Mechanism in Kinematics

In this tutorial, you will learn the basics of Mechanism, machine and various relative motions. Also, you will learn about various terminologies and concepts which are prerequisites in the learning of Theory of Machines.


  1. Mechanism and Machine
  2. Types of Relative Motion
  3. Types of Kinematic Links
  4. Classification of Joints
  5. Kinematic Pairs
  6. Degree of Freedom & Kutzbach’s Criterion
  7. Grubler’s Criterion
  8. Uses of Mechanism in Kinematics

Mechanism and Machine

A mechanism is a combination of a number of rigid bodies assembled or connected in such a manner that the motion of one or more than one bodies causes relative motion in the other bodies.

  • The function of a mechanism is to transfer and modify the type of motion. A mechanism is made up of different kinds of links, joints, and pairs which when assembled will constitute a mechanism.
  • The type of motion transfer is made possible by fixing one of the components in a mechanism and then determining the type of motion of other components with respect to the fixed one.
  • The simplest example of a mechanism is Single-slider Crank Mechanism.

It consists of the following components:

  • Slider: It is the part which reciprocates to and fro and it is the part which generates the first initial pressure in the entire mechanism. It is usually made up of hardened metals such as nickel alloy steel, stainless steel and is usually made in a cylindrical profile.
  • Connecting Rod: It is common linkage from the slider to the power source or crank of the mechanism. This component is responsible for the transfer of reciprocating motion to rotating type.
  • Crank: It is the backbone of most mechanisms. It is the component where the actual energy/work is generated and is then transferred to some transmission to get the desired work.

A single-slider crank mechanism is shown below:

Single-Slider Crank Mechanism

The components labelled are as follows:

  1. Connecting Rod
  2. Crank
  3. Slider

A machine is a mechanism or a combination of mechanism which not only transfers and modify the type of motion but also transfers the available chemical energy/torque into useful mechanical work.

  • The single-slider crank mechanism shown above can be used in various type of configurations by adding one or more than one mechanism to form a useful work-generating machine. Some examples are engines used in Automobiles and Steam-Engines used in the early days.

Types of Relative Motion

Relative motion is the motion of a body with reference to another frame of reference which is moving or stationary. In Kinematics, there are three types of relative motion which are used in the analysis of different mechanisms. They are as follows:

  • Completely Constrained Motion: Irrespective of the force applied, when the relative motion of a pair is confined/constrained to only one definite direction, then the motion is said to be Completely Constrained Motion. A square key in a square hole is an example of Completely Constrained Motion as shown below:
  • Square-key in a Square Hole

    The above diagram describes about the Completely Constrained Motion in which the square key is within the square hole. As a result, its rotational motion is hindered and hence it can only perform to and fro motion within the square hole.

  • Incomplete Constrained Motion: When the relative motion between a pair can take in more than one direction, then this type of motion is called Incomplete Constrained Motion. Circular Shaft in a Circular hole is an example of this type of motion as shown below:
  • Circular Shaft in a Circular Hole

    The above figure shows a Circular Shaft in a Circular Hole. Here, the shaft is able to perform rotational as well as translational motion.

  • Successfully Constrained Motion: When the relative motion of a pair is not confined by the pair itself, but by some other means such as the application of a force or load, then this kind of motion is known as Successfully Constrained Motion. An example of this kind of motion is explained below:
    Foot-Step Bearing

    The above diagram shows a Foot-Step bearing. It consists of a bearing hub and shaft which would be able to slide within the bearing and also rotate about its axis if there was no load on the shaft.

    So, the motion would not have been successfully constrained. But as we apply load over the shaft, the to and fro motion is constrained and the foot-step bearing becomes an example of Successfully Constrained Motion.

Hence, we can conclude that this type of motion requires a load/force to complete the motion. The valve springs used in the cylinder head of an Automobile is also an example of Successfully Constrained Motion where the load is derived from the motion of the pushrods or cams in the head of the engine.


Types of Kinematic Links

A link is an element or a combination of elements in a mechanism or a machine which connects other members and has relative motion within them. Various types of links are as follows:

  • Binary Link: A link in which two links can be connected is known as Binary Link. Binary links are the simplest type of links and they can be either temporary or permanent as per the requirement. Here, only one link is allowed to be in contact with the other.
  • Tertiary Link: A link to which three links can be connected is known as Tertiary Link. Such type of links is a combination of 2 Binary Joints.
  • Quaternary Link: A link to which four links can be connected is known as Quaternary Link. This type of links are a combination of 3 Binary Joints connected together. Quaternary joints can also be designed by attaching 4 different types of links to a common point.
Binary, Tertiary and Quaternary Links

The above figures describe about the different types of joints and their possible configurations.

Based on the level of deformability, links can also be classified as under:

  • Rigid Link: A link which does not deform (negligible if any) while in motion transfer is known as Rigid Link.
  • Flexible Link: A link which undergoes partial deformation is known as Flexible Link. It is to be noted that there must be no effect in motion transfer even if the link deforms partially. Belt Drive is an example of Flexible link as shown in the diagram below.
    Belt and Pulley Drive

    The above diagram shows a belt and pulley drive in which the belt is the Flexible Link.

  • Fluid Link: Such type of link is observed in hydraulic brakes and lifts where the fluid acts as a link by transmitting the pressure applied from one part of the machine to other. The fluid needs to be sealed properly to prevent any leaks thereby affecting the energy transfer.

    While studying different types of links, the concept of rigid body must be clear. Ideally, a rigid body is defined as a homogenous body in which the intermolecular force of attraction is too high and the distance between two consecutive molecules must remain constant under different load conditions.

    But in practice, the type of material, the manufacturing process the heattreatment affects the homogeneity of the material. Hence, a link must rather be resistant body so that it is capable of transmitting the forces without any deformation or negligible, if any.

Hence, a Kinematic Link must have the following characteristics:

  1. It must be a resistant body.
  2. It should always have relative motion.
  3. It must be able to transfer the motion with minimum energy losses.

Classification of Joints

Joint is the common point between two or more than two links in a mechanism. For a mechanism to work, temporary or permanent joints are a necessity. Without joints, a mechanism cannot be imagined.

The different types of joints are mentioned below:

  • Binary Joint: A joint in which two links are attached is known as Binary Joint. A scissor is an example of Binary Joint.
  • Tertiary Joint: A joint in which three kinematic links are attached is known as Tertiary Joint. A nail-clipper is an example of Tertiary Joint where the two cutting edges and a handle are attached to one single point.
  • Quaternary Joint: A joint in which four kinematic links are attached is known as a Quaternary Joint. Rotary IC Engines are a great example of Quaternary Joint where all the connecting rod from different cylinders are attached to one common point.

The following figures show the different types of joints.

Different types of Joints

The above figure shows classification of joints and their possible arrangements. Such joints are generally nut-bolt or riveted in type.

Kinematic Pairs

Two links or elements in a joint which have relative motion within them and are in contact with each other constitute a pair. If the relative motion between the joints is completely or successfully constrained and they are in a definite direction, then that pair will be known as a Kinematic Pair.

Pairs are widely classified into different types according to the conditions. Some of them are mentioned below:

  • According to the type of contact
    • Lower Pair: When any two mechanical elements are in surface/area contact with each other, that is, the surface of one element is in contact with the surface of another element, then that pair is known as a Lower Pair.
      Foot-Step Bearing, Screw-Nut combination are examples of Lower Pairs.
    • Higher Pair: When any two mechanical elements are in line/point contact with each other, then that pair is known as a Higher Pair.
      A wheel rolling on a surface, cam-follower arrangement are examples of Higher Pair.
  • According to the type of constraints
    • Self-Closed Pair: In a pair, when the links are connected by some mechanical means in such a manner that only the desired motion takes place, then that pair is known as Self-Closed Pair. All lower pairs are Self-Closed Pairs.
    • Force-Closed Pair: In a pair, when the links are not connected by any means, but are still kept in contact with each other by the action of temporary external forces, then that pair is known as Force-Closed Pair.

As shown in the figure below, Cam and Follower connected by spring and gravity is an example of Force-Closed Pair. The Cam which is elliptical in shape rotates rotate about its axis (shown by the dotted black circle) which makes the follower reciprocate about its position. The spring of the follower must be of a very high quality because a Cam and Follower arrangement is used in places where the RPM of the mechanism is very high and the follower can reciprocate up to 200 times a second.

CAM and Follower arrangement

The above figure shows a Cam with a reciprocating follower. Due to the rotation of the Cam, the Follower performs to and fro translational motion.

  • According to the type of contact
    • Rolling Pair: If one of the links in a pair has rolling motion with respect to each other, then that pair is known as a Rolling Pair. Ball Bearings, Roller Bearings, Hole and Shaft arrangement are examples of Rolling Pair.
    • Turning Pair: In a mechanism, if one link has turning or revolving motion in a fixed axis with respect to the other link, then that pair is known as a Turning Pair.

      Cycle wheels turning on the axle, Journal Bearing, Lathe Spindle turning on the Headstock, crankshaft are examples of Turning Pair.

    • Sliding Pair: If one of the elements of a pair are connected in such a way that one ink slides over the other, then that pair is called Sliding Pair. Piston-Cylinder in an automobile, Shaper machine are few examples of Sliding Pair.
    • Screw Pair: In such kind of pair, one link can turn or rotate about a fixed axis with the help of screw threads.
    • Spherical Pair: In such kind of pair, one link swivels around the other in a fixed cavity as per the requirement. Ball and Socket Joints, Rear View Mirror are examples of Spherical Pair.

Degree of Freedom & Kutzbach Criterion

Degree of Freedom is the number of independent co-ordinates used to describe the position or configuration of all the links in a mechanism with respect to the fixed link. The formula to determine Degree of Freedom is known as Kutzbach equation described below:

F = 3(N – 1) – 2J – H

N = number of links
J = number of binary joints
H = number of higher pairs

  • Here, (N-1) is the number of movable links in a given mechanism in a lower pair.
  • If F = 0, the mechanism has no movable links, hence it is known as a structure as shown below
  • If F = 1, the mechanism only one link is movable and only one co-ordinate is enough to specify the positions of all the links in the mechanism.
  • If F = 2, the mechanism has two movable links and hence two inputs are to be provided to get one output.
  • If F = -1, the mechanism has redundant constraints and it becomes a statically indeterminate structure.

We have already discussed Kutzbach equation used to determine DOF in lower pair. One higher pair case is mentioned below:

Consider the following diagram below:

A Higher Pair Arrangement

The following figure shows a wheel attached to the end of a link in such a manner that it moves to and fro about the base line. There are 3 binary joints and a Higher Pair between the wheel and the base surface. So,
N = 4, J = 3, H = 1.
Hence from the Kutzbach eq., we have,
F = 3 (4 – 1) – 2 × 3 -1
F = 2
Hence, the DOF of a Higher Pair is 2.

Grubler’s Criterion

This condition arises when the number of higher pair in a mechanism is zero. In that case, the DOF equation changes as described below:

  • F = 3(N-1) – 2J
    N = number of links in the mechanism
    J = number of binary joints in the mechanism
  • Now, it is to be noted that the grubler’s criterion will only be applicable to mechanisms of F = 1.
    Therefore, the equation becomes
    1 = 3(N – 1) – 2J
    or, 1 = 3N – 3 – 2J
    or, 3N – 2J – 4 = 0,
    This equation is known as Grubler’s equation.
  • The possible outcomes in these cases are a Four-Bar Mechanism shown below and a Single-Slider Crank Mechanism as shown in Figure 1.
    Four-Bar Mechanism

    The above diagram shows a Four-Bar Mechanism with 4 binary joints. As an input given to a link will cause similar motion to the opposite adjacent link.

  • It is to be noted that in cases where the movability is 1 for a given mechanism, it cannot have odd number of links.
  • Also, the minimum number of links required to form a closed chain mechanism is 4 since with the least even value of 2 links, a closed chain is not possible.

Uses of Mechanism in Kinematics

Mechanism plays the most significant role while analyzing any topic of Kinematics. Some uses are mentioned below:

  • Applications of relative motion can be seen in the field of physics.
  • The single-slider crank mechanism forms the basis of all the automobiles running across the globe.
  • Pairs such as spherical pairs are used as rear-view mirrors which prevents millions of accidents around the globe.
  • Joints such as ball and socket are an important part of human body which is again a spherical pair mechanism.
  • Brakes are examples of sliding pair which help in stopping a vehicle
  • The tailor machine used in tailoring industry is also an inversion of mechanism.
  • Manufacturing industries use various mechanism in different machine tools like lathe, shaper, milling machines to name a few.
  • Joints form the building blocks of all the mechanisms that we see in our life.

Key Points to Remember

Here is the list of key points we need to remember about “Mechanism in Kinematics”.

  • Theory of Machines is the branch of engineering science which deals with the analysis of relative motion between the various parts or structure of a mechanism or machine.
  • Computer Network is an interconnection of different computers to share resources through a communication medium between them.
  • One Tertiary Joint has 2 Binary Joints.
  • One Quaternary Joint has 3 Binary Joints.
  • In a higher pair, the contact surfaces of the two links are not similar
  • In a lower pair, the contact surfaces of the two links are similar in nature.
  • One Higher Pair is equal to 2 Lower Pairs.
  • Self-Closed Pair is also known as Closed Pair.

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Manish Bhojasia, a technology veteran with 20+ years @ Cisco & Wipro, is Founder and CTO at Sanfoundry. He lives in Bangalore, and focuses on development of Linux Kernel, SAN Technologies, Advanced C, Data Structures & Alogrithms. Stay connected with him at LinkedIn.

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