.. _working-notes: ************* Working Notes ************* This chapter captures some notes that I needed as this library was developed. .. _spur-gear-notes: ================== Spur Gear Geometry ================== .. index:: single: spur gear; module single: spur gear; pitch radius single: spur gear; profile curve single: spur gear; teeth count The gear's *module* (:math:`m`) and the *number of teeth* (:math:`z`) are the main inputs to a gear's size. Two meshing gears **must** have matching *module* and *profile curve* values. The *profile curve* has a role in the shape of the flank and is better explained by the references above. The default is a reasonable value. The *module* declares the length of the :index:`addendum`, that part of the cog above the :index:`pitch radius`, as well as the length of the :index:`dedendum`, the part of the cog below the :index:`pitch radius`. .. index:: single: spur gear; pitch diameter Two gears coincide at their pitch point and the diameter of the gear at that point is known as the *pitch diameter*. .. math:: D_{pitch} = m \times z .. tip:: If you want a physically larger gear, increase the module size. .. index:: spur gear; tip diameter As mentioned earlier the *module* is the length of the gear's cog above the pitch radius, so the diameter at the tip is, .. math:: D_{tip} = D_{pitch} + ( m \times 2 ) To construct a set of gears with a specific diameter start with the number of teeth and the target diameter, then derive the *module*, .. math:: :nowrap: \begin{eqnarray} D_{tip} & = & D_{pitch} + ( m \times 2 ) \\ & = & (m \times z) + ( m \times 2 ) \\ & = & m \times (z + 2) \\ m & = & \frac{D_{tip}}{z+2} \end{eqnarray} Spur gear on plane ~~~~~~~~~~~~~~~~~~ .. figure:: /images/eg-index_diameter_01.png :align: right :width: 300 Here is an example of creating a gear that fits into a specific dimension using geometry nodes. We start with the TARGET shown as a rectanglular grid that is 80mm square and we want to fit a 14-tooth spur gear onto it. .. figure:: /images/eg-index_diameter_02.png :align: center :width: 800 Geometry nodes for calculating the module from a target diameter. Remember that the very first step to working with these nodes is to set the Scene Properties to the proper Units. Refer to the :ref:`modeling-setup` section for more details. Angle at base ~~~~~~~~~~~~~ The shape of a gear is controlled by the *pressure angle*. The angle controls the width at the base as well as the spur tip. Gears with small pressure angles will have a wider root and thicker tip than those with a larger pressure angle. .. math:: :nowrap: \begin{eqnarray} Angle_{base} = 2 \times \frac{\pi}{(2 \times z)} + \tan\alpha - \alpha \end{eqnarray} Regarding clearance ~~~~~~~~~~~~~~~~~~~ The :index:`clearance` value for gears defines additional space added to the root between two cogs to allow for the passage of the tip of a cog of an opposing gear. When gears are cut, these are created somewhat naturally by the machine tool and usually form a circular shape. That is not strictly necessary and, in fact, it would create more topology than I was willing to add. The :index:`whole depth` is the total size of the gear's cog, from tip to root, .. math:: Depth_{whole} = (2 \times m) + (clearance \times m) The default clearance is usually sufficient. .. _bevel-gear-notes: =========== Bevel Gears =========== If you visualize spur gears as two cylinders rotating against each other, a pair of bevel gears can be seen as two cones rotating against each other. These cones are known as *pitch cones* because the pitch radius slides along the edge of the cone and, just like spur gears, they are sized according to their *module* and number of teeth. The first figure in the next section shows this visualization from a side view. .. _pitch-cone: The Pitch Cone ~~~~~~~~~~~~~~ .. figure:: /images/ref-cone-angle.png :width: 400 :align: center Reference Cone Angle .. index:: single: bevel gear; pitch cone single: bevel gear; cone angle This drawing shows a generalized bevel gear set with the pitch :index:`cone angles` (:math:`\delta_1` and :math:`\delta_2`) where the sum :math:`\sum` doesn't equal :math:`90^\circ`. When the number of teeth (:math:`z`) in the pinion is equal to the number of teeth in the bull gear (a gear ratio of :math:`1:1`), and the gears are set at a :math:`90^\circ` angle, they are known as miter gears. The sum of the pitch cone angles is known as the :index:`shaft angle`, usually annotated as :math:`\Sigma`, .. math:: :label: shaft-angle :nowrap: \begin{eqnarray} \Sigma = \delta_1 + \delta_2 \end{eqnarray} .. math:: :label: pitch-cone :nowrap: \begin{eqnarray} \tan \delta_1 & = & \frac{\sin\Sigma}{\frac{z_2}{z_1}+\cos\Sigma} \\ \tan \delta_2 & = & \frac{\sin\Sigma}{\frac{z_1}{z_2}+\cos\Sigma} \end{eqnarray} Bevel gears must be considered in pairs since the tooth counts will affect the pitch cone angle. For the miter gear described earlier (set at a right angle with :math:`z_1 = z_2`), the above calculation will result in a pitch cone angle of :math:`45^\circ`. It is typically necessary to provide the pitch cone angles during construction of the bevel gear, which can be determined with this derivation, .. math:: :label: pitch-cone-angle :nowrap: \begin{eqnarray} \tan \delta_1 & = & \frac{\sin\frac{\pi}{2}}{\frac{z_2}{z_1}+\cos\frac{\pi}{2}} \\ \delta_1 & = & \arctan{\frac{1}{\frac{z_2}{z_1}}} \\ \delta_1 & = & \arctan{\frac{z_1}{z_2}} \end{eqnarray} For bevel gearing, the :index:`reference diameter` :math:`d` is known as the pitch diameter. The equations should look familiar (:math:`m` is the module of the gear), .. math:: :label: ref-diameter :nowrap: \begin{eqnarray} d_1 & = & z_1 \times m \\ d_2 & = & z_2 \times m \end{eqnarray} The :index:`cone distance` (:math:`R`) defines the linear distance between the reference points at the intersection of the reference diameters, .. math:: :label: cone-distance :nowrap: \begin{eqnarray} R = \frac{d_2}{2\times\sin \delta_2} \end{eqnarray} .. index:: single: bevel gear; tooth profile Face Width ~~~~~~~~~~ The face width (:math:`b`) is the distance across the gear teeth and length should match for a bevel gear pair. Not much is said about this but the going recommendation is, .. math:: :label: face-width :nowrap: \begin{eqnarray} b < \frac{R}{3} \end{eqnarray} ========== Gear Racks ========== .. figure:: /images/gear-rack-detail.svg :align: center :width: 300 Basic dimensions of a gear rack Where, .. math:: :label: gear-rack :nowrap: \begin{eqnarray} m & = & module \\ \alpha & = & pressure\ angle \\ h & = & 2\times{m} \\ pitch & = & m\times\pi \end{eqnarray} As with other types of gears, sizing a rack depends on the :math:`m` value and the shape of the cog depends on :math:`\alpha`. As :math:`\alpha` increases the tip and root get narrower. The center of the tooth is the pitch line and is analogous to a circular gear's pitch diameter. Not shown in the diagram is a small clearance below the root (between the gears). This clearance is defined in terms of a factor multiplied by the module to set the depth and is hardcoded to :math:`0.25`. ========== References ========== * `Spur Gears wiki `_ * `Bevel Gears wiki `_ * `Wikipedia page on the Involute Gear `_ (nice animation here.) * `Wikipedia page on the Involute of a circle `_ * `tec-science `_