Gears Chapter 13

Dr S.Mekid

2013-2014

ME Department, KFUPM 1

OBJECTIVES:

1- Gear Geometry 2- Kinematic Relations 3- Force Transmission

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“ISSUS COLEOPTRATUS NYMPH” AS IT JUMPED, RESEARCHERS NOTICED THAT THE YOUNG INSECTS SPORTED CURVED STRIPS OF GEAR TEETH ON THEIR HIND LEGS.

3 Source: http://sciencefocus.com/news/mechanical-gears-insects-got-there-first?utm_source=Adestra&utm_medium=Email&utm_campaign=FOC_200913_Newsletter_Focus_Newsletters&utm_content=

Definition of Gears: Gears are toothed wheels used primarily to transmit: 1- motion and 2-power between rotating shafts. Definition of Gearing: An assembly of two or more gears. Reliability of Gears: The most durable of all mechanical drives, gearing can transmit high power at efficiencies approaching 0.99 and with a long service life. 4

To transmit power without slippage, a positive drive is required, a condition that can be fulfilled by properly designed teeth. Gears are thus a logical extension of the friction wheel concept.

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Specific functions of gears: Gears are spinning levers capable of performing three

important functions:

1. Provide a positive displacement coupling between shafts.

2. Increase, decrease, or maintain the speed of rotation with accompanying change in torque.

3. Change the direction of rotation and/or shaft arrangement (orientation).

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Types of gears: Relative shaft position - parallel, intersecting, or skew- accounts for three basic types of gearing. To function properly, gears assume various shapes to accommodate shaft orientation. If the shafts are parallel, the basic function wheels and gears developed from them assume the shape of cylinders.

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When the shafts are intersecting, the wheels become frustums of cones, and gears developed from these conical surfaces are called bevel gears.

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When the shafts cross (skew, one above the other), the friction wheels may be cylindrical or of hyperbolic cross section.

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Other classifications of gears Besides shaft position and tooth form, gears may be

classified according to: 1. System of measurement: Pitch (U.S.) or Module (SI,

European).

2. Pitch: Coarse or Fine.

3. Quality: Commercial, Precision, and Ultra precision.

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Basic Gear Terminology Pitch circle: A theoretical circle upon which all calculations are usually based; its diameter is the pitch diameter (d). The pitch circles of a pair of mating gears are tangent to each other. Pinion: The smaller of two mating gears. The larger one is often called the gear. The circular pitch, p: The distance measured on the pitch circle, from a point on one tooth to a corresponding point on an adjacent tooth.

11 Circular pitch (p) = Tooth thickness + Width of space

Module, m (European) The ratio of the pitch diameter (d) to the number of teeth (N). It is an index of tooth size in SI.

m = d / N d in millimetres. Diametral pitch, P (U.S.) The ratio of the number of teeth (N) on the gear to the pitch diameter (d). It is an index to the density of number of teeth (teeth per inch).

P = N / d d in inches. Circular pitch, p

p = ( π × d ) / N

p = (π × d ) / N = π × m (mm)

p = (π × d ) / N = π / P (inches) 12

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Always remember that: The pitch-line velocity of two mating gears without slipping (rolling on one another) is given by:

V = r1 × ω1 = r2 × ω 2

ω1 / ω2 = r2 / r1

Where r1 and r2 are the pitch radii of the pinion and gear respectively.

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Example: N1 = number of pinion teeth = 18 N2 = number of gear teeth = 30 Assume that the diametral pitch of the pinion and the gear = P = 2 (teeth per inch) The pitch diameters of the pinion and the gear respectively are:

d1 = N1 / P = 18 / 2 = 9 in d2 = N2 / P = 30 / 2 = 15 in

The center distance of the pinion and gear =

(d1 + d2) / 2 = (9+15) / 2 = 12 in. This means that the pinion and gear centers should be located 12 in. apart

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Conjugate Action of Gears Mating cam profiles that yield a constant angular displacement ratio are termed conjugate. Although an infinite number of profile curves will satisfy the Law of Gearing, only the cycloid and the involute have been standardized. The involute has several advantages; the most important is its ease of manufacture and the fact that the center distance between two involute gears may vary without changing the velocity ratio.

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Involute Gear Principles An involute curve is generated by a point moving in a definite relationship to a circle, called the base circle. The Fixed Base Circle is a method used to generate involute curves. In this method the base circle and the drawing plane in which the involutes are traced remain fixed. The radius of curvature, to any point (within the involute), is always tangent to the base circle and normal to one and only one tangent on the involute. This is the underlaying principle of involute compasses and involute dressers (where the motion of a diamond tool gives an involute profile to grinding wheels) 17

Involute Tooth Form The most widely used spur gear tooth form is the full-depth involute form.

As mentioned earlier, the involute is one of a class of geometric curves called Conjugate Curves. When two such gear teeth are in mesh and rotating, there is a constant angular velocity ratio between them: From the moment of initial contact to the moment of disengagement, the speed of the driving gear is in a constant proportion to the speed of the driven gear. The resulting action of the two gears is very smooth. If this were not the case, there would be some speeding up and slowing down during the engagement with the resulting accelerations causing vibrations, noise, and dangerous torsional oscillations in the system.

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Curved Involutes in Contact

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Statement of the law of gearing As demonstrated here, the gear teeth made in the involute tooth form obey the law. Of course, only the part of the gear tooth that actually comes into contact with the mating tooth needs to be in the involute form.

To transmit motion at a constant angular-velocity, the pitch point must remain fixed; that is, all the lines of action (common tangent) for every instantaneous point of contact must pass through the same point P. 20

The line of action (Common tangent) is also called the pressure line and the generating line. It represents the direction in which the resultant force (it passes through the contact point) acts between the gears. The angle Ø is called the pressure angle. The pressure angle has usually one of following standard values:

•14.5 deg •20 deg •25 deg

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Addendum, a: The radial distance between the top land and the pitch circle. Dedendum, b The radial distance from the bottom land to the pitch circle.

The whole depth of teeth, ht = a + b

Clearance circle A circle, which is tangent to the addendum circle of the mating gear. Clearance, c The amount by which the dedendum in a given gear exceeds the addendum of its mating gear.

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Whole depth of teeth

ht = a + b = hk +c

= (a+a)+c

where

hk = working depth = 2a

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Backlash If the tooth thickness is made identical in value to tooth space, as it is theoretically, the tooth geometry would have to be absolutely precise for the gears to operate, and there would be no space available for lubrication of the tooth surfaces. To alleviate these problems, practical gears are made with the tooth space slightly larger than the tooth thickness, the difference being called the BACKLASH. Backlash also prevents gears from jamming (making contact on both sides of their teeth simultaneously). It compensates for machining error and thermal expansion. To provide backlash, the cutter generating the gear teeth can be fed deeper into the gear blank than the theoretical value on either or both of the mating gears. Alternatively, backlash can be created by adjusting the centre distance to a larger value than the theoretical value.

The magnitude of backlash depends on the desired precision of the gear pair and on the size and pitch of the gears. 24

The pitch circles (having radii r1 and r2) should be tangent at point P, the pitch point. Line ab represents the common tangent. Draw line cd (Pressure line) at an angle Ø to line ab On each gear draw a circle tangent to the pressure line cd. These lines are the base circles. The radii of the base circles are determined based on the value of the pressure angle (14.5 or 20 or 25 deg).

As shown above, the radius of the base circle is : rb = r × cos ( Ø )

where r is the pitch circle. 25

As previously described, generate an involute on each circle (starting from the base circle). Remember that any line perpendicular to any point on the involute is tangent to the base circle (perpendicular to a line drawn from the origin of the base circle to the point of tangency).

The standard addendum (a) and dedendum (b) distances for standard teeth are 1 / P and 1.25 / P respectively.

a = 1 / P = 1 / 2 = 0.500 in b = 1.25 / P = 0.625 in

Using the values of a and b draw the addendum and the dedendum circle on the pinion and on the gear.

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The circular pitch = p = π / P = π / 2 = 1.57 in The tooth thickness= t = circular pitch / 2 = p / 2 = 1.57 / 2 =0.785 in

knowing the tooth thickness, t, the tooth could be drawn easily remembering that the tooth thickness is measured at the pitch circle. Remember also that the involute starts at the base circle. The profile of the gear tooth below the base circle, is a radial drawn line. The actual shape, however, will depend upon the kind of machine tool used to form the teeth in manufacture. After drawing the clearance circle (c = b – a = 0.625 – 0.5 = 0.125 in), the portion of the tooth between the clearance circle and the dedendum circle includes a fillet. So, the portion between the clearance circle and the base circle is a radial line. 27

Angle of approach The initial contact will take place when the flank of the drive comes into contact with the tip of the driven tooth. This happens when the addendum of the driven gear crosses the pressure line. If we construct tooth profiles (as shown before) through point of initial contact and draw lines from the intersections of these profiles with the pitch circles to the gear centers, we obtain the angle of approach of each gear. 28

Angle of recess As the teeth go into mesh, the point of contact will slide up the side of the driving tooth so that the tip of the driver will be in contact just before contact ends. The final point of contact will therefore be where the addendum circle of the driver crosses the pressure line. The angles of recess are then obtained in a similar manner to finding the angles of approach.

The sum of the angle of approach and the angle of recess for either gear is called the angle of action. The line extending between the start and the end points is called the line of action. 29

Limitations on Spur Gears Two spur gears will mesh properly, within wide limits, provided they have the same diametral pitch or module. Limitations are set by many factors, but two in particular are important: -Contact ratio and -Interference

To obtain the contact ratio, the length of action must first be introduced. The length of action, Lab or length of contact is the distance on an involute line of action through which the point of contact moves during the action of the tooth profiles. I t is the part of the line of action located between the two addendum circles or outside diameters. 30

Contact length and contact ratio As two gears rotate, smoothed continuous transfer of motion from one pair of meshing teeth to the following pair is achieved when contact of the first pair continues until the following pair has established contact. In fact, considerable overlapping is necessary to compensate for contact delays caused by tooth deflection, errors in tooth spacing, and center distance tolerances. To assure a smooth transfer of motion, overlapping should not be less than 20%. In power gearing, it is often 60 to 70 %. Contact ratio, mc, is another more common means of expressing overlapping tooth contact. On a time basis, the contact ratio, mc, is the number of pairs of teeth simultaneously engaged. If two pairs of teeth were in contact at all time, the ratio would be 2.0, corresponding to 100 % overlapping.

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Most gears are designed with contact ratios between 1.2 and 1.6. For example, a ratio of 1.4 means that one pair of teeth is always in contact, and a second pair is in contact 40 percent of the time. As was described earlier: When two gear teeth come into mesh, the initial point (starts) of contact occurs when the flank of the driver comes into contact with the tip of the driven gear. Similarly, the contact ends when the tip of the driver tooth comes into contact with the flank of the driven tooth. Because the tips of gear teeth lie on the addendum circle, contact between two gear teeth starts when the addendum circle of the driven gear intersects the pressure line.

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Calculation of the contact ratio CR The contact ratio is calculated as length of contact, Lab divided by the base pitch, pb: The length of contact, Lab can be derived as:

(rap2 - rbp

2)0.5 +(raG2 - rbG

2)0.5 – C × sin (Ø)

where rap & raG : addendum radii rbp & rbG : base circle radii C is the center distance = rgear + rpinion The base pitch, pb = p × cos (Ø) represents the theoretical minimum path of contact. The contact Ratio CR is:

mc= Lab/pb

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Extra discussion on the pressure Angle: It is the angle between the tangent to the pitch circles and the line drawn normal (perpendicular to the surface of the gear tooth). When two gear teeth are in mesh and transmitting power, the force transferred from the driver to the driven gear tooth acts in a direction along the line of action.

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Also, the actual shape of the gear tooth depends on the pressure angle as illustrated below. The teeth in the figure were drawn according to the proportions for a 20-tooth, 5-pitch gear having a pitch diameter of 4.000 in. All three teeth have the same tooth thickness, because the thickness of the pitch line depends only on the pitch. The difference between the three teeth is due to the different pressure angles, because the pressure angle determines the size of the base circle. Remember that the base circle is the circle from which the involute is generated. The line of action is always tangent to the base circle (Remember that rb = r × cos ( Ø ) or db = d × cos ( Ø ))

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Standard values of the pressure angle are established by gear manufacturers, and the pressure angle of two gears in mesh must be the same. The 14.5 deg pressure angle is not used anymore in design of gears. The advantages and disadvantages of the different values of pressure angle relate to the strength of the teeth, the occurrence of the interference, and the magnitude of forces exerted on the shaft. The effects will be discussed later.

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Interference Under certain conditions, tooth profiles overlap or cut into each other. This situation, termed interference, should be avoided because of excess wear, vibration or jamming. Generally, it involves contact between involute surfaces of one gear and noninvolute surfaces of the mating gears.

This gear shows clear evidence that the tip of its mating gear has produced an interference condition in the root section. Lacalized scoring has taken place, causing rapid removal in the root section. Generally, an interference of this nature causes considerable damage if not corrected.

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The figure here shows maximum length of contact being limited to the full length of the common tangent. Any tooth addendum extended beyond the tangent points T and Q, termed interference points, is useless and interferes with the root fillet area of the mating tooth. To operate without profile overlapping would require undercut teeth. But undercutting weakens a tooth (in bending) and may also remove part of the useful involute profile near the base circle.

Interference sets a geometric limitation on tooth profiles. For

standard tooth forms interference takes place for contact to the right of point T and to the lift of point Q

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To operate without interference (profile overlapping), either the pinion must be undercut or the

gear must have stub teeth.

Observe that each pair of tooth encounter interference twice every

time they come in contact, once during approach when the gear

tooth digs into the root section of he mating pinion tooth. During the recess this sequence is revered.

Undercutting is the process of cutting away the material at the

fillet or root of the gear teeth. 40

Interference can be defined as the contacts of portions of tooth profiles that are not conjugate. The driver, gears 2 turn clockwise. The initial and final points of contact are designated A and B (they are located on the pressure line of action) Notice that the points of tangency of the pressure line with the base circles C and D are located inside of points A and B. Interference is present.

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Contact occurs below the base circle of gear 2 on the noninvolute portion of the flank. The actual effect is that the invloute tip or face of the driven gear tends to dig out the noninvolute flank of the driver. In this case, the same effect occurs as the teeth leave contact. Contact should end at point D or before. Since it does not end until point B, the effect is for the tip of the driving tooth to dig out, or interfere with, the flank of the driven tooth. As shown before, the inteference could be avoided if undercutting takes place. The disadvantage of undercutting is that the tooth is weakened.

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Required number of teeth on pinions and gears to avoid interference: It is known that the interference condition intensifies as the number of teeth on the pinion decreases. The surest way to control interference is to control the minimum number of teeth in the pinion to the limiting values as shown in the table below. The right part of the table indicates the maximum number of gear teeth that can be used for a given number of pinion teeth to avoid interference. As noted earlier, the 14.5 deg system is considered to be obsolete. The data indicates one of its advantages: its potential for increasing interference. Interference intensifies as the number of teeth on the pinion decreases.

Number of pinion teeth

to ensure NO

interference

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The smallest number of teeth on a spur pinion and gear (for a one-o-one gear ratio), which can exist without interference is Np = Ng:

Where k = 1 for full-depth teeth k = 0.8 for stub teeth

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If the mating gear has more teeth than the pinion, NP ≠ NG Then, Np is defined as:

Where m = NG / NP In the case of a pinion and a rack, the smallest number of teeth on the spur pinion is given as:

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The largest gear with a specified pinion that is interference free is:

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The following methods can also be used to avoid interference: 1. Increasing the addendum of the pinion while decreasing

the addendum of the gear. The center distance can remain the same as its theoretical value for the number of teeth in the pair. But the resulting gears are of course nonstandard.

2. It is possible to make the pinion of the gear pair larger than

the standard while keeping the gear standard if the center distance for the pair is enlarged.

3. Using a large pressure angle. This results in a smaller base circle, so that more of the tooth profile becomes involute. The demand for smaller pinion with fewer teeth thus favors the use of a 25 deg pressure angle even though the frictional forces and bearing loads are increased and the contact ratio decreased. 47

Interference and contact ratio vary inversely with the pressure angle. When the pressure angle increases from Ø1 to Ø2, the involute section between the pitch line and the base line lengthens, tending to alleviate interference. The path of contact however, shortens, thereby effectively lowering the contact ratio.

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Tooth system for spur gears: A tooth system is a standard that specifies the relationships involving:

•Addendum •Dedendum •Working depth •Tooth thickness •Pressure angle

Table below shows contains the standard used for spur gears. Observe that the 14.5 deg system is not included.

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Fig 13-27 51

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Solve example 13-6

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