Gears are toothed mechanical components that can engage with each other, and their application is extremely widespread in mechanical transmission and the entire field of machinery. After reading today’s article, you will have a complete understanding of the past and present of gears.
The History of Gears
As early as 350 BC, the famous Greek philosopher Aristotle documented gears in his writings. Around 250 BC, mathematician Archimedes also described the use of a winch with a worm gear in his works. Gears from the same period have been preserved in the ruins of Ctesiphon, located in present-day Iraq.
The history of gears in China is also long and rich. According to historical records, as early as 400–200 BC, gears were already in use in ancient China. The bronze gear unearthed in Shanxi, China, is the oldest gear discovered to date. The guide carriage, a mechanical device reflecting ancient scientific achievements, used a gear mechanism at its core. In the late 15th century during the Italian Renaissance, the famous polymath Leonardo da Vinci made indelible contributions to the history of gear technology, in addition to his achievements in art and culture. More than 500 years later, modern gears still retain the prototypes sketched by him.
It wasn’t until the late 17th century that people began studying the correct tooth shapes for accurately transmitting motion.
After the European Industrial Revolution in the 18th century, the application of gear transmission became increasingly widespread. First, cycloidal gears were developed, followed by involute gears. By the early 20th century, involute gears had become dominant in applications. Afterward, new types such as modified gears, arc gears, bevel gears, and helical gears were developed.
Modern gear technology has achieved the following advancements: gear module sizes ranging from 0.004 to 100 millimeters; gear diameters from 1 millimeter to 150 meters; power transmission capabilities reaching up to 100,000 kilowatts; speeds up to 100,000 revolutions per minute; and maximum circumferential speeds of 300 meters per second.
Globally, power transmission gear systems are advancing toward miniaturization, high-speed, and standardization. Notable trends in gear design include the application of special gears, the development of planetary gear systems, and the creation of low-vibration, low-noise gear systems.
Types of Gears
There are many types of gears, and the most common method of classification is based on the orientation of the gear shafts. Generally, they are divided into three categories: parallel shafts, intersecting shafts, and skewed shafts.
There are many types of gears, and the most common method of classification is based on the orientation of the gear shafts. Generally, they are divided into three categories: Parallel axes, Intersecting axes, Nonparallel, nonintersecting axes.
- Parallel axes: This category includes spur gears, helical gears, internal gears, racks, and helical racks.
- Intersecting axes: These include straight bevel gears, spiral bevel gears, and zerol bevel gears.
- Nonparallel, nonintersecting axes: Examples are crossed helical gears, worm gears, and hypoid gears.
The efficiency listed in the table refers to transmission efficiency and does not account for losses from bearings or lubricating agitation. The meshing of gear pairs in parallel and intersecting shafts is primarily through rolling, with minimal relative sliding, resulting in high efficiency. In contrast, skewed shaft gears, such as crossed helical gears and worm gears, rely on relative sliding to produce rotation for power transmission. This results in significant friction, which lowers transmission efficiency compared to other gears. The efficiency of gears refers to their transmission efficiency under normal assembly conditions. If there are assembly errors, especially if the bevel gears are incorrectly spaced, causing a misalignment of the cone apexes, the efficiency will decrease significantly.
01 Parallel axes
1) Spur Gear
A spur gear is a cylindrical gear with teeth that are parallel to the axis of rotation. Due to its ease of manufacturing, it is the most widely used gear type in power transmission applications.
2) Rack
A rack is a linear gear that meshes with a spur gear. It can be considered a special case of a spur gear where the pitch circle diameter becomes infinitely large.
3) Internal Gear
An internal gear is a gear that has teeth machined on the inner side of a circular ring and meshes with a spur gear. It is primarily used in applications such as planetary gear transmission mechanisms and gear couplings.
4) Helical Gear
A helical gear is a cylindrical gear with teeth that are cut in a spiral pattern. Due to its higher strength and smoother operation compared to spur gears, it is widely used in various applications. However, helical gears generate axial thrust during transmission.
5) Helical Rack
A helical rack is a linear gear that meshes with a helical gear. It can be considered as the situation where the pitch diameter of the helical gear becomes infinitely large.
6) Herringbone gear
A gear composed of two helical gears with left-handed and right-handed tooth lines. It has the advantage of not generating thrust in the axial direction.
02 Intersecting Shaft Gears
1) Straight Bevel Gear
A straight bevel gear is a type of bevel gear where the tooth lines align with the generatrix of the pitch cone. Among bevel gears, this type is relatively easy to manufacture, making it widely used in various power transmission applications.
2) Spiral Bevel Gear
An spiral bevel gear is a type of bevel gear with curved tooth lines that have a helical angle. Although it is more difficult to manufacture compared to straight bevel gears, it is widely used due to its high strength and low noise characteristics.
3) Zerol Bevel Gear
A zerol bevel gear is a type of bevel gear with a zero-degree helical angle. It combines the characteristics of both straight and curved bevel gears, with the tooth surfaces experiencing similar loading conditions as those of straight bevel gears.
03 Skewed Shaft Gears
1) Cylindrical Worm Gear Pair
A cylindrical worm gear pair refers to the combination of a cylindrical worm and its corresponding worm wheel. One of its key features is the ability to achieve a large gear ratio with smooth operation using just a single pair of gears. However, it has the drawback of lower efficiency compared to other gear types.
2)Crossed Helical Gears
Crossed helical gears refer to a gear system used for power transmission between skewed shafts. This configuration can be applied in pairs of helical gears or in combinations of helical and spur gears. While operation is smooth, crossed helical gears are best suited for light-load applications due to their limited capacity to handle heavier loads.
04 Other Special Gears
1) Face Gear
A face gear is a disc-shaped gear that can mesh with either spur gears or helical gears. It is used for power transmission between intersecting or skewed shafts.
2) Drum Worm Gear Pair
A drum worm gear pair refers to the combination of a drum-shaped worm and its corresponding worm wheel. Although it is more challenging to manufacture, this gear type can handle larger loads compared to cylindrical worm gear pairs.
3) Hypoid Gear
A hypoid gear is a conical gear used for power transmission between skewed shafts. The larger gear is machined with an eccentric offset, making it similar to an arc bevel gear, but the meshing principle is quite complex. Hypoid gears are known for their smooth operation and ability to handle high loads while providing a compact design.
Gear Terminology
Gears have many specific terms and expressions unique to their design and function. To help everyone better understand gears, here are some commonly used basic terms related to gears.
1) Names of Gear Components
2) The Term That Represents the Size of Gear Teeth is the Module
The module is a universally recognized term denoted by symbols such as m1, m3, m8, which refer to module sizes of 1, 3, and 8, respectively. The module (m) indicates the size of the gear teeth in millimeters; the larger the number, the larger the teeth.
Additionally, in countries using imperial units (such as the United States), the size of gear teeth is represented using the symbol (Diametral Pitch, DP) along with a number that indicates the number of teeth on a gear with a pitch circle diameter of 1 inch, such as DP24, DP8, etc. There is also a relatively unique method of representing gear tooth sizes using the symbol (Circular Pitch, CP) and a number (in millimeters), for example, CP5, CP10.
The pitch (p), which is the length between two adjacent teeth, can be calculated by multiplying the module by π (pi). The formula is expressed as:
p=π×module=πmp = \pi \times \text{module} = \pi mp=π×module=πm
Comparing the sizes of gear teeth with different modules:
3) Pressure Angle
The pressure angle is a parameter that determines the tooth profile of a gear, specifically the angle of inclination of the tooth surface. The pressure angle (α) is typically set at 20°. Historically, gears with a pressure angle of 14.5° were quite common.
The choice of pressure angle affects the gear’s strength, tooth shape, and the smoothness of operation. A larger pressure angle generally results in stronger teeth and greater load-carrying capacity, but may also increase the sliding between meshing gears, which can affect efficiency.
The pressure angle is the angle formed between the radius line at a point on the tooth surface (commonly referred to as the pitch point) and the tangent to the tooth profile. In the diagram, α represents the pressure angle. Since α’ = α, it indicates that α’ is also a pressure angle.
This angle is crucial because it influences the tooth shape and the force transmission characteristics of the gear. A proper understanding of the pressure angle helps in designing gears that meet specific load and performance requirements.
When observing the meshing condition of gear A and gear B from the pitch point:
Gear A exerts a force on point B. The force at this moment acts along the common normal line shared by both gears. This means that the common normal line is the direction of force application and also the direction in which pressure is exerted, with α representing the pressure angle.
The module (m), pressure angle (α), and the number of teeth (z) are the three fundamental parameters of gears. These parameters serve as the basis for calculating the dimensions of various components of the gear.
4) Tooth Depth and Tooth Thickness
The depth of the gear tooth is determined by the module (m). The module serves as a fundamental measurement that influences the overall dimensions of the gear teeth.
The full tooth height hhh is calculated as:
h=2.25m(which equals the sum of the root height and the addendum height)h = 2.25m \quad (\text{which equals the sum of the root height and the addendum height})
h=2.25m(which equals the sum of the root height and the addendum height)
- Addendum Height (hah_aha): This is the height from the tooth tip to the pitch circle, defined as:ha=1m ha=1mh_a = 1m
- Root Height (hfh_fhf): This is the height from the tooth root to the pitch circle, defined as:hf=1.25m hf=1.25mh_f = 1.25m
- Tooth Thickness (sss): The reference for tooth thickness is half of the tooth pitch. It can be calculated as:s=2πm s=πm2s = \frac{\pi m}{2}
These dimensions are crucial for ensuring that gears mesh properly and function efficiently within their applications.
5) Gear Diameter
The parameter that determines the size of a gear is the pitch circle diameter (ddd). The pitch circle serves as the reference from which the tooth pitch, tooth thickness, tooth height, addendum height, and root height can be defined.
- Pitch Circle Diameter:d=zm d=zmd = zm Where zzz is the number of teeth and mmm is the module.
- Addendum Circle Diameter (dad_ada):da=d+2m da=d+2md_a = d + 2m
- Root Circle Diameter (dfd_fdf):df=d−2.5m df=d−2.5md_f = d – 2.5m
The pitch circle is not directly visible in actual gears, as it is an imaginary circle used to determine the size of the gear. Understanding these diameters is essential for the design and proper functioning of gear systems.
6) Center Distance and Tooth Clearance
When a pair of gears mesh at their pitch circles, the center distance is defined as half the sum of the diameters of the two pitch circles.
- Center Distance: a = (d1 + d2) / 2
In gear meshing, achieving a smooth engagement is essential, and tooth clearance is a crucial factor. Tooth clearance refers to the gap between the tooth faces of a pair of meshing gears.
Additionally, there is a gap in the direction of the tooth height, known as the Clearance. The clearance (c) is the difference between the root height of one gear and the addendum height of the mating gear.
- Clearance: c=1.25m−1m=0.25m
This clearance ensures that there is sufficient space for the gears to rotate without interference, contributing to the effective operation of the gear system.
7) Helical Gears
Helical gears are created by twisting the teeth of spur gears into a spiral shape. Most of the geometric calculations applicable to spur gears can also be applied to helical gears. Helical gears can be classified based on their reference surfaces into two types:
- End Face (Axial Right Angle) Reference: This refers to the calculations based on the modulus and pressure angle taken from the end face of the gear.
- Normal Face (Tooth Right Angle) Reference: This is based on the normal modulus and pressure angle taken from the tooth face.
The relationship between the end face modulus (mt) and the normal modulus (mn) is given by:
mt=mn/cosβ
8) Spiral Direction and Pairing
Gears with helical teeth, such as helical gears and hypoid gears, have a specific direction of helix and pairing. The spiral direction refers to the orientation of the gear teeth when viewing the gear from the front, with the central axis pointing up and down. In this context:
If the teeth of the gear point to the upper right, it is termed [Right-Hand Spiral]. If the teeth point to the upper left, it is termed [Left-Hand Spiral]. Proper pairing of gears is crucial for effective meshing and transmission of power. Below are examples of how various gears pair according to their spiral direction:
Right-Hand Helical Gear with Right-Hand Helical Gear: This pairing will operate smoothly. Right-Hand Helical Gear with Left-Hand Helical Gear: This pairing is not suitable as they will not mesh correctly. Left-Hand Helical Gear with Left-Hand Helical Gear: This pairing will operate smoothly as well.
Common Involute Tooth Profiles
When teeth are simply divided into equal pitches along the outer circumference of a friction wheel, and then protrusions are added for meshing, several issues can arise:
- Sliding at the Contact Point: The teeth may slide against each other at the contact point rather than rolling smoothly, leading to increased wear.
- Variable Velocity at the Contact Point: The speed of the moving contact point fluctuates, which can disrupt the smooth operation of the gear system and lead to inefficiencies.
- Vibration and Noise: The aforementioned sliding and variable velocities contribute to vibrations and noise during operation, negatively affecting the performance and lifespan of the machinery.
To mitigate these issues, the involute tooth profile is commonly used in gear design. This profile ensures that the gears mesh smoothly with minimal sliding, maintaining a constant velocity ratio, which results in reduced vibration and noise, ultimately leading to improved efficiency and reliability in mechanical systems.
1) What is an Involute?
An involute is formed by tying a pencil to one end of a string wrapped around the outer circumference of a cylinder. As the string is gradually unwound while keeping it taut, the curve traced by the pencil is the involute curve. The outer circumference of the cylinder is referred to as the base circle.
2) Example of an 8-Tooth Involute Gear
After dividing the cylinder into 8 equal parts, tie 8 pencils to it and draw 8 involute curves. Then, wrap the string in the opposite direction and draw another 8 curves using the same method. This results in a gear with an involute profile and 8 teeth.
3) Advantages of Involute Gears
- They can mesh correctly even with some errors in center distance.
- It is relatively easy to achieve the correct tooth profile, and they are easier to manufacture.
- They roll on the curve during meshing, allowing for smooth transmission of rotational motion.
- As long as the tooth sizes are the same, a single cutter can process gears with different numbers of teeth.
- The roots of the teeth are robust, providing high strength.
4) Base Circle and Pitch Circle
The base circle is the fundamental circle that forms the involute tooth profile. The pitch circle is the reference circle that determines the size of the gear. Both the base circle and the pitch circle are crucial geometric dimensions of a gear. The involute tooth profile is formed on the exterior of the base circle, where the pressure angle is zero degrees.
5) Involute Gear Meshing
Two standard involute gears mesh tangentially at the pitch circles under a standard center distance.
When the two gears are meshing, it appears as though they are friction wheels with pitch circle diameters of d1 and d2 transmitting motion. However, in reality, the meshing of involute gears depends on the base circle rather than the pitch circle.
The contact points of the tooth profiles of the two gears move along the meshing line in the order of P1—P2—P3. Note the yellow tooth in the driving gear. For a period of time after this tooth begins to mesh, the gears are in a two-tooth engagement (P1, P3). As the meshing continues and the contact point moves to the point P2 on the pitch circle, there is only one engaged tooth remaining. The meshing continues, and when the contact point moves to point P3, the next tooth starts to engage at point P1, once again forming a two-tooth engagement state. In this way, the two-tooth engagement and single-tooth engagement of the gears alternately repeat the transmission of rotational motion.
The common tangent A-B of the base circles is called the meshing line. The contact points of the gears are all located on this meshing line.
Use a visual diagram to represent it, just like a belt crossing around the outer circumference of two base circles to transmit power through rotational motion.
Gear Modification
Gear modification is divided into positive modification and negative modification. The tooth profile of the gears we usually use is typically a standard involute, but there are situations where gear teeth need to be modified, such as to adjust center distance or to prevent undercutting in small gears.
1) Number of Gear Teeth and Shape
The involute tooth profile curve varies with the number of teeth. The more teeth a gear has, the closer the tooth profile curve is to a straight line. As the number of teeth increases, the tooth root becomes thicker, thereby increasing the strength of the gear teeth.
From the above diagram, it can be observed that for a gear with 10 teeth, part of the involute tooth profile near the tooth root is cut away, resulting in undercutting. However, if a positive modification is applied to a gear with 10 teeth, increasing the tip diameter and the tooth thickness, it is possible to achieve a gear strength equivalent to that of a gear with 200 teeth.
2) Modified Gears
The diagram below shows a schematic of a gear with 10 teeth undergoing positive modification during gear cutting. The amount of radial movement of the cutter during cutting, xm (mm), is called the radial modification amount (referred to as the modification amount).
- xm = modification amount (mm)
- x = modification coefficient
- m = module (mm)
Through positive modification, the tooth thickness increases, and the outside diameter (tip diameter) also enlarges. By applying positive modification, the gear can avoid undercutting. Gear modification can also achieve other purposes, such as changing the center distance. Positive modification can increase the center distance, while negative modification can reduce it.
Both positive and negative modifications have limitations on the amount of modification that can be applied.
3) Positive Modification and Negative Modification
Modification includes positive and negative types. Although the tooth height remains the same, the tooth thickness differs. Gears with thicker teeth are positive modification gears, while those with thinner teeth are negative modification gears.
When the center distance between two gears cannot be changed, the smaller gear undergoes positive modification (to avoid undercutting), while the larger gear undergoes negative modification to maintain the same center distance. In this case, the absolute values of the modification amounts are equal.
4) Meshing of Modified Gears
Standard gears mesh when their reference circles are tangent to each other. In contrast, as shown in the diagram, modified gears mesh at the operating pitch circles. The pressure angle on the operating pitch circle is called the meshing angle. This meshing angle differs from the pressure angle on the reference circle (reference circle pressure angle). The meshing angle is an important factor when designing modified gears.
5) Effects of Gear Modification
Gear modification can prevent undercutting that may occur during machining due to a low number of teeth. It allows for the desired center distance to be achieved. In cases where there is a significant difference in the number of teeth between a pair of gears, positive modification can be applied to the smaller gear to increase tooth thickness, which helps reduce wear. Conversely, negative modification can be applied to the larger gear to decrease tooth thickness, thus making the lifespans of the two gears more comparable.
Gear Precision
Gears are mechanical components that transmit power and rotation. The main performance requirements for gears are:
- Greater power transmission capability
- Use of smaller-sized gears whenever possible
- Low noise
- Accuracy
To meet the aforementioned requirements, improving gear precision will be a necessary challenge to address.
1) Classification of Gear Precision
Gear precision can be broadly divided into three categories:
a) Accuracy of the involute tooth profile—Tooth profile precision
b) Accuracy of the tooth lines on the tooth surface—Tooth line precision
c) Accuracy of the position of the teeth/teeth slots
- Pitch accuracy of the gear teeth—Single tooth pitch accuracy
- Accuracy of the pitch—Cumulative pitch accuracy
- Deviation of the position of the measuring ball between two gears in the radial direction—Radial runout precision
2) Tooth Profile Error
3) Tooth Line Error
4) Pitch Error
Measure the pitch value on a circumference centered on the gear shaft.
Single tooth pitch deviation (fpt) is the difference between the actual pitch and the theoretical pitch.
Cumulative pitch total deviation (Fp) evaluates the pitch deviations across all teeth of the gear. The total amplitude of the cumulative pitch deviation curve represents the total pitch deviation.
5) Radial Runout (Fr)
Place a measuring probe (spherical or cylindrical) successively in the tooth slots and measure the difference between the maximum and minimum radial distances from the probe to the gear axis. The eccentricity of the gear shaft is a part of the radial runout.
6) Radial Total Composite Deviation (Fi′′)
So far, we have discussed methods for evaluating the precision of individual gear components, such as tooth profile, pitch, and tooth line accuracy. In contrast, there is also a method that evaluates gear precision through a two-tooth face meshing test after the gear is engaged with a measuring gear. The left and right tooth faces of the measured gear contact and mesh with the measuring gear while rotating through a full rotation. The variations in center distance are recorded. The diagram below shows the test results for a gear with 30 teeth. There are 30 waveforms corresponding to the single tooth radial composite deviation. The radial total composite deviation value is approximately the sum of the radial runout deviation and the single tooth radial composite deviation.
7) Relationship Between Various Gear Precisions
8) Conditions for High-Precision Gears
Thank you for reading. We are looking forward to serving you with our exceptional gear solutions. #BeyondGears
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