Analisis Y Sintesis De Maquinas

INSTITUTO TECNOLOGICO Y DE ESTUDIOS SUPERIORES DE MONTERREY
Campus Santa Fe

Análisis y Síntesis de máquinas
Sadegh Babaii Kochekseraii
Firts Partial Project

* Estrada Cancino Ma. Teresa A00965856
* José Rodolfo Rizzo A01012828

2010-02-10
Introduction
Gears are very useful tools used in many types of machinery. Gears have been used by men for over more than three hundredyears and until today they are an important element in all manner of machinery used in current times. Designing a gear is something really complicated and sophisticated such that can be consider an art. A lot of factors influence the way gears are designed and a long time according to needs they had evolved considering prices, quieter running, lighter weight and efficiency.
To define clearlywhat a gear is we can say that gears are toothed wheels that have been used to transmit circular motion or rotational force from one part of a machine to another. Almost all gears are used in pairs (minimum) and each gear is usually attached to a rotating shaft. When the teeth of two gears are meshed, rotation of one shaft and its gear causes the other gear and its shaft to rotate.
Making acomplex gear, which meshes together smoothly and efficiently at high speed, requires a good design work. For designing any gear the first step is to determine the base dimensions, in other words draw the root circle, all the teeth of the gear will stick out of the root circle. Then the next step is to determine the high of the teeth, for that we’ll need to draw another circle this one bigger than theprevious. To determine de distance between each tooth the easiest way is to determine the angle dividing 360° by de number of teeth. At this point the work seems very easy; the hardest part comes when we need to design the shape of each tooth. The shape of the teeth determines how efficient our gear will be.
What is an involute?
An involute is a geometric property of certain curve. Given thatmost of the times, such curve is described by a mathematical function; the involute of such function can also be described by the same means. The involute refers to an alternate description of the curve of consideration by taking the derivative of the curve at each point in its path of trajectory. Knowing that the derivative of a curve at certain point will always be a tangent path to the curve ofinterest, we would end up with as many tangent lines along the curve as the number of points considered in between the curve of study. Most importantly, the involute will result in the sketch of the length of such tangent lines at each point of the curve relative to the arc length of each of the points measured from the first starting point or origin of the formation of the curve of study. Thefurther we move from the starting point, the longer the projection of the tangent line at such point. Knowing that the length of the tangent lines will increase as we move away from the first point or origin, we can now imagine that the involute is the geometric description of the change in length of the geometric derivatives of a curve, which will eventually serve as an alternate description to thetrajectory of the path of consideration.
The clearest example to show the formation of an involute, and that contributes to the understanding of its importance in the design of gears is the circle involute. A circle involute will be obtained by finding the tangent lines of each of the point along the circle and tracing them with a length as a function of the arc length between the origin and thepoint along the path of the circle. The following image, taken from the digital source “Virtual dictionary of special plane curves”, helps visualize the formation of the involute of a circle.

In order to describe the trajectory of the involute, we also need to describe the trajectory of the curve, in this case the circle. Parametrizing the curve will help us parameterize as a function of time...