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Gears Gears are among the most important power transmission elements. A gear is a rotating machine element having cut teeth which mesh with another toothed part, usually having teeth of similar size and shape, in order to transmit power. Two or more gears working together are called a “transmission” (or gear set) and can produce “mechanical advantage” and thus may be considered a simple machine. The mechanical advantage is a measure of the force or torque amplification that is obtained using mechanical devices. When two gears mesh with one gear bigger than the other (the size of the teeth must match thus the bigger gear has more teeth), a mechanical advantage is obtained where the rotational speeds and the torques of the two gears will be different. Since the input and output power must be equal (ignoring friction losses), there is an inverse relationship between the speed and torque ratios (the small gear will have higher speed and lower torque and the larger gear will have lower speed and higher torque). A transmission (or gear set) can be used to change the speed, torque, direction of rotation, direction of a power source, or the type of motion. The most common configuration for a gear is to mesh with another gear, however, a gear can also mesh with a non-rotating toothed part, called a “rack”, thereby producing translation instead of rotation, as shown in the figure. Such arrangement is refered to as “rack and pinion” and it is commonly used in the steering systems of automobiles.

Types of Gears There are four principal types of gears:  Spur gears: The simplest type of gears. The teeth are parallel to the axis of rotation, as seen in the figure. It transmits rotation between parallel shafts.

MENG 204 - Mechanical Drawing Gears

Lecture Notes by: Dr. Ala Hijazi Page 1 of 12

 Helical gears: The teeth are inclined with respect to the axis of rotation, as seen in the figure. Same as spur gears, it transmits rotation between parallel shafts, but it is less noisy than spur gears because of the more gradual engagement of the teeth during meshing and thus it is more suitable for transmitting motion at higher speeds. In some cases, helical gears can also be used to transmit rotation between perpendicular shafts, as seen in the figure.  Bevel gears: The teeth are somehow similar to those of a spur gear but they are formed on conical surfaces instead of cylinders. Bevel gears transmit rotation between intersecting shafts. The gear shown in the figure has straight teeth where this is the simplest type. However, there are other types were the teeth form circular arcs and it is called spiral bevel gears, as shown in the figure. With spiral bevel gears, the teeth engagement will be more gradual (similar to helical gears) and thus it is less noisy and it is suitable for higher speeds.  Worms and worm gears: Transmit rotation between perpendicular shafts (not intersecting, there is an offset between them). The worm resembles a screw which can be right handed or left handed. Worm gear sets are usually used when high reduction in speed is desired (speed ratios of 3 or higher). It transmits rotation from the worm to the worm gear, but not the opposite.

Nomenclature Since spur gears are the simplest type, it will be used for illustration and to define the primary parameters of gears and their relations. The figure illustrates the terminology of spur gears.  Pitch circle: the theoretical circle upon which all gear calculations are based and its diameter is called the “pitch diameter”. MENG 204 - Mechanical Drawing Gears

Lecture Notes by: Dr. Ala Hijazi Page 2 of 12

   

o Pitch circles of mating gears are tangent to each other. o The “centers distance” between two mating gears is the sum of the pitch radiuses of the two gears. o The smaller of two mating gears is called the “pinion” and the larger is called the gear. Addendum and Dedendum circles: the circles defining the top and bottom faces of the teeth. Addendum “𝑎”: the radial distance from the pitch circle to the top surface of the teeth. Dedendum “𝑏”: the radial distance from the pitch circle to the bottom surface of the teeth. Clearance circle (or working depth circle): the circle tangent to the addendum circle of the mating gear. The radial distance between the addendum circle and the clearance circle is called the “working depth”.

 Clearance “𝑐 ”: the distance between the tooth top surface and the bottom surface of a mating gear.  Circular pitch “𝑝”: the distance measured on the pitch circle from a point on one tooth to the corresponding point on an adjacent tooth. The circular pitch is equal to the sum of "tooth thickness" and "width of space". The width of space is slightly larger than the tooth thickness such that mating teeth can engage easily without obstruction.  Module “𝑚”: is the ratio of pitch diameter to the number of teeth of a gear. Module

(mm per tooth)

Circular pitch

𝑚= 𝑝=

𝜋𝑑 𝑁

𝑑

Pitch diameter

𝑁

Number of teeth

=𝜋𝑚

 Diametral pitch “𝑃”: the ratio of the number of teeth of a gear to the pitch diameter (it is the inverse of the module and it is used with gears sized in inches; its unit is “teeth per inch” ). 𝑃=

MENG 204 - Mechanical Drawing Gears

𝑁 1 = 𝑚 𝑑

→

𝑝=

𝜋 𝑃 Lecture Notes by: Dr. Ala Hijazi Page 3 of 12

Important Notes:  The module (or the diametral pitch) determines the size of gear teeth.  In order for gears to be able to mesh (work together), they must have the same module (or diametral pitch).  The pitch diameter of a gear (i.e., its size) is determined based on its module (or diametral pitch) and the number of teeth: 𝑑 = 𝑚 𝑁

Spur Gears Teeth Profile Gears need to have constant angular velocity ratio during meshing. This property is important for smooth transmission of power with minimal speed or torque variations as pairs of teeth go into or come out of mesh. The smooth transmission is important for high-speed gearing (it is not very critical for low-speed gearing). In order to obtain a constant angular velocity ratio, gear teeth must have a special type of curved profile (not circular) that is called an "Involute" curve. Also, when the profile of gear teeth are made as involute curves, the mating teeth surfaces will simply roll on each other and there will be no sliding contact.  During the meshing of gears, the contact between a pair of gear teeth occurs at a single instantaneous point. This point of contact (point c) occurs where the two surfaces are tangent to each other, and the forces will be directed along the common normal (line ab) which is also called the “Line of Action” or the “Pressure Line”.  Rotation of the gears causes the location of the contact point (point c) to move across the respective tooth surfaces. For gear teeth having involute profiles, as the point of contact moves, it will move along the pressure line which remains at a constant angle.  The angle between the pressure line and the pitch line is called the “Pressure Angle (∅)”.

MENG 204 - Mechanical Drawing Gears

Lecture Notes by: Dr. Ala Hijazi Page 4 of 12

Drawing Gear Teeth Involute Profile An involute profile can be simply generated using a cord wrapped around a circle, as seen in the figure.  One end of the cord is fixed at a point on the circumference of the circle (point c).  A pin is attached at the other end of the cord (point d).  By moving the pin while the cord is tensioned, an involute profile is generated (profile ab). The same idea of generating an involute profile can be applied graphically (without using a cord) as illustrated by the figure:  The circle from which the involute profile starts is called the Base circle.  A series of radial lines with constant angular increments between them are drawn to points A0, A1, .. , A4 on the circumference of the circle.  Tangent lines are drawn at points A1, .. , A4.  Points B1, .. , B4 are defined along the tangent lines knowing that the length of the line segments A1 - B1, .. , A4 - B4 are as follows:  Length of line A1 - B1 = Length of arc segment between A1 - A0  Length of line A2 - B2 = Length of arc segment between A2 - A0  Length of line A3 - B3 = Length of arc segment between A3 - A0  Length of line A4 - B4 = Length of arc segment between A4 - A0  The involute curve is drawn starting from point A0 and passing through points B1, .. , B4. The radius of a base circle “𝑟𝑏 ” used to generate teeth involute profiles depends on the radius of the pitch circle “𝑟” and the pressure angle “∅” (the angle between the pressure line and the common tangent of the pitch circles of mating gears), as illustrated in the figure. 𝑟𝑏 = 𝑟 cos ∅ MENG 204 - Mechanical Drawing Gears

Lecture Notes by: Dr. Ala Hijazi Page 5 of 12

 Standard values of ∅ are 14.5°, 20° and 25° (20° is the most commonly used) Important Note: mating gears must have the same pressure angle to be able to mesh correctly. A tooth profile starts from the dedendum circle and ends at the addendum circle where the standard values of dedendum and addendum are defined as a function of the module “𝑚” (or the diametral pitch “𝑃”): o Dedendum:

𝑏 = 1.25 𝑚 =

o Addendum:

𝑎=𝑚=

1.25 𝑃

1

𝑃

 If the dedendum circle is smaller than the base circle (this happens when a small pressure angle is used), the remaining portion of a tooth profile, which is not part of the involute profile, is drawn as a tangent arc.

Contact Ratio The contact ratio “ 𝑚𝑐 ” defines the average number of teeth pairs in contact during meshing.  If 𝑚𝑐 = 1 it means that only one pair of teeth is in contact at a time.

 To reduce noise and possibility of impact it is recommended that 𝑚𝑐 ≥ 1.2

 The contact ratio increases with the number of teeth of a gear and for this reason, gears having less than ten teeth are not commonly used.

Interference The contact of portions of teeth profiles that are not involute is called interference.  Interference happens in gear pairs because the dedendum circle is smaller than the base circle and thus the involute portion of teeth profile is small.  Thus, to reduce interference larger pressure angles 20° or 25° are commonly used (i.e., making the base circle smaller).

MENG 204 - Mechanical Drawing Gears

Lecture Notes by: Dr. Ala Hijazi Page 6 of 12

Gears Standards Usually, gears are designed based on standards established by ISO (for metric sizes) or ANSI (for inch sizes).  The standards specify the different parameters used to define the geometry of gears (such as, module, addendum, dedendum, tooth thickness, pressure angle, etc.) and the relationships between these parameters.  The standards were developed to attain interchangeability of gears having the same pressure angle and module (or diametral pitch) regardless of the number of teeth.  The table gives the preferred values of Module and Diametral Pitch according to the ISO and ANSI standards.

 Gear manufacturers make a large variety of standard sized gears and such gears can usually be available as off the shelf components.  When a standard gear is chosen, the gear is usually not shown in full detail in technical drawings, as shown in the figure. Instead, a schematic representation is used, and the gear is defined using its geometric parameters. Detailed drawings of a gear are made when a special gear is designed or when gears in assembly must be shown.  The geometry of a standard spur gear is defined by four parameters; module (or diametral pitch), pressure angle, number of teeth, and face width.  A standard ISO spur gear having a module of 4 mm, 25 teeth, 20° pressure angle, and 30 mm face width is designated as: ISO- Spur gear 4M 25T 20PA 30FW. MENG 204 - Mechanical Drawing Gears

Lecture Notes by: Dr. Ala Hijazi Page 7 of 12

 When the diameter of a gear is large, usually it is not made as a solid gear, instead it is made in the shape of a rim, as seen in the figure, in order to save material and reduce its weight. Also, when the face width of a gear is small compared to its diameter, usually the gear is made with a hub in order to improve its stability (as seen in the figure).

Gear Ratio and Gear Trains The gear ratio “𝑅” (also known as speed ratio) of a gear set is defined as the ratio of the angular velocity of the input gear (driver gear) relative to that of the output gear (driven gear): 𝑅 = 𝑛𝑑𝑟𝑖𝑣𝑒𝑟 ⁄𝑛𝑑𝑟𝑖𝑣𝑒𝑛  The gear ratio is simply equal to the ratio of the number of teeth (or pitch diameters) of the driven gear relative to driver gear: 𝑅=

𝑛𝑑𝑟𝑖𝑣𝑒𝑟 𝑁𝑑𝑟𝑖𝑣𝑒𝑛 𝑑𝑑𝑟𝑖𝑣𝑒𝑛 = = 𝑛𝑑𝑟𝑖𝑣𝑒𝑛 𝑁𝑑𝑟𝑖𝑣𝑒𝑟 𝑑𝑑𝑟𝑖𝑣𝑒𝑟

Consider a pinion “2” driving a gear “3”, the speed of the gear is: 𝑛

𝑁

𝑛3 = | 2 | = | 2 𝑛2 | 𝑁 𝑅 3

The absolute-value sign is to give freedom in choosing positive or negative directions. A Gear Train is a gear set that consists of more than two gears. In general, there can be several driving gears (input) and several driven gears (output).  Any gear that is not giving any input torque nor taking any output torque is called an “idler” gear.  For a gear train, the gear ratio is found as: 𝑅=

𝑃𝑟𝑜𝑑𝑢𝑐𝑡 𝑜𝑓 𝑑𝑟𝑖𝑣𝑒𝑛 𝑔𝑒𝑎𝑟𝑠 𝑡𝑜𝑜𝑡ℎ 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 𝑃𝑟𝑜𝑑𝑢𝑐𝑡 𝑜𝑓 𝑑𝑟𝑖𝑣𝑒𝑟 𝑔𝑒𝑎𝑟𝑠 𝑡𝑜𝑜𝑡ℎ 𝑛𝑢𝑚𝑏𝑒𝑟𝑠

For the gear train shown in the figure, pinion “2” is the input gear and gear “6” is the output gear (other gears are idlers), the gear ratio is found to be: MENG 204 - Mechanical Drawing Gears

Lecture Notes by: Dr. Ala Hijazi Page 8 of 12

𝑁3 𝑁4 𝑁6 𝑅 =𝑁 𝑁 𝑁 2 3 5  Note that gears 2, 3 & 5 are drivers while gears 3, 4 & 6 are driven And the speed of gear “6” is: The sign is decided manually.

𝑛6 = − Planetary gear trains

𝑛2 𝑅

In this type of trains, the axes of some of the gears rotate about other gears, as seen in the figure .  Planetary trains include three main components: a sun gear, a planet carrier with one or more planet gears on it, and a ring gear.  Planetary trains have two or more degrees of freedom and thus have two or more inputs.  Planetary gear trains are commonly used for obtaining different gear ratios by fixing one of the elements (the sun, the ring, or the planet carrier) and using one of the remaining two elements as an input and the other as an output.

Parallel Helical Gears Helical gears are mostly used to transmit motion between parallel shafts.  The teeth are inclined with respect to the axis of rotation and the angle of inclination of teeth is called the “Helix Angle” . The most commonly used value of the helix angle is 30°.  The shape of the teeth is an involute helicoid, and it can be produced by wrapping a piece of paper shaped as parallelogram on the base cylinder.  Contact of the teeth starts as a point then extends into a diagonal line across the face of the tooth as teeth come into more engagement.  Because of the gradual engagement of the teeth, helical gears can transmit heavy loads at high speeds.  A helical gear can have a right-hand or a lefthanded helix, as shown in the figure.  In order for two helical gears to mesh, both gears must have the same helix angle, but one gear MENG 204 - Mechanical Drawing Gears

Lecture Notes by: Dr. Ala Hijazi Page 9 of 12

must have a right-hand helix and the other a left-hand helix, as seen in the figure.  Unlike spur gears which produce only radial load, a helical gear produces both radial and thrust loads on the shaft supporting it.  It is possible to cancel the resulting thrust loading by using two opposite-hand helical gears mounted side by side on each shaft. 

Another way to eliminate thrust load is to use “Herringbone gears”. A herringbone gear is a special type of double helical gear that combines two helical gears of opposite hands, as seen in the figure.

 Because of the inclination of the teeth, the module, the pressure angle, and the circular pitch, can be defined in two different directions; the normal direction (normal to the teeth direction) and the transverse direction (parallel to the gear face), as seen in the figure. 

The figure illustrates the nomenclature of helical gears:  Helix angle: 𝜓  Transverse circular pitch: 𝑝𝑡  Normal circular pitch: 𝑝𝑛 𝑝𝑛 = 𝑝𝑡 cos 𝜓 o The same relation applies to the module in the normal and transverse directions: 𝑚𝑛 = 𝑚𝑡 cos 𝜓  Normal and transverse pressure angles 𝜙𝑛 & 𝜙𝑡 are related to the helix angle as: cos 𝜓 =

tan 𝜙𝑛 tan 𝜙𝑡

 According to standards, the values of the module and pressure angle are specified in the normal direction, while, of course, the module in the transverse direction is used for determining the diameter of the pitch circle (𝑑 = 𝑚𝑡 𝑁).  A helical gear is designated by its hand, module (normal), number of teeth, helix angle, pressure angle (normal), and face width. Example: RH Helical gear 4M 25T 30HA 20PA 30FW o If we have a spur gear and a helical gear with the same module and number of teeth, would the diameters of the two gears be equal? MENG 204 - Mechanical Drawing Gears

Lecture Notes by: Dr. Ala Hijazi Page 10 of 12

Straight Bevel Gears Bevel gears are used to transmit motion between intersecting shafts. The shafts usually make 90° angle with each other but also other angles are possible.  Similar to helical gears, bevel gears produce both radial and thrust loads on the shaft.  Because bevel gears are cut on conical surfaces (not cylindrical surfaces as spur and helical gears), they have a large end and a small end.  The figure illustrates the terminology of bevel gears:  The pitch diameter is measured at the large end of the tooth.  The module and circular pitch are calculated the same as in spur gears.  the face width is measured along the conical surface. For considerations related to teeth deflection, the face width of a bevel gear is usually kept small relative to its diameter.  The pitch angles are defined as shown in the figure and they are related to the number of teeth as follows: Pinion:

tan 𝛾 =

𝑁𝑝

𝑁𝐺

Gear:

tan Γ =

𝑁𝐺 𝑁𝑃

Important Note: The pitch angle for the pinion or the gear depend on the number of teeth of both the pinion and the gear. Thus, unlike spur and helical gears, in order to determine the geometry for any bevel gear, the number of teeth of the mating gear needs to be known.  Since the face width of bevel gears is kept relatively small, bevel gears usually come with a hub to give better stability. Also, the hub helps in controlling the mounting distance between the gear (or pinion) and the mating element.  The Mounting Distance is the distance from the back surface of the gear (or pinion) to the centerline of the mating element, as seen in the figure above.  A straight bevel gear is designated by its, module, number of teeth, number of teeth of the pinion (or mating gear), pressure angle, and face width. MENG 204 - Mechanical Drawing Gears

Lecture Notes by: Dr. Ala Hijazi Page 11 of 12

Example: Straight Bevel gear 4M 25GT 15PT 20PA 18FW  Because bevel gears are used on intersecting shafts, it is extremely difficult to have both gears mounted between two bearings (straddle mounted). In most cases, one of the two gears have to be mounted outboard of the bearings (outboard mounted), as shown in the figure.

Worm Gears The work principle of a worm and a worm gear is somehow similar to that of a power screw moving a nut. The worm is similar to a screw and as the worm rotates it bushes the teeth of the mating worm gear causing it to rotate.  Unlike other types of gears, in a worm gear set, the input shaft must be connected to the worm (not the worm gear) since the worm gears are selflocking (the worm gear cannot rotate the worm).  The friction losses in worm ge...