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What is gearing for? Types and principles of operation of gears. Machine parts. Lecture notes. Basic requirements for the design of machine parts
As a result of the study, the student should know:
Scope of gears;
- classification of gears.
4.1.1 The role and importance of gears in mechanical engineering
Gears are the most common types of mechanical transmissions. They are widely used in all branches of mechanical engineering, in particular in machine tools, automobiles, tractors, agricultural machines, etc., in instrument making, the watch industry, etc. They are used to transmit power from fractions to tens of thousands of kilowatts at peripheral speeds up to 150 m / s and gear ratios up to several hundred and even thousands, with wheel diameters from fractions of a millimeter to 6 m or more.
Gear transmission refers to gears with direct contact with a pair of gears. The smaller of the transmission wheels is called a gear, and the larger one is called a wheel. The gear train is designed primarily to transmit rotational motion.
4.1.2 Benefits of gears
1) high load capacity;
2) small dimensions;
3) high reliability and durability (40000 h);
4) the constancy of the gear ratio;
5) high efficiency (up to 0.97 ... 0.98 in one stage);
6) Easy to operate.
4.1.3 Disadvantages of gears
1) increased requirements for the accuracy of manufacturing and installation;
2) noise at high speeds;
3) high rigidity, which does not allow compensating dynamic loads.
4.1.4. Gear classification
1. According to the mutual arrangement of the geometric axes of the shafts, gears are distinguished:<>br - with parallel axes - cylindrical (Fig. 2.3.1.a-d);
- with intersecting axes - conical (Fig. 2.3.1.d; e);
- with crossed axes - cylindrical screw (Fig. 2.3.1.g);
- conical hypoid and worm (Fig. 2.3.1.h);
- rack and pinion (Fig. 2.3.1.i).
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Figure 2.3.1 Types of gears
2. Depending on the relative position of the gears:
- with external gearing (gear wheels rotate in opposite directions);
- with internal gearing (the direction of rotation of the wheels is the same).
3. According to the location of the teeth on the surface of the wheels, gears are distinguished:
- spur; helical; chevron; with round teeth.
4. According to the shape of the tooth profile, gears are distinguished:
- involute;
- with the engagement of M. L. Novikov;
- cycloidal.
5. According to the circumferential speed, gears are distinguished:
- low-speed ();
- medium speed
4. The main types of mechanical gears.
mechanical transmission called a device for transmitting mechanical movement from the engine to the executive bodies of the machine. It can be carried out with a change in the value and direction of the speed of movement, with the transformation of the type of movement. The need to use such devices is due to the inexpediency, and sometimes the impossibility of direct connection of the working body of the machine with the motor shaft. Mechanisms of rotational movement allow for continuous and uniform movement with the least energy loss to overcome friction and the least inertial loads.
Mechanical transmissions of rotational motion are divided into:
According to the method of transferring movement from the leading link to the slave link for gears friction(friction, belt) and engagement(chain, gear, worm);
According to the ratio of the speeds of the driving and driven links on slowing down(reducers) and accelerating(animators);
According to the mutual arrangement of the axes of the driving and driven shafts for gears with parallel, intersecting And intersecting shaft axes.
gears
gear train a three-link mechanism is called, in which two moving links are gears, or a wheel and a rack with teeth that form a rotational or translational pair with a fixed link (body).
The gear train consists of two wheels, through which they interlock with each other. A gear with fewer teeth is called gear, with a large number of teeth - wheel.
planetary gears
planetary transmissions containing gears with moving axles are called. The transmission consists of a central gear with external teeth, a central gear with internal teeth, a planet carrier and satellites. Satellites rotate around their axes and together with the axis around the central wheel, i.e. move like planets.
Worm gears
Worm-gear used to transfer rotation from one shaft to another when the axes of the shafts intersect. The crossing angle in most cases is 90º. The most common worm gear consists of the so-called Archimedean worm, i.e. screw having a trapezoidal thread with a profile angle in the axial section equal to the double engagement angle (2 α = 40), and a worm wheel.
Wave mechanical transmissions
Wave transmission is based on the principle of transformation of motion parameters due to wave deformation of the flexible link of the mechanism.
Wave gears are a type of planetary gears in which one of the wheels is flexible.
Friction gears
Gears, the operation of which is based on the use of friction forces arising between the working surfaces of two bodies of rotation pressed against each other, are called friction gears.
Belt drives
Belting consists of two pulleys mounted on shafts and a belt covering them. The belt is put on pulleys with a certain tension, providing friction between the belt and the pulleys, sufficient to transfer power from the drive pulley to the driven one.
Depending on the shape of the cross section of the belt, there are: flat belt, V-belt and round belt
chain drives
chain drive consists of two wheels with teeth (asterisks) and a chain covering them. The most common gears are bush-roller chain and toothed chain. Chain gears are used to transfer medium power (not more than 150 kW) between parallel shafts in cases where the center distances are large for gears.
Transmission screw-nut
Transmission screw-nut serves to convert rotational motion into translational. The widespread use of such gears is determined by the fact that, with a simple and compact design, it is possible to carry out slow and precise movements.
In the aircraft industry, the screw-nut transmission is used in aircraft control mechanisms: to move the take-off and landing flaps, to control trimmers, rotary stabilizers, etc.
The advantages of the transmission include simplicity and compactness of the design, a large gain in strength, and accuracy of movement.
The disadvantage of the transmission is a large friction loss and the associated low efficiency.
Cam mechanisms
Cam mechanisms(Fig. 2.26) in terms of breadth of application they are second only to gears. They are used in machine tools and presses, internal combustion engines, machines for the textile, food and printing industries. In these machines, they perform the functions of approaching and retracting tools, feeding and clamping material in machines, pushing, turning, moving products, etc.
Types of mechanical gears and transmission mechanisms
Rotational motion in machines is transmitted by friction, gear, belt, chain and worm gears. We will conditionally call a pair that performs rotational movement wheels. The wheel from which the rotation is transmitted is called the driving wheel, and the wheel receiving the movement is called the driven wheel.
Any rotational movement can be measured in revolutions per minute. Knowing the RPM of the drive wheel, we can determine the RPM of the driven wheel. The number of revolutions of the driven wheel depends on the ratio of the diameters of the connected wheels. If the diameters of both wheels are the same, then the wheels will spin at the same speed. If the diameter of the driven wheel is larger than the driving wheel, then the driven wheel will spin more slowly, and vice versa, if its diameter is smaller, it will make more revolutions. The number of revolutions of the driven wheel so many times less than number revolutions of the drive, how many times its diameter is greater than the diameter of the drive wheel.
The dependence of the number of revolutions on the diameters of the wheels.
In engineering, when designing machines, it is often necessary to determine the diameters of the wheels and their number of revolutions. These calculations can be done on the basis of simple arithmetic proportions. For example, if we conventionally denote the diameter of the drive wheel as D 1, the diameter of the driven through D 2, the number of revolutions of the drive wheel through n 1, the number of revolutions of the driven wheel through n 2, then all these quantities are expressed by a simple relation:
D 2 / D 1 \u003d n 1 / n 2
If we know three quantities, then by substituting them into the formula, we can easily find the fourth, unknown quantity.
In technology, one often has to use the expressions: "gear ratio" and "gear ratio". The gear ratio is the ratio of the number of revolutions of the drive wheel (shaft) to the number of revolutions of the driven one, and the gear ratio is the ratio between the numbers of revolutions of the wheels, regardless of which one is leading. Mathematically, the gear ratio is written like this:
n 1 / n 2 = i or D 2 / D 1 = i
Where i- gear ratio. The gear ratio is an abstract value and has no dimension. The gear ratio can be anything - both integer and fractional.
friction gear
With frictional transmission, the rotation from one wheel to another is transmitted using friction force. Both wheels are pressed against each other with some force and, due to the friction between them, rotate one another. Disadvantage of friction gear: a large force pressing on the wheels, causing additional friction, and therefore requiring additional force to rotate. In addition, the wheels during rotation, no matter how they are pressed against each other, give slippage. Therefore, where an exact ratio of the number of revolutions of the wheels is required, the friction transmission does not justify itself.
Advantages of friction transmission:
Easy to manufacture rolling elements;
Uniform rotation and quiet operation;
Possibility of stepless speed control and on/off transmission on the go;
Due to the possibility of slipping, the transmission has safety properties.
Disadvantages of friction gear:
Slippage leading to inconsistent gear ratio and loss of energy;
The need for clamping.
Application of friction gear:
In mechanical engineering, stepless friction gears are most often used for stepless speed control.
Friction gears:
a - frontal gear, b - angular gear, c - cylindrical gear.
In homemade devices, friction gear can be widely used. Cylindrical and frontal gears are especially acceptable. Gear wheels can be made of wood. For better grip, the working surfaces of the wheels should be "sheathed" with a layer of soft rubber 2-3 mm thick. Rubber can be either nailed with small cloves or glued with glue.
Gear
In gears, the rotation from one wheel to another is transmitted by means of teeth. Gear wheels rotate much easier than friction wheels. This is explained by the fact that here pressing the wheel on the wheel is not required at all. For proper engagement and easy work wheels, the tooth profile is made along a certain curve, called an involute.
v transmit rotational motion;
v change the rpm;
v increase or decrease the force of rotation;
v change direction of rotation.
Depending on the shape of the wheels and their relative position, the following are distinguished: types of gears : cylindrical, conical, worm, rack, planetary.
Cylindrical gear consists of two or more cylindrical wheels mounted on parallel shafts.
Rice. 215 Cylindrical gear
Bevel gear consists of two conical wheels located on two shafts, the axes of which intersect. The angle of intersection can be any, but usually it is 90º.
Rice. 216 Bevel gear
Worm gear (gear screw gear) - mechanical transmission, carried out by engaging the worm and the worm wheel associated with it. Worm gear is used for crossing but not intersecting shafts. The worm gear consists of a screw (worm) and a gear.
Rice. 217 Worm gear
Worm gear has a number of unique properties. Firstly, it can only be used as a driving gear, and cannot be used as a driven gear. This is very convenient for mechanisms that are needed to lift and hold a load without loading the engine. There are many possible applications this property of the worm gear, for example, in many types of cranes and loaders, railway barriers, drawbridges, winches. The LEGO worm gear is very widely used in the design of the grip for the robotic arm.
Secondly, a characteristic feature of the worm gear is that it has a large gear ratio. Therefore, worm gears are used as a reduction gear whenever there is a very high torque.
Conclusion: Worm gear has a number of advantages:
v Takes up little space.
v Has the property of self-braking.
v Reduces rpm by many times.
v Increases drive force.
v Changes the direction of rotation by 90°.
rack and pinion - a mechanical transmission that converts the rotational movement of the gear into forward movement rails and vice versa. The rack can be considered as a circle of a large gear wheel elongated in a straight line.
It should be noted that there are ring gears and internal gears in LEGO sets.
ring gear - this is a special type of gears, their teeth are on the side surface. Such a gear works, as a rule, in tandem with a spur gear.
Rice. 220 Connections crown gear and spur gears with 8 and 24 teeth
Gears with internal gearing have teeth cut from the inside. When they are used, one-way rotation of the driving and driven gears occurs. This gear train has less friction costs, which means higher efficiency *. Gears with internal gearing are used in mechanisms of limited dimensions, in planetary gears, in the drive of a manipulator robot.
Rice. 221 Internal gear
The special feature of the LEGO internal gear is that it has teeth on the outside so it can be used in gears like a 56 tooth spur gear.
Rice. 222 Ways of connecting a wheel with internal gearing to a cylindrical gear, a wheel with a crown and a "worm"
Rice. 223 Method of connecting a wheel with internal gearing to a motor
planetary gear
planetary gear (differential gear) - a mechanical system consisting of several planetary gears (gears) rotating around a central, sun, gear. Usually planetary gears are fixed together with a planet carrier. The planetary gear may also include an additional external annular (crown) gear having internal engagement with the planetary gears.
Such a transmission has found wide application, for example, it is used in kitchen appliances or an automatic transmission of a car.
The main elements of the planetary gear can be considered as follows:
v Sun gear: located in the center;
v Carrier: rigidly fixes the axes of several planetary gears (satellites) of the same size relative to each other, which are engaged with the sun gear;
v Ring gear: an external gear that meshes internally with the planetary gears.
Rice. 224 An example of a planetary gear: the carrier is motionless, the sun is leading, the crown is driven
In a planetary gear, torque is transmitted using any (depending on the selected gear) of its two elements, of which one is the master, the second is the slave. The third element is stationary (table 8).
Table 8. Elements of the planetary gear
|
Fixed |
Leading |
Slave |
Broadcast |
|
Crown |
lowering |
||
|
Boosting |
|||
|
Sun |
lowering |
||
|
Boosting |
|||
|
carrier |
Reverse, lowering |
||
|
Reverse boost |
Reverse - change the course of the mechanism to reverse, opposite.
Rice. 225 An example of the design of a planetary gear: the crown is stationary, the carrier is leading, the sun is driven
Mechanical transmissions with flexible elements
To transfer motion between shafts relatively far apart, mechanisms are used in which the force from the driving link to the driven one is transmitted using flexible links. Belts, cords, chains of various designs are used as flexible links.
Gears with flexible links can provide a constant and variable gear ratio with a step or smooth change in its value.
Belting
The belt drive consists of two pulleys mounted on shafts and a belt covering these pulleys. The load is transmitted due to the friction forces that arise between the pulleys and the belt due to the tension of the latter. The belt drive is not very sensitive to the relative position of the drive and driven shafts. They can even be turned at right angles to each other or put on a belt in the form of a crossed loop, and then the direction of rotation of the driven shaft will change.
Rice. 226 Belt drive
chain drive
Rice. 227 Chain drive
friction gear
Rice. 228 Friction gear
With frictional transmission, the rotation from one wheel to another is transmitted using friction force. Both wheels are pressed against each other with some force and due to the friction that occurs between them, one rotates the other.
Friction gears are widely used in cars. Disadvantage of friction gear: a large force pressing on the wheels, causing additional friction in the car, and therefore requiring additional force to rotate.
In addition, the wheels during rotation, no matter how they are pressed against each other, give slippage. Therefore, where an exact ratio of the number of revolutions of the wheels is required, the friction transmission does not justify itself.
Project "Automatic barrier":
1. Design a model of an automatic barrier.
Specifications:
b) the design uses a worm gear;
c) automatic raising and lowering of the barrier boom must be carried out using an ultrasonic sensor.
4. As part of the robotics circle, make an automatic barrier.
6. Write a description of the automatic barrier in your workbook.
Project "Turnplatform":
1. Design a turntable model.
Specifications:
b) the design uses a gear with internal gearing;
c) automatic rotation of the platform occurs with the help of a touch sensor (light sensor).
4. As part of the robotics circle, make a turntable.
6. Write a description of the turntable in your workbook.
Project "Slidingautomatic doors":
1. Design a model of sliding automatic doors.
Specifications:
a) the model includes one servomotor, NXT microcontroller;
b) rack and pinion is used in the design;
c) automatic door opening occurs with the help of an ultrasonic sensor (light sensor).
2. Sketch the model in the workbook.
3. Discuss the project with the teacher.
4. As part of the robotics circle, make a model of automatic sliding doors.
5. Using the NXT-G programming language, write a program to control the model.
6. Write a description of the automatic sliding door model in your workbook.
Gear assignment – transfer motion from one shaft to another with a change in angular velocities and moments in magnitude and direction. This transmission consists of two wheels. The transmission of torque in the gear train is due to the pressure of the teeth in engagement of one wheel on the teeth of the other. Gear transmissions are widely used in Russia and abroad due to their advantages over other mechanical transmissions.
Advantages: great durability and high reliability; high efficiency (up to 0.98); constancy of the gear ratio; possibility of application in a wide range of torques, speeds and gear ratios; small dimensions; ease of operation.
Flaws: the presence of noise; the impossibility of a smooth change in the gear ratio; the need for high precision manufacturing and installation, which increases their cost.
According to the original contour, gears are divided:
- on involute - mainly common in industry;
- with a circular profile (M. L. Novikov's engagement) - are used for gears with heavy loads.
In involute engagement, the working surface of the tooth has an involute profile. In what follows, we will consider only gears with involute gearing.
Gears include cylindrical, bevel, planetary, wave, etc.
Cylindrical gears
spur gearis called a parallel axle transmission. They come with a straight tooth (Fig. 4.13, A), oblique tooth, (Fig. 4.13, b) and chevron, (Fig. 4.13, V)(β is the angle of inclination of the tooth). It is recommended not to exceed the maximum gear ratios in one stage, otherwise the overall dimensions of the mechanisms increase compared to a two-stage transmission with the same gear ratio.
Advantages gears with a chevron and helical tooth compared to a straight one: greater bending strength of the tooth (more
Rice. 4.13
load capacity); greater smoothness of gearing and low noise, as well as smaller dynamic loads.
Flaws , the presence of axial force in helical gears; great manufacturing complexity.
Helical gears are used at peripheral speeds m / s; chevron gears - mainly in heavily loaded gears.
Kinematics and geometry of spur gears. Gear ratio, where is the angular frequency of rotation of the i-th shaft.
For external gearing (see Fig. 4.4, A- wheel rotation in different directions) i is taken with the "-" sign, for internal (see Fig. 4.4, b- rotation in one direction) with a "+" sign. From the kinematic condition - equality of velocities at the point of contact of the teeth of the wheels, , we obtain ,
where is the frequency of rotation of the i-ro wheel; is the pitch diameter of the gear.
Taking ( is the number of teeth of the i-th wheel) and taking into account relation (4.3), we obtain
(4.4)
where is the gear ratio (always a positive value). It is customary to call the smaller of the gears in a pair gear and designate "sh" or "1", and more - wheel("k" or "2"),
There are downshifts (Fig. 4.14, A), which lower the speed and are used in gearboxes;
Rice. 4.14
overdrives (rps. 4.14, b), which increase the rotational speed and are used in multipliers.
Gears are mainly used with involute gearing, which provides a constant gear ratio, low speeds of sliding in the gearing and easy manufacturing. Since rolling friction predominates in the transmission, and sliding friction is small, it has a high efficiency. This engagement is not very sensitive to the deviation of the center distance. In involute engagement, the working surface of the tooth has the shape of an involute. involute call the curve described by the point-forming line N–N, rolling without slipping along the main circle of diameter. The generatrix of the straight line is always perpendicular to the involute, and the segment is its radius of curvature (Fig. 4.15).
Let's move on to the consideration of the geometry of involute gears.
On fig. 4.16 shows a helical gear, for which the normal pitch is determined by the formula
where - circumferential pitch - the distance between the same profiles of adjacent teeth, measured along the arc of the pitch circle of the gear; - the angle of inclination of the tooth.
Rice. 4.15
Rice. 4.16
The district module is a value that is at times less than the district step:
Dividing formula (4.5) by π, we obtain
where is the normal module, specified according to GOST, which makes it possible to use a standard tool, for example, modular cutters.
The modulus is the main parameter of the gearing.
The pitch circle length of the gear is determined by the formula
Dividing both sides of the equation by π, we obtain an expression for determining the pitch diameter
which confirms the relation adopted in formula (4.4).
Gears are cut with a tool rack. The circle of the gear wheel, on which the pitch p and the angle of engagement, respectively, are equal to the pitch and angle of the profile a of the tool rack, is called divisive ( d). On On a rail, the dividing plane is the plane on which the thickness of the teeth is equal to the width of the cavity. Conjugated pairs of gears touch each other at the gearing pole. Circles passing through the pole of engagement R and rolling over one another without slipping are called initial(Figure 4.17, A, where, are the diameters of the initial circles; is the engagement angle). Line segment AB the engagement line, limited by the circles of the tops of the gear and wheel teeth, is called the active section of the engagement line. This line determines the beginning of the entry of a pair of teeth into engagement and exit from it.
The distance between the initial and dividing circles is called the offset of the initial contour. The ratio of this offset to T called coefficient
Rice. 4.17
offsets (Fig. 4.18). The dividing and initial diameters are equal. When the tooth is cut, which is eliminated by introducing a positive offset. If you set the offset, then the total offset coefficient will be equal to
In this case, the teeth of the wheels have the same height, but the height of the head and root of the tooth, the diameters of the circles of the vertex
Rice. 4.18
tires and troughs are different. The thickness of the gear teeth increases, and the wheel decreases. If the condition is not you
is filled, then you need to enter the equalization bias factor .
The main geometric characteristics of a helical external gear with X= O are shown in fig. 4.17, b:
Pitch diameter
The gear engagement area is shown in fig. 4.19, where is the width of the teeth of the gear and wheel; is the working width of the tooth on which their contact occurs:
where is the relative width of the tooth (greater value for large loads);
(4.12)
- center distance ("+" - for external gearing, "-" - for internal).
Rice. 4.19
Geometrical parameters of the equivalent wheel for helical gear. Analytical determination of bending stresses in the dangerous section of oblique teeth is difficult due to their curvilinear shape and oblique arrangement of contact lines. Therefore, they move from helical gears to involute gears with a straight tooth. Stresses, as for straight teeth, can be determined by considering the normal section of oblique teeth (Fig. 4.20).
In a normal section, we get an ellipse with semiaxes A And b:
Using the expression known from geometry, we determine the radius of the circle of the ellipse at the point of contact R with mating wheel:
Pitch Diameter of Equivalent Gear
Taking we get the formula . Substituting into it, we determine the number of teeth of the equivalent wheel
Calculations of helical gears for strength are made for equivalent cylindrical spur gears with a pitch circle diameter and a number of teeth.
Manufacturing of gears. There are two methods of cutting teeth: copying and running.
Copy method consists in cutting the cavities between the teeth with modular disk cutters (Fig. 4.21a) or finger (Fig. 4.21, b). After cutting each
Rice. 4.20
Rice. 4.21
hollows, the workpiece is rotated by the engagement step. The root profile is a copy of the profile of the cutter's cutting edges. For cutting gears different number teeth require a different tool. The copying method is inefficient and less accurate than when running.
When grinding, the cutter is replaced with a grinding wheel of the corresponding profile.
Break-in method is based on reproducing the engagement of a gear pair, one of the elements of which is cutting tool- worm cutter (Fig. 4.22, A), dolbyak (Fig. 4.22, b) or rack comb (Fig. 4.22, V). When cutting with a gear comb, the workpiece rotates around its axis, and the tool rail 1 performs a reciprocating movement parallel to the axis of the workpiece 2 and translational movement parallel to the tangent to the rim of the workpiece. Combs cut spur and helical gears with a large engagement module. When cutting with a worm cutter, which has the shape of a tool rail in axial section, the workpiece and cutter rotate around their axes, ensuring the continuity of the process. The dolbyak has the shape of a gear with a cutting edge. It reciprocates along the workpiece axis and rotates with the workpiece. For cutting cylindrical wheels
Rice. 4.22
with an external arrangement of teeth, a cutter and a comb are used; for cutting wheels with an internal and external arrangement of teeth, cutters are used.
gear materials. If machining is performed after heat treatment, then the hardness of the gears should be HB 350. This material is used in fine-module gears and in gears with a module T< 2. To reduce the size of gears (usually when t> 2) it is necessary to strengthen the working surface of the tooth, which increases the allowable contact stress. Bulk hardening is used for medium carbon steels (for example, 40Kh, 40KhN, etc.) up to hardness HRCa > 45÷55. Such hardening makes the core less ductile, which contributes to tooth breakage. In modern gears, a ductile core is retained, and only the working surface of the tooth is strengthened by thermal (surface hardening of HFC), chemical-thermal methods (cementation and nitriding), high-energy physical impact (laser hardening, ion nitriding), etc. When cementing 12KhNZA steels , 18Kh2NMA, 15KhF surface hardness 56–62 HRC3; when nitriding steels 38Kh2Yu, 38Χ2ΜΙΟΛ - 50–55 HRC3; with ion nitriding – 80–90 HRCe; with laser hardening – 56–60 HRCe; with surface hardening of the working surface of the tooth, the mass of the gearbox is reduced by 1.5–2 times and, accordingly, its overall dimensions are reduced.
Gear accuracy. The standard provides gearing accuracy grades 1-12 (from most accurate to least accurate). The following accuracies are most widespread: 6 - increased accuracy (up to v= 20 m/s); 7 - normal accuracy (up to v= 12 m/s); 8 - reduced accuracy (up to v= 6 m/s); 9 - coarse accuracy (up to v= 3 m/s). Values of the highest permissible speeds v are given for spur gears, and for helical gears they must be increased by about 1.5 times. The degree of accuracy is assigned taking into account the operating conditions of the transmission and the requirements for it.
The degree of accuracy is characterized by the following main indicators:
- the norm of the kinematic accuracy of the wheel, which establishes the value of the total error in the angle of rotation of the gears per revolution. It is an important indicator for high-precision dividing mechanisms;
- the rate of smooth operation of the wheel, which determines the magnitude of the components of the total error of the angle of rotation of the gear wheel, which are repeated many times in one revolution of the transmission. It is associated with manufacturing inaccuracies in the step π of the profile and causes additional dynamic loads in the engagement;
- the contact norm, which characterizes the completeness of the fit of the side surfaces of the mating teeth. It is estimated by a trace on the working surface of the tooth after contact with a rotating wheel, the teeth of which are smeared with paint (Fig. 4.23).
The degree of accuracy must correspond to the circumferential speed in the mesh: the higher it is, the higher the transmission accuracy must be. Depending on the degree of accuracy and dimensions, tolerances are established for individual elements of engagement and transmission.
The lateral clearance between the teeth (Fig. 4.24, where is the tolerance; are the minimum and maximum lateral clearances) must ensure the free rotation of the wheels and eliminate jamming. It is determined by the type of pairing of wheels from L before N. The largest gap A, and the smallest N. For gears with module t> 1 set types of mates A, B, C, D, E, H. Commonly used pairing IN, and for reverse gears WITH. For fine gears (T < 1) виды сопряжений D, E, F, G, H. More often used E, and in reverse gears F. It is allowed to apply once
Rice. 4.23
Rice. 4.24
personal degrees of accuracy on individual indicators, for example, when T≥ 1 7-6-7-V (7 is the norm of kinematic accuracy, 6 is the norm of smoothness, 7 is the norm of contact), and with the same accuracy in all indicators (7-7-7-V), 7-V is recorded.
Types of tooth decay. During the operation of cylindrical gears, various damage to the gear teeth is possible: mechanical and molecular-mechanical wear, as well as tooth breakage.
Mechanical wear. It includes:
- chipping working surfaces (Fig. 4.25, A). This is the most common cause of failure in lubricated gears. The failures are of a fatigue nature. Cracks develop to chipping, mainly on the root of the teeth in places of irregularities left after the final processing. In the process of work from the loading of the tooth, the number of pits increases and their sizes increase. The tooth profile is distorted, the surface becomes uneven, dynamic loads increase. The chipping process intensifies, and the working surface on the tooth stem is destroyed. Dangerously progressive chipping - cracks from pits can spread and affect the entire surface of the legs. If there is no lubricant or its quantity is small, chipping is rarely observed, since the resulting damage is smoothed out. The resistance to chipping increases with increasing hardness of the tooth surface, cleanliness of processing and the correct selection of lubricant;
- wear, teeth (Fig. 4.25, 6) - wear of the working surfaces of the teeth, which increases with increasing contact stresses and specific slip. Wear distorts the involute profile, increasing dynamic
Rice. 4.25
loads. Since the greatest slip occurs at the initial and final points of contact of the teeth, the greatest wear is observed on the roots and heads of the teeth. Wear increases greatly due to irregularities on the working surfaces of the tooth, after processing, as well as when the gear is contaminated with abrasive particles (abrasive wear). It is observed when working with open mechanisms. If the roughness is less than the thickness of the oil film, wear is reduced, and with insufficient lubrication, it increases. It can be reduced by reducing the contact stresses σΗ, increasing the wear resistance of the tooth surface (increase the hardness of the working surfaces of the teeth, choose the right lubricant).
Molecular mechanical wear. This wear appears like a jam(Fig. 4.25, c) under the action of high pressures in a zone where there is no oil film. The mating surfaces of the teeth interlock with each other so strongly that particles of the softer tooth surface are welded to the tooth surface of the other wheel. The formed outgrowths on the teeth are applied to the working surfaces of the other teeth of the furrow. Jamming is especially intense in a vacuum or when the working surfaces of the tooth are subjected to high pressure. Jamming is prevented by an increase in hardness and a decrease in surface roughness, by the correct selection of extreme pressure oils.
To prevent chipping of the working surfaces of the teeth, it is necessary to calculate the transmission for contact strength.
Tooth breakage. This is the most dangerous type of damage. It has a fatigue character and is usually absent in gear wheels of gearboxes when their working surfaces are not hardened. The fracture of the teeth is a consequence of the repeated alternating stresses arising in them from bending during overloads. Fatigue cracks form at the base of the tooth on the side where the greatest tensile stresses arise from bending. The fracture occurs in the section at the base of the tooth.
Breakage is prevented by calculating the strength of the bending stresses.
Forces in the engagement of cylindrical gears. The force applied to the tooth of the helical gear F can be decomposed into three components F t , F r , F a (Fig. 4.26):
where is the circumferential force (G is the calculated torque on the wheel); is the radial force; axial force; - engagement angles in the end and normal sections.
A spur gear has no axial force, i.e.
Estimated forces in engagement. When transferring the load in the mesh, in addition to the static, an additional dynamic component of the force occurs, and there is also an uneven distribution of the load across the width of the tooth and the distribution of the load between the teeth. All changes in the load compared to the original take into account the load factors and
Specific, district and design forces. In calculations for contact endurance is determined by the formula
(4.17)
In calculations for bending endurance
Rice. 4.26
- coefficient of load in bending; - coefficient of load distribution between the teeth; - coefficient taking into account the uneven distribution of the load along the width of the tooth; - coefficient taking into account the additional dynamic load on the teeth during bending.
During drive operation, dynamic external loads increase forces and moments. In strength calculations it is necessary to use the design force Fu design moment T:
where is the dynamic coefficient of the external load; - rated force and torque.
Specific circumferential dynamic loads acting on the teeth of the wheels occur during the interaction of the teeth in engagement due to inaccuracies in manufacturing in pitch and their deformation. These forces are determined taking into account the error of engagement in step, depending on the degree of accuracy in terms of smoothness and transmission module.
Specific circumferential dynamic load for cylindrical gears when calculated for contact strength
(4.21)
where is a coefficient that takes into account the hardness of the working surfaces and the angle of inclination of the tooth (Table 4.6); - coefficient taking into account the gearing error in step
Table 4.6
Table 4.7
|
Module 171, mm |
The degree of accuracy according to the standards of smoothness GOST 1643-81 |
|||||||||
(Table 4.7); - circumferential speed in engagement, m / s; - center distance, mm; And- the gear ratio of the gear pair; - the limiting value of the circumferential dynamic force, N / mm (see table. 4.7).
In calculations tooth bending strength cylindrical gears
(4.22)
The values are the same as in the verification calculation for contact strength (see Table 4.7), and the values \u200b\u200bare given in Table. 4.6.
With an increase in the degree of accuracy in terms of transmission smoothness, additional dynamic loads are reduced. The same happens when moving from straight teeth to oblique ones. With an increase in the hardness of the teeth, the load can be increased. Note that the dynamic load increases with increasing speed, but up to a certain limit.
Coefficients of internal dynamic load on the teeth. For calculations for contact and bending strength, these coefficients are determined by the formulas
(4.23)
where ; is the circumferential force in engagement; is the working width of the tooth.
The coefficients take into account the distribution on
loads between teeth in calculations for contact and bending strength. These coefficients are related to the manufacturing error. For spur gears; for helical gears, they depend on the accuracy of engagement and the hardness of the working surface of the teeth: (Table 4.8), since at least two pairs of teeth are simultaneously engaged in helical gears. Without load, a gap appears in one of the pairs, which is eliminated with an increase in load due to elastic deformations.
The coefficients take into account the uneven distribution of the load across the width of the gear rims, associated with the deformation of shafts, bearings and with the error of their manufacture. Shaft deflections at the locations of the wheels lead to their misalignment and uneven distribution of the load along the contact line. The load concentration depends on the dis-
Table 4.8
|
Odds |
Degree of accuracy |
|||
|
TO On, Xfa at HB< 350 |
||||
|
TO ia , TO Go at HB> 350 |
||||
the position of the supports and the hardness of the material. The values of the coefficients are almost the same when calculating the contact and bending strengths:
where for straight teeth, for oblique teeth; is the coefficient of relative hardness of the contact surfaces, taking into account the running-in of the teeth:
- coefficient taking into account the influence of the shaft deflection, which is affected by the location of the wheels relative to the supports: with a symmetrical arrangement, with an asymmetric>, with a cantilever.
The greatest misalignment under loading occurs for shafts with a cantilever arrangement of supports, and the smallest for a symmetrical one.
contact stresses. The nature of the interface of some machine parts is characterized by the fact that the load transmitted by them over a small surface in the contact zone causes high stresses. Contact stresses are typical for gears and rolling bearings. Contact can be point (ball on a plane) and linear (cylinder on a plane). Under loading, deformation occurs and the contact zone expands to a region bounded by a circle, rectangle or trapezoid, in which contact stresses arise. At high contact stresses exceeding the permissible ones, damage to the surfaces is possible on the contact surface, which appear in the form of dents, grooves, cracks. Such damage can occur in gears and bearings, the contact stresses of which vary with time but in an intermittent cycle. Variable stresses are the cause of fatigue failure of the working surface of the teeth: chipping, wear, seizing. At high contact stresses, static loading can cause plastic deformation and the appearance of dents on the surface.
Solution of the contact problem. The solution of the contact problem was obtained by G. Hertz. When solving it, the following assumptions were used: the materials of the contacting bodies are homogeneous and isotropic, the contact area is very small, the acting forces are directed normally to the contact surface, the loads create only elastic deformations in the contact zone and obey Hooke's law. In real structures, not all the formulated conditions are met, however experimental studies confirmed the possibility of using the Hertz formula for engineering calculations. Consider the contact stresses when two cylinders are compressed (Fig. 4.27, A). The specific load acts on the cylinders
Where F- Normal strength; h is the width of the cylinders.
In the contact zone on a section of width 4, the highest contact stress is determined (at V ≠ v 2) according to the formula
(4.26)
where is the reduced radius of curvature for cylinders with radii and are Poisson's ratios for cylinders; are the elastic moduli of cylinder materials;; is the specific circumferential force (Fig. 4.28).
Rice. 4.27
Rice. 4.28
Reduced modulus of elasticity and radius
(4.27)
In the formula for the "+" sign is placed at the contact of two convex surfaces; the "-" sign is for one concave and the other convex surface (Fig. 4.27, b).
If the Poisson's ratios of the cylinders are equal, then formula (5.26) can be written as follows:
(4.28)
Formula (4.28) is called Hertz's formula.
Expressions (4.26) or (4.28) are used in deriving formulas for contact stresses.
Verification calculation of a cylindrical spur gear for contact strength
Estimated contact stresses To determine the highest contact stresses, the Hertz formula (4.28) is taken as the initial one. Substituting the values into expressions (4.27), we obtain
Substituting into the Hertzian formula, we have
(4.29)
(the "+" sign is used for external gearing, and "-" for internal gearing). Here Z,- coefficient taking into account the shape of the mating surfaces of the teeth in the gearing pole,
(for straight teeth , at , a - engagement angles in the end plane for helical and spur gears, respectively), the values for helical gears are given in Table. 4.9; coefficient that takes into account the mechanical properties of materials of mating gears. For steel teeth MPa1/2.
Table 4.9
The Z coefficient takes into account the total length of the contact lines: for straight teeth, and for oblique teeth, where is the end overlap coefficient. It is equal to the ratio of the active site AB engagement lines to the circumferential pitch (see Fig. 4.17, i). It is determined by the number of teeth of the wheels that are simultaneously in contact (one pair is in engagement, and then one, then two). The coefficient εα affects the smoothness of the transmission. For spur gears, it must be greater than one (), otherwise the transmission may be disrupted (movement will not be transmitted). The coefficient can be approximately determined by the formula
(4.30)
where is the number of gear teeth.
Here the "+" sign is used for external engagement, and "-" for internal engagement.
To calculate helical gears, you can take the average value of I.
Limit contact stresses. The endurance curve for limiting contact stresses in logarithmic coordinates is shown in fig. 4.29, where - pre-
Rice. 4.29
specific contact stresses for the design durability for the number of cycles of variable loading. Endurance curve within
(section L/)) where is the limit of contact endurance at the base number of loading cycles , and is assigned from the condition of the absence of plastic flow of the material or brittle fracture on the working surface of the tooth at, is described by the formula:
(4.32)
Note that , a , which is associated with the zero loading cycle on the tooth surface and with the local action of the load. The values of limiting stresses are selected according to table. 4.10.
Table 4.10
The hardness of the gear material is made greater than that of the wheel, by 10–50 HB. The basic number of cycles of stress changes for steel wheels is determined by the formula
The number of cycles of change in contact stresses on the tooth surface, where is the time of the cycle; With- the number of contacts of one tooth surface in one revolution; P is the rotation frequency, rpm; is the number of loading cycles.
When the tooth is working on two sides of the profile for reverse gears, the time of operation during the cycle of one of the sides, where the load is greater, is taken into account, since the contact stresses act only near the surface of the tooth and the load of one working surface does not affect the other (Fig. 4.30, A, where is the loading time of one side of the tooth in one cycle; is the loading cycle time), and when rotating in one direction, is the total loading time (Fig. 4.30, b). If a resource is given, then
In the presence of a reverse, and with one-way rotation
After determining the values, they are substituted into inequality (4.31). If the value of the function, then should be accepted, if, then. We choose from two values for the gear σ//Pt i and the minimum wheel.
Permissible contact stresses are determined by the formula
where is the margin of safety when calculating the tooth for
contact strength. For mechanisms with high reliability, larger values should be taken
Rice. 4.30
Contact strength condition:
If the strength condition is not met and , then with a small deviation (less than 10%), the load on the tooth can be reduced by increasing the width of the wheels: , where are the primary and refined values of the width of the gear rim. With a larger deviation, you need to increase the modulus and repeat the calculations.
Design calculation of a spur gear based on contact stresses
From the formulas for the verification calculation for contact stresses (4.29), (4.34), expressing the specific circumferential force in terms of torque, we obtain an expression for the approximate value of the center distance:
(4.35)
where is the calculated torque on the gear, N ∙ mm. In the formula, the "+" sign is for external gearing, the "-" sign is for internal gearing.
If both wheels are steel, MPa, then
(4.36)
When carrying out the design calculation, the speed is unknown, and therefore, in the first approximation, . In the future, when conducting a verification calculation, if it differs by more than 20%, then it is necessary to re-determine with an updated value included in
After determining the center distance, the tooth engagement module is approximately determined by the formula
and refine it to the value T according to GOST 9563–80 (Table 4.11). Then all the geometric characteristics of the gear rims for the gear and wheel are determined using formulas (4.9) - (4.12).
Table 4.11
|
Tooth modules, mm |
Tooth modules, mm |
Tooth modules, mm |
||||||
Usually, the width of the ring gear of the cylindrical gear is made somewhat larger than that of the wheel (to increase the bending strength of the teeth).
Another calculation option is also possible, when instead of the center distance from the formula (4.36), the pitch diameter of the gear is determined
Having determined |, find the module, refine it to the value T but GOST 9563-80 and determine all the geometric parameters of gears.
Verification calculation for bending strength
Estimated bending stresses. Consider a cylindrical gear with a straight tooth. The calculation is carried out to prevent tooth breakage. The maximum stresses occur in the seal (at the base of the tooth), when the force is at the circle of the vertices and is transmitted by one pair of teeth. The tooth will be considered as a cantilever beam. The most dangerous point A, since fatigue cracks and fractures start from the stretched side of the teeth. A force acts on the tooth at the apex F, which we decompose into two components (Fig. 4.31):
In calculations, we use not commemorative, but calculated forces, which are determined by introducing the coefficient ■; respectively, we obtain the normal bending stresses at the base of the tooth from the bending moment and the compressive stress from the force:
where is the moment of resistance in bending; - sectional area at the base of the tooth.
At the dangerous point, the bending stresses will be equal to
where is the theoretical stress concentration factor at the base of the tooth.
After replacing with and introducing for helical gears the coefficients and the formula for will take the form
where is the specific circumferential force; - coefficient taking into account the overlap of the teeth; - coefficient taking into account the inclination of the tooth (obtained experimentally); – tooth shape factor:
For external engagement;
For internal engagement. (4.39)
When calculating helical gears according to formula (4.38), the coefficients . For spur gears
Rice. 4.31
Permissible tooth bending stresses. First, we determine the limit of the limited bending endurance of the teeth for the zero cycle. The limiting bending stresses with a one-sided application of a load (a cycle with an asymmetry coefficient) for steel gears are determined from the inequality
where are the maximum limiting bending stresses that do not cause residual deformations or brittle fracture. Such stresses correspond to the number of loading cycles:
(priipri); - the endurance limit of the bending stresses of the tooth at the base number of loading cycles, it depends on the hardness
material and type of heat treatment (Table 4.12).
For gears made of steel
(4.41)
where is the coefficient of durability; /" = 9 for cement wheels
tinted and nitrided with an unpolished transitional surface at the base of the tooth; in other cases t = 6;
Table 4.12
is the number of loading cycles in bending. With a given number of cycles (see Fig. 4.30, A) or (see Fig. 4.30, b); for a given resource number of cycles
Permissible voltage in dangerous section AB is determined by the formula
where is a coefficient that takes into account the effect of surface roughness at the root of the tooth (with unpolished teeth; with ground teeth); bending safety factor ().
To obtain the probability of failure-free operation of the transmission, it is necessary to take
Bending strength test condition
The check is carried out separately for the gear 1 and wheels 2.
Spur Gear Calculation Procedure
Initial data. Kinematic scheme, gear ratio and number of teeth; rated torque on the drive shaft; coefficient of dynamism; frequency of rotation of the drive shaft; loading schedule (cyclogram); guaranteed operating time (resource) in hours or in the number of loading cycles; operating conditions (temperature range, vibrations, external loads, etc.).
Design calculation. The calculation is performed in the following sequence:
Check calculation. When making a calculation:
Design of cylindrical gears. Gears are made from round rolled products (rods) and blanks obtained by forging, stamping and casting. The gear is made integral with the shaft (shaft - gear), if its diameter is close to the diameter of the shaft. The teeth are cut on the protruding crown (Fig. 4.32). When the diameter of the crown is greater than or equal to the diameter of the shaft, the teeth go deep into the body of the shaft partially or completely. Cylindrical gears mounted on a shaft can be made with a hub and in the form of a solid disk, where the workpiece is stamped or turned (Fig. 4.33). To connect the wheels to the shaft, a key or spline (gear) connection is used. With a large wheel diameter, 4-6 holes are made in the disk with a diameter, which reduces its weight. In addition to the dimensions of the gear rim, determined by calculation, you can use the following recommendations for choosing the dimensions of other elements of the cylindrical gear
Rice. 4.32
Rice. 4.33
th wheel (see Fig. 4.33):
The designs of spur gear reducers, see fig. 4.8 and 4.9.
1. Gears
1.1 Designs
2. Wear and repair of gears
2.1 Replacing and repairing gears
2.2 Gear repair methods
List of used literature
1. GEARS
1.1 Constructions
Gears are used in almost all mechanisms that metallurgical shops are equipped with (cranes and hoists, roller tables, winches of changeover devices, mill drives, etc.)
The main parts of gears are gears (gears). They serve to transfer rotation from one shaft to another when the shafts are not on the same axis.
Depending on the relative position of the shafts, gears are used: cylindrical, conical and helical.
A spur gear serves to transfer rotation from one to another parallel shaft (Fig. 1, a).
The bevel gear is used to transfer rotation from the shaft to the shaft, located at the intersection of the axes (Fig. 1.6).
A helical gear is used to transfer rotation from a shaft to a shaft located with intersecting but not intersecting axes (Fig. 1, c).
Rice. 1. Gears: a - cylindrical: b - bevel: c - screw: g-chevron gear.
The gear wheel and the rack are used to convert the rotational motion into a reciprocating motion.
The teeth of cylindrical wheels can be straight (Fig. 1, a and b), oblique and chevron (Christmas tree) - fig. 1, Mr.
The chevron gear consists, as it were, of two gears with oblique teeth connected together.
During the operation of gears with straight teeth, one or two teeth are simultaneously engaged, as a result of which the transmission operation is accompanied by some shocks.
Smoother operation of the gear train is achieved by using helical or chevron teeth, since the number of teeth involved in the engagement increases.
Gears are made from steel forgings, steel castings and rolled products, or from cast iron. For critical gears (for example, hoisting machines), the use of cast iron gears is not allowed.
Classification of gears. Depending on the purpose of the gear, type of tooth and speed of rotation, gears are divided into four classes of gear accuracy according to manufacturing and assembly tolerances (Table 119).
Table 1 Classification of gears
| Class | Permissible | |||
| exactly- | gear type | Type | district speed | Note |
| sti | tooth | height, m/s | ||
| 4 | Cylindrical | Straight | Up to 2 | Applicable where accuracy |
| oblique | » 3 | and do not have smoothness | ||
| values, as well as | ||||
| conical | Straight | " 1 | manual and unloaded | |
| broadcasts | ||||
| 3 | Cylindrical | Straight | » 6 | |
| oblique | " 8 | |||
| conical | Straight | » 2 | ||
| oblique | " 5 | |||
| 2 | Cylindrical " | Straight | " 10 | |
| oblique | " 18 | |||
| conical | Straight | " 5 | ||
| oblique | " 10 | |||
| 1 | Cylindrical | Straight | Above 8 | 1 When required, pain |
| oblique | " 15 | 1 shoy transfer smoothness | ||
| conical | Straight | " 5 | whether, as well as counting- | |
| oblique | " 10 | mechanisms |
Gears are made open, semi-open and closed.
Open gears are those that do not have a casing (reservoir) for an oil bath; such gears are periodically lubricated with grease. Typically, these gears are low-speed and are used mainly in simple machines and mechanisms.
Semi-open gears differ from open gears by the presence of a reservoir for a liquid oil bath.
Closed gears are called, which, together with bearings, are mounted in special housings.
The gearbox gears are lubricated different ways:
1) at circumferential speeds of gears above 12--14 m / sec-jet method with supply, jets into the zone of the beginning of gear engagement;
2) at circumferential speeds of gears below 12 m / s - by dipping.
For dip lubrication, the following must be observed:
a) the larger gear of the pair must be immersed in oil by two to three times the height of the tooth;
b) if the gearbox has several stages, then the oil level is determined taking into account the speed of the gears.
In the latter case, level b (Fig. 2) is allowed when the gear wheel 1 of the low-speed stage rotates at low speed. In gearboxes with medium and large
Rice. 2. Jet lubrication of gears.
Rice. 3. Scheme of gear lubrication by dipping.
speeds of low-lying wheels, the latter are immersed by two to three times the height of the tooth larger wheel, and the oil is poured to level a. lubrication of the first stage put an auxiliary gear wheel 3 with a narrow tooth, which supplies lubricant to the impeller.
The viscosity of the oil poured into the gearbox is selected depending on the speed and load - usually from 4 to 12 ° E at a viscosity determination temperature of 50 ° C. At the same time, the temperature conditions in which the unit operates are also taken into account; when the temperature rises, a higher viscosity oil is used; when the temperature drops, a lower viscosity is used.
Open gears are usually lubricated with greases (solid oil, constaline, etc.).
The stuffing of the seals provided (drawings) in the bearings and along the junction line of the gearbox housing must be carried out very carefully to avoid oil leakage and dust ingress into the gearbox.
2. Wear and repair of gears
Gears fail for two main reasons: tooth wear and tooth breakage.
Wear is usually the result of: 1) incomplete clutch and 2) increased friction (gradual wear).
Wear in the first case is mainly the result of poor assembly and with correct assembly (strict observance of the radial clearance) is usually absent. However, a change in the radial clearance can also be a consequence of the wear of the bearing shells, and as a result of the wear of the bearings, there can be both an increase in the radial clearance and its decrease (work in thrust).
If the load on the liners is transferred to the sides opposite to the clutch during operation, as the liners are worn out, an increase in the radial clearance is possible.
If the load on the liners is transferred to the side of the cordon (for example, at the gear wheels of crane runners, during operation, as the liner is developed (in this example slider bushing) it is possible to reduce the radial clearance.
In both cases, after changing the liners, the radial clearance is restored.
Gradual wear from increased friction depends on a number of conditions, including the hardness of the material from which the gears are made, heat treatment, the correct selection of lubricant, insufficient oil cleanliness and untimely oil change, transmission overload, etc.
Correct installation and good supervision during operation are the main conditions for long and trouble-free operation of the equipment.
Gear tooth breakage occurs for the following reasons: gear overload, one-sided (from one end of the tooth) load, tooth undercutting, imperceptible cracks in the workpiece material and microcracks as a result of poorly performed heat treatment, poor resistance of the metal to shocks (in particular, as a result of failure to anneal castings and forgings), increased impacts, hard objects getting between the teeth, etc.
Rice. 4. Repair of teeth with the help of screw drivers with subsequent welding
As a rule, gear wheels with worn and broken teeth are not subject to repair, but to be replaced, and it is recommended to replace both wheels included in this engagement at the same time. However, when the large wheel in engagement is many times larger than the size of the small one, it is necessary to replace the small wheel in a timely manner, which wears out faster than the large one by about a gear ratio of times. Timely replacement of the small wheel will protect the large wheel from wear.
The wear of the gear teeth should not exceed 10-20% of the tooth thickness, counting along the pitch circle arc. In low-responsibility gears, tooth wear is allowed up to 30% of the tooth thickness, in gears of critical mechanisms it is much lower (for example, for load lifting mechanisms, wear should not exceed 15%: tooth thickness, and for gear wheels of crane lifting mechanisms transporting liquid and hot metal - to 10%").
Gears with case-hardened teeth should be replaced when the case-hardening layer is worn more than 80%1 of its thickness, as well as when the case-hardened layer is cracked, chipped, or peeled off.
If the teeth are broken, but not more than two in a row in gears that are not particularly important (for example, crane movement mechanisms), it is allowed to restore them, which is done in the following way: the broken teeth are cut down to the base, two or three holes are drilled along the width of the tooth and threads are cut into them, studs are made and tightly screwed into the prepared holes, the studs are welded to the gear and the metal is welded by electric welding, giving it the shape of a tooth, on a gear-cutting, milling or planing machine or by filing manually, the deposited metal is shaped into a tooth, after which the restored profile is checked by adhesion to the mating part and by template.
GEARS
P lan l e c t i o n
1. General information.
2. Classification of gears.
3. Geometric parameters of gears.
4. Accuracy of parameter conversion.
5. Dynamic relationships in gears.
6. Wheel design. Materials and allowable stresses.
1. General information
Gear- This is a mechanism that, with the help of gearing, transmits or converts motion with a change in angular velocities and moments. The gear train consists of wheels with teeth that interlock with each other, forming a series of successively working cam mechanisms.
Gears are used to convert and transmit rotational motion between shafts with parallel, intersecting or crossing axes, as well as to convert rotational motion into translational and vice versa.
Advantages of gears:
1. Ratio constancy i .
2. Reliability and durability of work.
3. Compactness.
4. Large range of transmitted speeds.
5. Little pressure on the shafts.
6. High efficiency.
7. Ease of maintenance.
Disadvantages of gears:
1. The need for high precision manufacturing and installation.
2. Noise when running at high speeds.
3. The impossibility of stepless regulation of the gear ratio
solutions i .
2. Classification of gears
Gears used in mechanical systems, varied. They are used for both lowering and increasing angular velocity.
The classification of gear converter designs groups gears according to three criteria:
1. Type of tooth engagement. IN technical devices gears are used with external (Fig. 5.1, but), with internal (Fig. 5.1, b) and rack and pinion (Fig. 5.1, c) engagement.
Gears with external gearing are used to convert rotary motion with a change in direction of motion. The gear ratio ranges from -0.1 i -10. Internal gearing is used if it is required to convert rotational motion while maintaining direction. Compared to external gearing, the gear has smaller overall dimensions, a larger overlap ratio and increased strength, but is more difficult to manufacture. Rack engagement is used when converting rotational motion into translational and vice versa.
2. According to the mutual arrangement of the axes of the shafts distinguish gears with cylindrical wheels with parallel axes of the shafts (Fig. 5.1, A ), bevel wheels with intersecting axles (Fig. 5.2), wheels with crossed axles (Fig. 5.3). Gears with bevel wheels have a lower gear ratio (1/6 i 6), are more difficult to manufacture and operate, have additional axial loads. Screw wheels work with increased slip, wear out faster, and have a low load capacity. These gears can provide different gear ratios for the same wheel diameters.
3 . According to the location of the teeth relative to the generatrix of the wheel rim
there are spur gears (Fig. 5.4, a), helical (Fig. 5.4, b), chevron (Fig. 5.5) and with circular teeth.
Helical gears have pain |
||||
smoother gearing, less |
||||
technologically |
are equivalent |
|||
spur, but in transmission there are |
||||
additional |
loads. |
|||
Double helical |
counter |
|||
teeth tilt (chevron) transmission |
||||
cha has all the benefits of helical |
||||
and balanced axial forces. But |
||||
transmission is somewhat more difficult to manufacture |
||||
leniya and installation. Curvilinear |
||||
teeth are most often used in horse- |
||||
broadcasts |
raise |
|||
load capacity, |
smoothness |
|||
work at high speeds. |
||||
3. Geometric parameters of gears
TO the main geometric parameters of gears (Fig. 5.6) include: tooth pitch P t, module m (m = P t /), number of teeth Z, diameter d of the pitch circle, height h a of the dividing head of the tooth, height h f of the dividing leg of the tooth, diameters d a and d f of the circles of tops and troughs, width of the gear rim b.
df 1 |
db 1 |
|||
dw 1 (d1 ) |
||||
da 1 |
||||
df 2 |
dw 2 (d2) |
da 2 |
||
db2 |
||||
Pitch circle diameter d = mZ. With a dividing circle, the wheel tooth is divided into a dividing head and a dividing leg, the ratio of the sizes of which is determined by the relative position of the wheel blank and the tool in the process of cutting teeth.
With a zero offset of the original contour, the height of the dividing head and the leg of the tooth of the wheel corresponds to those of the original contour, i.e.
ha = h a * m; hf = (h a * + c* ) m,
where h a * is the coefficient of the height of the tooth head; c * – coefficient of radial
For wheels with external teeth, the diameter of the circle of the vertices
da = d + 2 ha = (Z + 2 h a * ) m.
Cavity circle diameter
df \u003d d - 2 hf \u003d (Z - 2 h a * - 2 c * ) m.
At m ≥ 1 mm h a * = 1, c * = 0.25, d a = (Z - 2.5)m.
For wheels with internal teeth, the diameters of the circles of the tops and bottoms are as follows:
da \u003d d - 2 ha \u003d (Z - 2 h a * ) m;
df = d + 2 hf = (Z + 2 h a * + 2 c * ) m.
For wheels cut with an offset, the diameters of the tops and troughs are determined taking into account the value of the offset coefficient according to more complex dependencies.
If two wheels, cut without displacement, are engaged, then their pitch circles will touch, i.e., coincide with the initial circles. The angle of engagement in this case will be equal to the angle of the profile of the original contour, i.e., the initial legs and heads will coincide with the dividing legs and heads. The center distance will be equal to the pitch center distance, determined through the diameters of the pitch circles:
aw = a = (d1 + d2 )/2 = m(Z1 + Z2 )/2.
For wheels cut with an offset, there is a difference for the initial and pitch diameters, i.e.
d w 1 ≠ d 1 ; d w 2 ≠ d 2 ; a w ≠ a ; αw = α.
4. Parameter Conversion Accuracy
IN During the operation of the gear train, the theoretically constant gear ratio undergoes continuous changes. These changes are caused by the inevitable manufacturing errors in the size and shape of the teeth. The problem of manufacturing gears with low sensitivity to errors is solved in two directions:
a) application special types profiles (for example, hour gearing);
b) limitation of manufacturing errors.
IN unlike such simple parts as shafts and bushings, gears are complex parts, and the errors in the execution of their individual elements not only affect the mating of two individual teeth, but also affect the dynamic and strength characteristics of the gear as a whole, as well as the accuracy transmission and transformation of rotational motion.
The errors of gears and gears, depending on their impact on transmission performance, can be divided into four groups:
1) errors affecting the kinematic accuracy, i.e., the accuracy of the transmission and transformation of rotational motion;
2) errors that affect the smooth operation of the gear;
3) tooth contact spot errors;
4) errors leading to a change in the side clearance and affecting the backlash of the transmission.
In each of these groups, complex errors can be distinguished, which most fully characterize this group, and element-by-element, partially characterizing transmission performance.
Such a division of errors into groups is the basis for the standards for tolerances and deviations of gears: GOST 1643–81 and GOST 9178–81.
To assess the kinematic accuracy of transmission, smoothness of rotation, characteristics of the contact of the teeth and backlash in the standards under consideration, 12 degrees of accuracy in the manufacture of gears are established.
And gears. Degrees of accuracy in descending order are indicated by numbers 1–12. Degrees of accuracy 1 and 2 according to GOST 1643–81 for m > 1 mm and according to GOST 9178–81 for 0.1< m < 1 являются перспективными, и для них в стандартах численные значения допусков нормируемых параметров не приводятся. Стандартом устанавливаются нормы кинематической точности, плавности, пятна контакта и бокового зазора, выраженные в допустимых погрешностях.
It is allowed to use gears and gears, the error groups of which may belong to different degrees of accuracy. However, a number of errors belonging to different groups in terms of their influence on the transmission accuracy are interrelated, therefore, restrictions are set on the combination of accuracy standards. Thus, the norms of smoothness can be no more than two degrees more accurate or one degree rougher than the norms of kinematic accuracy, and the norms of tooth contact can be assigned in any degree more accurate than the norms of smoothness. The combination of accuracy standards allows the designer to create the most economical transmissions, while choosing such degrees of accuracy for individual indications.
tels that meet the performance requirements for a given transmission without overestimating the cost of manufacturing the transmission. The choice of degrees of accuracy depends on the purpose, the field of application of the wheels and the circumferential speed of rotation of the teeth.
Let us consider in more detail the errors of gears and gears that affect their quality.
5. Dynamic relationships in gears
Gears transform not only motion parameters, but also load parameters. In the process of converting mechanical energy, part of the power P tr supplied to the input of the converter is spent on overcoming rolling and sliding friction in the kinematic pairs of gears. As a result, the output power is reduced. To estimate the loss
power, the concept of efficiency is used, defined as the ratio of the power at the output of the converter to the power supplied to its input, i.e.
η = P out / P in. |
If the gear train converts rotational motion, then, respectively, the input and output powers can be defined as
Let's denote ωout / ωin through i, and the value T out /T in through i m, which we call the gear ratio of the moments. Then expression (5.3) takes the form
η \u003d i m.
The value of η ranges from 0.94 to 0.96 and depends on the type of transmission and the transmitted load.
For a cylindrical gear transmission, the efficiency can be determined from the dependence
η = 1 – cf π(1/Z 1 + 1/Z 2 ), |
where c is a correction factor that takes into account the decrease in efficiency with a decrease in the transmitted power;
20T out 292mZ 2 |
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20T out 17.4mZ 2 |
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where Тout – output torque, H mm; f is the coefficient of friction between the teeth. To determine the actual forces on the gear teeth,
Let's take a look at the load transformation process (Fig. 5.7). Let the driving input torque T 1 be applied to the driving gear wheel 1 with pitch circle diameter d w l , and the moment of resistance T 2 of the driven wheel 2 be directed in the direction opposite to the rotation of the wheel. In involute gearing, the point of contact is always on a line that is a common normal to the contacting profiles. Therefore, the pressure force of the tooth F of the drive wheel on the tooth of the driven will be directed along the normal. We transfer the force along the line of action to the pole of the link P and decompose it into two components.
Ft’ |
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Ft’ |
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The tangent component F t is called |
district force. She |
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performs useful work, overcoming the moment of resistance T and setting the wheels in motion. Its value can be calculated by the formula
F t = 2T /d w .
The vertical component is called radial force and denoted F r . This force does no work, it only creates an additional load on the shafts and transmission supports.
When determining the magnitude of both forces, the forces of friction between the teeth can be neglected. In this case, the following dependencies exist between the total pressure force of the teeth and its components:
F n = F t /(cos α cos); |
F r = F t tg α/ cos , |
where α is the engagement angle.
The engagement of cylindrical spur gears has a number of significant dynamic disadvantages: limited values of the overlap coefficient, significant noise and impacts at high speeds. To reduce the dimensions of the transmission and reduce the smoothness of operation, spur gearing is often replaced with helical gearing, the side tooth profiles of which are involute helical surfaces.
In helical gears, the total force F is directed perpendicular to the tooth. Let us decompose this force into two components: F t is the circumferential force of the wheel and F a is the axial force directed along the geometric axis of the wheel;
F a = F t tg β, |
where is the angle of inclination of the tooth.
Thus, in a helical engagement, in contrast to a spur engagement, there are three mutually perpendicular forces F a , F r , F t , of which only F t performs useful work.
6. Wheel design. Materials and allowable stresses
Wheel design. When studying the principles of designing gears, the main goal is to master the methodology for determining the shape and basic parameters of wheels according to the conditions of performance and operation. Achieving this goal is possible by solving the following tasks:
a) selection of optimal wheel materials and determination of permissible mechanical characteristics;
b) calculation of wheel sizes according to the conditions of contact and bending strength;
c) development of the design of gears.
Gears are typical converters for which quite a lot of justified design concepts have been developed. best options. A general design diagram of a gear wheel can be represented as a combination of three main structural elements: a ring gear, a hub and a central disk (Fig. 5.9). The shape and dimensions of the gear wheel are determined depending on the number of teeth, the module, the shaft diameter, as well as the material and manufacturing technology of the wheels.
On fig. 5.8 shows examples of designs of gear wheels of mechanisms. Wheel dimensions are recommended to be taken in accordance with the instructions of GOST 13733-77.
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