Modeling Spur and Helical Gears in CATIA

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MIET 2136 Mechanical Design I

RMIT University - Thierry Perret-Ellena – mail: [email protected]

In order for the fundamental law of gearing to be true (“the angular velocity ratio between the gears of a gearset must remain constant throughout the mesh”), the gear tooth contours on mating teeth must be conjugates of one another. There is an infinite number of possible conjugate pairs that could be used, but only a few curves have seen practical application as gear teeth. The cycloid is still used as tooth form in some watches and clocks, but most gears use the involute of a circle for their shape.

The involute of a circle is a curve that can be generated by unwrapping a taut string from a cylinder. The cylinders from which the strings are unwrapped are called the base circles (𝒅𝒃𝒄) and are smaller than the pitch circles (𝒅), which are the diameters of the original rolling cylinders. The gear tooth must project both below and above the rolling-cylinder surface (pitch circle), and the involute only exists outside of the base circle. - 𝒅𝒃𝒄 = 𝒅cosϕ, where ϕ is the gearset pressure angle - 𝒅 = 𝒎𝑵, where 𝑵 represents the number of teeth and 𝒎 is called the module and define the tooth size.

2 RMIT University - Thierry Perret-Ellena – mail: [email protected]

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Application I want to draw a gearset with the following properties:

- Velocity ratio: 𝒎𝑽 =𝑵𝒑𝒊𝒏𝒊𝒐𝒏

𝑵𝒈𝒆𝒂𝒓 = 𝟐𝟏

𝟓𝟓 = 𝟎. 𝟑𝟖

- Module: 𝒎 = 𝟐𝒎𝒎 - Pitch Circle Diameter pinion and gear: 𝒅𝒑 = 𝒎𝑵𝒑𝒊𝒏𝒊𝒐𝒏 = 𝟐 ∗ 𝟐𝟏 = 𝟒𝟐𝒎𝒎, 𝒅𝒈 = 𝟏𝟏𝟎𝒎𝒎

- Base Circle Diameter pinion and gear: 𝒅𝒃𝒄𝒑 = 𝒅𝒑 cosϕ = 𝟒𝟐 ∗ cos 𝟐𝟎 = 𝟑𝟗. 𝟒𝟕𝒎𝒎, 𝒅𝒃𝒄𝒈 = 𝟏𝟎𝟑. 𝟑𝟕𝒎𝒎

CATIA - Click File/New/Part, rename it “Gearset 1 – Pinion”, Deselect “Enable hybrid design” and Click Ok - Select Sketch Icon and click the desired reference plane

- In Sketch tools toolbar, deselect “Snap to point” option

- Using Circles tool, draw 2 circles coincident with the point reference system

- Click Constraint icon and modify both diameters to the pitch and base circles diameter values


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- Click the axis icon and draw 5 lines as follow. The first should be vertical and they should all be coincident with the base circle diameter.

- Using constraint tool, create angle dimensions between each line => 10deg. The angle value is independent to the number of teeth and the circle diameters. The smaller the angle is, the better the spline curvature will be.

- For gear with greater diameters and modules, you may need more than 5 lines to create an involute that project above the outside diameter.

- Keep in mind that your sketch should always be fully constraint. i.e. the elements (lines, circles, points, curves) should be green.


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- In order to draw the involute, we need strings tangent to the base circle. A tangent to the involute is always normal to the string, which is the instantaneous radius of curvature of the involute curve.

- Select the axis tool and click on the axis end point as starting point. - Select both the line and the base circle with Ctrl and click “Constraints defined in Dialog Box” - Tick on “tangency” and click ok to complete - Click Constraint icon and give a length to the string

- The string’s length is equal to the arc length over this section. 𝒍 = 𝝅𝒅𝒃𝒄𝟑𝟔𝟎

𝑺𝒆𝒄𝒕𝒊𝒐𝒏 𝒂𝒏𝒈𝒍𝒆 = 𝟑. 𝟒𝟒𝟓𝒎𝒎


- Repeat the steps described above for the next 3 strings.

- The string’s lengths are respectively equal to 2*3.445, 3*3.445 and 4*3.445

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- Click The Spline icon and select the strings end points. You should double click the last point to end the curve selection.

- The curve should be green (fully constraint). If not, you probably missed one or more end points during the selection. You can solve this problem using constraint tools.


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- The tooth height is defined by the addendum (added on) and the dedendum (subtracted from), which are referenced to the normal pitch circle. The dedendum is slightly larger than the addendum to provide a small amount of clearance between the tip of one mating tooth (addendum circle/outside diameter) and the bottom of the tooth space of the other (dedendum circle/inside diameter).

- Standard, full-depth gear teeth have equal addenda on pinion and gear. The tooth dimensions are defined in terms of the module m.

Interference - The involute tooth form is only defined outside of the base circle. In some cases, the dedendum will be

large enough to extend the base circle. If so, then the portion of tooth below the base circle will not be an involute and may interfere with the tip of tooth on the mating gear, which is an involute.

- Interference can be prevented simply be avoiding gears with too few teeth. If a pinion has a larger number of teeth, they will be small compared to its diameter. As the number of teeth is reduced for a fixed diameter pinion, the teeth must become larger. At some point, the dedendum will exceed the radial distance between the base circle and the pitch circle, and interference will occur.

- The opposite table shows the minimum number of full-depth pinion teeth that can be used against a selection of full-depth gears of various sizes (for ϕ = 𝟐𝟎𝒅𝒆𝒈) to avoid interference.

Unequal-Addendum tooth Forms - In order to avoid interference on small pinions, the tooth form can be changed from the standard, full-

depth shapes that have equal addenda on both pinion and gear to an involute shape with a longer addendum on the pinion and a shorter one on the gear. These are called profile-shifted gears. The net effect is to shift the pitch circles away from the pinion’s base circle and eliminate that noninvolute portion of pinion tooth below the base circle. The limit of this approach occurs when the pinion tooth becomes pointed.

Specifications

Full-Depth Gearset 25% Long-Addendum Teeth Addendum: 𝒂𝒑𝒊𝒏𝒊𝒐𝒏 = 𝒂𝒈𝒆𝒂𝒓 = 𝒎 Addendum: 𝒂𝒑𝒊𝒏𝒊𝒐𝒏 = 𝟏. 𝟐𝟓𝒎; 𝒂𝒈𝒆𝒂𝒓 = 𝟎. 𝟕𝟓𝒎

Dedendum: 𝒃𝒑𝒊𝒏𝒊𝒐𝒏 = 𝒃𝒈𝒆𝒂𝒓 = 𝟏. 𝟐𝟓𝒎 Dedendum: 𝒃𝒑𝒊𝒏𝒊𝒐𝒏 = 𝒎; 𝒂𝒈𝒆𝒂𝒓 = 𝟏. 𝟓𝒎


Minimum Pinion Teeth

Maximum Gear Teeth

17 1309

16 101

15 45

14 26

13 16


Application - The number of teeth for my pinion is 21, thus no interference are expected and I can draw a standard, full-depth gearset. - 𝒂 = 𝟐𝒎𝒎, 𝒃 = 𝟐. 𝟓𝒎𝒎

CATIA - Draw 2 circles concentric with pitch and base diameters and create 2 offset dimensions using Constraint tool.


- The dedendum extends below the base circle and a line should be created to link the tooth profile to the root diameter.

- Draw a line coincident with the beginning of the curve and on one end and the dedendum on the other.

- Click Constraint icon

- Select the line and the curve

- Right Click and select tangency

- The line should become green (fully constraint)


- Backlash is defined as the gap between mating teeth measured along the circumference of the pitch circle. Manufacturing tolerances preclude a zero backlash, as all teeth cannot be exactly the same dimensions, and all must mesh without jamming. So, there must be some small difference between the tooth thickness and the width of space.

- We have:

𝒄𝒊𝒓𝒄𝒖𝒍𝒂𝒓 𝒑𝒊𝒕𝒄𝒉 𝒑 = 𝒕𝒐𝒐𝒕𝒉 𝒕𝒉𝒊𝒄𝒌𝒏𝒆𝒔𝒔 𝒕 + 𝒘𝒊𝒅𝒕𝒉 𝒐𝒇 𝒔𝒑𝒂𝒄𝒆 𝒘 = 𝝅𝒅

𝑵

< = > 𝒄𝒊𝒓𝒄𝒖𝒍𝒂𝒓 𝒑𝒊𝒕𝒄𝒉 𝒑 = 𝟐 ∗ 𝒕𝒐𝒐𝒕𝒉 𝒕𝒉𝒊𝒄𝒌𝒏𝒆𝒔𝒔 (𝒕) + 𝒃𝒂𝒄𝒌𝒍𝒂𝒔𝒉 =𝝅𝒅

𝑵

< = > 𝒕𝒐𝒐𝒕𝒉 𝒕𝒉𝒊𝒄𝒌𝒏𝒆𝒔𝒔 𝒕 =𝝅𝒅

𝟐𝑵− 𝒃𝒂𝒄𝒌𝒍𝒂𝒔𝒉

Application - For my pinion, I have selected a backlash of 0.15mm. Hence:

𝒕𝒐𝒐𝒕𝒉 𝒕𝒉𝒊𝒄𝒌𝒏𝒆𝒔𝒔 𝒕 =𝝅𝒅

𝟐𝑵− 𝒃𝒂𝒄𝒌𝒍𝒂𝒔𝒉 =

𝝅∗𝟒𝟐

𝟐∗𝟐𝟏− 𝟎. 𝟏𝟓 = 𝟐. 𝟗𝟗𝒎𝒎 (arc length measuring at pitch circle

diameter) - Unfortunately, I can’t constrain arc length on CATIA sketch workbench, so I will use the tooth angle at pitch

circle instead.

- The length of an arc of a circle is: 𝑨𝒓𝒄 𝒍𝒆𝒏𝒈𝒕𝒉 =𝜶𝝅𝒅

𝟑𝟔𝟎

- The tooth angle at pitch circle is then: 𝜶 =𝟑𝟔𝟎𝒕

𝝅𝒅=

𝟑𝟔𝟎∗𝟐.𝟗𝟗

𝝅∗𝟒𝟐= 𝟖. 𝟏𝟔𝒅𝒆𝒈


CATIA - Click the Axis icon

- The tooth angle is measured at pitch circle diameter, so you need to draw an axis starting from the intersection of the involute tooth form and the pitch circle.

The other end is the system reference center.

Pitch Circle

Involute form

Intersection point

Axis


- We need to draw another axis for the tooth profile symmetry

- Click the Axis icon

- Select the system center point, press Shift button and click the end point somewhere on the sketch top section. (The shift button will make unavailable the automatic constraint creation)

- Select the Constraint icon and select both axis. Change the value to your tooth angle at pitch circle divided by 2. Here 8.16/2=4.08deg


- Select both the involute curve and the line with Ctrl

- Click the symmetry icon

- Select the mirror axis


- The Pitch and Base circle diameters should be kept in your sketch and used as construction elements. They help you to fully constrain your sketch. If you delete them, you sketch would become underc-onstained as you would loose most of your parametric dimensions.

- Select both circle using Ctrl and swith their properties from “standard” to “Construction” elements using this icon available in sketch tool toolbox.

- Double Click the Quick Trim icon

- Trim all the geometric elements you don’t need to create a closed profile. (Top end of involute, Addendum, dedendum)

- Exit the sketch


1) Click the Pad icon , select sketch.1 and enter your Face Width value into length option.

2) Select the Edge Fillet icon , select the 2 intersection lines between the tooth profile and the root diameter, and change the fillet value to 0.3*m. (0.6mm in this example)

3) Select the circular pattern icon , change Parameters to “complete crown”, instances is your number of teeth and right click on reference element and select “x axis” for the pattern orientation.


- Click the Plane icon

- Select an “Offset from plane” type and yz plane as reference. The offset value is your Face width.

- Click the sketch icon and select the newly created plane as reference. You enter the sketch design workbench.

- Select all the elements from sketch.1 using Ctrl and hit the “Project 3d elements” option. These projections are yellow. You cannot move these elements.


- Multi select the yellow elements using Ctrl

- Click the Rotate tool, select the system reference point as rotation center and change the angle value to your rotation angle φ.

- The rotation angle φ corresponds to a fraction of the helical lead 𝑳:

𝝋 = 𝟑𝟔𝟎𝒅𝒆𝒈 ∗𝒇𝒂𝒄𝒆 𝒘𝒊𝒅𝒕𝒉

𝑳, with 𝑳 =

𝝅𝒅

tan ψ, Ψ being the helix angle.

- Mutli select again the yellow elements and change their properties to construction element

- Exit the sketch

Application My helix angle is 15deg and my rotation angles for pinion and gear are:

𝝋𝒑𝒊𝒏𝒊𝒐𝒏 =𝟑𝟔𝟎 ∗ 𝒇𝒂𝒄𝒆 𝒘𝒊𝒅𝒕𝒉 ∗ tanψ

𝝅𝒅=𝟑𝟔𝟎 ∗ 𝟐𝟎 ∗ tan(𝟏𝟓𝒅𝒆𝒈)

𝟒𝟐𝝅= 𝟏𝟒. 𝟔𝟐𝒅𝒆𝒈

𝝋𝒈𝒆𝒂𝒓 =𝟑𝟔𝟎 ∗ 𝒇𝒂𝒄𝒆 𝒘𝒊𝒅𝒕𝒉 ∗ tanψ

𝝅𝒅=𝟑𝟔𝟎 ∗ 𝟐𝟎 ∗ tan(𝟏𝟓𝒅𝒆𝒈)

𝟏𝟏𝟎𝝅= 𝟓. 𝟓𝟗𝒅𝒆𝒈


- Click the Multi-Sections Solid icon

- Select both sketches and be sure to get the same closing point and direction on each section. You can change a closing point position if your right click on it in the workspace or toolbar and select “Replace”

- Click ok to complete

- Repeat step 2) and 3) describe on page 15


- Save your pinion and create a new product (File/New/Product)

- If you are not yet in assembly design workbench, click Start/Mechanical Design/Assembly Design

- Right click product1/Components/Existing component… and load the pinion in the assembly.

- Using “Fix component” icon in constraints toolbox, fix the pinion in space.

- Create a new part in assembly design


- Create a new part in assembly design. Right Click Product1/Components/New Part

- Click Yes in the pop up window.

- Your assembly contains now 2 parts. Part1 being empty in the moment.

- Expand the Part1 under the specification tree and double click “part1”. You just enter the mechanical design workbench and will be able to create/modify the part.

- The blue rectangle means that your are working on part1. You can come back to assembly design workbench by double clicking Product1

- Right Click Part1/Properties/Product and change the Part Number to “Gearset 1 – Gear”.

- “Save as” the new part


- Select yz plane and hit the sketch icon

- Draw 2 circles tangent to each other. They represent the pitch circle diameter of the pinion and the gear.


- Using the same process described above, draw the base circle, the involute tooth form of the gear and the addendum and dedendum. The only difference is that your first axis line shouldn't be vertical. We are going to constrain the gear’s involute profile tangent to the pinion’s involute profile.

Not vertical

- Once the involute is modelled, click the constraint icon, select the curve and the pinion profile, right click and select Tangency


- As you can notice, the base circle is smaller than the dedendum.


- Exit the sketch and finish the gear design as described previously.

Spur Gear Helical Gear

Modeling Spur and Helical Gears in CATIA

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