Filter bank - Revision history

en>Bob K: /* Conclution and Application */ typo

2014-12-19T14:30:43Z

Conclution and Application: typo

← Older revision		Revision as of 16:30, 19 December 2014
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	~~[[Image:Non-circular gear.PNG\|thumb\|Non-circular gear example]]~~		Hi, everybody! <br>I'm Portuguese male :). <br>I really love Handball!<br><br>Feel free to surf to my site: www.hostgator1centcoupon.info ([http://dawonls.dothome.co.kr/db/?document_srl=404137 http://dawonls.dothome.co.kr/db/?document_srl=404137])
	~~[[File:Non-circular gear.svg\|thumb\|Another non-circular gear]]~~

	A '''non-circular gear''' ('''NCG''') is a special [[gear]] design with special characteristics and purpose. While a regular gear is optimized to transmit [[torque]] to another engaged member with minimum noise and wear and with maximum [[Mechanical efficiency\|efficiency]], a non-circular gear's main objective might be [[ratio]] variations, axle displacement [[oscillation]]s and more. Common applications include textile machines,<~~ref name="test1"~~>~~[http://prozamet.pl/art_2008_3_08.pdf Zarebski~~ I~~., Salacinski T.~~: ~~Designing of non-circular gears]</ref> [[potentiometer]]s, CVTs ([[continuously variable transmission]]s~~)~~,<ref>[http://next-1b.manuf.bme.hu/Laczik/Noncircular~~.~~pdf Laczik- Involute Profile of Non-Circular Gears]~~<~~/ref~~> ~~window shade panel drives, mechanical presses and high torque hydraulic engines.<ref name="test1" />~~

	A regular gear pair can be represented as two [[circle]]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 looking like regular gears are also possible (small ratio variations result from meshing area modifications).

	Generally NCG should meet all the requirements of regular gearing, but in some cases, for example variable [[axle]] distance, could prove impossible to support and such gears require very tight manufacturing tolerances and assembling problems arise. Because of complicated [[geometry]], NCGs are most likely [[spur gear]]s and [[Molding (process)\|molding]] or [[electrical discharge machining]] 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~~ <~~math~~>~~r_1(\theta_1)~~<~~/math~~> be the radius of the first gear wheel as a function of angle from the axis of rotation <math>\theta_1</math>, and let <math>r_2(\theta_2)</math> be the radius of the second gear wheel as a function of angle from its axis of rotation <math>\theta_2</math>. If the axles remain fixed, the distance between the axles is also fixed:<ref name="hexagon">[http://www.hexagon.de/pdf/noncgear.pdf Laczik - Design and Manufacturing of Non-Circular Gears by Given Transfer Function]</ref>

	~~:<math>r_1(\theta_1)+r_2(\theta_2)=a\,</math>~~

	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:<ref name="hexagon"/>~~

	~~:<math>r_1\,d\theta_1=r_2\,d\theta_2</math>~~

	Of course, 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.<ref name="hexagon"/>

	~~It is more convenient~~ to use the circular variable <math>z=e^{i\theta}</math> when analyzing this problem. Assuming the radius of the first gear wheel is known as a function of ''z'', and using the relationship <math>dz=iz\,d\theta</math>, the above two equations can be combined to yield the differential equation:

	~~:<math>\frac{dz_2}{z_2}=\frac{r_1(z_1)}{a-r_1(z_1)}\,\frac{dz_1}{z_1}</math>~~

	~~where <math>z_1</math> and <math>z_2</math> describe the rotation of the first and second gears respectively. This equation can be formally solved as:~~

	~~:<math>\ln(z_2)=\ln(K)+\int\frac{r_1(z_1)}{a-r_1(z_1)}\,\frac{dz_1}{z_1}</math>~~

	~~where <math>\ln(K)</math> is a constant of integration.~~

	~~==References==~~
	~~{{reflist}}~~

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

	~~==External links==~~
	*[http://www.~~youtube~~.~~com/watch?v=mkQ2pXkYjRM Historic Video of Non-Circular Gears on YouTube]~~
	*[http://travellingcurves.com/ The Eye of an Artist]
	*[http://kmoddl.library.cornell.edu/index.php Kinematic Models for Design Digital Library (~~KMODDL)]~~
	*[http://~~www~~.~~gearoscillator~~.~~net The Gear Oscillator]~~
	*[http://gearexpert.free.fr/~~fichiers_pdf~~/~~Non%20circular%20gear.pdf Laczik- Involute Profile of Non-Circular Gears]~~
	*[http://books.google.com/books?~~id=XUSHSYwdiV8C&lpg=PR14&pg~~=~~PA318 "Gear geometry and applied theory" by Faydor L. Litvin and Alfonso Fuentes]~~
	*[http://~~prozamet.pl/art_2008_3_08~~.~~pdf A paper on designing of non-circular gears]~~
	*[http://www.~~mauricelacroix~~.~~com~~/en/~~News/All_News/Masterpiece_Roue_carree.html Maurice Lacroix Masterpiece Regulateur Roue Carree. Squaring the Circle.]~~

	~~{{Gears}}~~

	~~{{DEFAULTSORT:Non-Circular Gear}}~~
	~~[[Category:Gears]~~]

en>Bob K: /* FFT filter banks */ WikiLink to "upsampling"

2014-01-28T13:04:59Z

FFT filter banks: WikiLink to "upsampling"

New page

[[Image:Non-circular gear.PNG|thumb|Non-circular gear example]]
[[File:Non-circular gear.svg|thumb|Another non-circular gear]]

A '''non-circular gear''' ('''NCG''') is a special [[gear]] design with special characteristics and purpose. While a regular gear is optimized to transmit [[torque]] to another engaged member with minimum noise and wear and with maximum [[Mechanical efficiency|efficiency]], a non-circular gear's main objective might be [[ratio]] variations, axle displacement [[oscillation]]s and more. Common applications include textile machines,<ref name="test1">[http://prozamet.pl/art_2008_3_08.pdf Zarebski I., Salacinski T.: Designing of non-circular gears]</ref> [[potentiometer]]s, CVTs ([[continuously variable transmission]]s),<ref>[http://next-1b.manuf.bme.hu/Laczik/Noncircular.pdf Laczik- Involute Profile of Non-Circular Gears]</ref> window shade panel drives, mechanical presses and high torque hydraulic engines.<ref name="test1" />

A regular gear pair can be represented as two [[circle]]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 looking like regular gears are also possible (small ratio variations result from meshing area modifications).

Generally NCG should meet all the requirements of regular gearing, but in some cases, for example variable [[axle]] distance, could prove impossible to support and such gears require very tight manufacturing tolerances and assembling problems arise. Because of complicated [[geometry]], NCGs are most likely [[spur gear]]s and [[Molding (process)|molding]] or [[electrical discharge machining]] 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 <math>r_1(\theta_1)</math> be the radius of the first gear wheel as a function of angle from the axis of rotation <math>\theta_1</math>, and let <math>r_2(\theta_2)</math> be the radius of the second gear wheel as a function of angle from its axis of rotation <math>\theta_2</math>. If the axles remain fixed, the distance between the axles is also fixed:<ref name="hexagon">[http://www.hexagon.de/pdf/noncgear.pdf Laczik - Design and Manufacturing of Non-Circular Gears by Given Transfer Function]</ref>

:<math>r_1(\theta_1)+r_2(\theta_2)=a\,</math>

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:<ref name="hexagon"/>

:<math>r_1\,d\theta_1=r_2\,d\theta_2</math>

Of course, 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.<ref name="hexagon"/>

It is more convenient to use the circular variable <math>z=e^{i\theta}</math> when analyzing this problem. Assuming the radius of the first gear wheel is known as a function of ''z'', and using the relationship <math>dz=iz\,d\theta</math>, the above two equations can be combined to yield the differential equation:

:<math>\frac{dz_2}{z_2}=\frac{r_1(z_1)}{a-r_1(z_1)}\,\frac{dz_1}{z_1}</math>

where <math>z_1</math> and <math>z_2</math> describe the rotation of the first and second gears respectively. This equation can be formally solved as:

:<math>\ln(z_2)=\ln(K)+\int\frac{r_1(z_1)}{a-r_1(z_1)}\,\frac{dz_1}{z_1}</math>

where <math>\ln(K)</math> is a constant of integration.

==References==
{{reflist}}

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

==External links==
*[http://www.youtube.com/watch?v=mkQ2pXkYjRM Historic Video of Non-Circular Gears on YouTube]
*[http://travellingcurves.com/ The Eye of an Artist]
*[http://kmoddl.library.cornell.edu/index.php Kinematic Models for Design Digital Library (KMODDL)]
*[http://www.gearoscillator.net The Gear Oscillator]
*[http://gearexpert.free.fr/fichiers_pdf/Non%20circular%20gear.pdf Laczik- Involute Profile of Non-Circular Gears]
*[http://books.google.com/books?id=XUSHSYwdiV8C&lpg=PR14&pg=PA318 "Gear geometry and applied theory" by Faydor L. Litvin and Alfonso Fuentes]
*[http://prozamet.pl/art_2008_3_08.pdf A paper on designing of non-circular gears]
*[http://www.mauricelacroix.com/en/News/All_News/Masterpiece_Roue_carree.html Maurice Lacroix Masterpiece Regulateur Roue Carree. Squaring the Circle.]

{{Gears}}

{{DEFAULTSORT:Non-Circular Gear}}
[[Category:Gears]]