Gears are machine elements that transmit motion by means of successively engaging teeth. The gear teeth act like small levers.
Gears may be classified according to the relative position of the axes of revolution. The axes may be
Here is a brief list of the common forms. We will discuss each in more detail later.
The left pair of gears makes external contact, and the right pair of gears makes internal contact
Figure 7-2 shows two mating gear teeth, in which
Although the two profiles have different velocities V1 and V2 at point K, their velocities along N1N2 are equal in both magnitude and direction. Otherwise the two tooth profiles would separate from each other. Therefore, we have
We notice that the intersection of the tangency N1N2 and the line of center O1O2 is point P, and
Thus, the relationship between the angular velocities of the driving gear to the driven gear, or velocity ratio, of a pair of mating teeth is
Point P is very important to the velocity ratio, and it is called the pitch point. Pitch point divides the line between the line of centers and its position decides the velocity ratio of the two teeth. The above expression is the fundamental law of gear-tooth action.
For a constant velocity ratio, the position of P should remain unchanged. In this case, the motion transmission between two gears is equivalent to the motion transmission between two imagined slipless cylinders with radius R1 and R2 or diameter D1 and D2. We can get two circles whose centers are at O1 and O2, and through pitch point P. These two circle are termed pitch circles. The velocity ratio is equal to the inverse ratio of the diameters of pitch circles. This is the fundamental law of gear-tooth action.
The fundamental law of gear-tooth action may now also be stated as follow (for gears with fixed center distance) (Ham 58):
The common normal to the tooth profiles at the point of contact must always pass through a fixed point (the pitch point) on the line of centers (to get a constant velocity ration).
To obtain the expected velocity ratio of two tooth profiles, the normal line of their profiles must pass through the corresponding pitch point, which is decided by the velocity ratio. The two profiles which satisfy this requirement are called conjugate profiles. Sometimes, we simply termed the tooth profiles which satisfy the fundamental law of gear-tooth action the conjugate profiles.
Although many tooth shapes are possible for which a mating tooth could be designed to satisfy the fundamental law, only two are in general use: the cycloidal and involute profiles. The involute has important advantages -- it is easy to manufacture and the center distance between a pair of involute gears can be varied without changing the velocity ratio. Thus close tolerances between shaft locations are not required when using the involute profile. The most commonly used conjugate tooth curve is the involute curve (Erdman & Sandor 84).
The following examples are involute spur gears. We use the word involute because the contour of gear teeth curves inward. Gears have many terminologies, parameters and principles. One of the important concepts is the velocity ratio, which is the ratio of the rotary velocity of the driver gear to that of the driven gears.
The SimDesign file for these gears is simdesign/gear15.30.sim. The number of teeth in these gears are 15 and 30, respectively. If the 15-tooth gear is the driving gear and the 30-teeth gear is the driven gear, their velocity ratio is 2.
Other examples of gears are in simdesign/gear10.30.sim and simdesign/gear20.30.sim
The curve most commonly used for gear-tooth profiles is the involute of a circle. This involute curve is the path traced by a point on a line as the line rolls without slipping on the circumference of a circle. It may also be defined as a path traced by the end of a string which is originally wrapped on a circle when the string is unwrapped from the circle. The circle from which the involute is derived is called the base circle.
In Figure 7-3, let line MN roll in the counterclockwise direction on the circumference of a circle without slipping. When the line has reached the position M'N', its original point of tangent A has reached the position K, having traced the involute curve AK during the motion. As the motion continues, the point A will trace the involute curve AKC.
Figure 7-4 shows some of the terms for gears.
In the following section, we define many of the terms used in the analysis of spur gears. Some of the terminology has been defined previously but we include them here for completeness. (See (Ham 58) for more details.)
That is, the product of the diametral pitch and the circular pitch equals .
Instead of using the theoretical pitch circle as an index of tooth size, the base circle, which is a more fundamental circle, can be used. The result is called the base pitch pb, and it is related to the circular pitch p by the equation
Figure 7-5 shows two meshing gears contacting at point K1 and K2.
To get a correct meshing, the distance of K1K2 on gear 1 should be the same as the distance of K1K2 on gear 2. As K1K2 on both gears are equal to the base pitch of their gears, respectively. Hence
To satisfy the above equation, the pair of meshing gears must satisfy the following condition:
Gear trains consist of two or more gears for the purpose of transmitting motion from one axis to another. Ordinary gear trains have axes, relative to the frame, for all gears comprising the train. Figure 7-6a shows a simple ordinary train in which there is only one gear for each axis. In Figure 7-6b a compound ordinary trainis seen to be one in which two or more gears may rotate about a single axis.
We know that the velocity ratio of a pair of gears is the inverse proportion of the diameters of their pitch circle, and the diameter of the pitch circle equals to the number of teeth divided by the diametral pitch. Also, we know that it is necessary for the to mating gears to have the same diametral pitch so that to satisfy the condition of correct meshing. Thus, we infer that the velocity ratio of a pair of gears is the inverse ratio of their number of teeth.
For the ordinary gear trains in Figure 7-6a, we have
These equations can be combined to give the velocity ratio of the first gear in the train to the last gear:
Thus, it is not difficult to get the velocity ratio of the gear train in Figure 7-6b:
Planetary gear trains, also referred to as epicyclic gear trains, are those in which one or more gears orbit about the central axis of the train. Thus, they differ from an ordinary train by having a moving axis or axes. Figure 7-8 shows a basic arrangement that is functional by itself or when used as a part of a more complex system. Gear 1 is called a sun gear , gear 2 is a planet, link H is an arm, or planet carrier.
The SimDesign file is simdesign/gear.planet.sim. Since the sun gear (the largest gear) is fixed, the DOF of the above mechanism is one. When you pull the arm or the planet, the mechanism has a definite motion. If the sun gear isn't frozen, the relative motion is difficult to control.
Take the planetary gearing train in Figure 7-8 as an example. Suppose N1 = 36, N2 = 18, 1 = 0, 2 = 30. What is the value of N?
With the application of the velocity ratio equation for the planetary gearing trains, we have the following equation:
(7-19) From the equation and the given conditions, we can get the answer: N = 10