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Mechanical Design 337

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FUNDAMENTALS OF GEARS 3 - 2

308803 MECHANICAL DESIGN 337 SEMESTER 2, 2009

1.0 INTRODUCTION In many engineering applications, it is desired to transmit power or motion from one rotating shaft to another. There is a range of alternatives to accomplish this task. The available choices include flat belts, V-belts, toothed timing belts, chain drives, friction wheel drives, and gear drives. Selection of a particular transmission system depends on factors like distance between rotating members, power requirements, system efficiency requirements, vibration limitations and budget. For example, belt drives and chain drives are often less costly, and may be used to advantage where input and output shafts are widely spaced.

On the other hand, if smooth slip-free uniform motion, high speed, light weight, precise timing, high efficiency, or compact design are important design criteria, the selection of an appropriate system of gears will, in nearly all cases, fulfil these criteria better than any of the other alternatives. Gears are toothed wheels used to transmit motion and power from one shaft to the next. Gears can transmit power with up to 98% efficiency [2]. Design of gears involves selection of materials, strength, and wear characteristics.

This lecture covers the study of fundamentals of gear design with particular emphasis on spur gears. In this regards, the following topics are discussed in these notes: - Fundamentals of gears - Law of gearing - Conjugate action and involute properties - Gear-tooth geometry and nomenclature - Pressure angle - Interference and undercutting - Contact ratio - Gear-tooth systems - Types of gears



2.0 GEARING The simplest means of transferring rotary motion from one shaft to another is a pair of rolling cylinders (external or internal), as shown in Figure 2.1a. This mechanism works quite well as long as sufficient friction is available at the rolling interface. However, such arrangements are limited by their load (torque) carrying capacity and possibility of slip.

Figure 2.1: Motion transmission mechanisms [3]

Furthermore, these drives are not suitable for applications, which require absolute phasing of the input and output shafts for timing purposes. Such applications require adding some meshing teeth to the rolling cylinders. Then, this mechanism becomes gears, as shown in Figure 2.1b.

Note: When two gears are in mesh to form a gearset (gear train), the smaller of the two is referred to as the pinion and the other is called as the gear.

VELOCITY RATIO Usually, a pair of gears is used to get a speed change. This is accomplished by employing different sizes of the gears in mesh. The velocity ratio is defined as the ratio of rotational speed of the input gear to that of the output gear for a single pair of gears.



3.0 FUNDAMENTAL LAW OF GEARING The most basic requirement of a pair of gears is that the angular velocity ratio between them remains constant. The gear teeth must be carefully formed so they do not interfere with each other as the gears rotate, and so that the angular velocity ratio between the driving and the driven members neither increases nor decreases at any instant as successive teeth pass through the ‘mesh’ (region of tooth contact). If these conditions are met, the gears are said to fulfil the fundamental law of gearing. It is expressed as

CONSTANT=1

2

ωω

This requirement must be met if measured for one or more turns of gear, or if measured over 1/100th of a turn. If this is not constant throughout the ‘mesh’, inertia loading from the driving and driven components may seriously increase loads, causing damage or noise problems.

The mechanism, as shown in Figure 2.2, involves two friction cylinders to transfer rotary motion from one shaft to another. Provided there is no slip, the magnitudes of the tangential velocity are equal for the two contacting members; hence

2211 ωω rr =

CONSTANTrr==

1

2

2

1

ωω

Figure 2.2: Friction cylinders [4]

If r1/r2 is not constant, w1/w2 is not constant. For the angular velocity ratio to remain constant, the contours of the gear teeth have to satisfy the ‘fundamental law of gearing’:



As the gears rotate, the common normal to the surfaces at the point of contact must always intersect the line of centres at the same point P (Figure 2.3), called the pitch point.

When the tooth profiles, or cams, are designed so as to satisfy this requirement, these are said to have conjugate action. This is illustrated in Figure 2.3.

Figure 2.3: Cam-follower and gear-tooth profiles ensuring conjugate action [1, 5]

The above discussion is based on the assumption that the gear-teeth are perfectly formed, perfectly smooth, and absolutely rigid. Such an assumption is, of course, unrealistic, because manufacturing inaccuracies and tooth deflections will cause slight deviations in velocity ratio, but acceptable tooth profiles are based on theoretical curves that meet this criterion.



4.0 GEAR-TOOTH PROFILE In order for the fundamental law of engineering to be true, the gear tooth contours on mating teeth must be conjugates of one another. There are an infinite number of possible conjugate pairs that could be used, but only a few curves have seen practical application as gear teeth. Two of them are briefly discussed here.

CYCLOIDAL GEARING:

Cycloidal gearing was once extensively used. In this type, the curve is traced by a point on the circumference of a generating circle, as it rolls without slipping along the inside and outside of the pitch circle of the gear, as illustrated in Figure 2.4.

Figure 2.4: Generation of epicycloid and hypocycloid

The more costly tools required to manufacture cycloidal gearing have led to it being superseded by another form. The other problem with cycloidal tooth forms is that tooth action is not correct unless the centre distance is accurately maintained.

The cycloid is still used as tooth form in some watches and clocks, but most gears use the involute of a circle for their shape. This is discussed below.



THE INVOLUTE CURVE

The involute is one of a category of geometric curves called conjugate curves. When two involute gear teeth are in mesh and rotating, there is constant angular velocity ratio between them. It means that right from the initial contact to the disengagement, the speed of the driving gear is in a constant proportion to the speed of the driven gear. This results in a very smooth action between mating teeth.

The involute curve can be generated using a pencil attached to a taut string, which is wrapped around a fixed circle. The development of the involute of a circle is illustrated in Figure 2.5.

Figure 2.5: The development of the Involute of a circle [3]

The Involute curve generation process is characterised by the following features [3]:

a. The string is always tangent to the base circle.

b. The centre of curvature of the involute is always at the point of tangency of the string with the base circle

c. A tangent to the involute is always normal to the string, which is the instantaneous radius of curvature of the involute curve.

Figure 2.6 illustrates two involute curves generated on separate cylinders in contact (or ”in mesh”). The cylinders from which the strings are unwrapped are called the base circles of the respective gears. It is important to note that:

- The base circles are smaller than the pitch circles, which are at the radii of the original rolling cylinders, rp and rg.

- The gear tooth must project both below and above the rolling-cylinder surface (pitch circle), and the involute only exists outside of the base circle



- Common normal (Line of action) is tangent to both base circles. It always passes through the pitch point regardless of where in the mesh the two teeth are contacting.

- The pitch-point has the same linear velocity in both pinion and gear. It is known as the pitch-line velocity.

Figure 2.6: Contact geometry and pressure angle of involute gear teeth [3]

The two biggest advantage of the involute tooth shape are:

- Tools required to ‘generate’ the shape are simpler to make and re-sharpen (A rack cutter or a ‘hub’ both have straight sides.)

- The fundamental law of gearing is not broken if the centre distance between gears is changed. This is discussed later.



5.0 GEAR-TOOTH NOMENCLATURE Figure 2.7 demonstrates two teeth of a gear with the standard nomenclature defined [2, 3, 6]: The basic terminologies are explained below. PITCH DIAMETER (D): It is the diameter of an imaginary circle on which gear tooth is designed. Pitch circles of two spur gears are tangent to each other. Mathematically,

PND =

Where, N = Number of teeth P = Dimetral pitch NUMBER OF TEETH (N): The total number of teeth on a gear.

DPN =

DIAMETRAL PITCH (P): A ratio equal to the number of teeth to a unit length of pitch diameter.

DNP =

CIRCULAR PITCH (p): It is the arc length along the pitch circle circumference measured from the point of one tooth to the corresponding point on the adjacent tooth. Mathematically,

PNDp /1416.31416.3 ==

Figure 2.7: Gear-teeth terminology [1]



ADDENDUM (a): It is the amount of tooth that sticks out above the pitch circle (or it is the radial distance from the pitch circle to the top of the tooth). The standard addendum is equal to the module ‘m’ in millimetres.

Pa 1=

DEDENDUM (b): It is the radial distance from the pitch circle to the bottom of the tooth. The standard dedendum is equal to module x 1.25 (i.e. 1.25 m) in millimetres.

Pb 250.1=

CENTRE DISTANCE (C): The distance between the axes of two mating gears.

2___ diapitchofsumC =

WHOLE DEPTH (ht): It is the full height of the tooth. It equals to the sum of addendum and dedendum.

Pbaht /250.2=+= WORKING DEPTH (hk): It is defined as the distance that a tooth occupies in the mating space. It is equal to the two times the addendum.

Pahk /22 == CLEARANCE: It is the radial distance between the top of a tooth and the bottom of the mating tooth space. It is also the difference between the addendum and dedendum.

Pabc /250.0=−= OUTSIDE DIAMETER (Do): It is the overall diameter of the gear. It is equal to the pitch diameter plus two addendums

aDDo 2+= ROOT DIAMETER (Dr): It is the diameter of a circle that coincides with the bottom of the tooth spaces.

bDDr 2−=



CIRCULAR THICKNESS (t): It is defined as the length of an arc between the two sides of a gear tooth measured on the pitch circle. It is also known as tooth thickness.

Pt /5708.1= CHODRAL THICKNESS (tc): It is the straight line thickness of a gear tooth measured on the pitch circle (Figure 2.8)

)/90sin( NDtc=

CHODRAL ADDENDUM (ac): The height from the top of the tooth to the line of the chodral thickness (Figure 2.8).

Dtaac 4/2+=

Figure 2.8: Gear-teeth terminology [2]

BASE CIRCLE DIAMETER (DB): It is the diameter of a circle from which the involute tooth form is generated.

)cos(φDDB = TOOTH THICKNESS: The tooth thickness is measured on the pitch circle from one side of a tooth to the other side. It is also known as circular thickness. Theoretically, t = p/2 SPACE WIDTH: The tooth space (space width) is also measured on the pitch circle, from the right-side of one tooth to the left-side of the next tooth. Theoretically, it is equal to the tooth thickness. But for practical reasons, the tooth space is made slightly larger than the tooth thickness.



BACKLASH: The difference between tooth thickness and tooth space is called the backlash. Some backlash is need for lubrication and proper functioning of gears in mesh. To provide backlash: • the cutter generating the gear teeth can be fed more deeply than the theoretical value on

either or both of the mating gears, or • the centre distance can be adjusted to a larger value than the theoretical value MODULE (m): It is essentially the reciprocal of diametral pitch (P) and is defined as the pitch diameter in millimetres divided by the number of teeth. In other words, it is the number of millimetres of pitch diameter per tooth. Mathematically,

NDm =

THE PRESSURE ANGLE (Ø): The pressure angle Ø (as shown in Figure 2.9) in a gearset is defined as the angle between the line of action (common normal) and the direction of velocity at the pitch point such that the line of action is rotated Ø degrees in the direction of rotation of the driven gear.

Figure 2.9: The pressure angle Ø [1]



Technically speaking, any pressure angle Ø is possible but it has been standardised at a few values by the gear manufactures. These values are defined at the nominal centre distance for the gearset as cut. Only three have been in common use:

a) 14.5° in use for many years but now obsolete.

b) 20° Most common in DP and Module form

c) 25° in USA only.

Figure 2.10: Tooth-profiles for three pressure angles [3]

Note that the actual pressure angle (i.e. operating pressure angle) is a function of the installed centre distance. This is explained in the following sections:



6.0 OPERATION OF GEARS AT EXTENDED CENTRE DISTANCE Centre distance is related to a pair of gears in mesh. It is measured from the centre of the pinion (smaller gear) to the centre of the gear; the sum of the pitch radii of two gears in mesh (Figures 2.11 and 2.12). Mathematically,

𝐶𝐶 =𝐷𝐷𝐺𝐺2

+𝐷𝐷𝑃𝑃2

=12

(𝐷𝐷𝐺𝐺 + 𝐷𝐷𝑃𝑃)

Where,

C = Centre distance

DG = Pitch diameter of gear

DP = Pitch diameter of pinion

Design of gearsets is based on theoretical centre distances, which is calculated from the theoretical pitch diameters of the gears in mesh. However, the final installation of gears may result in different value of centre distance. This variation in the centre distance will directly affect the pressure angle (Ø), as shown in Figure 2.11.

Figure 2.11: Changing centre distance of involute gears [3]



Despite the variation in the centre distance, the speed ratio remains unaffected. This fact is further explained by considering two gears in mesh at the design centre distance and then at extended (actual) centre distance (Figure 2.12).

Figure 2.12: Operation of gears at extended centre distance

Therefore,

Design Distance Apart Figure 2.12a Design pressure angle (Ø) Pitch Circle Diameters Dp1 = m N1 Dp2 = m N2

2/2/)cos(

2

2

p

b

DD

=φ

2/2/)cos(

1

1

p

b

DD

=φ

1

2

1

2

2

1

p

p

b

b

DD

DD

==ωω

1

2

1

2

1

2

1

2

2

1

NN

mNmN

DD

DD

p

p

b

b ====ωω

p

b

DD

=)cos(φ

Also, the angular speed ratio is given as,

2

2

')'cos(

p

b

DD

=φ

1

2

2

1

''

p

p

DD

=ωω

)'cos(/)'cos(/

1

2

2

1

φφ

ωω

b

b

DD

=

1

2

2

1

b

b

DD

=ωω

Extended Distance Apart Figure 2.12b In this case, different point of involute is in contact. Pressure angle is changed, pitch circle diameters are changed although they retain the same proportions to each other.

The angular speed ratio remains unchanged, whereas the pressure angle has been changed.



Note that increasing the centre distance results in increasing the pressure angle. The fundamental law of gearing still holds in the increased centre distance case because the common normal is still tangent to the two base circles and still passes through the pitch point. Clearly, involute shape gears comply with the fundamental law of gearing – even if the centre distance between the gears is changed.

It is also important to note that the actual pressure angle (operating pressure angle) is a function of the installed centre distance.

𝐶𝐶𝐶𝐶𝐶𝐶 (∅′) = 𝐷𝐷𝑃𝑃2𝐷𝐷𝑃𝑃1

(Refer to Figure 2.12b)

EXAMPLE: Determine the operating pressure angle for a gearset that has the following specifications. Design pressure angle = 20°, module = 5 mm, number of teeth on pinion = 20, number of teeth on gear = 60, and installed at 201 mm centre distance.

SOLUTION

Design pressure angle (Ф) = 20° Module = 5 mm Number of teeth on pinion (N1) = 20 Number of teeth on gear (N2) = 60 Centre distance = 201 mm

Operating pressure angle (Ф’) =?

DB1 = D1cosФ

DB1 = mN1cos20° DB1 = 93.97 mm

DB2 = D2cosФ

DB2 = mN2cos20° DB2 = 281.91 mm

𝑐𝑐 = 𝐷𝐷12

+ 𝐷𝐷22

= 12� 𝐷𝐷𝐵𝐵1𝐶𝐶𝐶𝐶𝐶𝐶(∅′ )

+ 𝐷𝐷𝐵𝐵2𝐶𝐶𝐶𝐶𝐶𝐶(∅′ )

� = 12 𝐶𝐶𝐶𝐶𝐶𝐶(∅′ )

(93.97 + 281.91)

201 =1

2𝐶𝐶𝐶𝐶𝐶𝐶(∅′)(93.97 + 281.91)

∅′ = 𝐶𝐶𝐶𝐶𝐶𝐶−1(375.88

402)

∅′ = 20.77°



7.0 GEAR TOOTH SYSTEMS A tooth system is a standard that specifies the relationships involving addendum, dedendum, working depth, tooth thickness, and pressure angle. There are two systems – Diametral Pitch System and Metric Module System:

7.1 Diametral Pitch System This is the most commonly used system in the United States. It is based on the number of teeth per inch of pitch diameter. Mathematically,

DN

DimatereCirclePitchTeethOfNumberP ==

____

(in inches)

7.2 Diametral Pitch System The metric system – uses ‘Modules’: ‘m’ and millimetres

ND

TeethOfNumberDimatereCirclePitchm ==__

__ (in millimetres)

The two systems are not compatible in any way. With metric system, commonly used standard values of modules are: 0.2 to 1.0 by increments of 0.1 1.0 to 4.0 by increments of 0.25 4.0 to 5.0 by increments of 0.5 Increasing module results in larger teeth and increasing the pressure angle Ø results in smaller teeth. The designer usually specifies teeth of the smallest size which will carry the load. This produces:

- The best contact ratio and therefore smoother gear mesh - Less likelihood of interference

8.0 GEAR MANUFACTURING Involute gear teeth may be cut using a ‘form’ tool – i.e. one which is the exact shape to remove the material between two teeth, leaving involute form flanks. However, it is usual to generate the involute using either a ‘rack’ cutter or a ‘hob.’ Gear generation by a rack-cutter is shown in Figure 2.13, whereas the hobbing method of gear cutting is illustrated in Figure 2.14.

The rack is a gear of infinite diameter (i.e. it is straight) and the involute profile, in this case, is a straight line. Gear teeth, generated from a straight sided rack or hob, are of perfect involute form.



Figure 2.13: Gear cutting with a rack cutter [1]

Figure 2.14: Gear-teeth cutting by the hobbing method [7]

Having straight sides on the cutting tool facilitates cutter manufacture and resharpening. Figure 2.15 shows the steps in gear tooth generation, using a reciprocating rack cutter.



Figure 2.15: Steps in tooth generation



Manufacturing process will not be dealt with here – however it is possible in some circumstances not to generate an involute tooth profile – if the cutter extends below the base circle. An involute cannot exist beneath the base circle.

8.1 INTERFERENCE Interference occurs if either of the addendum circles extends beyond tangent points a and b, as shown in Figure 2.16. This prevents rotation of the mating parts. The points a and b are called interference points.

Figure 2.16: Interference points a and b [1]

As shown in Figure 2.17, it is evident that the gears will not operate without modification. The gear tooth can be corrected to overcome this problem. The preferred approach is to remove the interfering tip, shown shaded in Figure 2.17.



Figure 2.17: Interference preventing gears to operate [1]

8.2 UNDERCUTTING Undercutting refers to the process of cutting away the material at the fillet or root of the gear teeth, thus alleviating the interference.

Figure 2.18: Undercutting of a gear tooth [8]



Caution must be exercised as undercutting of gear teeth:

• Weakens the gear teeth (Figure 2.18) • Results in non-conjugate action below the base circle

The alternate options include modification of the addendum on the pinion or the gear, or modification of the centre distance.

NOTE: When gear teeth are manufactured with a rack cutter, as described above, the gear teeth are automatically undercut if they would interfere with a rack.

MAXIMUM POSSIBLE ADDENDUM

From Figures 2.16 and 2.17,

arra +=

Where

ra = addendum circle

r = pitch circle radius

a = addendum

The equation for the maximum possible addendum circle radius without interference is given as

φ222(max) sincrr ba +=

Where

ra(max) = maximum non-interfering addendum circle radius of pinion or gear

c = centre distance

Ø = pressure angle (actual, not nominal, value)

It is also noted that:

I. Interference is more likely to involve the tips of the gear teeth than the tips of the pinion teeth

II. Inference is promoted by having small number of pinion teeth, a large number of gear teeth, and a small pressure angle.



8.3 MINIMUM NUMBER OF TEETH REQUIRED TO AVOID UNDERCUTTING

A gear having few teeth, generated by a rack or hob cutter, with undercut flanks is shown in Figure 2.19a. For such gears, the end of the cutting tool extends inside the interference point A (also called as the point of tangency of base circle and pressure line) and removes excessive amount of metal (undercutting)

Undercutting can be avoided by using more and smaller teeth in the gear. No undercutting can occur if the addendum is reduced so that it does not extend inside the interference point A. The tooth with reduced addendum is called a stub tooth (such teeth can provide 80% of the usual working depth). In this case, the addendum will be 0.8 x module (0.8 m).

To generalise it, let addendum

mka ×= (1)

From the geometry of the Figure 2.19a (further illustrated in Figure 2.19b), it can be determined that the minimum amount of addendum required to avoid undercutting is

φ2cosrra −= (2)

Therefore, equations 1 and 2 will give

φ2cosrrmk −=×

)cos1( 2 φ−= rkm (3)

We also know that the pitch diameter is given as

NmrD ×=×= 2

2Nmr ×

=

Therefore,

)cos1(2

2 φ−=mNkm

So the minimum number of teeth will be

φφ 22 sin2

cos12 kkNmm =

−=



Figure 2.19a: Undercutting due to insufficient number of gear teeth [4]

Figure 2.19b: Amount of addendum

Now calculating the number of teeth for the commonly used values of pressure angle will produce the following results (Table 2.1) .



Table: 2.1: Minimum number of teeth

PRESSURE ANGLE (Ø) φ2sin

2kNmm = MINIMUM NUMBER OF TEETH (Nmm)

k

14.5° 31.9 32 k=1 for full depth

teeth. 20° 17.097 18 (17 is also used)

25° 11.2 12

20° 13.68 14 k=.8 (stub teeth)

Note that increasing pressure angle allows smaller number of teeth.

8.4 CONTACT RATIO For a smooth operation of a pair of gears, it is an essential requirement that the tooth profiles are proportioned so that a second pair of mating teeth comes into contact before the first pair is out of contact. The average number of teeth in contact as the gears rotate is the contact ratio (CR). It is also defined as the ratio of the length of the line-of-action to the base pitch for the gear. It is calculated from the following equation [1].

b

bgagbpap

PcrrrrCR φsin2222 −−+−

=

Where rap, rag = addendum radii of mating pinion and gear rbp, rbg = base circle radii of the mating pinion and gear The base pitch Pb is

NDP bb /π= Where N = Number of teeth Db = Diameter of the base circle The following relationships can be derived from Figure 2.20

φcosdDb =

φcosrrb =

φcospPb =



Figure 2.20: Gear and pinion in mesh [1]

Generally, the greater the contact ratio, the smoother and quieter the operation of the gears. A contact ratio of 2 or more means that at least two pairs of teeth are theoretically in contact at all times. A recommended minimum contact ratio is 1:2 and typical spur gear combinations often have values of 1.5 or higher.

8.5 GEAR MATERIALS Gear materials are expected to have good strength (especially fatigue strength), good wear resistance, high resistance to surface fatigue, high resilience, good damping ability, and good machinability. Commonly used gear materials are:

– Steel and steel alloys

– Gray and alloy cast iron

– Brass

– Bronze

– Polymeric materials



9.0 GEAR TYPES*

There are many types of gears, as shown in Figure 3.1. Selection of the best type of gearing for a particular application depends upon several factors, including geometric arrangement proposed for the machine, reduction ratio required, power to be transmitted, speeds of rotation, efficiency goals, noise-level limitations and cost constraints.

Figure 3.1: Commonly used types of gears [7]

Generally, the following three shafting orientations are encountered by the designer when considering the transmission of power or motion from rotating shaft to another. - Applications in which the shaft axes are parallel - Applications in which the shaft axes intersect - Applications in which shaft axes are neither parallel nor do they intersect Gears may be grouped according to the type of shafting arrangement required for a particular application. Following are the commonly used categories:

* The information provided in this section is based on the book titled”Mechanical Design of Machine Elements and Machines (2003) by Jack A. Collins, published by John Wiley & Sons. For further details, read chapter 15 of this book.



9.1 GEARS FOR PARALLEL SHAFTS Figure 3.2 shows types of gears in applications with parallel shafts. Different types are further explained below.

Figure 3.2: Gears for use in applications with parallel shafts [7]



SPUR GEARS Spur gears connect parallel shafts. Spur gear teeth are straight and parallel to the gear shaft axis. They are further classified as external (Figure 3.3) and internal (Figure 3.2b) spur gears. Spur gears are:

– Relatively simple to design, manufacture and check for precision

– Relatively inexpensive

– Impose only radial loads on supporting bearings.

– Noisy in high speed applications

Tooth-profiles are ordinarily involute in shape, and therefore tolerate small variations in centre distance.

Figure 3.3 : Spur gears [2]

HELICAL GEARS Helical gears, as shown in Figure 3.4, are very similar to spur gears except that their teeth are angled with respect to the axis of shaft to form parallel helical spirals. The helixes for the two mating external gears must have the same helix angle, but for parallel shaft applications the hand of the helix on the pinion must be opposite to the hand of the gear. Because of the angled teeth, helical gears:

- Allow more than one tooth to be in contact at the same time.

- Impose both radial and thrust (axial) loads on supporting bearings.

Helical gears tolerate small variations in centre distance, and are designed to carry more loads when compared to spur gears. The angled teeth provide gradual-engagement between mating gears, which produces smoother and quieter operation than with straight-tooth spur gears. Helical gears are further classified as:

- External helical gears (Figures 3.2c and 3.4)

- Internal helical gears (Figure 3.2d)



- Single helical gears (Figure 3.4a)

- Double helical gears (also known as Herringbone gears). As shown in Figure 3.4b, these gears have double helixes. This type is specifically designed to eliminate end thrust and provide long life under heavy loads.

Figure 3.4: Helical gears [2]

RACK AND PINION This is a special case of gearing used for parallel shafts. It is called rack-and-pinion drive, as shown in Figure 3.5. It is a combination of a spur gear (pinion) operating on a flat straight bar rack (may be a considered as a segment of a gear having infinite pitch radius). This type of arrangement is used to convert rotary motion to reciprocating or linear motion. The gear-teeth may be either straight spur or helical type.

Figure 3.5: Rack and pinion [2]



9.2 GEARS FOR SHAFTS HAVING INTERSECTING AXES Figure 3.6 shows types of gears for use in applications requiring shafts with intersecting axes. The gears types in this category are described below.

Figure 3.6: Gears for use in applications with intersecting shaft axes [7]



BEVEL GEARS Straight bevel gears, as shown in Figure 3.7, represent the simplest type of gearing used for intersecting shafts. Bevel gears are conical in shape, allowing the axes of the shafts of the gear and pinion to intersect at 90º or at any desired angle. The shape of the teeth on bevel gear is the same as the shape of the teeth on spur gear except they are tapered (toward the apex of the cone) in both thickness and height, from large tooth profile at one end to a smaller tooth profile at the other end.

Figure 3.7: Bevel gears [2]

Bevel gears impose both radial and thrust loads on supporting bearings. Sub-categories of bevel gears are:

Zerol†

bevel gears, as shown in Figure 3.6b, are similar to straight bevel gears except that they have teeth curved in their length-wise direction. This provides a slight engagement overlap to produce smoother operation than straight bevel gears.

MITER GEARS: Miter gears, a special case of bevel gears, have the same size for gear and pinion and are used when shafts must intersect at 90º without speed reduction.

SPIRAL BEVEL GEARS:

Spiral bevel gears, as shown in Figure 3.6c, are designed to provide gradual engagement along the tooth face (just like helical gears). This provides smoother operation.

Face Gears (Figure 3.6d) have the functionality features similar to bevel gears but have gear teeth cut on an annular ring at the outer edge of a gear “face.”

† Registered trademark of the Gleason Works, Rochester, NY



9.3 GEARS FOR SHAFTS WHERE SHAFT AXES ARE NEITHER PARALLEL NOR INTERSECTING

This type of gearing applications is shown in Figure 3.8. The gears used in such type of applications are Hypoid gears, spiroid gears, crossed-helical gears, and worm gears.

Figure 3.8: Gears for use in applications where shafts are neither parallel nor intersecting [7]



HYPOID GEARS Hypoid gears also have the same design as spiral bevel gears except the gear shaft axes are offset and do not intersect, as shown in Figure 3.9. The offset is measured along the perpendicular common to the two axes. If hypoid gears had zero offset, they would be spiral bevel gears.

Figure 3.9: Hypoid gears [2]

SPIROID GEARS This combination of gears, as shown in Figure 3.8b, involves a face gear with teeth spirally curved along their length, and a mating tapered pinion. Offsets for spiroid gearing are larger than for hypoid gearsets, and the pinion somewhat resembles a worm. This combination provides high reduction ratios with a compact design. The load-carrying is also good.

CROSSED HELICAL GEARS (Figure 3.8c)

Both members of this gearing system have a cylindrical shape, as opposed to worm gearsets in which one or both members are throated. Usually, the non-intersecting axes are 90 degree to each, but almost any angle can be accommodated. Load carrying capacity is low.

WORM GEARS This is another very popular gearing system used for high reduction ratios in small space. The worm, as shown in Figure 3.10, has teeth similar to the threads of a power screw. The worm gear (also known as worm wheel) has teeth similar to those of a helical gear except that they are contoured to envelop the worm.



Figure 3.10: Worm and worm gear [2]

SINGLE ENVELOPING (CYLINDRICAL)

This occurs when only the worm wheel teeth are contoured (throated) to envelop a cylindrical worm, as shown in Figure 3.8f.

DOUBLE ENVELOPING

In this type of enveloping, the worm profile is also throated to envelop the gear, as shown in Figure 3.8e.



10.0 REFERENCES 1. Juvinall, R.C. and K.M. Marshek, Fundamentals of Machine Component Design. Fourth

ed. 2006: John Wiley & Sons, Inc. 2. David A. Madsen, David P. Madsen, and J.L. Turpin, Engineering Drawing and Design.

4th ed. 2007: Thomson Delmar Learning. 3. Norton, R.L., Machine Design: An Integrated Approach. Third ed. 2006: Pearson Prentice

Hall. 4. Spotts, M.F., T.E. Shoup, and L.E. Hornberger, Design of Machine Elements. Eighth ed.

2004: Pearson Prentice Hall. 5. Budynas, R.G. and J.K. Nisbett, Shigley's Mechanical Engineering Design. Eighth ed.

2008: McGraw Hill. 6. Cecil H. Jensen and J.D. Helsel, Interpreting Engineering Drawings. 7th ed. 2007:

Thomson Delmar Learning. 7. Collins, J.A., Mechanical Design of Machine Elements and Machines. 2003: John Wiley &

Sons. 8. Mott, R.L., Machine Elements in Mechanical Design. 4th ed. 2006, Singapore: Prentice

Hall (Pearson Education South Asia Pte Ltd).

PRACTICE QUESTIONS Q1: Describe law of gearing.

Q2: What are the problems associated with undercutting?

Q3: What is the difference between spur and helical gears? Draw sketches to support your answer.

Q4: Draw a sketch to define pressure angle. How would it affect the pressure angle if the installed centre distance is greater than the theoretical centre distance?

Q5: What is the cause of interference in gears?

Q6: What type of gearing is suitable for converting rotary motion to linear motion?

Q7: Define contact ratio.

Q8: What is the difference between full-depth and stub-teeth gears?

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