Non-circular Gear
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A non-circular gear (NCG) is a special
gear A gear or gearwheel is a rotating machine part typically used to transmit rotational motion and/or torque by means of a series of teeth that engage with compatible teeth of another gear or other part. The teeth can be integral saliences or ...
design with special characteristics and purpose. While a regular gear is optimized to transmit
torque In physics and mechanics, torque is the rotational analogue of linear force. It is also referred to as the moment of force (also abbreviated to moment). The symbol for torque is typically \boldsymbol\tau, the lowercase Greek letter ''tau''. Wh ...
to another engaged member with minimum noise and wear and with maximum
efficiency Efficiency is the often measurable ability to avoid making mistakes or wasting materials, energy, efforts, money, and time while performing a task. In a more general sense, it is the ability to do things well, successfully, and without waste. ...
, a non-circular gear's main objective might be
ratio In mathematics, a ratio () shows how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8:6, which is equivalent to the ...
variations, axle displacement
oscillation Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. Familiar examples of oscillation include a swinging pendulum ...
s and more. Common applications include textile machines,Zarebski I., Salacinski T.: Designing of non-circular gears
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potentiometer A potentiometer is a three- terminal resistor with a sliding or rotating contact that forms an adjustable voltage divider. If only two terminals are used, one end and the wiper, it acts as a variable resistor or rheostat. The measuring instrum ...
s, CVTs (
continuously variable transmission A continuously variable transmission (CVT) is an automated Transmission (mechanical device), transmission that can change through a continuous range of gear ratios, typically resulting in better fuel economy in gasoline applications. This contr ...
s),Laczik- Involute Profile of Non-Circular Gears
/ref> window shade panel drives, mechanical presses and high torque hydraulic engines. A regular gear pair can be represented as two
circle A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is cal ...
s rolling together without slip. In the case of non-circular gears, those circles are replaced with anything different from a circle. For this reason NCGs in most cases are not round, but round NCGs that look like regular gears are also possible (small ratio variations result from meshing area modifications). Generally, NCGs should meet all the requirements of regular gearing but in some cases, for example variable
axle An axle or axletree is a central shaft for a rotation, rotating wheel and axle, wheel or gear. On wheeled vehicles, the axle may be fixed to the wheels, rotating with them, or fixed to the vehicle, with the wheels rotating around the axle. In ...
distance, could prove impossible to support, and such gears require very tight manufacturing tolerances and assembling problems arise. Because of complicated
geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, NCGs are most likely spur gears and molding or
electrical discharge machining Electrical discharge machining (EDM), also known as spark machining, spark eroding, die sinking, wire burning or wire erosion, is a metal fabrication process whereby a desired shape is obtained by using electrical discharges (sparks). Material is ...
technology is used instead of generation.


Mathematical description

Ignoring the gear teeth for the moment (i.e. assuming the gear teeth are very small), let r_1(\theta_1) be the radius of the first gear wheel as a function of angle from the axis of rotation \theta_1, and let r_2(\theta_2) be the radius of the second gear wheel as a function of angle from its axis of rotation \theta_2. If the axles remain fixed, the distance between the axles is also fixed:Laczik - Design and Manufacturing of Non-Circular Gears by Given Transfer Function
/ref> :r_1(\theta_1)+r_2(\theta_2)=a\, Assuming that the point of contact lies on the line connecting the axles, in order for the gears to touch without slipping, the velocity of each wheel must be equal at the point of contact and perpendicular to the line connecting the axles, which implies that: :r_1\,d\theta_1=r_2\,d\theta_2 Each wheel must be cyclic in its angular coordinates. If the shape of the first wheel is known, the shape of the second can often be found using the above equations. If the relationship between the angles is specified, the shapes of both wheels can often be determined analytically as well. It is more convenient to use the circular variable z=e^ when analyzing this problem. Assuming the radius of the first gear wheel is known as a function of ''z'', and using the relationship dz=iz\,d\theta, the above two equations can be combined to yield the differential equation: :\frac=\frac\,\frac where z_1 and z_2 describe the rotation of the first and second gears respectively. This equation can be formally solved as: :\ln(z_2)=\ln(K)+\int\frac\,\frac where \ln(K) is a constant of integration.


References


Further reading

* Noncircular Gears: Design and Generation by Faydor L. Litvin, Alfonso Fuentes-Aznar, Ignacio Gonzalez-Perez, and Kenichi Hayasaka


External links


Historic Video of Non-Circular Gears on YouTube
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Kinematic Models for Design Digital Library (KMODDL)
*
Laczik- Involute Profile of Non-Circular Gears"Gear geometry and applied theory" by Faydor L. Litvin and Alfonso FuentesA paper on designing of non-circular gears
* {{DEFAULTSORT:Non-Circular Gear Gears