Proell governor | Construction | Working | Equation of height

A porter governor is known as a proell governor is the two balls are fixed on the upward extensions of the lower links which are in the form of bent links BAE and CDF as shown in the figure below. 

Now, considering the equilibrium of the link BAE which is under the action of following characteristics. 

• The weight of the ball, mg
• The tension in the link AO
• The horizontal reaction of the sleeve
• The weight of sleeve and friction 1/2 (mg+f) or 1/2 (mg-f)
• The centrifugal force, mr’ᾠ² 

I is the instantaneous centre of the link BAE and take moments about I to find out the height relation for proell governor. 

N² = 895×a/h×e [ 2mg + ( mg + or – f ) ( 1+k) / 2mg ] 

Porter Governor | Construction | Working | Equation of height

Porter governor is same as Watt governor. If the watt governor sleeve is loaded with a heavyweight, it becomes a Porter governor. 

Let M = Mass of the sleeve
m = Mass of each ball
f = Force of friction at the sleeve

Porter Governor

The frictional force always acts in the opposite direction to the motion, so when the sleeve moves up, the frictional force acts in the downward direction, and the downward force acting on the sleeve is Mg+f. Likewise, the force on the sleeve will be Mg-f when the sleeve moves down. Depending on whether the sleeve is going up or down, the net force acting on the sleeve is mg+f or Mg-f.

Forces acting on the sleeve and on each ball have been shown in the figure. 

Let h = height of the governor
r = distance of the centre of each ball from the axis of rotation

The instantaneous rotation centre of AB link is at I. It is because of the motion of its two points A and B relative to the link. The point A oscillates about the point O and B moves in the verticle direction parallel to the axis. 

Now, the equilibrium of the left-hand half of the governor and taking moments about I. We equate the equation form given below to find the height of governor. 

N² = 895/h × [ 2mg + ( mg + or -  f ) ( 1 + k ) / 2mg ] 

This equation would provide two values of N for the governor for the same height depending upon the sleeve movement.  

Types of mechanical vibration

There are mainly three basic types of vibration according to the axis of body move : 

1. Longitudinal vibration 
2. Transverse vibration 
3. Torsional vibration 

Now we can check it out in detail below. Let us check out the different types of mechanical vibration here in this article. Before you start you can check some basic introduction of mechanical vibration. 

• Longitudinal vibration :

If the shaft is elongated and shortened so that the same moves up and down resulting in tensile and compressive stresses in the shaft, the vibrations are said to be longitudinal. The different particles of the body move parallel to the axis of the body. 

• Transverse vibration :

When the shaft is bent alternately and tensile and compressive stresses due to bending result, the vibrations are said to be transverse. The particles of the body move approximately perpendicular to its axis. 

• Torsional vibration :

When the shaft is twisted and untwisted alternately and torsional shear stresses are induced, the vibrations are known as torsional vibrations. 

There are also some more vibrations that are following below :

1. Free and Forced vibration 
2. Linear and non-linear vibration 
3. Damped and Un-damped vibration 
4. Deterministic and Random vibration 

Now we can check it out in detail below : 

• Free vibration : 

After distributing the system, the external excitation is removed, then the system vibrates on its own. This type of vibration is known as free vibration. 

Example - Simple pendulum 

• Forced vibration 

The vibration which is under the influence of external force is called forced vibration. 

Example - Machine tools, electric bells

• Linear vibration : 

In a system, if mass, spring and damper behave in a linear manner, the vibrations caused are known as linear in nature. 

They are governed by linear differential equations. 

They follow the law of superposition. 

• Non-linear vibration : 

If any of the basic components of a vibratory system behaves non-linearly that type of vibration is called non-linear in nature. 

Linear vibration becomes non-linear for a very large amplitude of vibrations. 

They do not follow the law of superposition. 

• Damped and Un-damped vibration : 

If the vibratory system has a damper and the motion of the system will be opposed by the damper also the energy of the system will be dissipated in friction is called damped vibration. 

On the other hand, the system that has no damper is known as un-damped vibration. 

• Deterministic and Random vibration :

In the vibratory system, the amount of external excitation is known in magnitude, it causes deterministic vibration is called deterministic vibration.

The non-deterministic vibration is known as random vibrations. 

Introduction to mechanical vibration

A body is said to vibrate if it has a to and fro motion. A pendulum swinging on either side of a mean position does so under the action of gravity. When the pendulum swings through the mid position, its centre of mass is at the lowest point and it possesses only kinetic energy. At each extremity of its swing, it has only potential energy. In the absence of any friction, the motion continues indefinitely. It can be shown that if the swings on either side of the mean position are very small, it approximates to simple harmonic motion. 

Usually, vibrations are due to elastic forces. Work is done on the elastic constraints of the forces on the body when a body is displaced from its equilibrium position and is stored as strain energy. Now, if the body is released, the internal forces cause the body to move towards its equilibrium position. If the motion is frictionless, the strain energy stored in the body is converted into kinetic energy during the period the body reaches the equilibrium position at which it has maximum kinetic energy. The body passes through the mean position, the kinetic energy is utilized to overcome the elastic forces and is stored in the form of strain energy. 

The vibration is mainly three types :

1. Longitudinal vibrations
2. Transverse vibrations
3. Torsional vibrations 

Terms related to vibration :

1. Free vibration 
2. Damped vibration 
3. Forced vibration 
4. Period
5. Cycle
6. Frequency 
7. Resonance 

All the terms are used in vibrations are explain below :

Free vibration is also called natural vibration. Elastic vibration in which there are no friction and external forces after the initial release of the body is known as free vibration.

Damped vibrations are when the energy of a vibrating system is gradually dissipated by friction and other resistances is called damping. The vibrations gradually cease and the system rests in its equilibrium position.

Forced vibration is when a repeated force continuously acts on a system. The vibration is said to be forced. The frequency of the vibrations is that of the applied force and is independent of their own natural frequency of vibrations. 

The period is the time taken by a motion to repeat itself and is measured in seconds. 

The cycle is the motion completed during one time period. 

Frequency is the number of cycles of motion completed in one second. It is expressed in hertz ( Hz ) and is equal to one cycle per second. 

Resonance is the frequency of the external forces is the same as that of the natural frequency of the system. A state of resonance is said to have been reached. 

It results in large amplitudes of vibrations and this may be dangerous. 

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1. Types of mechanical vibration

Watt governor | Working | Limitations | Height calculation

Watt governor is the simplest form of a centrifugal governor. It is also called a simple conical pendulum governor. It is basically a conical pendulum with a link attached to a sleeve of negligible mass. This governor is used by James Watt in his steam engine and it also has many applications. 

Working of Watt governor :

The upper sides of arms are pivoted with the governor balls so the governor balls can move upward and downward as they revolve with a vertical spindle. Bevel gears drive the engine. A bevel gear is driven by the spindle. The lower arms are connected to the sleeves and are keyed to the spindle in such a way that revolves around the spindle. At that time, it can slide up and down according to the spindle speed. Two stoppers are provided at the bottom and top of the spindle to limit the movement of the sleeve.

If the load on the engine decreases, the speed of the engine and then the angular velocity of the governor spindle increase. The centrifugal force on the ball increases that tends balls to move outward and the sleeve to move upward thus sleeve actuates a mechanism that operates the throttle valve at the end of the bell crank lever to decrease the fuel supply. Thus the power output is reduced.

If the speed of the engine decreases as the load on the engine increase, the centrifugal force decreases. So that the inward movement fly balls and downward movement of the sleeve cause a wide opening of the throttle valve. Engine speed increase with increasing the fuel supply.

Types of Watt governor :

The Watt governor was classified based on the position of upper arms. The arms can be connected by the way we describe below :

1. Pivot is on the axis of the spindle 
2. Pivot is offset from spindle
3. The pivot is offset, and arms cross the axis.

Based on this Watt governors are classified into three types :

1. Simply pinned type governor 
2. Open arm type governor 
3. Crossed arm type governor 

Simply pinned type governor: The upper arms are joined to a point O on the axis of the spindle, where both arms intersect the spindle axis.

Open arm type governor: The upper arm of Watt governor is hinged on a collar attached to the spindle or joined by a horizontal link as shown in fig below instead of connecting directly to the spindle. The arms, when produced, meet the axis of the spindle at O. 

Crossed arm type governor: The upper arms o governor in hinged on a collar on the axis of the spindle or arms are joined through a fixed horizontal link as shown in fig below. The arms intersect the axis at a point O.

Limitations of Watt governor :

• Its use is limited up to vertical position applications. 
• It is used in a very slow speed engines because, at a higher speeds, the sensitivity of the governor will decrease. 

Height Calculations :

The vertical distance from the plane of rotation of the balls to the point of intersection of the upper arms along the axis of the spindle is called the height of the governor. 

The height of the governor decreases with an increase in speed and increases with a decrease in speed. 

Let, 

m = mass of each ball

h = height of governor

w = weight of each ball ( w = mg )

ω = angular velocity of the balls, arms and the sleeve

T = Tension in the arm

r = radial distance of ball-centre from spindle-axis 

For the finding h, the height of governor the equilibrium of the mass provides 

Tcosθ = mg and Tsinθ = mrω²

Tanθ = mrω² / mg = rω² / g

r / h = rω² / g

h = g / ω² = g / ( 2πN / 60 ) = ( 60 / 2π )² * 9.81 / N²

h = 859 / N² m

h = 859000 / N² mm

Thus, the height of a Watt governor is inversely proportional to the square of the speed. 

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1. Application of Watt governor

Hunting of governor | Definition | Process

Definition of hunting of governor:

You know that the function of the governor is to keep the speed constant so the hunting of a governor is the condition in which the speed of the engine can fluctuate continuously above and below the mean speed and which are controlled by the governor.

In other words, hunting is said to occur in a governor when it is not able to find a stable position for a change in load at a new value of speed and oscillates around it due to the tendency of balls and sleeve to restore original speed and inertia effect leading to an overshoot of the desired position.

Process of hunting:

The sensitiveness of a governor and the hunting of a governor both are desirable qualities. If a governor is too sensitive, it may fluctuate continuously, because when the load on the engine falls, the sleeve rises rapidly to a maximum position. This shuts off the fuel supply to the extent of effect a sudden fall in speed. As the speed falls below the mean value, the sleeve again moves rapidly and falls to a minimum position to increase the fuel supply. The speed subsequently rises and becomes more than the average with the result that the sleeve again rises to reduce the fuel supply. This process is continuing and it is known as hunting of the governor. 

Hunting is directly proportional to sensitiveness. If more sensitive the governor, more is the hunting.

Sensitiveness of governor | Definition | Formula

Definition of sensitiveness : 

The ratio of the difference between the maximum and minimum equilibrium speed to the mean equilibrium speed is called sensitiveness of governor.

A governor is said to be sensitive when it readily responds to a small change of speed. The movement of the sleeve for a fractional change of speed is the measure of sensitivity. 

Explanation of sensitiveness :

As a governor is used to limit the change of speed of the engine between minimum to full-load conditions, the sensitiveness of a governor is also defined as the ratio of the difference between the maximum and the minimum speeds to the mean equilibrium speed. 

Sensitiveness = Range of speed / Mean speed 

                       = N₂ - N₁ / N 

                       

                       = 2 ( N₂ - N₁ ) / N₁ + N₂ 

Where, 

N = Mean speed

N₁ = Minimum speed corresponding to full load condition 

N₂ = Maximum speed corresponding to no-load condition 

Types of governor | Centrifugal | Intertia |

Governor is known as speed manager OR speed controller. 

They are used in automobile engines to measure and regulate the speed of an engine. It uses gears and flyweights inside the crankcase to detect changes in the load and adjusts the throttle accordingly.

Governor can broadly classify into two types :

1. Centrifugal governor 
2. Inertia governor 

Now let us know brief details about these two types of the governor.

• Centrifugal governor :

This is the most common type of governor. Its action depends on the change in speed. It has a pair of masses, known as governor balls, which rotate with a spindle. The spindle is driven by an engine through bevel gears. 

The action of this governor depends upon the centrifugal effects produced by the governor ball. With the increases in speed, the balls tend to rotate at a greater radius from the axis. This causes the sleeve to slide up on the spindle and this movement of the sleeve is communicated to the throttle through a bell crank lever. This closes the throttle valve to the required extent. When the speed decreases, the governor balls rotate at a smaller radius and the valve is opened according to the requirements. 

Centrifugal governor again classified into two groups :

• Pendulum type governor :

Pendulum type governor is Watt governor. 

• Loaded type governor :

Loaded type governor is Porter governor, Proell governor, Hartnell governor, Wilson Hartnell governor, Hartung governor and Pickering governor. 

• Inertia governor :

In this type of governor, action depends on acceleration. The position of the balls is affected by the forces set up by angular acceleration or deceleration of the given spindle in addition to centrifugal forces on the governor balls. Using suitable linkage and springs, the change in position of the balls is made to open or close the throttle valve. 

Thus, the governor balls are operated by the actual change of engine speed in the case of centrifugal governors, it is by the rate of change of speed in the case of inertia governor. Therefore, the response of inertia governors is faster than that of the centrifugal governor.

What is mechanical governor | Function | Working

What is the governor?

In simple words, the governor controls engine speed. It uses gears and flyweights inside the crankcase to detect changes in the load and adjusts the throttle accordingly.

The function of a governor?

To maintain the speed of an engine within specified limits whenever there is a variation of load occurs.

How it works?

The governor system is like a cruise control system in an automobile. It maintains the speed of your lawnmower or outdoor power products. 

In general, the speed of an engine varies in two ways, during each revolution and over a number of revolutions. For the first case, it is due to variation in the output torque of the engine during a cycle and can be regulated by mounting a suitable flywheel on the shaft. While in the second case, it is due to variation of the load upon the engine and requires a governor to maintain the speed.

If the load on the shaft increases, the speed of the engine decreases unless the supply of fuel is increased by opening the throttle valve. 

If the load on the shaft decreases, the speed of the engine increases unless the fuel supply is decreased by closing the valve sufficiently to slow the engine to its original speed. 

Thus, the throttle valve is operated by the governor through a mechanism for the purpose. 

Use of governor?

A governor is used to measure and regulate the speed of a machine, such as an engine. It is also called a speed limiter or controller. 

There are two types of governor used in any automobile application. One is centrifugal governor and another is inertia governor. 

What is Unbalance

In simple word, we can say that make someone unsteady so that they fall is called unbalance. 

In theoretical language, the condition which exists in a rotor when vibratory force or motion is imparted to its bearings as a result of centrifugal forces is called unbalance or the uneven distribution of mass about rotor's rotating line.  

Dynamic balancing

What is dynamic balancing?

A system is in dynamic balance when there does not exist any resultant centrifugal force and also a resultant couple. 

For example Several rotating masses 

When several masses rotate in different planes, the centrifugal forces in addition to being out of balance also from the couple. 

As shown in the figure above the product of mr an mrl usually have been referred to as force and couple respectively as it is more convenient to draw force and couple polygons with these quantities. 

If m₁ and m₂ are two masses revolving diametrically opposite to each other in different planes such that m₁r₁ = m₂r₂.

The centrifugal forces are balanced, but an unbalanced couple of magnitude m₁r₁l₁ = m₂r₂l₂ is introduced. 

The couple acts in a plane that contains the axis of rotation and the two masses. Thus, the couple is of constant magnitude but variable direction. 

Static balancing

Before we start to learn about static balancing you know about balancing. Balancing is the process of designing or modifying machinery so that the unbalance is reduced to an acceptable level and if possible is eliminated entirely. 

What is static balancing?

The combined mass centre of the system lies on the axis of rotation then a system of rotating mass is said to be in static balance.

Explanation of static balancing :

The figure shows a rigid rotor revolving with a constant angular velocity of ω rad/s. A number of masses are depicted by point masses at different radii in the same transverse plane. They may represent different kinds of rotating masses such as turbine blades, eccentric discs etc. 

If m₁, m₂ and m₃ are the masses revolving at radii r₁, r₂ and r₃ respectively in the same the plane, then each mass produces a centrifugal force acting radially outwards from the axis of rotation. 

Let F be the vector sum of these forces.

F = m₁r₁ω² + m₂r₂ω² + m₃r₃ω² 

The rotor is said to be statically balanced if the vector sum F is zero. 

The rotor is said to be unbalanced if vector sum F is not zero. 

Graphical solution :

In graphical solution, vectors, m₁r₁, m₂r₂, m₃r₃ are added. If they close in a loop, the system is balanced. Otherwise, the closing vector will be giving m_cr_c.

Its direction identifies the angular position of the countermass relative to the other masses. 

Introduction of balancing

Balancing of forces is most important in any machinery industries. Unbalance of forces is produced in rotary or reciprocating machinery due to the inertia forces associated with the moving masses. 

What is balancing?

Balancing is the process of designing or modifying machinery so that the unbalance is reduced to an acceptable level and if possible is eliminated entirely. 

Many serious problems encountered in high-speed machinery are the direct result of unbalanced forces. These forces exerted on the frame by moving machine members are time-varying impact vibratory motion to the frame and produce noise. There are some human discomfort and detrimental effects on machine performance and the structural integrity of the machine foundation. 

The most common approach to balancing is by redistributing the mass which may be accomplished by addition or removal of mass from various machine members. 

There are two basic types of unbalance :

1. Rotating unbalance
2. Reciprocating unbalance 

They both occur separately or in combination too. 

There are also two types of balancing system : 

1. Static balancing 
2. Dynamic balancing 

If we consider balancing whether it is static or dynamic but balancing can help to extend the service life, quality and accuracy of your machine while unbalanced parts can lead to breaking down to your machine. There is some difference between static and dynamic balancing
 also. 

Types of chain drive

Chains have been classified into three groups like :

• Hosting chains
• Conveyor chains
• Power-transmission chains 

Each type has been discussed below :

Hosting chains :

Hosting chains include an oval link and stud link chains. An oval-link chain is a common form of a hosing chain. It consists of oval links and is also known as coil chain. 

Hosting chains are used for lower speeds only. 

Conveyor chains :

Conveyor chains may be of the detachable or hook-joint type or of the closed-end pintle type. The sprocket teeth are so shaped and spaced that the chain should run onto the off the sprocket smoothly and without interference. The motion of this type of chain is not very smooth. 

Conveyor chains are used for low-speed agricultural machinery. The material of the links is usually malleable cast iron. 

Power transmission chains :

Power transmission chains are made of steel in which the wearing parts are hardened. They are accurately machined and run on carefully designed sprockets. 

They classified into three types :

• Blockchain: It is mostly used for transmission of power at low speeds. 

• Roller chain: Roller chain is fixed to the inner link whereas the outer link has a pin fixed to it. There is only sliding motion between pin and bushing. The roller is made of hardened material and is free to turn on the bushing. A good roller chain is quite and wears less as compared to a blockchain. 

• Silent chain: It is also known as Inverted Tooth Chain. This type of chain is used where maximum quietness is desired. The silent chain doesn't have rollers. The links are shaped as to engage directly with the sprocket teeth. The included angle is either 60 or 75 degree. 

Law of belting

To transmit power from one shaft to another pulley are mounted on the two shafts. A Belt drive is a mechanism in which power is transmitted by the movement of a continuous flexible belt. 

so now let us check it out some information of how to belt drive works? to know more about the law of belting. 

The law of belting states that the centre line of the belt when it approaches a pulley must lie in the midplane of that pulley OR A pulley in that plane must contain the point at which the belt leaves the other pulley. However, when a belt leaving a pulley may be drawn out of the plane of the pulley. 

By following this law of belting, non-parallel shafts may be connected by a flat belt. From the above figure, two shafts with two pulleys are at right angles to each other. Both of these can be observed that the centre line of the belt approaching the larger pulley and it lies in its plane which is also true for the smaller pulley. Also, the points at which the belt leaves a pulley are contained in the plane of the other pulley. 

By observation, it should not possible to operate the belt in the reverse direction without violating the law of belting. Thus, for non-parallel shafts, motion is possible only in one direction. Otherwise, the belt is thrown off the pulley. However, it is possible to run a belt in either direction on the pulleys of two intersecting or two non-parallel shafts with the help of guide pulleys. Thus, the law of belting is still satisfied. 

Types of follower motion

Though the follower can be made to have any type of desired motion, knowledge of existing motion program saves time and labour while designing the cams.

Following are some basic displacement programmes :

1. Simple Harmonic Motion ( SHM )
2. Constant Acceleration Deceleration ( Parabolic )
3. Constant Velocity 
4. Cycloidal 

Mechanical advantage definition

What is a mechanical advantage?

The ratio of the output force or torque to the input force or torque at any instant OR 
The ratio of force produced by a machine to the force applied to it is called the mechanical advantage of mechanism. 

For any linkage, if friction and inertia forces are ignored and the input torque is applied on one link and resisting torque is on another link then formula for finding mechanical advantage :

Power input = Power output 

 T₁w₁ = T₂w₂

So, Mechanical Advantage = T₁/T₂ = w₁/ w₂

Thus mechanical advantage is written as the reciprocal of the velocity ratio or the reciprocal of the torque ratio.  

Classification of gears

Gears can be classified according to the relative positions of their shaft axes as follows :

Parallel shafts :

In the parallel shaft, the manner of contact and uniform rotary motion between two parallel shafts is equivalent to the rolling of two cylinders. 

The following are the main types of gears to join parallel shafts :

• Spur gears :

In this type of gear, the teeth are straight and parallel to the axes and thus are not subjected to the axial trust due to load. 

• Spur rack and pinion :

Spur rack is a special case of a spur gear where it is made of infinite diameter so that a pitch surface is a plane. 

The spur rack and pinion combination convert rotary motion into translatory motion or vice-versa. 

This type of gear usually used in a lathe in which rack transmits motion to the saddle.

• Helical gears or Helical spur gears :

In this type of gear, teeth are curved, each being helical in shape. Two mating gears have the same helix angle but have teeth of opposite hands. 

In the helical gear, at the beginning of engagement contact occurs only at the point of the leading edge of the curved teeth. As the gear rotates, the contact extends along with a diagonal line across the teeth. Thus, the load application is gradual which results in low impact stresses and it can be used for higher velocities than the spur gears. 

• Double helical gears or Herringbone gears :

A double-helical gear is equivalent to a pair of helical gears secured together, one having a right-hand helix an the other a left-hand helix. 

Axial thrust which occurs in case of single helical gear is eliminated in double helical gears. 

If the left and the right inclinations of a double helical gear meet at a common apex and there is no groove in between, the gear is known as herringbone gear. 

Intersecting shafts :

Kinematically, the motion between two intersecting shafts is equivalent to the rolling of two cones, assuming no slipping. This type of gear is known as bevel gears.

• Straight bevel gears :

The teeth are straight, radial to the point of intersection of the shaft axes vary in cross-section throughout their length. 

Usually, they are used to connect shafts at right angles which runs at low speeds. Gears of the same size and connecting two shafts at right angles to each other are known as mitre gears. 

• Spiral bevel gears :

When the teeth of bevel gears are inclined at an angle to the face of the bevel, they are known as spiral bevels or helical bevels. 

They are smoother in action and quieter than straight tooth bevels as there is gradual load application and low impact stresses. 

• Zerol bevel gears :

Spiral bevel gears with curved teeth but with a zero degree spiral angle are known as zerol bevel gears. 

They are quieter in action than the straight bevel type as the teeth are curved. 

Skew shafts :

In case of parallel and intersecting shafts, a uniform rotary motion is possible by pure rolling contact. But in case of skew shafts, this is not possible.

• Crossed helical gears :

The use of crossed helical or spiral gears is limited to light loads. By a suitable choice of helix angle for the mating gears, the two shafts can be set at any angle. 

• Worm gears :

Worm gear is special case of spiral gear in which the larger wheel, usually has a hollow or concave shape such that a portion of the pitch diameter of the other gear is enveloped on it. 

The smaller of the two wheels is called the worm which also has a larger spiral angle. 

• Hypoid gears :

Hypoid gears are approximations of hyerboloids though they look like spiral gears. 

The hypoid pinion is larger and stronger than a spiral bevel pinion. 

A hypoid pair has a quite and smooth action. 

Involute tooth profile

What is involute?

The locus of the point on a straight line which rolls without slipping on the circumference of a circle is called the involute.

In other words, it is the path traced out by the end of a piece of the taut cord being unwound from the circumference of a circle. The circle on which the straight line rolls or from which the cord is unwound is known as the base circle.

Formation of involute tooth :

From the above figure, an involute generated by a line rolling over the circumference of a base circle with centre at O. At the start, the tracing point is at A. As, the line rolls on the circumference of the circle, the path ABC traced out by the point A is the involute. 

D can be regarded as the instantaneous centre of rotation of B, the motion of B is perpendicular to BD. Since BD is tangent to the base circle, the normal to the involute is a tangent to the base circle.  

A short length EF of the involute drawn from A can be utilized to make the profile of an involute tooth. The other side HJ of the tooth has been taken from the involute drawn from G in the reverse direction. The profile of an involute tooth is made up of a single curve, and teeth, usually, are termed as single curve teeth. 

Notes :

Because of ease of standardization and manufacture, and low cost of production, the use of involute teeth has become universal by entirely superseding the cycloidal shape. Only one cutter or tool is necessary to manufacture a complete set of interchangeable gears. The cutter is in the form of a rack as all gears will gear with their corresponding rack. The cutters of this form can be made to a higher degree of accuracy as the teeth of an involute rack are straight. 

Key points :

• Points of contact lie on the line of action which is the common tangent to the two base circles. 
• The contact is made when the tip of a tooth of the driven wheel touches the flank of a tooth of the driving wheel and the contact is broken when the tip of the driving wheel touches the flank of the driven wheel. 
• If the direction of angular movement of the wheels is reversed, the points of contact will lie on the other common tangent to the base circles. 
• Initial contact occurs where the addendum circle of the driven wheel intersects the line of action. 
• Final contact occurs at a point where the addendum circle of the driver intersects the line of action. 
• For a pair of involute gears, the velocity ratio is inversely proportional to the pitch circle diameters as well as base circle diameters.