Cycloidal tooth profile

What is cycloid?

The cycloid is defined as the locus of a point on the circumference of a circle that rolls without slipping on a fixed straight line.
In this type, the faces of the teeth are epicycloids and the flanks are the hypocycloids. 

Now one question arises in your mind that what is epicycloids and hypocycloids? Let we check 

What is epicycloid?
An epicycloid is the locus of a point on the circumference of a circle that rolls without slipping on the circumference of another circle. 

What is hypocycloid?
A hypocycloid is the locus of a point on the circumference of a circle that rolls without slipping inside the circumference of another circle. 

Formation of cycloidal tooth :

A circle O₁ rolls inside another circle APB is called pitch circle. At the start, the point of contact of the two circles is at A. As the circle O₁ rolls inside the pitch circle, the locus of the point A on the circle O₁ traces the path ALP which is called hypocycloid. A small portion of this curve near the pith circle is used for the flank of the tooth.  

The line joining the generating point A with the point of contact of the two circles is normal to the hypocycloid when the circle O₁ touches the pitch circle at D, the point A is at C and CD is normal to the hypocycloid ALP. 

In the same way, if the circle O₂ rolls outside the pitch circle, starting from P, an epicycloid PFB is obtained.

A small portion of the curve near the pitch circle is used for the face of the tooth also. 

Some key points :

• The path of approach is equal to the arc of approach. 
• The path of contact is equal to the arc of contact. 
• In case of cycloidal teeth, the pressure angle varies from the maximum at the beginning of the engagement to zero when the point of contact coincides with pitch point and then again increased to a maximum in the reverse direction. 
• Since cycloidal teeth are made up of the two curves, it is very difficult to produce accurate profiles. This has rendered this system obsolete. 

Forms of tooth

Types of tooth :

Two curves of any shape that fulfil the law of gearing can be used as the profiles of teeth. 

In other words, an arbitrary shape of one of the mating teeth can be taken and applying the law of gearing the shape of the other can be determined. Such gear is said to have conjugate teeth. 

However, it will be very difficult to manufacture such gears and the cost will be high. It will be very difficult to replace them with the available gears. Thus, arises the need to standardize gear tooth. 

Common forms of teeth that also satisfy the law of gearing are the following :

• Cycloidal tooth profile
• Involute tooth profile  

Contact ratio of gears

Arc of contact :

The distance travelled by a point on either pitch circle of the two wheels during the period of contact of a pair of teeth is called the arc of contact. 

It is the length of the pitch circle traversed by a point on it during the mating of a pair of teeth. 

Thus, all teeth laying in between the arc of contact will be meshing with the teeth on the other wheel. 

Number of teeth in contact n = Arc of contact / Circular pitch 

As we mentioned above the ratio of the arc of contact to the circular pitch is also the contact ratio, the number of teeth is also expressed in terms of contact ratio. 

For continuous transmission of motion, at least one tooth of one wheel must be in contact with another tooth of the second wheel. Therefore, n must be greater than unity.

If n lies between 1 and 2, the number of teeth in contact at any time will not be less than one and never more than two. 

If n is between 2 and 3, it is never less than pairs of teeth and not more than three pairs, and so on. 

If n is 1.6, one pair of teeth are always in contact whereas two pairs of teeth are in contact for 60% of the time. 

Introduction of gears

Gears are used to transmit motion from one shaft to another or between a shaft and a slide. This is done by successively engaging the teeth. 

Gears use no intermediate link or connector and transmit the motion by direct contact. 

The two bodies have either a rolling or sliding motion. Motion is along with the tangent at the point of contact. No motion is possible along the common normal as that will either break the contact or one body will tend to penetrate into the other.  

To transmit a definite motion of one disc to the other or to prevent slip between the surfaces, projections and recesses on the two-disc can be made which can mesh with each other. This leads to the formation of teeth on the discs and the motion between the surfaces changes from rolling to sliding. The discs with teeth are known as gears or gear wheels. 

What is rope drive

The rope drive is widely used where a large amount of power is to be transmitted. For power transmission by rope, grooved pulleys are used. 

The rope drives use the following two types of ropes :

• Fibre ropes 
• Wire ropes 

The rope is gripped on its sides as it bends down in the groove reducing the chances of slipping. Pulleys with several grooves can also be employed to increase the capacity of power transmission. 

These may be connected in the two ways following below :

• Using a continuous rope passing from one pulley to the other and back again to the same pulley in the next groove, and so on. 
• Using one rope for each pair of grooves.

Number of rope required = Total power transmitted / Power transmitted per rope 

Rope drives are, usually, preferred for long centre distances between the shafts.

Rope drive are cheaper as compared to belt drive.

Types of belt drive

A belt drive is one of the most common and effective means of transmission of motion from one shaft to another shaft. There are different belt drive used for different applications. Now we can see the different types of belt drive :

Types of belt drive :

According to the power transmitted :

1. Light belt drive 
2. Medium belt drive 
3. Large belt drive 

According to the arrangement of the belt :

1. Open belt drive
2. Crossbelt drive
3. Quarter twist drive
4. Right angle drive
5. Stepped pulley drive
6. Fast and loose pulley drive 
7. Compound drive 

What is belt drive

Introduction :

To transmit power from one shaft to another pulley are mounted on the two shafts. The pulleys are then connected by an endless belt or rope passing over the pulleys. The connecting belt or rope is kept in tension so that the motion of one pulley is transferred to the other without slip. The speed of the driven shaft can be varied by wearing the diameter of the two pulleys. 

What is belt drive?

Belt the drive is a mechanism in which power is transmitted by the movement of a continuous flexible belt. 

A belt may be a rectangular section is known as a flat belt or of the trapezoidal section known as V-belt. In case of a flat belt, the rim of the pulley is slightly crowned which helps to keep the belt running centrally on the pulley rim. The groove on the rim of the pulley of a V-belt is made deeper to take advantage of the wedge action. The belt does not touch the bottom of a groove. Owing to wedging action V-belt needs a little adjustment and transmit more power without sleep as compared to flat belt also multiple V-belt a system using more than one belt in the two pulleys can be used to increase the power transmitting capacity generally these are more suitable for shorter centre distance. Thus, belt drive works on the law of belting. 

Open belt drive :

An open belt drive used when the driven pulley is desired to be rotated in the same direction as the driving pulley.  

Generally, the centre distance for an open belt drive is 14 to 16 metre. If the centre distance is too large, the belt whips vibrate in a direction perpendicular to the direction of motion. For very short at the centre distance, the belt sleep increasing. Both these phenomena limit the use of belts for power transmission.

Crossbelt drive :

A the crossed-belt drive is adopted when the driven pulley is to be rotated in the opposite direction to that of the driving pulley.

A cross belt drive can transmit more power than an open belt drive as the angle of wrap is more. However, the belt has to be a band in two different planes and it we are out more. 

Advantages of belt drive :

• A belt drive is simple and economical.
• Wide range of speeds is available.
• In belt drive don't need parallel shaft.
• Noise and vibration are damped out.
• Machinery life is increased because load fluctuations are shock-absorbed.
• Less maintenance cost because no lubrication are required.
• Belts permit flexibility ranging from high horsepower drives to slow speed and high speed drives so it is highly efficient in use.
• A flat belt is best for very high-speed drives.
• This drive is very economical even when the distance between the shaft is very large.
• Belts will slip under overload conditions this leads the biggest advantages that preventing mechanical damage to shafts, keys, and other machine parts.
• All the belt drives do not need the pulleys to be maintained at the same height.

Disadvantages of belt drive :

• Operating temperature is restricted up to 80 to 85⁰C.
• Heat buildup occurs.
• Belts can't be used where exact timing or speed is required because of slippage.
• Because of slipping and stretching the angular velocity ratio is not necessarily equal or constant to pulley diameter ratio.
• In belt, drive belts are damaged easily by abrasives or heat or some chemicals.
• Some adjustment of centre distance or use of an idler pulley is necessary for wearing and stretching of belt drive compensation.
• Speed is limited to usually 35 meters per second.
• Power transmission is limited to 370 kilowatts.

What is linkage

What is linkage :

A linkage is obtained if one of the links of a kinematic link is fixed to the ground.

The action of linking or the state of being linked two links is known as linkage.

In other words, the linkage is a system of links.

What is kinematic pair

Kinematic pair :

A kinematics pair of simply a pair is a joint of two kinematic links that have relative motion with respect to each other. 
When two links in a machine are in contact with each other, they form a pair. Each individual links of a mechanism form a pairing element. 

A degree of freedom of kinematic pair is given by the number of independent coordinates required to completely specify the relative movement between the pair of two links.

Types of kinematic pair :

Kinematic pair can be classified according to :

• The nature of the contact.
• The nature of mechanical constraint.
• The nature of relative motion.

Kinematics pairs according to nature of contact :

• Lower pair 
• Higher pair 

Kinematic pairs according to the nature of mechanical constraint :

• Closed pair 
• Unclosed pair 

Kinematics pair according to nature of relative motion :

• Sliding pair
• Turning pair
• Rolling pair 
• Screw pair
• Spherical pair 

Types of kinematic pairs

Types of kinematic pair :

Kinematic pair can be classified according to :

• The nature of the contact.
• The nature of mechanical constraint.
• The nature of relative motion.

Kinematics pairs according to nature of contact :

Lower pair: A pair of links having surface area contact between the member is known as a lower pair.

In the lower pair, the contact surfaces of the two links are similar.

For example : 

• Nut turning on a screw shaft.
• Shaft rotating in a bearing.
• All pairs of slider-crank mechanism.
• Universal joint.

Higher pair: A pair has a point or line contact between the links is known as a higher pair.

In the higher pair, the contact surfaces of the two links are dissimilar.

For example : 

• Rotating on a surface 
• Cam and follower pair 
• Tooth gear 
• Ball-bearing  
• Roller bearings 

Kinematic pairs according to the nature of mechanical constraint :

Closed pair: When the elements of a pair are held together mechanically it is known as a closed pair.
The two elements are geometrically identical one is solid and full and other is hollow or open. The letter not only envelops the former but also and closed it. 
All the lower pairs and some of the higher pairs are closed pair.
A screw pair belong to the closed pair category also. 
  

Unclosed pair: When two links of a pair are in contact but due to some spring action or force of gravity, they constitute an unclosed pair.
In this, the links are not held together mechanically. 

Kinematics pair according to nature of relative motion :

Sliding pair: If two links have a sliding motion related to each other, they form the sliding pair. 

For example - A rectangular rod in a rectangular hole in a prism

Turning pair: When one link has a turning on revolving motion relative to others, they constituted a turning pair.

They also called revolving pair.

For example - In a slider-crank mechanism all pairs except the slider and guide pair are turning pair.

A circular shaft revolving inside a bearing.

Rolling pair: When the link of a pair has a rolling motion relative to each other, they form a rolling pair.

For example - Rolling wheel on a flat surface

Ball and roller bearing in a ball bearing 

The ball and shaft constitutive and rolling pair where is the ball and the bearing is a second rolling pair.

Screw pair : If two mating links have a turning as well as sliding motion between them, they form of screw pair. This can be achieved by cutting matching threads on the two links.

For example - The lead screw and the nut of a late is a screw pair.

Spherical pair : When one link in the form of sphere turn inside a fixed link, it is a spherical pair.

For example - The ball and socket joint

Types of constrained motion

Types of constrained motion :

• Completely constrained motion 

When the motion between two elements of a pair is indefinite direction irrespective of the direction of the force applied it is known as completely constrained motion.

The constrained motion may be linear or rotary.

Example of completely constrained motion :

• The sliding pair 
• The turning pair 

In sliding pair, the inner prism can only slide inside the hollow prism.

In case of the turning pair, the inner shaft can have only rotary motion due to the collar at the ends. 
In each case of force has to be applied in a particular direction force required motion.

• Incompletely constrained motion 

When the motion between two elements of a pair is possible in more than one direction and depend upon the direction of the force applied that motion is known as incompletely constrained motion. 

Example of incompletely constrained motion :

• The inner shaft may have sliding or rotary motion depending upon the direction of the force applied if the turning pair does not have a collar.

Each motion is independent of the other.

• Successfully constrained motion 

When the motion between two elements of a pair is possible in more than one direction but with the using of some external means motion is made to have only in one direction so that motion is known as successfully constrained motion.

Example of a successfully constrained motion :

• A shaft in a footstep bearing may have vertical motion apart from rotary motion but due to the load applied on the shaft, it is a constraint to move in that direction and thus is successfully constrained motion.
•  A Piston in a cylinder of an internal combustion engine is made to have only reciprocating motion and no rotary motion due to constraining of the piston pin. 
• The value of an IC engine is kept on the seat by the force of a spring.

Types of joints in kinematics

Types of joints :

• Binary joint 
• Ternary joint 
• Quaternary joint 

Binary joint :

If two links are joint at the same connection it is known as a binary joint.

Ternary joint :

If three links are joint at a connection it is known as a ternary joint. 
It is considered equivalent to the two binary joints since fixing of anyone link constitute two binary joints with each of the other two links.

Quaternary joint :

If four links are joint at a connection it is known as a quaternary joint.
It is considered equivalent to three binary joints since fixing of anyone link constitutes three binary joints.

If n number of links are connected at a joint, it is equivalent to ( n -1 ) binary joints. 

What is kinematic link

What is the kinematic link?

A mechanism is made of a number of resistant bodies out some may have motions relative to each other. A resistant body or a group of resistant bodies with rigid connections preventing their relative movement is known as a link.
This link is also known as kinematic link or element.

The link can be classified into binary, ternary and quaternary depending upon the end on which revolute or turning pairs can be placed.

A link may also be defined as a member or a combination of members of a mechanism, connecting other member and having motion relative to them. Thus, a link may consist of one or more resistant bodies. 

For example, a slider-crank mechanism consists of four links: frame and guides, crank, connecting rod and slider. however, the frame may consist of bearings of for the crankshaft. The crank link may have a crankshaft and flywheel also, forming one link having no relative motion of this.

Mechanism and machine

In this article, we will discuss the mechanism and machine.

What is the mechanism :

When one of the links of a kinematic link is fixed, then the chain is known as a mechanism.

A mishmash of a number of bodies assembles in such a way that it causes the constrained and predictable motion to the other is known as a mechanism.
Thus, the function of the mechanism is to transmit and modify a motion.

What is a machine :

Machine is a mechanism which receives energy and transforms it into some useful work.

A collection of mechanisms which, transmits and modifies available mechanical energy into some kind of desired work and also imparting definite motions to the parts is called machine.

It is neither a source of energy nor a producer of work that helps in proper utilization of the same. The motive power has to be derived from external sources.

Example :

A slider-crank mechanism converts the reciprocating motion of a slider into rotary motion of the crank or vice-versa. however, when it is used as an automobile engine by adding valve mechanism it is it becomes a machine which converts the available energy into the desired energy.  

Some other examples of mechanisms are typewriter, clocks, watches, spring toys etc. 
In each of these force or energy provided is not more than what is required to overcome the friction of the path and which is utilized just to get the desired motion of the mechanism not to obtain any useful work.

Some useful notes :

A machine is tangible, but the mechanism is not. Only the effect of the mechanism is observable.

In a simple way, a mechanism is a system that is followed by a machine to achieve the particular function while a machine is a combination of tools and parts that performs specific functions at the expense of energy.

For Example :

We think a car is a machine. It made of various mechanisms such as the wipers, piston and crankshaft, differential.
All this parts are simple a mechanism following below : 

• Wipers is a 4 link crank lever mechanism
• Piston and crankshaft is a slider crank mechanism
• Differential is a gear mechanism. 

Thus a machine is made of number of mechanism to carryout a particular task.

Mechanism and structure

Mechanism :

Linkage is obtained if one of the links of kinematic chain is fixed to the ground. If the motion of any of these movable ink results in definite motion of the other, the linkage is known as the mechanism.

However, this distance between a mechanism and linkage is hardly followed and it can be referred in place of others.

In a mechanism, links are connected with temporary fasteners. The links can move relative to each other.

This is facilitated by the presence of joints between the links. So a degree of freedom is >=1

Structure :

If one of the links of the redundant chain is fix it is known as a structure.
It is also known as a locked system.
To obtain constrain or definite motion of some of the links of linkage, it is necessary to know how many inputs are needed. In some mechanism, only one input is necessary that determine the motion of other links and it is said to have one degree of freedom. While other mechanisms, two inputs necessary to determine to constrain motion of the other links so we said they have two degrees of freedom. 

In a structure again a combination of links connected to each other but no relative motion exists between them. So a degree of freedom is zero.

A structure who have a negative degree of freedom is known as a superstructure.

A mechanism is a force manipulating device, while the structure is a load bearing device.

Degree of freedom in kinematics

Degree of freedom :

Degree of freedom is related to motion possibilities of rigid bodies.

An unconstrained rigid body moving in space can describe the following independent motions :

• Translational motion along any three manual perpendicular axis x y and Z.
• Rotational motion about this axis.

Thus, rigid body processes 6 degrees of freedom.

The concept of degree of freedom in the kinematics of machines is used in three ways :

• A body is relative to a reference frame.
• Kinematic joints
• A mechanism.

The connection of a link with another imposes certain constraints on their relative motion.
The number of restraint can never be zero or six.

Degree of freedom of a pair is defined as the number of independent relative motions, both translational and rotational a pair can have. 

Degree of freedom = 6 -  Number of restraints 

The general equation to find out degrees of freedom of a planar mechanism is given below. This equation is also known as Kuthbach equation.

D.O.F = 3 ( N-1 ) - 2L_p - H_p

Here N = Total number of links in the mechanism. 

L_P and H_P = Number of lower pairs and higher pairs respectively.

If the mechanism is 3 dimensional in nature, you could easily derive an equation for mobility using the same concept. So the equation for the degree of freedom would be following below :

D.O.F = 6 ( N-1 ) - 5P₅ - 4P₄ - 3P₃ - 2P₂ - 1P₁

Where P_n = number of pairs which block 'n' degrees of freedom. 

The main thing here will be the determination of nature of the kinematic pair. 

Cam terminology

CAM NOMENCLATURE :

Definitions of terms that used in cam profile nomenclature :

• Base circle :

It is the smallest circle drawn a tangent to the cam profile from the centre.

• Tracepoint :

It is a reference point on the follower to trace the cam profile such as the knife-edge of a knife-edge follower and centre of the roller follower.

• Pitch curve :

It is the curve drawn by the tracepoint assuming that the cam is fixed and the tracepoint of the follower rotates around the cam.

• Pressure angle :

It is the angle between the normal to the pitch curve at a point and the direction of the follower motion.

Pressure angle representing the steepness of the cam profile.

• Pitch point :

It is the point on the pitch curve at which the pressure angle is maximum.

• Pitch circle :

It is the circle passing through the pitch point and concentric with the base circle.

• Prime circle :

The smallest circle drawn a tangent to the pitch curve is known as the prime circle.

Displacement diagram :

As a cam rotates about the axis, it imparts a specific motion to the follower which is repeated with each revolution of the cam. It is enough to know the motion of follower for only one revolution.

The motion of the cam can be represented on a graph the x-axis represents the can rotation and the y-axis represents the displacement of the follower. Now we discuss the follower displacement diagram and its terms :

• The angle of ascent :

It is the angle through which the cam turns during the timing of the follower rising. 

In the above figure angle of ascent represent by Ɵ_ri.

• The angle of dwell :

It is the angle through which the cam turns at the same time follower remains stationary at the highest or the lowest position.

In figure angle of dwell represent for the highest position by Ɵ_d1.

• The angle of descent :

It is the angle through which the cam turns during the timing when the follower returns to its initial position.

In figure angle of descent represent by Ɵ_re.

• Angle of action :

It is the angle through which the cam turns during the time between the beginning of rise and the end of the return of the follower. In fact angle of action is the total angle moved by the cam.

In figure angle of action represent by Ɵ.

For variation of cam types and different types of followers motions there are also different different displacement diagram are made.