Friction, force that resists the sliding or rolling of one solid object over another. Frictional forces provide the traction needed to walk without slipping, but they also present a great measure of opposition to motion. Types of friction include kinetic friction, static friction, and rolling friction.In engineering applications friction is desirable and undesirable. We can walk on the ground because of friction. Friction is useful in power transmission by belts. It is useful in appliances like brakes, bolts, screw jack, etc. It is undesirable in bearing and moving machine parts where it results in loss of energy and, thereby, reduces efficiency of the machine. In this unit, you will study screw jack, clutches
This document discusses friction and provides an example problem. It defines friction as the force resisting the relative motion of surfaces in contact. There are two types of friction: static and kinetic. The key elements of friction are defined, including the coefficient of friction μ and angle of friction φ. Laws of friction relate φ to the angle of an applied force θ. An example problem then applies these concepts to calculate the minimum force P required to start motion of a 400 lb block on a rough surface with a coefficient of friction of 0.40.
The document provides an overview of applied mechanics, including definitions of mechanics, engineering, applied mechanics, and their various branches and topics. It also covers fundamental concepts such as units, scalars, vectors, and trigonometry functions that are important to mechanics. Examples of static force analysis using vector operations like resolution and resultant are presented.
D'Alembert's Principle states that the resultant of all external forces and inertia forces acting on a body is zero for the body to be in dynamic equilibrium. Inertia forces are represented as minus mass times acceleration. The principle allows equations of static equilibrium to be applied to bodies undergoing translational motion by considering an imaginary inertia force equal and opposite to actual inertia. Several example problems are provided applying the principle to analyze motion of connected bodies over pulleys, motion on inclined planes, and motion within elevators.
The document discusses systems of forces acting on a body. It defines different types of force systems including coplanar forces that act in the same plane, and non-coplanar forces whose lines of action are not in the same plane. Within these categories, forces can be further classified as collinear if they act along the same line, concurrent if they intersect at a single point, parallel if their lines of action are parallel, and like or unlike based on direction. Examples are provided of different force system configurations such as concurrent coplanar forces intersecting in a plane, and non-concurrent non-coplanar forces that do not intersect and act in different planes.
The document discusses Deepak's academic and professional background, including an MBA from IE Business School in Spain and experience founding perfectbazaar.com. It also provides an overview of the topics to be covered in the Strength of Materials course, such as stresses, strains, Hooke's law, and analysis of bars with varying cross-sections. The grading policy and syllabus are outlined which divide the course into 5 units covering various strength of materials concepts.
This document discusses the divergence of a vector field and the divergence theorem. It begins by defining the divergence of a vector field as a measure of how much that field diverges from a given point. It then illustrates the divergence of a vector field can be positive, negative, or zero at a point. The document expresses the divergence in Cartesian, cylindrical, and spherical coordinate systems. It proves the divergence theorem, which states that the outward flux of a vector field through a closed surface is equal to the volume integral of the divergence of the field over the enclosed volume. The document provides two examples applying the divergence theorem to calculate outward fluxes.
This document provides an introduction and overview of mechanics of materials. It defines key terms like stress, strain, normal stress, shear stress, factor of safety, and allowable stress. It also gives examples of calculating stresses in structural members subjected to various loads. The document is an introductory reading for a mechanics of materials course that will analyze the relationship between external forces and internal stresses and strains in structural elements.
This document discusses friction and provides an example problem. It defines friction as the force resisting the relative motion of surfaces in contact. There are two types of friction: static and kinetic. The key elements of friction are defined, including the coefficient of friction μ and angle of friction φ. Laws of friction relate φ to the angle of an applied force θ. An example problem then applies these concepts to calculate the minimum force P required to start motion of a 400 lb block on a rough surface with a coefficient of friction of 0.40.
The document provides an overview of applied mechanics, including definitions of mechanics, engineering, applied mechanics, and their various branches and topics. It also covers fundamental concepts such as units, scalars, vectors, and trigonometry functions that are important to mechanics. Examples of static force analysis using vector operations like resolution and resultant are presented.
D'Alembert's Principle states that the resultant of all external forces and inertia forces acting on a body is zero for the body to be in dynamic equilibrium. Inertia forces are represented as minus mass times acceleration. The principle allows equations of static equilibrium to be applied to bodies undergoing translational motion by considering an imaginary inertia force equal and opposite to actual inertia. Several example problems are provided applying the principle to analyze motion of connected bodies over pulleys, motion on inclined planes, and motion within elevators.
The document discusses systems of forces acting on a body. It defines different types of force systems including coplanar forces that act in the same plane, and non-coplanar forces whose lines of action are not in the same plane. Within these categories, forces can be further classified as collinear if they act along the same line, concurrent if they intersect at a single point, parallel if their lines of action are parallel, and like or unlike based on direction. Examples are provided of different force system configurations such as concurrent coplanar forces intersecting in a plane, and non-concurrent non-coplanar forces that do not intersect and act in different planes.
The document discusses Deepak's academic and professional background, including an MBA from IE Business School in Spain and experience founding perfectbazaar.com. It also provides an overview of the topics to be covered in the Strength of Materials course, such as stresses, strains, Hooke's law, and analysis of bars with varying cross-sections. The grading policy and syllabus are outlined which divide the course into 5 units covering various strength of materials concepts.
This document discusses the divergence of a vector field and the divergence theorem. It begins by defining the divergence of a vector field as a measure of how much that field diverges from a given point. It then illustrates the divergence of a vector field can be positive, negative, or zero at a point. The document expresses the divergence in Cartesian, cylindrical, and spherical coordinate systems. It proves the divergence theorem, which states that the outward flux of a vector field through a closed surface is equal to the volume integral of the divergence of the field over the enclosed volume. The document provides two examples applying the divergence theorem to calculate outward fluxes.
This document provides an introduction and overview of mechanics of materials. It defines key terms like stress, strain, normal stress, shear stress, factor of safety, and allowable stress. It also gives examples of calculating stresses in structural members subjected to various loads. The document is an introductory reading for a mechanics of materials course that will analyze the relationship between external forces and internal stresses and strains in structural elements.
This document provides an overview of the content covered in the Basic Civil Engineering course. It discusses the following topics:
1. Mechanics of Rigid Bodies and Mechanics of Deformable Bodies, which make up Parts I and II of the course.
2. Concepts in mechanics of solids including resultant and equilibrium of coplanar forces, centroids, moments of inertia, kinetics principles, stresses and strains.
3. Five textbooks recommended as references for the course.
4. Definitions of terms like particle, force, scalar, vector, and rigid body.
5. Methods for resolving forces into components, obtaining the resultant of coplanar forces, and solving mechanics problems
Moment of inertia concepts in Rotational Mechanicsphysicscatalyst
Moment of inertia is a measure of an object's resistance to changes in its angular acceleration due to an applied torque. It depends on how the object's mass is distributed relative to its pivot point. The moment of inertia of a rigid body can be calculated by imagining it divided into particles, multiplying each particle's mass by the square of its distance from the axis of rotation, and summing these values. Important theorems for calculating moment of inertia include the perpendicular axis theorem and parallel axis theorem. Examples are given for calculating the moment of inertia of a solid disk and sphere about their central axes.
This document discusses approximate methods for determining natural frequencies of structures, including Rayleigh's method and Dunkerley's method. Rayleigh's method involves estimating the mode shape and using the Rayleigh quotient to calculate an upper bound for the fundamental frequency. Dunkerley's method provides a lower bound by assuming the structure vibrates as separate components. Examples are provided to illustrate both methods and how they can provide good estimates of natural frequencies.
This document provides an overview of mechanics of solids, including fundamentals of statics such as laws for analyzing coplanar concurrent force systems using the parallelogram, triangle, and polygon laws. It also discusses analytical methods for resolving and composing concurrent coplanar forces using components, as well as graphical methods. Additional topics covered include coplanar non-concurrent forces, moments, couples, Varignon's theorem, and equilibrium conditions.
Dealing with Notations and conventions in tensor analysis-Einstein's summation convention covariant and contravariant and mixed tensors-algebraic operations in tensor symmetric and skew symmetric tensors-tensor calculus-Christoffel symbols-kinematics in Riemann space-Riemann-Christoffel tensor.
A force is an external agent acting on another body. This force may moves or tends to move the body in the direction of its action. The force is a vector quantity since it is represented by its magnitude and direction. The force may be of pulling or pushing type. Copy the link given below and paste it in new browser window to get more information on Principle Of Transmissibility:-
http://www.transtutors.com/homework-help/mechanical-engineering/force-systems-and-analysis/principle-of-transmissibility.aspx
Lecture 4 3 d stress tensor and equilibrium equationsDeepak Agarwal
* Shear stress, τ = 50 N/mm2
* Shear modulus, C = 8x104 N/mm2
* Strain energy per unit volume = τ2/2C
= (50)2 / 2(8x104)
= 0.3125 J/mm3
Therefore, the local strain energy per unit volume stored in the material due to shear stress is 0.3125 J/mm3.
The document discusses determining the forces acting on a rigid body in static equilibrium. It provides three key points:
1) For a rigid body to be in static equilibrium, the external forces and moments acting on it must balance so there is no translational or rotational motion.
2) The conditions for static equilibrium are that the resultant force and couple from all external forces equals zero.
3) Resolving each force and moment into rectangular components provides six scalar equations that also express the static equilibrium conditions.
The document discusses numerical methods for solving structural mechanics problems, specifically the Rayleigh Ritz method. It provides an overview of the Rayleigh Ritz method, indicating that it is an integral approach that is useful for solving structural mechanics problems. The document then provides a step-by-step example of using the Rayleigh Ritz method to determine the bending moment and deflection at the mid-span of a simply supported beam subjected to a uniformly distributed load over the entire span.
This document discusses different types of forces and stresses. It defines surface forces as forces distributed over a body's surface, like hydrostatic pressure, and body forces as forces distributed throughout a body's volume, like gravitational force. It also defines stress as a measure of force per unit area within a body, and explains that stress can be broken down into normal and shear components based on their orientation relative to the plane they act on.
This document discusses friction, including the limiting force of friction, coefficient of friction, angle of friction, and angle of repose. It defines static and dynamic friction, with dynamic friction further divided into sliding and rolling friction. The laws of static and kinetic friction are also outlined. Several example problems are provided to calculate values like the coefficient of friction given information about the applied forces and weights of objects on horizontal or inclined planes.
This document discusses different coordinate systems used to describe points in 2D and 3D space, including polar, cylindrical, and spherical coordinates. It provides the key formulas for converting between Cartesian and these other coordinate systems. Examples are given of converting points and equations between the different coordinate systems. The key points are that polar coordinates use an angle and distance to specify a 2D point, cylindrical coordinates extend this to 3D using a z-value, and spherical coordinates specify a 3D point using a distance from the origin, an angle, and an azimuthal angle.
Lecture 9 shear force and bending moment in beamsDeepak Agarwal
The document discusses stresses in beams. It covers topics like shear force and bending moment diagrams, bending stresses, shear stresses, deflection, and torsion. Beams are structural members subjected to transverse forces that induce bending. Stresses and strains are created within beams when loaded. Shear forces and bending moments allow determining these internal stresses and maintaining equilibrium. Formulas are provided for calculating shear forces and bending moments in different beam configurations like cantilevers, simply supported beams, and beams with various load types.
This document discusses friction, including the definition and types of friction, laws of friction, coefficients of friction, and applications involving friction such as ladders, wedges, and connected blocks. It defines dry friction and fluid friction, and explains the six laws of dry friction. It discusses static and kinetic coefficients of friction, friction cones, and how friction forces depend on normal forces and surface properties. Several example problems are included that apply the principles of friction to analyze systems involving ladders, wedges, and blocks on inclined planes.
Scalars represent physical quantities at a point, like pressure. Vectors track magnitude and direction of quantities like force and velocity. Tensors track three pieces of information, like stress which has magnitude, direction, and acting plane. A scalar has one value, a vector has three components, and a tensor has nine components. The continuity equation relates the accumulation of mass in a region to the net flux through the boundary, and for incompressible fluids reduces to the divergence of velocity being zero.
1) Coplanar concurrent forces are forces that act in the same plane and meet at a single point.
2) For a system of coplanar concurrent forces to be in equilibrium, the algebraic sum of the forces along two perpendicular directions must be zero.
3) There are analytical and graphical methods to find the resultant force of a coplanar concurrent forces system, including the parallelogram law, resolution of forces, triangle law, and polygon law of forces.
Bending Stresses are important in the design of beams from strength point of view. The present source gives an idea on theory and problems in bending stresses.
This document discusses various topics in mechanics including:
- Mechanics deals with forces and their effects on bodies at rest or in motion. It includes statics, dynamics, and the mechanics of rigid and deformable bodies.
- Forces can be analyzed using concepts such as free body diagrams, components, resultants, and equilibrium conditions. Friction and trusses are also analyzed.
- Kinematics examines the motion of particles and rigid bodies without considering forces. It relates time, position, velocity, and acceleration. Dynamics analyzes forces and acceleration using concepts like work, energy, impulse, and momentum.
This document provides an overview of friction, including:
1) It defines friction as the resistance to motion when two surfaces are in contact, and introduces the concepts of static and kinetic friction. Charles-Augustin de Coulomb conducted experiments on friction and distinguished between static and kinetic friction.
2) Laws of friction include: friction opposes motion; friction is parallel to the contact surfaces; friction is independent of contact area; static friction is proportional to normal force; and kinetic friction is also proportional to normal force but is slightly less than static friction.
3) Coefficients of friction - the ratio of static friction to normal force is the coefficient of static friction, while the same ratio for kinetic friction is the
This document provides information about the recruitment exam guide and handbook for junior engineers. It includes the following sections: engineering mechanics and strength of materials, theory of machines and machine design, thermal engineering, fluid mechanics and machinery, and production engineering. It also provides information on different engineering mechanics concepts like forces, force systems, equilibrium, moments, friction, kinematics, dynamics, and Newton's laws of motion. Additionally, it covers topics like stress, strain, stress-strain relationships, shear stress, and strain tensor. Examples of stress analysis for different structural components are also given.
This document provides an overview of the content covered in the Basic Civil Engineering course. It discusses the following topics:
1. Mechanics of Rigid Bodies and Mechanics of Deformable Bodies, which make up Parts I and II of the course.
2. Concepts in mechanics of solids including resultant and equilibrium of coplanar forces, centroids, moments of inertia, kinetics principles, stresses and strains.
3. Five textbooks recommended as references for the course.
4. Definitions of terms like particle, force, scalar, vector, and rigid body.
5. Methods for resolving forces into components, obtaining the resultant of coplanar forces, and solving mechanics problems
Moment of inertia concepts in Rotational Mechanicsphysicscatalyst
Moment of inertia is a measure of an object's resistance to changes in its angular acceleration due to an applied torque. It depends on how the object's mass is distributed relative to its pivot point. The moment of inertia of a rigid body can be calculated by imagining it divided into particles, multiplying each particle's mass by the square of its distance from the axis of rotation, and summing these values. Important theorems for calculating moment of inertia include the perpendicular axis theorem and parallel axis theorem. Examples are given for calculating the moment of inertia of a solid disk and sphere about their central axes.
This document discusses approximate methods for determining natural frequencies of structures, including Rayleigh's method and Dunkerley's method. Rayleigh's method involves estimating the mode shape and using the Rayleigh quotient to calculate an upper bound for the fundamental frequency. Dunkerley's method provides a lower bound by assuming the structure vibrates as separate components. Examples are provided to illustrate both methods and how they can provide good estimates of natural frequencies.
This document provides an overview of mechanics of solids, including fundamentals of statics such as laws for analyzing coplanar concurrent force systems using the parallelogram, triangle, and polygon laws. It also discusses analytical methods for resolving and composing concurrent coplanar forces using components, as well as graphical methods. Additional topics covered include coplanar non-concurrent forces, moments, couples, Varignon's theorem, and equilibrium conditions.
Dealing with Notations and conventions in tensor analysis-Einstein's summation convention covariant and contravariant and mixed tensors-algebraic operations in tensor symmetric and skew symmetric tensors-tensor calculus-Christoffel symbols-kinematics in Riemann space-Riemann-Christoffel tensor.
A force is an external agent acting on another body. This force may moves or tends to move the body in the direction of its action. The force is a vector quantity since it is represented by its magnitude and direction. The force may be of pulling or pushing type. Copy the link given below and paste it in new browser window to get more information on Principle Of Transmissibility:-
http://www.transtutors.com/homework-help/mechanical-engineering/force-systems-and-analysis/principle-of-transmissibility.aspx
Lecture 4 3 d stress tensor and equilibrium equationsDeepak Agarwal
* Shear stress, τ = 50 N/mm2
* Shear modulus, C = 8x104 N/mm2
* Strain energy per unit volume = τ2/2C
= (50)2 / 2(8x104)
= 0.3125 J/mm3
Therefore, the local strain energy per unit volume stored in the material due to shear stress is 0.3125 J/mm3.
The document discusses determining the forces acting on a rigid body in static equilibrium. It provides three key points:
1) For a rigid body to be in static equilibrium, the external forces and moments acting on it must balance so there is no translational or rotational motion.
2) The conditions for static equilibrium are that the resultant force and couple from all external forces equals zero.
3) Resolving each force and moment into rectangular components provides six scalar equations that also express the static equilibrium conditions.
The document discusses numerical methods for solving structural mechanics problems, specifically the Rayleigh Ritz method. It provides an overview of the Rayleigh Ritz method, indicating that it is an integral approach that is useful for solving structural mechanics problems. The document then provides a step-by-step example of using the Rayleigh Ritz method to determine the bending moment and deflection at the mid-span of a simply supported beam subjected to a uniformly distributed load over the entire span.
This document discusses different types of forces and stresses. It defines surface forces as forces distributed over a body's surface, like hydrostatic pressure, and body forces as forces distributed throughout a body's volume, like gravitational force. It also defines stress as a measure of force per unit area within a body, and explains that stress can be broken down into normal and shear components based on their orientation relative to the plane they act on.
This document discusses friction, including the limiting force of friction, coefficient of friction, angle of friction, and angle of repose. It defines static and dynamic friction, with dynamic friction further divided into sliding and rolling friction. The laws of static and kinetic friction are also outlined. Several example problems are provided to calculate values like the coefficient of friction given information about the applied forces and weights of objects on horizontal or inclined planes.
This document discusses different coordinate systems used to describe points in 2D and 3D space, including polar, cylindrical, and spherical coordinates. It provides the key formulas for converting between Cartesian and these other coordinate systems. Examples are given of converting points and equations between the different coordinate systems. The key points are that polar coordinates use an angle and distance to specify a 2D point, cylindrical coordinates extend this to 3D using a z-value, and spherical coordinates specify a 3D point using a distance from the origin, an angle, and an azimuthal angle.
Lecture 9 shear force and bending moment in beamsDeepak Agarwal
The document discusses stresses in beams. It covers topics like shear force and bending moment diagrams, bending stresses, shear stresses, deflection, and torsion. Beams are structural members subjected to transverse forces that induce bending. Stresses and strains are created within beams when loaded. Shear forces and bending moments allow determining these internal stresses and maintaining equilibrium. Formulas are provided for calculating shear forces and bending moments in different beam configurations like cantilevers, simply supported beams, and beams with various load types.
This document discusses friction, including the definition and types of friction, laws of friction, coefficients of friction, and applications involving friction such as ladders, wedges, and connected blocks. It defines dry friction and fluid friction, and explains the six laws of dry friction. It discusses static and kinetic coefficients of friction, friction cones, and how friction forces depend on normal forces and surface properties. Several example problems are included that apply the principles of friction to analyze systems involving ladders, wedges, and blocks on inclined planes.
Scalars represent physical quantities at a point, like pressure. Vectors track magnitude and direction of quantities like force and velocity. Tensors track three pieces of information, like stress which has magnitude, direction, and acting plane. A scalar has one value, a vector has three components, and a tensor has nine components. The continuity equation relates the accumulation of mass in a region to the net flux through the boundary, and for incompressible fluids reduces to the divergence of velocity being zero.
1) Coplanar concurrent forces are forces that act in the same plane and meet at a single point.
2) For a system of coplanar concurrent forces to be in equilibrium, the algebraic sum of the forces along two perpendicular directions must be zero.
3) There are analytical and graphical methods to find the resultant force of a coplanar concurrent forces system, including the parallelogram law, resolution of forces, triangle law, and polygon law of forces.
Bending Stresses are important in the design of beams from strength point of view. The present source gives an idea on theory and problems in bending stresses.
This document discusses various topics in mechanics including:
- Mechanics deals with forces and their effects on bodies at rest or in motion. It includes statics, dynamics, and the mechanics of rigid and deformable bodies.
- Forces can be analyzed using concepts such as free body diagrams, components, resultants, and equilibrium conditions. Friction and trusses are also analyzed.
- Kinematics examines the motion of particles and rigid bodies without considering forces. It relates time, position, velocity, and acceleration. Dynamics analyzes forces and acceleration using concepts like work, energy, impulse, and momentum.
This document provides an overview of friction, including:
1) It defines friction as the resistance to motion when two surfaces are in contact, and introduces the concepts of static and kinetic friction. Charles-Augustin de Coulomb conducted experiments on friction and distinguished between static and kinetic friction.
2) Laws of friction include: friction opposes motion; friction is parallel to the contact surfaces; friction is independent of contact area; static friction is proportional to normal force; and kinetic friction is also proportional to normal force but is slightly less than static friction.
3) Coefficients of friction - the ratio of static friction to normal force is the coefficient of static friction, while the same ratio for kinetic friction is the
This document provides information about the recruitment exam guide and handbook for junior engineers. It includes the following sections: engineering mechanics and strength of materials, theory of machines and machine design, thermal engineering, fluid mechanics and machinery, and production engineering. It also provides information on different engineering mechanics concepts like forces, force systems, equilibrium, moments, friction, kinematics, dynamics, and Newton's laws of motion. Additionally, it covers topics like stress, strain, stress-strain relationships, shear stress, and strain tensor. Examples of stress analysis for different structural components are also given.
Friction arises due to interlocking of minutely projecting particles when two surfaces are in contact and one surface moves relative to the other. There are two main types of friction: static friction and dynamic (kinetic) friction. Static friction acts when a body is at rest, while dynamic friction acts when a body is in motion. Friction can be classified further as dry friction between unlubricated surfaces and fluid friction between lubricated surfaces. The coefficient of friction is defined as the ratio between limiting friction force and normal reaction force. A screw jack uses the principle of an inclined plane to lift loads, with torque required to overcome friction proportional to the load, coefficient of friction, and pitch of the screw.
1) The document discusses linear impulse, linear momentum, and their relationship to forces via Newton's second law. It defines key terms like impulse, momentum, and coefficient of restitution.
2) Examples of direct central impact and oblique central impact between objects are described. Equations for analyzing 1D and 2D impacts are provided.
3) Five example problems involving collisions between objects and calculating velocities, angles, and masses are presented.
Friction is a force that opposes motion between two surfaces in contact. There are two types of friction: static friction and kinetic friction. Static friction has a greater maximum force than kinetic friction. The laws of friction state that frictional force is proportional to the normal force and depends on the coefficient of friction, which varies based on the materials in contact. Problems involving blocks on inclined planes can be solved using the static friction force and applying equations of equilibrium. Wedges can be used to move heavy objects by applying a smaller input force, with the mechanical advantage determined by the wedge angle.
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Friction is a tangential force that acts parallel to surfaces in contact and moving relative to each other. There are two types of friction: static and kinetic. Static friction resists motion of stationary objects, while kinetic friction resists motion of objects already moving. Friction depends on the normal force perpendicular to the surfaces and the coefficient of friction, which varies based on the nature of the surfaces. Laws of static friction state that friction is proportional to the normal force, independent of contact area, and the coefficient of static friction is greater than kinetic friction.
The document discusses balancing of reciprocating masses in engines. It describes:
1. The various forces acting on reciprocating parts and how the inertia force is balanced by an opposing force on the crankshaft, leaving an unbalanced force.
2. Methods to partially balance the primary unbalanced force using a balancing mass on the crank, which changes the direction of the maximum unbalanced force.
3. How balancing is applied to two-cylinder locomotives, reducing variation in tractive force, swaying couple, and hammer blow.
The document provides information about mechanics of solids-I, including:
1) It describes different types of supports like simple supports, roller supports, pin-joint supports, and fixed supports. It also describes different types of loads like concentrated loads, uniformly distributed loads, and uniformly varying loads.
2) It discusses shear force as the unbalanced vertical force on one side of a beam section, and bending moment as the sum of moments about a section.
3) It explains the relationship between loading (w), shear force (F), and bending moment (M) for an element of a beam. The rate of change of shear force is equal to the loading intensity, and the rate of change of bending
1. The document discusses structures, loads, stresses, strains and material properties related to mechanics of materials.
2. It defines key terms like stress, strain, elastic modulus and explains stress-strain relationships. Common stress types like tensile, compressive, shear and their effects are described.
3. Examples of different structures like cylinders, spheres, arches, towers and bridges are provided to illustrate stress distributions and effects of loads. Material properties of common materials are also listed.
This document contains a series of sections written by Prof. A B Karpe on topics related to mechanical engineering. It discusses turning moment diagrams, friction, types of friction including static and dynamic friction, laws of static friction, coefficient of friction, angle of friction, and applications of friction in mechanisms like turning pairs and connecting rods in internal combustion engines. Each section is attributed to Prof. Karpe and covers definitions and concepts regarding the given topic.
This document discusses friction, clutches, and brakes. It begins with an introduction to friction, describing the different types (dry, skin/greasy, film), laws of friction, and coefficient of friction. It then discusses motion up and down inclined planes, defining the angle of friction and efficiency. Specific topics covered include screw threads, pivot and collar friction, friction clutches, brakes, brake classification, and vehicle braking systems. Problems with friction, clutches, and brakes are also mentioned.
Force and its application for k12 studentsArun Umrao
Force changes the state of body. If body is in rest and a force is applied on it, body came in motion. Similarly, a force bring a body to rest from its motion if applied force is in opposite direction to the direction of momentum of the body. Unit of force is Kg m/s2 . Second unit of force is Newton represented by N, honoring to Sir James Newton. Mass of a body is m and force F is applied on it then mass force relation is F = ma (1) While we discuss the physics’ rule, we always take ideal conditions not real one. For ex- ample, in Newton’s force law, “body” means tiny, round, symmetrical particle of sufficient large mass but not too much small in size. Its center of mass lies at its center. As the “size” of body increases, the environmental phenomenon shall affect the motion of body in several ways, by means of frictional or drag force.
Principle of Force Application - Physics - explained deeply by arun kumarssuserd6b1fd
This notes is about force and force applications for CBSE Class X and XI students. Suitable for quick revision before exam, i.e. last minute preparation.
The document discusses static and dynamic force analysis of mechanisms. It defines key terms like static equilibrium, inertia force, inertia torque, and D'Alembert's principle. It explains the conditions for a body to be in equilibrium under different force configurations. Dynamic force analysis considers inertia forces to determine input torque required. Equivalent masses and Klein's construction diagram are discussed for dynamic analysis of reciprocating engines. Correction couple and torque are also summarized.
Okay, here are the step-by-step workings:
a) Plot the Mohr's circle with σx = 20 kPa, σy = 30 kPa, τxy = 10 kPa
b) Determine the pole
c) Draw a line through the pole at 30° from the horizontal
d) It intersects the circle at σa = 26 kPa, τa = 8.66 kPa
e) The major and minor principal stresses are the intersections of the circle with the σ axes:
σ1 = 30 kPa
σ3 = 20 kPa
f) The major principal plane is parallel to the σy axis. The minor principal plane is parallel to the σ
This document discusses mechanics of solid members subjected to torsional loads. It describes how torsion works, generating shear stresses in circular shafts. The key equations for relating applied torque (T) to shear stress (τ) and angle of twist (θ) are developed. For a solid circular shaft under torque T, the maximum shear stress τmax occurs at the outer surface and is equal to T/J, where J is the polar moment of inertia of the cross section. Power transmitted by a shaft is also defined as 2πNT, where N is rotational speed in revolutions per minute. Shear stress distribution and failure modes under yielding are also briefly covered.
This document provides an overview of dry friction, including:
- Dry friction occurs between unlubricated solid surfaces and always opposes motion or impending motion. It depends on the normal force and roughness of the surfaces.
- Static friction is less than or equal to the maximum static friction force (Fmax), which is proportional to the normal force by the static coefficient of friction (μs).
- Kinetic friction occurs once motion begins and is proportional to the normal force by the kinetic coefficient of friction (μk), which is usually less than μs.
- Friction angles (θs and θk) can be defined in terms of the coefficients based on the direction of the total reaction force.
Similar to Friction and type,laws,angle,coefficient (20)
How to Manage Your Lost Opportunities in Odoo 17 CRMCeline George
Odoo 17 CRM allows us to track why we lose sales opportunities with "Lost Reasons." This helps analyze our sales process and identify areas for improvement. Here's how to configure lost reasons in Odoo 17 CRM
How to Add Chatter in the odoo 17 ERP ModuleCeline George
In Odoo, the chatter is like a chat tool that helps you work together on records. You can leave notes and track things, making it easier to talk with your team and partners. Inside chatter, all communication history, activity, and changes will be displayed.
LAND USE LAND COVER AND NDVI OF MIRZAPUR DISTRICT, UPRAHUL
This Dissertation explores the particular circumstances of Mirzapur, a region located in the
core of India. Mirzapur, with its varied terrains and abundant biodiversity, offers an optimal
environment for investigating the changes in vegetation cover dynamics. Our study utilizes
advanced technologies such as GIS (Geographic Information Systems) and Remote sensing to
analyze the transformations that have taken place over the course of a decade.
The complex relationship between human activities and the environment has been the focus
of extensive research and worry. As the global community grapples with swift urbanization,
population expansion, and economic progress, the effects on natural ecosystems are becoming
more evident. A crucial element of this impact is the alteration of vegetation cover, which plays a
significant role in maintaining the ecological equilibrium of our planet.Land serves as the foundation for all human activities and provides the necessary materials for
these activities. As the most crucial natural resource, its utilization by humans results in different
'Land uses,' which are determined by both human activities and the physical characteristics of the
land.
The utilization of land is impacted by human needs and environmental factors. In countries
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9
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1. 51
Friction
UNIT 2 FRICTION
Structure
2.1 Introduction
Objectives
2.2 Types of Friction
2.3 Laws of Dry Friction
2.4 Static and Kinetic Friction
2.5 Coefficient of Friction
2.6 Angle of Repose
2.7 Least Force Required to Drag a Body on a Rough Horizontal Plane
2.8 Horizontal Force Required to Move the Body
2.9 Screw and Nut Friction
2.10 Self-Locking Screws
2.11 Condition for Maximum Efficiency
2.12 Screw Jack
2.13 Pivot and Collar Friction
2.13.1 Flat Pivot
2.13.2 Conical Pivot
2.13.3 Collar Bearing
2.14 Clutch
2.15 Types of Clutches
2.15.1 Conical Clutch
2.15.2 Single Plate Clutch
2.15.3 Multi Plate Clutch
2.16 Journal Bearing
2.17 Rolling Friction
2.18 Ball and Roller Bearings
2.19 Summary
2.20 Key Words
2.21 Answers to SAQs
2.1 INTRODUCTION
When a body moves or tends to move on another body, a force appears between the
surfaces. This force is called force of friction and it acts opposite to the direction of
motion. Its line of action is tangential to the contacting surfaces. The magnitude of this
force depends on the roughness of surfaces.
In engineering applications friction is desirable and undesirable. We can walk on the
ground because of friction. Friction is useful in power transmission by belts. It is useful
in appliances like brakes, bolts, screw jack, etc. It is undesirable in bearing and moving
machine parts where it results in loss of energy and, thereby, reduces efficiency of the
machine.
In this unit, you will study screw jack, clutches and different type of bearings.
2. 52
Theory of Machines Objectives
After studying this unit, you should be able to
know application of friction force,
know theory background of screw jack,
know different type of bearings,
analyse bearings, and
know function of cluthes.
2.2 TYPES OF FRICTION
There are two types of friction :
(a) Friction in un-lubricated surfaces or dry surfaces, and
(b) Friction in lubricate surfaces.
The friction that exists when one dry surface slides over another dry surface is known as
dry friction and the friction.
If between the two surfaces a thick layer of an oil or lubricant is introduced, a film of
such lubrication is formed on both the surfaces. When a surface moves on the other, in
effect, it is one layer of oil moving on the other and there is no direct contact between
the surfaces. The friction is greatly reduced and is known as film friction.
2.3 LAWS OF DRY FRICTION
The laws of dry friction, are based on experimental evidences, and as such they are
empirical in nature :
(a) The friction force is directly proportional to the normal reaction between
the surfaces.
(b) The frictional force opposes the motion or its tendency to the motion.
(c) The frictional force depends upon the nature of the surfaces in contact.
(d) The frictional force is independent of the area and the shape of the
contacting surfaces.
(e) For moderate speeds, frictional force is independent of the relative
velocities of the bodies in contact.
SAQ 1
On which factors friction force depends?
2.4 STATIC AND KINETIC FRICTION
Suppose a block of weight W rests on a plane surface as shown in Figure 2.1. The
surface offers a normal reaction RN equal to the weight W. Suppose, now, a pull P1 is
applied to the block such that it actually does not move but instead tends to move. This
will be opposed by frictional force F1, equal to P1. The resultant reaction will be R1
3. 53
Frictioninclined at an angle 1 with the normal reaction. If the pull is increased to P2, the
frictional force will increase to F2 and the resultant reaction will increase to R2 inclined
at an angle of 2. Thus, with increase of the pull or attractive force, the frictional force;
the resultant reaction and its inclination will increase.
Figure 2.1 : Static and Kinetic Friction
Figure 2.1 shows a block of mass m resting on a plane surface with application of force
P1, P2 and P3 at which the body impends sliding, the self adjusting frictional forces will
increase from F1, to F2 and finally F3 when the body tends to move. Thus, in the limiting
condition, the resultant active force will be R and reactive force RR.
The frictional resistance offered so long as the body does not move, is known as static
friction. F1 and F2 are the static frictional forces. It may be noted that the direction of the
resultant reaction RR is such that it opposes the motion.
The ultimate value of static friction (F) when the body just tends to move is called
limiting friction or maximum static friction or friction of impending slide. The condition,
when all the forces are just in equilibrium and the body has a tendency to move, is called
limiting equilibrium position. When a body moves relative to another body, the resisting
force between them is called kinetic or sliding friction. It has been experimentally found
that the kinetic friction is less than the maximum static friction.
SAQ 2
What is the value of limiting friction?
2.5 COEFFICIENT OF FRICTION
The ratio between the maximum static frictional force and the normal reaction RN
remains constant which is known as coefficient of static friction denoted by Greek
letter.
Coefficient of friction
Maximum static frictional force
Normal reaction
P
mgR
F F2 F1 P1 P2 P
1
2
RN
R1 R2
F1
F2
F
Resultant
Reaction RR
Plane
Surface
Resultant
Action
Block
m
4. 54
Theory of Machines
or
N
F
R
. . . (2.1)
The maximum angle which the resultant reaction RN makes with the normal reaction RN
is known as angle of friction. It is denoted by .
From Figure 2.1 tan
N
F
R
1
tan
. . . (2.2)
The coefficient of friction is different for different substances and even varies for
different conditions of the same two surfaces.
Approximate values of static coefficient of friction for dry (un-lubricated) and greasy
lubricated surfaces are given as below :
Materials Dry Greasy
Hard steel on hard steel 0.42 0.029-0.108
Mild steel on mild steel 0.57 0.09-0.19
Wooden wood 0.2-0.5 0.133
Mild steel on cast iron 0.24 0.09-0.116
Wood on metal 0.2-0.6
Glass on glass 0.4
Metal on stone 0.3-0.7
Metal on leather 0.3-0.6
Wood on leather 0.2-0.5
Earth on earth 0.1-1
Cast iron on cast iron 0.3-3.4 0.065-0.070
Rubber on concrete 0.65-0.85
Rubber on ice 0.05-0.2
2.6 ANGLE OF REPOSE
Consider a mass m resting on an inclined plane. If the angle of inclination is slowly
increased, a stage will come when the block of mass m will tend to slide down
(Figure 2.2). This angle of the plane with horizontal plane is known as angle of repose.
For satisfying the conditions of equilibrium all the forces are resolved parallel to the
plane and perpendicular to it.
Figure 2.2 : Angle of Repose
mg
P
RN
RN sin
RN cos
5. 55
Frictionsin NP R
cos Nmg R
tan
P
mg
But tan
N
P F
mg R
tan tan
Angle of repose = angle of friction .
2.7 LEAST FORCE REQUIRED TO DRAG A BODY
ON A ROUGH HORIZONTAL PLANE
Suppose a block, of mass m, is placed on a horizontal rough surface as shown in
Figure 2.3 and a tractive force P is applied at an angle with the horizontal such that the
block just tends to move.
Figure 2.3
For satisfying the equilibrium conditions the forces are resolved vertically and
horizontally.
For V = 0
sin Nmg P R
or ( sin ) NR mg P . . . (2.3)
For H = 0
cos ( sin ) NP F R mg P
[substituting the value of RN from Eq. (2.3)]
sin
cos ( sin )
cos
P mg P
cos cos sin sin sin P mg P
cos cos sin sin P P mg
(cos cos sin sin ) sin P mg
cos ( ) sin P mg
sin
cos ( )
mg
P . . . (2.4)
For P to be least, the denominator cos ( ) must be maximum and it will be so if
cos ( ) 1 or 0
F = RN
P cos
P sin P
m
mg
RN
6. 56
Theory of Machines = for least value of P
Pleast = mg sin . . . (2.5)
Hence, the force P will be the least if angle of its inclination with the horizontal : is
equal to the angle of friction .
2.8 HORIZONTAL FORCE REQUIRED TO MOVE
THE BODY
Up the Inclined Plane
The force P has been applied to move the body up the plane. Resolving all the
forces parallel and perpendicular to the plane and writing equations, we get
cos sin NP R mg
sin cos NP R mg
sin cos NR P mg
Figure 2.4
Substituting the value of RN, we get
cos ( sin cos ) sin P P mg mg
Assuming tan
where is angle of friction.
cos tan ( sin cos ) sin P P mg mg
or (cos tan sin ) (tan cos sin ) P mg
or
sin
cos sin
cos
sin
cos sin
cos
P mg
(sin cos sin cos )
(cos cos sin sin )
mg
sin ( )
tan ( )
cos ( )
mg mg . . . (2.6)
Down the Plane
In this case body is moving down the plane due to the application of force P.
Resolving the forces parallel and perpendicular to the plane and writing the
equations, we get
RN
P
mg
RN
7. 57
Frictioncos sin NP R mg
sin cos NP mg R
cos sin NR mg P
Figure 2.5
Substituting for RN, we get
cos ( cos sin ) sin P mg P mg
Since tan
cos tan ( cos sin ) sin P mg P mg
or (cos tan sin ) (tan cos sin ) P mg
or
sin
cos sin
cos
sin
cos sin
cos
P mg
(sin cos cos sin )
(cos cos sin sin )
mg
or
sin ( )
tan ( )
cos ( )
P mg mg . . . (2.7)
This means for force P to be applied < .
For < body will move without applying force P.
2.9 SCREW AND NUT FRICTION
Now consider screw and nut assembly. Both of them have threads in the form of helix.
Screw has external thread and nut has internal thread. The pitch of threads (p) for both is
same. When nut is rotated by one turn the screw traverses linear distance equal to the
pitch (p). The nut at the same time traverses by one turn. These distances so traveled
have been represented in Figure 2.6(a). Let mean diameter of the screw be ‘dm’.
Therefore, tan
p
dm
The theory of the inclined plane discussed in the preceding article can be applied for
determining the effort to be applied.
RN
P
mg
RN
8. 58
Theory of Machines
(a) (b)
Figure 2.6 : Screw and Nut Friction
For Upward Motion
Effort 0 tanP mg without friction [= 0].
Effort tan (P mg with friction.
Efficiency
tan
tan (
upe
For Downward Motion
tan ( ) P mg
2.10 SELF-LOCKING SCREWS
If > , the mass placed on screw will start moving downward by its own weight and
force P shall have to be applied to hold it. To guard against this undesirable effect, the
screws angle is always kept less than angle . Such a screw is known as self-locking
screw.
2.11 CONDITION FOR MAXIMUM EFFICIENCY
Efficiency of screw :
tan
tan ( )
upe
For determining condition of maximum efficiency,
2 2
2
sec tan ( ) sec ( ) tan
0 0
tan ( )
upd e
d
2 2
sec tan ( ) sec ( ) tan
sin ( ) cos ( ) sin cos
sin 2 ( ) sin 2
2 ( ) ( 2 )
or
4 2
m P
mg
p
xdm
Helix
Nut
Screw
Pitch
9. 59
FrictionSubstituting the value of in equation of efficiency.
2 2
o
max 2 2
o
1 tan
2
tan 45 1 tan cos sin1 tan
2 2 2 22
1 tantan 45 1 tan cos sin22 2 2 2
1 tan
2
e
1 2 sin cos
1 sin2 2
1 sin1 2 sin cos
2 2
max
1 sin
1 sin
e
SAQ 3
How self locking is provided in screw jack?
2.12 SCREW JACK
With Square Threads
A screw jack with its spindle, having square threads, is shown in Figure 2.7. The
theory discussed till now is directly applicable to this case.
Figure 2.7 : Screw Jack
m
Cup
Tommy
Bar
Nut
Square Threaded
Spindle
Stand
Hole
L
E
10. 60
Theory of Machines Let m = Mass on the jack,
P = Force applied at the screw tangentially in a horizontal plane,
Pe = Horizontal force applied tangentially at the end E of a tommy bar in a
horizontal plane, and
L = Horizontal distance between central axis of the screw and the end E of
the bar as shown.
In this screw jack nut is stationary and the screw is rotated with the help of tommy
bar.
eP L P r
e
P r
P
L
(tan tan )
tan ( )
1 tan tan
mg
P mg
But tan and tan
m
p
d
Substituting for tan and tan
( )
( )1
m m
m
m
p
mg
d mg p d
P
p d p
d
Hence,
( )
( )
m
m
mg p d
P
d p
( )
( )
m
e
m
mg r
p d
LP
d p
Mechanical advantage of the jack with tommy bar
Load lifted
Force applied
( )
( )
m
e m
d pmg L
P r p d
Velocity ratio with tommy bar :
Distance covered by
Distance covered by load in one revolution
eP
2
2 tan tan
m m
L L
r r
2 L
V. R.
p
With V-threads
The square threads, by their nature, take the axial load mg perpendicular to them
where as in V-threads, the axial load does not act perpendicular to the surface of
the threads as shown in Figure 2.8. The normal reaction RN between the threads
and the screw should be such that its axial component is equal and opposite to the
axial load mg.
11. 61
Friction
Figure 2.8 : V-threads
Let 2 be the angle included between the two sides of the thread.
cos NR mg
cos
N
mg
R
Frictional force which acts tangential to the surface of the threads = RN
1
cos
mg
mg
where 1 may be regarded as virtual coefficient of friction :
1
cos
While treating V-threads for finding out effort F or ‘e’, etc. may be substituting
by 1 in all the relevant equations meant for the square threads.
It may be observed that force required to lift a given load with V-threads will be
more than that with square threads.
Screw threads are also used for transmission of power such as in lathes (lead
screw), milling machines, etc. The square threads will transmit power without any
side thrust but is difficult to cut. The Acme threads, though not as efficient as the
square threads are easier to cut.
Example 2.1
Outside diameter of a square threaded spindle of a screw jack is 40 mm. The
screw pitch is 10 mm. If the coefficient of friction between the screw and the nut
is 0.15, neglecting friction between the nut and the collar, determine :
(a) Force required to be applied at the end of tommy bar 1 m in length to
raise a load of 20 kN.
(b) Efficiency of the screw.
mg
RN
2
12. 62
Theory of Machines Solution
Outside diameter of the screw : D = 40 mm
Inside diameter of the screw :
d = 40 mm – 10 = 30 mm
Mean diameter of the screw :
40 30
35 mm
2
md
The force required for raising the load.
(a)
(tan tan )
tan ( )
2 1 tan tan 2
m md dmg
P mg
L
But tan 0.15
10 7
tan 0.091
22 35
m
p
d
Substituting the values,
(0.091 0.15) 30
20 73 N
1 0.091 0.15 2 1000
P
We know that,
0
30
tan 20 0.091 15 27.3
2 1000
P mg
(b) Efficiency 0 27.3
100 37.4%
73
up
P
e
P
2.13 PIVOT AND COLLAR FRICTION
The shafts of ships, steam and water turbines are subjected to axial thrust. In order to
take up the axial thrust, they are provided with one or more bearing surfaces at right
angle to the axis of the shaft. A bearing surface provided at the end of a shaft is known
as a pivot and that provided at any place along with the length of the shaft with bearing
surface of revolution is known as a callar. Pivots are of two forms : flat and conical. The
bearing surface provided at the foot of a vertical shaft is called foot step bearing.
Due to the axial thrust which is conveyed to the bearings by the rotating shaft, rubbing
takes place between the contacting surfaces. This produces friction as well as wearing of
the bearing. Thus, power is lost in over-coming the friction, which is ultimately to be
determined in this unit.
Obviously, the rate of wearing depends upon the intensity of thrust (pressure) and
relative velocity of rotation. Since velocity is proportional to the radius, therefore,
Rate of wear pr.
Assumptions Taken
(a) Firstly, the intensity of pressure is uniform over the bearing surface. This
assumption only holds good with newly fitted bearings where fit between
the two contacting surfaces is assumed to be perfect. After the shaft has run
for quite sometime the pressure distribution will not remain uniform due to
varying wear at different radii.
13. 63
Friction(b) Secondly, the rate of wear is uniform. As given previously, the rate of wear
is proportional to pr which means that the pressure will go on increasing
radially inward and at the centre where r = 0, the pressure will be infinite
which is not true in practical sense. However, this assumption of uniform
wear gives better practical results when bearing has become older.
The various types of bearings mentioned above will be dealt with separately for
each assumption.
2.13.1 Flat Pivot
A flat pivot is shown in Figure 2.9.
Figure 2.9 : Flat Pivot
Let W = Axial thrust or load on the bearing,
R = External radius of the pivot,
p = Intensity of pressure, and
= Coefficient of friction between the contacting surfaces.
Consider an elementary ring of the bearing surfaces, at a radius r and of thickness dr as
shown in Figure 2.9.
Axial load on the ring
2 dW p r dr
Total load
0
2
R
W p r dr . . . (2.8)
Frictional force on the ring
2 dF dW p r dr
Frictional moment about the axis of rotation
2
2 dM dF r p r dr
Total frictional moment
2
0 0
2
R R
M dM p r dr . . . (2.9)
W
R
R dr
14. 64
Theory of Machines Uniform Pressure
If the intensity of pressure p is assumed to be uniform and hence constant.
From Eq. (2.8)
2 2
0 0
2 2 2
2 2
RR
r R
W p r dr p p
2
W p R
And from Eq. (2.9)
2
0
2
R
M p r dr
3
3
0
2
2
3 3
R
r
M p p R
But 2
p R W
2 2
3 3
M = WR = W R
The friction force W can be considered to be acting at a radius of
2
3
R .
Uniform Rate of Wear
By Eq. (2.8),
0
2
R
W p r dr
As the rate of wear is taken as constant and proportional to pr = a constant say c.
Substituting for pr = c in the above equation.
0
2 2
R
W c d r R c
2
W
c
R
. . . (2.10)
By Eq. (2.9), total frictional moment
2
0 0
2 2
R R
M p r dr c r dr
2 2
2 2
2 2 2
R W R
c
R
2
R
M = W . . . (2.11)
Thus, the frictional force : W acts at a distance
2
R
from the axis.
15. 65
Friction2.13.2 Conical Pivot
A truncated conical pivot is shown in Figure 2.10(a).
Let 2 = The cone angle,
W = The axial load/thrust,
R1 = The outer radius of the cone,
R2 = The inner radius of the cone, and
p = Intensity of pressure which will act normal to the inclined surface of the
cone as shown in Figure 2.10(b).
(a) Conical Pivot (b) Enlarged View of the Ring
Figure 2.10 : Truncated Conical Pivot
Consider an elementary ring of the cone, of radius r thickness dr and of sloping length dl
as shown. Enlarged view of the ring is shown in Figure 2.10(b).
Normal load on the ring
2 2
sin
n
dr
dW p r dl p r
Axial load on the ring
sin ndW dW
2 sin 2
sin
dr
p r p r dr
Total axial load on the bearing
1
2
2
R
R
W p r dr . . . (2.12)
Frictional force on the elementary ring
2 2
sin
n
dr
dF dW p r dl p r
Moment of the frictional force about the axis of rotation
2
2
sin
dr
dM p r
W
R
dr r
dI 2 dw
dwn
dw
dwn
dI
dr r
16. 66
Theory of Machines Total frictional moment
1
2
2
2
sin
R
R
dr
M p r . . . (2.13)
Uniform Pressure
Uniform pressure, p is constant
From Eq. (2.12),
11
2 2
2
2 2
2
RR
R R
r
W p r dr p
or 2 2 2 2
1 2 1 2
2
( ) ( )
2
W p R R p R R . . . (2.14)
And from Eq. (2.13),
1
2
2 3 3
1 2
2 2 1
( )
sin sin 3
R
R
p p
M r dr R R
But from Eq. (2.14)
2 2
1 2( )W p R R
2 2
1 2( )
W
p
R R
Substituting for p above,
3 3
1 2
2 2
1 2
( )
sin ( )
R RW
M
R R
Thus, frictional force
sin
W
acts at a radius of
3 3
1 2
2 2
1 2
( )2
3 ( )
R R
R R
For a full conical pivot shown in Figure 2.11, R2 = 0
Figure 2.11 : Full Conical Pivot
r = 0
2
1W p R
and 1
sin
W
M R
In this case the frictional force
sin
W
acts at a radius
2
3
from the axis.
R
2
17. 67
FrictionUniform Wear
From Eq. (2.12),
2
R
r
W p r dr
Since the rate of wear is uniform.
Therefore, pr = constant c
Substituting for pr
1 1
2 2
1 22 2 2 ( )
R R
R R
W c dr c dr c R R
1 22 ( )
W
c
R R
. . . (2.15)
From Eq. (2.13), frictional moment
1
2
2
2
sin
R
R
dr
M p r
1
2
2
sin
R
R
dr
r c [since pr = c]
1
2
2
sin
R
R
c
r dr
2 2
2 21 2
1 2
( )2
( )
sin 2 sin
R Rc c
R R
But
1 22 ( )
W
c
R R
. . . (2.16)
( )1 2
sin 2
R RW
M
[Thus, the frictional force
sin
W
acts at a radius 1 2( )
2
R R
]
For the full conical pivot (Figure 2.10), R2 = 0
1
sin 2
RW
M
Thus, the friction force
sin
W
acts at a radius 1
2
R
.
2.13.3 Collar Bearing
A collar bearing which is provided on to a shaft, is shown in Figure 2.12.
Let W = The axial load/thrust,
R1 = External radius of the collar, and
R2 = Internal radius of the collar.
Consider an elementary ring of the collar surface, of radius r and of thickness dr as
shown in Figure 2.12.
18. 68
Theory of Machines
Figure 2.12 : Collar Bearing
Axial load on the ring
2 dW p r dr
Total axial load
1
2
2
R
R
W p r dr . . . (2.17)
Frictional force on the ring
2 dF p r dr
Frictional moment of the ring
2
2 dM p r dr
Total frictional moment
1
2
2
2
R
R
M p r dr . . . (2.18)
Uniform Pressure
Uniform pressure, p is constant
From Eq. (2.17),
1
2
2 2
1 2( )
2 2
3
R
R
R R
W p r dr p
2 2
1 2( )W p R R . . . (2.19)
From Eq. (2.18),
1
2
3 3
2 1 2( )
2 2
3
R
R
R R
M p r dr p
W
R2
R1
Collar
R1 r1
R2
dr1
Axial Thrust
19. 69
FrictionSubstituting for p from above
3 3
1 2
2 2
1 2
( )2
3 ( )
R R
M W
R R
. . . (2.20)
Uniform Wear
From Eq. (2.17),
Since the rate of wear is uniform
pr = constant c
Substituting for pr above,
1
2
1 22 2 2 ( )
RR
r R
W c dr c dr c R R
1 22 ( )
W
c
R R
. . . (2.21)
From Eq. (2.18),
1 1
2 2
2
2 2
R R
R R
M p r dr c r dr
1 2( )
2
2
R R
c
Substituting for c from Eq. (2.21),
1 2( )
2
R R
M W
. . . (2.22)
There is a limit to the bearing pressure on a single collar and it is about 40 N/cm2
.
Where the axial load is more and pressure on each collar is not to be allowed to
exceed beyond the designed limit, then more collars are provided as shown in
Figure 2.12.
Number of collars :
Total axial load
Permissible axial load on each collar
n
It may be pointed out there is no change in the magnitude of frictional moments
with more number of collars. The number of collars, as given above, only limit the
maximum intensity of pressure in each collar.
Table 2.1 : Pivot and Collars Summary of Formulae
Sl. No. Particular Frictional Moments : M
Uniform Pressure Uniform Wear
1. Flat pivot
2
3
W R
2
R
W
2.
Conical pivot
(a) Truncated
(b) Full conical
3 3
1 2
2 2
1 2
( )2
sin 3 ( )
R RW
R R
2
3
W R
1 2( )
sin 2
R RW
2
R
W
3. Collar
3 3
1 2
2 2
1 2
( )2
3 ( )
R R
W
R R
1 2( )
2
R R
W
20. 70
Theory of Machines SAQ 4
When collar or pivot bearing becomes older which assumption is more suitable
and why?
2.14 CLUTCH
It is a mechanical device which is widely used in automobiles for the purpose of
engaging driving and the driven shaft, at the will of the driver or the operator. The
driving shaft is the engine crankshaft and the driven shaft is the gear-box driving shaft.
This means that the clutch is situated between the engine flywheel mounted on the
crankshaft and the gear box.
In automobile, gears are required to be changed for obtaining different speeds. It is
possible only if the driving shaft of the gear box is stopped for a while without stopping
the engine. These two objects are achieved with the help of a clutch.
Broadly speaking, a clutch consists of two members; one fixed to the crankshaft or the
flywheel of the engine and the other mounted on a splined shaft, of the gear box so that
this could be engaged or disengaged as the case may be with the member fixed to the
engine crankshaft.
2.15 TYPES OF CLUTCHES
Clutches can be classified into two types as follows :
(a) conical clutch, and
(b) the place or disc clutches can be of single plate or of multiple plates.
Figure 2.13 : The Single Cone Clutch
2.15.1 Conical Clutch
A conical clutch is shown in Figure 2.14. It consists of a cone A mounted on engine
crankshaft. Cone B has internal splines in its boss which fit into the corresponding
splines provided in the gear box shaft. Cone B could rotate the gear box shaft as well as
may slide along with it. The outer surface of cone B is lined with friction material. In the
normal or released position of the clutch pedal P, cone B fits into the inner conical
surface of cone A and by means of the friction between the contacting surfaces, power is
transmitted from crankshaft to the gear box shaft. When the clutch pedal is pressed,
pivot D being the fulcrum provided in it, the collar E is pressed towards the right side,
thus disengaging cone B from cone A and keeping the compression spring S compressed.
On releasing the pedal, by the force of the spring, the cone B is thrusted back to engage
cone A for power transmission.
Friction
Clutch
Conical Clutch
Plate or Disc
21. 71
Friction
Figure 2.14 : Conical Clutch
For calculating frictional moments or torque transmitted on account of friction in
clutches, unless otherwise specifically stated uniform rate of wear is assumed. For
torque transmitted formulae of conical pivot can be used.
2.15.2 Single Plate Clutch
A single plate clutch is known as single disc clutch. It is shown in Figure 2.15. It has two
sides which are driving and the driven side. The driving side comprises of the driving
shaft or engine crankshaft A. A boss B is keyed to it to which flywheel C is bolted as
shown. On the driven side, there is a driven shaft D. It carries a boss E which can freely
slide axially along with the driven shaft through splines F. The clutch plate is mounted
on the boss E. It is provided with rings of friction material – known as friction linings,
on the both sides indicated H. One friction lining is pressed on the flywheel face and the
other on the pressure plate I. A small spigot, bearing J, is provided in the end of the
driving shaft for proper alignment.
Figure 2.15 : Single Plate Clutch
The pressure plate provides axial thrust or pressure between clutch plate G and the
flywheel C and the pressure plate I through the linings on its either side, by means of the
springs, S.
The pressure plate remains engaged and as such clutch remains in operational position.
Power from the driving shaft is transferred to the driven shaft from flywheel to the clutch
Friction Brake Lining
Cone A
Enginge Crank Shaft
Stop K
Cone B
Gear Box Shaft
Spring S
Collar E
Pivot D
Clutch Pedal
Spring. S
Clutch Plate. G
Splines. F
Pressure Plate, I
Driven Shaft, D
Sleeve, K
Boss, E
Crank Shaft, A
Boss, B
Fly Wheel, C
J
H
22. 72
Theory of Machines plate through the friction lining between them. From pressure plate the power is
transmitted to clutch plate through friction linings. Both sides of the clutch plate are
effective. When the clutch is to be disengaged the sleeve K is moved towards right hand
side by means of clutch pedal mechanism (it is not shown in the figure). By doing this,
there is no pressure between the pressure plate, flywheel and the clutch plate and no
power is transmitted. In medium size and heavy vehicles, like truck, single plate clutch is
used.
2.15.3 Multi Plate Clutch
As already explained in a plate clutch, the torque is transmitted by friction between one
or more pairs of co-axial annular faces kept in contact by an axial thrust provided by
springs. In a single plate clutch, both sides of the plate are effective so that it has two
pairs of surfaces in contact or n = 2.
Obviously, in a single plate clutch limited amount of torque can be transmitted. When
large amount of torque is to be transmitted, more pair of contact surfaces are needed and
it is precisely what is obtained by a multi-plate clutch.
SAQ 5
(a) Clutches are used for which purpose?
(b) Where do we use single plate clutch. Name the vehicles?
2.16 JOURNAL BEARING
The portion of a shaft, which revolves in the bearing and subjected to load at right angle
to the axis of the shaft, is known as journal as clearly indicated in Figure 2.16. The
whole unit consisting of the journal and its supporting part (or bearing) is known as
journal bearing.
Figure 2.16 : Journal Bearing
2.17 ROLLING FRICTION
The frictional resistance arises only when there is relative motion between the two
connecting surfaces. When there is no relative motion between the connecting surfaces
or stated plainly when one surface does not slide over the other question of occurrence of
frictional resistance or frictional force does not arise.
Journal
Oil Hole
Shaft
Bearing
23. 73
FrictionWhen a wheel rolls over a flat surface, there is a line contact between the two surfaces,
parallel to the central axis of the cylinder. On the other hand when a spherical body rolls
over a flat surface, there is a point contact between the two. In both the above mentioned
cases there is no relative motion of slip between the line or point of contact on the flat
surface because of the rolling motion. If while rolling of a wheel or that of a spherical
body on the flat surface there is no deformation of depression of either of the two under
the load, it is said to be a case pure rolling.
In practice it is not possible to have pure rolling and it can only be approached.
How-so-ever hard the material be, either the rolling body will be deformed as happens in
case of a car or cycle tyre, indicated in Figure 2.17(a) or the flat surface gets depressed
or deformed. When the road roller passes over unsettled road or kacha road as shown in
Figure 2.17(b). The surface is depressed.
(a) Rolling Body Deformed (b) Flat Surface Deformed
Figure 2.17 : Rolling Friction
At times when a load or a heavy machine or its part is to be shifted from one place to
another place, for a short distance, and no suitable mechanical lifting device is available,
the same is placed on a few rollers in the form of short pieces of circular bars or pipes as
shown in Figure 2.18 and comparatively with a less force the load is moved. The rollers,
roll and the one which becomes free at the rear side is again placed in the front of the
load and so on. Because of the reduced friction, it requires less force.
Figure 2.18 : Ball and Roller Bearing
When a shaft revolves in a bush bearing, there is sliding motion between the journal and
the bearing surface, resulting in loss of power due to friction. If between the journal and
the bearing surface, balls of rollers are provided, instead of sliding motion, rolling
motion will take place. To reduce the coefficient of rolling friction the balls or rollers are
made of chromium steel or chrome-nickel steel and they are further heat treated with a
view to make them more hard. They are finally ground and polished with high precision.
Such an arrangement as mentioned above is provided in Ball and Roller bearings.
2.18 BALL AND ROLLER BEARINGS
Ball Bearing
A ball bearing is shown in Figure 2.19. As may be seen, it consists of mainly the
following parts :
(a) Inner ring,
Flat Surface
O
C
Pr
W
Rolling Bodies
O
C
Pr
W
Rolling Bodies
Rolling Surface
P
h
W
Load W
24. 74
Theory of Machines (b) Outer ring,
(c) Anti-friction element in the form of balls placed between the two
races, and
(d) A cage which separates the balls from one another.
Figure 2.19 : Radial Ball Bearing
The inner ring is tight press-fitted to the shaft and the outer ring is press-fitted into
a fixed housing which supports the whole bearing.
The cage is meant for keeping balls at fixed and equal distance from one another.
Balls and races are made of best quality chromium steel containing high
percentage of carbon with a view to reduce the value of coefficient of rolling
friction, to the barest minimum, to make the bearing really an anti-friction.
Roller Bearing
A roller bearing is shown in Figure 2.20. There is no difference between ball
bearing and the roller bearing except that in roller bearing rollers, instead of balls
are used. In roller bearings, there is a line contact instead of point contact as in
ball bearings. The roller bearings are widely used for more load carrying capacity
than that of ball bearings.
Figure 2.20 : Roller Bearing
Width Outer ring
Balls Race
Outer Ring
Inner Ring
Ball Race
Balls Cage
Ball
Bore
OuterDia
Width
Outer Ring
Roller
Cage
Inner Ring
OuterDia
Bore
25. 75
FrictionThe rollers may be cylindrical, straight or tapered. When tapered rollers are used, the
bearings are called tapered roller bearings and those in which needles are used are
called needle bearings.
Tapered Roller Bearings
Parts of a tapered roller bearing as well as its assembly are shown in Figure 2.21.
The rollers are in the form of frustrum of a cone. The contact angle is between 12o
to 16o
for thrusts of moderate magnitude and 28o
to 30o
for heavy thrust.
Figure 2.21 : Tapered Roller Bearing Parts
Needle Bearings
The rollers of small diameter are known as needless. It is : 2 to 4 mm. Ratio
between length and diameter of needle is 3-10 : 1. Needle bearings are not
provided with cage.
SAQ 6
Which type of bearing (journal bearing or ball bearing) has less friction?
2.19 SUMMARY
Friction force is generated when there is motion or tendency of motion of one body
relative to other body in contact. It is useful in power transmission like belt drive, brakes
and clutches. It is undesirable in bearings, etc. where it results in power loss.
Experiments were conducted to formulate laws of dry friction. According to the laws of
dry friction, maximum force of friction depends on normal reaction and it is independent
of area of contact and rubbing velocity. The ratio of the force of friction to the normal
reaction is constant for the given surfaces and it is called coefficient of friction. The
angle made by the resultant reaction to the normal reaction is called angle of friction.
26. 76
Theory of Machines If we unwind one helix of the screw it becomes an inclined plane. The theoretical
expression of the screw jack has been derived by using this inclined plane. To make the
screw self-locking, the helix angle should be less than the friction angle.
In some machines, shaft is subjected to the axial load. To support this load, pivot or
collar bearing has to be used. When bearing is new, pressure distribution is uniform.
Since wear is proportional to the product of pressure and radius, more wear takes place
at larger radius. This results in non-uniform clearance between bearing and shaft and
thereby reduction of pressure at larger radius and increase in pressure at the centre. Due
to this, change, rate of wear becomes uniform. More power is lost in friction when
pressure is uniform as compared to the case when ratio of wear is uniform.
To connect engine shaft with gear box, clutches are used. Two types of clutches are
generally used in medium and heavy vehicle and they are – cone clutch and single plate
clutch. In cars, buses and trucks, single plate clutch is used. Power transmitted by cone
clutch is more than single plate clutch. Therefore, it is used where torque transmitted is
more.
To support load at the supports, bearings are used. It can be journal bearing, ball bearing
or roller bearing. In journal bearing there is a film of the lubricant between shaft and the
bearing. The load is supported by the pressure generated in the layer of the lubricant. In
case of ball and roller bearing, contact may be through a point or a line. The power loss
due to friction will be very low in ball and roller bearing as compared to journal bearing.
2.20 KEY WORDS
Dry Friction : It is the friction between two dry surfaces.
Lubricated Surfaces : The lubricant which is put between surfaces forms
a film between surfaces to reduce friction.
Static Friction : It is the friction between the two static surfaces.
Kinetic Friction : It is the friction between the surfaces having
motion.
Coefficient of Friction : It is the ratio of force of limiting friction to the
normal reaction.
Limiting Friction : It is the maximum force of friction which is
developed when slide is impending.
Angle of Repose : It is the angle of the inclined plane to the
horizontal at which body slides on its own.
Screw Jack : It is machine to lift or lower the load.
Pivots : It is a surface which bears axial thrust of shaft.
Collars : Collars are like plates which support axial thrust.
Clutch : It is a friction coupling which is used between
engine and gear box in an automobile.
Journal Bearing : In these bearings, lubricant layer separate metal
surfaces to lower the friction.
Ball Bearing : In this bearing, balls are used to separate rotating
surface from stationary surface.
Roller Bearing : In this bearing rollers are used to separate rotating
surface from stationary surface.
27. 77
Friction
2.21 ANSWERS TO SAQs
Refer the relevant preceding text in the unit or other useful books on the topic listed in
section ‘Further Reading’ to get the answer of the SAQs. These questions are helpful to
prepare you for the examination.
2