constraints. The

z_{1} a n d z_{2}

$z_{1} a n d z_{2}$ always need to be positive, and respectively, the ring gear teeth (

z_{3}

$z_{3}$ ) must be negative. Also, for the correct meshing of the gears, the following equation for

z_{1}

$z_{1}$ and

z_{3}

$z_{3}$ needs to be accomplished:

m o d ({(z}_{1} - z_{3}) / N) = 0

$m o d ({(z}_{1} - z_{3}) / N) = 0$

The modulo of the ratio between the difference of the sun and ring number of teeth and the number of planets needs to be zero.

Another constraint for the planetary set, without affecting the input choices, is that the operating center distance for the (sun-planet) and (planet-ring) gearsets need to be equal. This does not mean that the reference center distance is also the same due to the profile shift. The following equation for the working pitch diameters of the gears is always true:

| d_{w 1} + d_{w 2} | = | d_{w 22} + d_{w 3} |

$|d_{w 1} + d_{w 2}| = | d_{w 22} + d_{w 3} |$

For planetary gear sets, the gear ratio

u

$u$ does not always define the transmission ratio for the gear system (

i

$i$ ). For the case that the carrier part is connected to ground, the planets work as idler gears so the transmission ratio (

i

$i$ ) is the same as the gear ratio (

u

$u$ ).

i = u = \frac{ω_{1}}{ω_{3}}

$i = u = \frac{ω_{1}}{ω_{3}}$

In the case that the ring gear is connected to the ground, the transmission ratio between the sun gear and the carrier is calculated as follows:

i_{1 c} = (1 - u) = \frac{ω_{1}}{ω_{C}}

$i_{1 c} = (1 - u) = \frac{ω_{1}}{ω_{C}}$

Finally, in the case that the sun is fixed, the transmission ratio is modified as follows:

i_{3 c} = (1 - \frac{1}{u}) = \frac{ω_{3}}{ω_{c}}

$i_{3 c} = (1 - \frac{1}{u}) = \frac{ω_{3}}{ω_{c}}$