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Lý thuyết của máy bởi R.S.KHURMI Theory of Machines by R.S.KHURMI Lý thuyết của máy bởi R.S.KHURMI Theory of Machines by R.S.KHURMILý thuyết của máy bởi R.S.KHURMI Theory of Machines by R.S.KHURMILý thuyết của máy bởi R.S.KHURMI Theory of Machines by R.S.KHURMI
CONTENTS Introduction .17 Definition Sub-divisions of Theory of Machines Fundamental Units Derived Units Systems of Units C.G.S Units F.P.S Units M.K.S Units International System of Units (S.I Units) 10 Metre 11 Kilogram 12 Second 13 Presentation of Units and their Values 14 Rules for S.I Units 15 Force 16 Resultant Force 17 Scalars and Vectors 18 Representation of Vector Quantities 19 Addition of Vectors 20 Subtraction of Vectors Kinematics of Motion .823 Introduction Plane Motion Rectilinear Motion Curvilinear Motion Linear Displacement Linear Velocity Linear Acceleration Equations of Linear Motion Graphical Representation of Displacement with respect to Time 10 Graphical Representation of Velocity with respect to Time 11 Graphical Representation of Acceleration with respect to Time 12 Angular Displacement 13 Representation of Angular Displacement by a Vector 14 Angular Velocity 15 Angular Acceleration 16 Equations of Angular Motion 17 Relation between Linear Motion and Angular Motion 18 Relation between Linear and Angular Quantities of Motion 19 Acceleration of a Particle along a Circular Path Kinetics of Motion Introduction Newton's Laws of Motion Mass and Weight Momentum Force Absolute and Gravitational Units of Force Moment of a Force Couple Centripetal and Centrifugal Force 10 Mass Moment of Inertia 11 Angular Momentum or Moment of Momentum 12 Torque 13 Work 14 Power 15 Energy 16 Principle of Conservation of Energy 17 Impulse and Impulsive Force 18 Principle of Conservation of Momentum 19 Energy Lost by Friction Clutch During Engagement 20 Torque Required to Accelerate a Geared System 21 Collision of Two Bodies 22 Collision of Inelastic Bodies 23 Collision of Elastic Bodies 24 Loss of Kinetic Energy During Elastic Impact (v) .2471 Simple Harmonic Motion 7293 Introduction Velocity and Acceleration of a Particle Moving with Simple Harmonic Motion Differential Equation of Simple Harmonic Motion Terms Used in Simple Harmonic Motion Simple Pendulum Laws of Simple Pendulum Closely-coiled Helical Spring Compound Pendulum Centre of Percussion 10 Bifilar Suspension 11 Trifilar Suspension (Torsional Pendulum) Simple Mechanisms .94118 Introduction Kinematic Link or Element Types of Links Structure Difference Between a Machine and a Structure Kinematic Pair Types of Constrained Motions Classification of Kinematic Pairs Kinematic Chain 10 Types of Joints in a Chain 11 Mechanism 12 Number of Degrees of Freedom for Plane Mechanisms 13 Application of Kutzbach Criterion to Plane Mechanisms 14 Grubler's Criterion for Plane Mechanisms 15 Inversion of Mechanism 16 Types of Kinematic Chains 17 Four Bar Chain or Quadric Cycle Chain 18 Inversions of Four Bar Chain 19 Single Slider Crank Chain 20 Inversions of Single Slider Crank Chain 21 Double Slider Crank Chain 22 Inversions of Double Slider Crank Chain Velocity in Mechanisms (Instantaneous Centre Method) .119142 Velocity in Mechanisms (Relative Velocity Method) .143173 Introduction Space and Body Centrodes Methods for Determining the Velocity of a Point on a Link Velocity of a Point on a Link by Instantaneous Centre Method Properties of the Instantaneous Centre Number of Instantaneous Centres in a Mechanism Types of Instantaneous Centres Location of Instantaneous Centres Aronhold Kennedy (or Three Centres-in-Line) Theorem 10 Method of Locating Instantaneous Centres in a Mechanism Introduction Relative Velocity of Two Bodies Moving in Straight Lines Motion of a Link Velocity of a Point on a Link by Relative Velocity Method Velocities in a Slider Crank Mechanism Rubbing Velocity at a Pin Joint Forces Acting in a Mechanism Mechanical Advantage (vi) Acceleration in Mechanisms .174231 Introduction Acceleration Diagram for a Link Acceleration of a Point on a Link Acceleration in the Slider Crank Mechanism Coriolis Component of Acceleration Mechanisms with Lower Pairs .232257 Introduction Pantograph Straight Line Mechanism Exact Straight Line Motion Mechanisms Made up of Turning Pairs Exact Straight Line Motion Consisting of One Sliding Pair (Scott Russels Mechanism) Approximate Straight Line Motion Mechanisms Straight Line Motions for Engine Indicators Steering Gear Mechanism Davis Steering Gear 10 Ackerman Steering Gear 11 Universal or Hookes Joint 12 Ratio of the Shafts Velocities 13 Maximum and Minimum Speeds of the Driven Shaft 14 Condition for Equal Speeds of the Driving and Driven Shafts 15 Angular Acceleration of the Driven Shaft 16 Maximum Fluctuation of Speed 17 Double Hookes Joint 10 Friction .258324 Introduction Types of Friction Friction Between Unlubricated Surfaces Friction Between Lubricated Surfaces Limiting Friction Laws of Static Friction Laws of Kinetic or Dynamic Friction Laws of Solid Friction Laws of Fluid Friction 10 Coefficient of Friction 11 Limiting Angle of Friction 12 Angle of Repose 13 Minimum Force Required to Slide a Body on a Rough Horizontal Plane 14 Friction of a Body Lying on a Rough Inclined Plane 15 Efficiency of Inclined Plane 16 Screw Friction 17 Screw Jack 18 Torque Required to Lift the Load by a Screw Jack 19 Torque Required to Lower the Load by a Screw Jack 20 Efficiency of a Screw Jack 21 Maximum Efficiency of a Screw Jack 22 Over Hauling and Self Locking Screws 23 Efficiency of Self Locking Screws 24 Friction of a V-thread 25 Friction in Journal Bearing-Friction Circle 26 Friction of Pivot and Collar Bearing 27 Flat Pivot Bearing 28 Conical Pivot Bearing 29 Trapezoidal or Truncated Conical Pivot Bearing 30 Flat Collar Bearing 31 Friction Clutches 32 Single Disc or Plate Clutch 33 Multiple Disc Clutch 34 Cone Clutch 35 Centrifugal Clutches 11 Belt, Rope and Chain Drives .325381 Introduction Selection of a Belt Drive Types of Belt Drives Types of Belts Material used for Belts Types of Flat Belt (vii) Drives Velocity Ratio of Belt Drive Velocity Ratio of a Compound Belt Drive Slip of Belt 10 Creep of Belt 11 Length of an Open Belt Drive 12 Length of a Cross Belt Drive 13 Power Transmitted by a Belt 14 Ratio of Driving Tensions for Flat Belt Drive 15 Determination of Angle of Contact 16 Centrifugal Tension 17 Maximum Tension in the Belt 18 Condition for the Transmission of Maximum Power 19 Initial Tension in the Belt 20 V-belt Drive 21 Advantages and Disadvantages of V-belt Drive Over Flat Belt Drive 22 Ratio of Driving Tensions for V-belt 23 Rope Drive 24 Fibre Ropes 25 Advantages of Fibre Rope Drives 26 Sheave for Fibre Ropes 27 Wire Ropes 28 Ratio of Driving Tensions for Rope Drive 29 Chain Drives 30 Advantages and Disadvantages of Chain Drive Over Belt or Rope Drive 31 Terms Used in Chain Drive 32 Relation Between Pitch and Pitch Circle Diameter 33 Relation Between Chain Speed and Angular Velocity of Sprocket 34 Kinematic of Chain Drive 35 Classification of Chains 36 Hoisting and Hauling Chains 37 Conveyor Chains 38 Power Transmitting Chains 39 Length of Chains 12 Toothed Gearing .382427 Introduction Friction Wheels Advantages and Disadvantages of Gear Drive Classification of Toothed Wheels Terms Used in Gears Gear Materials Condition for Constant Velocity Ratio of Toothed Wheels-Law of Gearing Velocity of Sliding of Teeth Forms of Teeth 10 Cycloidal Teeth 11 Involute Teeth 12 Effect of Altering the Centre Distance on the Velocity Ratio For Involute Teeth Gears 13 Comparison Between Involute and Cycloidal Gears 14 Systems of Gear Teeth 15 Standard Proportions of Gear Systems 16 Length of Path of Contact 17 Length of Arc of Contact 18 Contact Ratio (or Number of Pairs of Teeth in Contact) 19 Interference in Involute Gears 20 Minimum Number of Teeth on the Pinion in Order to Avoid Interference 21 Minimum Number of Teeth on the Wheel in Order to Avoid Interference 22 Minimum Number of Teeth on a Pinion for Involute Rack in Order to Avoid Interference 23 Helical Gears 24 Spiral Gears 25 Centre Distance for a Pair of Spiral Gears 26 Efficiency of Spiral Gears 13 Gear Trains .428479 Introduction Types of Gear Trains Simple Gear Train Compound Gear Train (viii) Design of Spur Gears Reverted Gear Train Epicyclic Gear Train Velocity Ratio of Epicyclic Gear Train Compound Epicyclic Gear Train (Sun and Planet Wheel) 10 Epicyclic Gear Train With Bevel Gears 11 Torques in Epicyclic Gear Trains 14 Gyroscopic Couple and Precessional Motion .480513 Introduction Precessional Angular Motion Gyroscopic Couple Effect of Gyroscopic Couple on an Aeroplane Terms Used in a Naval Ship Effect of Gyroscopic Couple on a Naval Ship during Steering Effect of Gyroscopic Couple on a Naval Ship during Pitching Effect of Gyroscopic Couple on a Navel during Rolling Stability of a Four Wheel drive Moving in a Curved Path 10 Stability of a Two Wheel Vehicle Taking a Turn 11 Effect of Gyroscopic Couple on a Disc Fixed Rigidly at a Certain Angle to a Rotating Shaft 15 Inertia Forces in Reciprocating Parts .514564 Introduction Resultant Effect of a System of Forces Acting on a Rigid Body D-Alemberts Principle Velocity and Acceleration of the Reciprocating Parts in Engines Kliens Construction Ritterhauss Construction Bennetts Construction Approximate Analytical Method for Velocity and Acceleration of the Piston Angular Velocity and Acceleration of the Connecting Rod 10 Forces on the Reciprocating Parts of an Engine Neglecting Weight of the Connecting Rod 11 Equivalent Dynamical System 12 Determination of Equivalent Dynamical System of Two Masses by Graphical Method 13 Correction Couple to be Applied to Make the Two Mass Systems Dynamically Equivalent 14 Inertia Forces in a Reciprocating Engine Considering the Weight of Connecting Rod 15 Analytical Method for Inertia Torque 16 Turning Moment Diagrams and Flywheel Introduction Turning Moment Diagram for a Single Cylinder Double Acting Steam Engine Turning Moment Diagram for a Four Stroke Cycle Internal Combustion Engine Turning Moment Diagram for a Multicylinder Engine Fluctuation of Energy Determination of Maximum Fluctuation of Energy Coefficient of Fluctuation of Energy Flywheel Coefficient of Fluctuation of Speed 10 Energy Stored in a Flywheel 11 Dimensions of the Flywheel Rim 12 Flywheel in Punching Press (ix) 565611 17 Steam Engine Valves and Reversing Gears .612652 Introduction D-slide Valve Piston Slide Valve Relative Positions of Crank and Eccentric Centre Lines Crank Positions for Admission, Cut off, Release and Compression Approximate Analytical Method for Crank Positions at Admission, Cut-off, Release and Compression Valve Diagram Zeuner Valve Diagram Reuleaux Valve Diagram 10 Bilgram Valve Diagram 11 Effect of the Early Point of Cut-off with a Simple Slide Valve 12 Meyers Expansion Valve 13 Virtual or Equivalent Eccentric for the Meyers Expansion Valve 14 Minimum Width and Best Setting of the Expansion Plate for Meyers Expansion Valve 15 Reversing Gears 16 Principle of Link Motions-Virtual Eccentric for a Valve with an Off-set Line of Stroke 17 Stephenson Link Motion 18 Virtual or Equivalent Eccentric for Stephenson Link Motion 19 Radial Valve Gears 20 Hackworth Valve Gear 21 Walschaert Valve Gear 18 Governors .653731 Introduction Types of Governors Centrifugal Governors Terms Used in Governors Watt Governor Porter Governor Proell Governor Hartnell Governor Hartung Governor 10 Wilson-Hartnell Governor 11 Pickering Governor 12 Sensitiveness of Governors 13 Stability of Governors 14 Isochronous Governor 15 Hunting 16 Effort and Power of a Governor 17 Effort and Power of a Porter Governor 18 Controlling Force 19 Controlling Force Diagram for a Porter Governor 20 Controlling Force Diagram for a Spring-controlled Governor 21 Coefficient of Insensitiveness 19 Brakes and Dynamometers .732773 Introduction Materials for Brake Lining Types of Brakes Single Block or Shoe Brake Pivoted Block or Shoe Brake Double Block or Shoe Brake Simple Band Brake Differential Band Brake Band and Block Brake 10 Internal Expanding Brake 11 Braking of a Vehicle 12 Dynamometer 13 Types of Dynamometers 14 Classification of Absorption Dynamometers 15 Prony Brake Dynamometer 16 Rope Brake Dynamometers 17 Classification of Transmission Dynamometers 18 Epicyclic-train Dynamometers 19 Belt Transmission Dynamometer-Froude or Throneycraft Transmission Dynamometer 20 Torsion Dynamometer 21 Bevis Gibson Flash Light Torsion Dynamometer (x) 20 Cams .774832 Introduction Classification of Followers Classification of Cams Terms used in Radial cams Motion of the Follower Displacement, Velocity and Acceleration Diagrams when the Follower Moves with Uniform Velocity Displacement, Velocity and Acceleration Diagrams when the Follower Moves with Simple Harmonic Motion Displacement, Velocity and Acceleration Diagrams when the Follower Moves with Uniform Acceleration and Retardation Displacement, Velocity and Acceleration Diagrams when the Follower Moves with Cycloidal Motion 10 Construction of Cam Profiles 11 Cams with Specified Contours 12 Tangent Cam with Reciprocating Roller Follower 13 Circular Arc Cam with Flatfaced Follower 21 Balancing of Rotating Masses .833857 Introduction Balancing of Rotating Masses Balancing of a Single Rotating Mass By a Single Mass Rotating in the Same Plane Balancing of a Single Rotating Mass By Two Masses Rotating in Different Planes Balancing of Several Masses Rotating in the Same Plane Balancing of Several Masses Rotating in Different Planes 22 Balancing of Reciprocating Masses .858908 Introduction Primary and Secondary Unbalanced Forces of Reciprocating Masses Partial Balancing of Unbalanced Primary Force in a Reciprocating Engine Partial Balancing of Locomotives Effect of Partial Balancing of Reciprocating Parts of Two Cylinder Locomotives Variation of Tractive Force Swaying Couple Hammer Blow Balancing of Coupled Locomotives 10 Balancing of Primary Forces of Multi-cylinder In-line Engines 11 Balancing of Secondary Forces of Multi-cylinder In-line Engines 12 Balancing of Radial Engines (Direct and Reverse Crank Method) 13 Balancing of V-engines 23 Longitudinal and Transverse Vibrations Introduction Terms Used in Vibratory Motion Types of Vibratory Motion Types of Free Vibrations Natural Frequency of Free Longitudinal Vibrations Natural Frequency of Free Transverse Vibrations Effect of Inertia of the Constraint in Longitudinal and Transverse Vibrations Natural Frequency of Free Transverse Vibrations Due to a Point Load Acting Over a Simply Supported Shaft Natural Frequency of Free Transverse Vibrations Due to Uniformly Distributed Load Over a Simply (xi) .909971 Supported Shaft 10 Natural Frequency of Free Transverse Vibrations of a Shaft Fixed at Both Ends and Carrying a Uniformly Distributed Load 11 Natural Frequency of Free Transverse Vibrations for a Shaft Subjected to a Number of Point Loads 12 Critical or Whirling Speed of a Shaft 13 Frequency of Free Damped Vibrations (Viscous Damping) 14 Damping Factor or Damping Ratio 15 Logarithmic Decrement 16 Frequency of Underdamped Forced Vibrations 17 Magnification Factor or Dynamic Magnifier 18 Vibration Isolation and Transmissibility 24 Torsional Vibrations .9721001 Introduction Natural Frequency of Free Torsional Vibrations 3.Effect of Inertia of the Constraint on Torsional Vibrations Free Torsional Vibrations of a Single Rotor System Free Torsional Vibrations of a Two Rotor System Free Torsional Vibrations of a Three Rotor System Torsionally Equivalent Shaft Free Torsional Vibrations of a Geared System 25 Computer Aided Analysis and Synthesis of Mechanisms .10021049 Introduction Computer Aided Analysis for Four Bar Mechanism (Freudensteins Equation) Programme for Four Bar mechanism Computer Aided Analysis for Slider Crank Mechanism Coupler Curves Synthesis of Mechanisms Classifications of Synthesis Problem Precision Points for Function Generation 10 Angle Relationship for function Generation 11 Graphical Synthesis of Four Bar Mechanism 12 Graphical synthesis of Slider Crank Mechanism 13 Computer Aided (Analytical) synthesis of Four Bar Mechanism 14 Programme to Co-ordinate the Angular Displacements of the Input and Output Links 15 Least square Technique 16 Programme using Least Square Technique 17 Computer Aided Synthesis of Four Bar Mechanism With Coupler Point 18 Synthesis of Four Bar Mechanism for Body Guidance 19 Analytical Synthesis for slider Crank Mechanism 26 Automatic Control .10501062 Introduction Terms Used in Automatic Control of Systems Types of Automatic Control System Block Diagrams Lag in Response Transfer Function Overall Transfer Function Transfer Function for a system with Viscous Damped Output Transfer Function of a Hartnell Governor 10 Open-Loop Transfer Function 11 Closed-Loop Transfer Function Index .10631071 (xii) GO To FIRST CONTENTS CONTENTS Chapter : Introduction l 1 Introduction Features Definition Sub-divisions of Theory of Machines Fundamental Units 1.1 Definition M.K.S Units The subject Theory of Machines may be defined as that branch of Engineering-science, which deals with the study of relative motion between the various parts of a machine, and forces which act on them The knowledge of this subject is very essential for an engineer in designing the various parts of a machine International System of Units (S.I Units) Note:A machine is a device which receives energy in some available form and utilises it to some particular type of work Derived Units Systems of Units C.G.S Units F.P.S Units 10 Metre 11 Kilogram 12 Second 13 Presentation of Units and their Values 14 Rules for S.I Units 15 Force 16 Resultant Force 17 Scalars and Vectors 18 Representation of Vector Quantities 19 Addition of Vectors 20 Subtraction of Vectors 1.2 Sub-divisions of Theory of Machines The Theory of Machines may be sub-divided into the following four branches : Kinematics It is that branch of Theory of Machines which deals with the relative motion between the various parts of the machines Dynamics It is that branch of Theory of Machines which deals with the forces and their effects, while acting upon the machine parts in motion Kinetics It is that branch of Theory of Machines which deals with the inertia forces which arise from the combined effect of the mass and motion of the machine parts Statics It is that branch of Theory of Machines which deals with the forces and their effects while the machine parts are at rest The mass of the parts is assumed to be negligible CONTENTS CONTENTS l 1.3 Theory of Machines Fundamental Units The measurement of physical quantities is one of the most important operations in engineering Every quantity is measured in terms of some arbitrary, but internationally accepted units, called fundamental units All physical quantities, met within this subject, are expressed in terms of the following three fundamental quantities : Length (L or l ), Mass (M or m), and Stopwatch Simple balance Time (t) 1.4 Derived Units Some units are expressed in terms of fundamental units known as derived units, e.g., the units of area, velocity, acceleration, pressure, etc 1.5 Systems of Units There are only four systems of units, which are commonly used and universally recognised These are known as : C.G.S units, 1.6 F.P.S units, M.K.S units, and S.I units C.G.S Units In this system, the fundamental units of length, mass and time are centimetre, gram and second respectively The C.G.S units are known as absolute units or physicist's units 1.7 F.P S Units P.S In this system, the fundamental units of length, mass and time are foot, pound and second respectively 1.8 M.K.S Units In this system, the fundamental units of length, mass and time are metre, kilogram and second respectively The M.K.S units are known as gravitational units or engineer's units 1.9 Inter na tional System of Units (S.I Units) Interna national The 11th general conference* of weights and measures have recommended a unified and systematically constituted system of fundamental and derived units for international use This system is now being used in many countries In India, the standards of Weights and Measures Act, 1956 (vide which we switched over to M.K.S units) has been revised to recognise all the S.I units in industry and commerce * It is known as General Conference of Weights and Measures (G.C.W.M.) It is an international organisation, of which most of the advanced and developing countries (including India) are members The conference has been entrusted with the task of prescribing definitions for various units of weights and measures, which are the very basic of science and technology today 1000 l Theory of Machines EXERCISES A shaft of 100 mm diameter and metre long is fixed at one end and the other end carries a flywheel of mass tonne The radius of gyration of the flywheel is 0.5 m Find the frequency of torsional vibrations, if the modulus of rigidity for the shaft material is 80 GN/m2 [Ans 8.9 Hz] The flywheel of an engine driving a dynamo has a mass of 180 kg and a radius of gyration of 30 mm The shaft at the flywheel end has an effective length of 250 mm and is 50 mm diameter The armature mass is 120 kg and its radius of gyration is 22.5 mm The dynamo shaft is 43 mm diameter and 200 mm effective length Calculate the position of node and frequency of torsional oscillation C = 83 kN/mm2 [Ans 205 mm from flywheel, 218 Hz] The two rotors A and B are attached to the end of a shaft 500 mm long The mass of the rotor A is 300 kg and its radius of gyration is 300 mm The corresponding values of the rotor B are 500 kg and 450 mm respectively The shaft is 70 mm in diameter for the first 250 mm ; 120 mm for the next 70 mm and 100 mm diameter for the remaining length The modulus of rigidity for the shaft material is 80 GN/m2 Find : The position of the node, and The frequency of torsional vibration [Ans 225 mm from A ; 27.3 Hz] Three rotors A, B and C having moment of inertia of 2000 ; 6000 ; and 3500 kg-m2 respectively are carried on a uniform shaft of 0.35 m diameter The length of the shaft between the rotors A and B is m and between B and C is 32 m Find the natural frequency of the torsional vibrations The modulus of rigidity for the shaft material is 80 GN/m2 [Ans 6.16 Hz ; 18.27 Hz] A motor generator set consists of two armatures P and R as shown in Fig 24.20, with a flywheel between them at Q The modulus of rigidity of the material of the shaft is 84 GN/m2 The system can vibrate with one node at 106.5 mm from P, the flywheel Q being at antinode Using the data of rotors given below, find: The position of the other node, The natural frequency of the free torsional vibrations, for the given positions of the nodes, and The radius of gyration of the rotor R Data of rotors Rotor P Q R Mass, kg Radius of gyration, mm 450 250 540 300 360 Fig 24.20 [Ans 225 mm from R ; 120 Hz ; 108 mm] An electric motor rotating at 1500 r.p.m drives a centrifugal pump at 500 r.p.m through a single stage reduction gearing The moments of inertia of the electric motor and the pump impeller are 400 kg-m2 and 1400 kg-m2 respectively The motor shaft is 45 mm in diameter and 180 mm long The pump shaft is 90 mm in diameter and 450 mm long Determine the frequency of torsional oscillations of the system, neglecting the inertia of the gears [Ans 4.2 Hz] The modulus of rigidity for the shaft material is 84 GN/m2 Two parallel shafts A and B of diameters 50 mm and 70 mm respectively are connected by a pair of gear wheels, the speed of A being times that of B The flywheel of mass moment of inertia kg-m2 is mounted on shaft A at a distance of 0.9 m from the gears The shaft B also carries a flywheel of mass moment of inertia 16 kg-m2 at a distance of 0.6 m from the gears Neglecting the effect of the shaft and gear masses, calculate the fundamental frequency of free torsional oscillations and the position of node Assume modulus of rigidity as 84 GN/m2 [Ans 22.6 Hz ; 0.85 m from the flywheel on shaft A] A centrifugal pump is driven through a pair of spur wheels from an oil engine The pump runs at times the speed of the engine The shaft from the engine flywheel to the gear is 75 mm diameter and 1.2 m long, while that from the pinion to the pump is 50 mm diameter and 400 mm long The moment of inertia are as follows: Flywheel = 1000 kg-m2, Gear = 25 kg m 2, Pinion = 10 kg-m2, and Pump impeller = 40 kg-m2 Find the natural frequencies of torsional oscillations of the system Take C = 84 GN/m2 [Ans 3.4 Hz ; 19.7 Hz] Chapter 24 : Torsional Vibrations l 1001 DO YOU KNOW ? Derive an expression for the frequency of free torsional vibrations for a shaft fixed at one end and carrying a load on the free end Discuss the effect of inertia of a shaft on the free torsional vibrations How the natural frequency of torsional vibrations for a two rotor system is obtained ? Describe the method of finding the natural frequency of torsional vibrations for a three rotor system What is meant by torsionally equivalent length of a shaft as referred to a stepped shaft? Derive the expression for the equivalent length of a shaft which have several steps Establish the expression to determine the frequency of torsional vibrations of a geared system OBJECTIVE TYPE QUESTIONS The natural frequency of free torsional vibrations of a shaft is 1 q q.I (d) I where q = Torsional stiffness of the shaft, and I = Mass moment of inertia of the disc attached at the end of the shaft At a nodal point in a shaft, the amplitude of torsional vibration is (a) zero (b) minimum (c) maximum Two shafts A and B are shown in Fig 24.21 The length of an equivalent shaft B is given by (a) q I (b) (c) l (c) (b) l (a) l = l1 + l2 + l3 d l1 l2 d2 q.I (d) l l1 d2 d3 l1 l2 d1 d2 Fig 24.21 A shaft carrying two rotors as its ends will have (a) no node (b) one node (c) two nodes A shaft carrying three rotors will have (a) no node (b) one node (c) two nodes l3 d1 d3 (d) three nodes (d) three nodes ANSWERS (c) (a) (d) (b) (c) GO To FIRST CONTENTS CONTENTS 1050 l Theory of Machines 26 Automatic Control Fea tur es eatur tures Introduction Terms Used in Automatic control of Systems Types of Automatic Control System Block Diagrams Lag in Response Transfer Function Overall Transfer Function Transfer Function for a System with Viscous Damped Output Transfer Function of a Hartnell Governor 10 Open-Loop Transfer Function 11 Closed-Loop Transfer Function 26.1 Intr oduction Introduction The automatic control of system (or machine) is a very accurate and effective means to perform desired function by the system in which the human operator is replaced by a device thereby relieving the human operator from the job thus saving physical strength The automatic control systems are also called as self-activated systems The centrifugally actuated ball governor which controls the throttle valve to maintain the constant speed of an engine is an example of an automatically controlled system The automatic control systems are very fast, produces uniform and quality products It reduces the requirement of human operators thus minimising wage bills 26.2 Ter ms used in Automa tic Contr ol of erms utomatic Control Systems The following terms are generally used in automatic control of systems : Command The result of the act of adjustment, i.e closing a valve, moving a lever, pressing a button etc., is known as command Response The subsequent result of the system to the command is known as response Process control The automatic control of variables i.e change in pressure, temperature or speed etc in machine is termed as process control 1050 CONTENTS CONTENTS Chapter 26 : Automatic Control l 1051 Process controller The device which controls a process is called a process controller Regulator The device used to keep the variables at a constant desired value is called as regulator Kinetic control The automatic control of the displacement or velocity or acceleration of a member of a machine is called as kinetic control Feed back It is defined as measuring the output of the machine for comparison with the input to the machine Error detector A differential device used to measure the actual controlled quantity and to compare it continuously with the desired value is A rail-track maintenance machine called an error detector It is also known Note : This picture is given as additional information and is not as deviation sensor a direct example of the current chapter Transducer It is a device to change a signal which is in one physical form to a corresponding signal in another physical form A Bourdon tube is an example of transducer because it converts a pressure signal into a displacement, thereby facilitating the indication of the pressure on a calibrated scale The other examples of transducer are a loud speaker (because it converts electrical signal into a sound) and a photoelectric cell (because it converts a light signal into an electric signal) Similarly, the primary elements of all the many different forms of thermometers are transducers 10 Amplification It is defined as increasing the amplitude of the signal without affecting its waveform For example, an error detector itself has insufficient power output to actuate the correcting mechanism and hence the error signal has to be amplified This is generally done by employing mechanical or hydraulic or pneumatic amplifying elements like levers, gears and venturimeters etc 26.3 Types of Automa tic Contr ol System utomatic Control The automatic control systems are of the following two types : Open-loop or unmonitored system When the input to a system is independent of the output from the system, then the system is called an open-loop or unmonitored system It is also called as a calibrated system Most measuring instruments are open-loop control systems, as for the same input signal, the readings will depend upon things like ambient temperature and pressure Following are the examples of open-loop system : (a) A simple Bourdon tube pressure gauge commonly used for measuring pressure (b) A simple carburettor in which the air-fuel ratio adjusted through venturi remains same irrespective of load conditions (c) In traffic lights system, the timing of lights is preset irrespective of intensity of traffic Closed-loop or monitored system When output of a system is measured and is continuously compared with the required value, then it is known as closed-loop or monitored system In this system, the output is measured and through a feedback transducer, it is sent to an error detector which detects any error in the output from the required value thus adjusting the input in a way to get the required output Following are the examples of a closed-loop system : (a) In a traffic control system, if the flow of traffic is measured either by counting the number of vehicles by a person or by counting the impulses due to the vehicles passing over a pressure pad and then setting the time of signal lights 1052 l Theory of Machines (b) In a thermostatically controlled water heater, whenever the temperature of water heater rises above the required point, the thermostate senses it and switches the water heater off so as to bring the temperature down to the required point Similarly, when the temperature falls below the required point, the thermostate switches on the water heater to raise the temperature of water to the required point 26.4 Block Diagrams Fig 26.1 Block diagram of a single carburettor The block diagrams are used to study the automatic control systems in a simplified way In this, the functioning of a system is explained by the interconnected blocks where each block represents a labelled rectangle and is thought of as a block box with a definite function These blocks are connected to other blocks by lines with arrow marks in order to indicate the sequence of events that are taking place Fig 26.1 shows the diagram of a simple carburettor The reduction of a control system to a block diagram greatly facilitates the analysis of the system performance or response 26.5 Lag in Response We know that response is the subsequent result of the system to the command In any control system, there is a delay in response (output) due to some inherent cause and it becomes difficult to measure the input and output simultaneously This delay in response is termed as lag in response For example, in steam turbines, with the sudden decrease in load, the hydraulic relay moves in the direction to close the valve But unless the piston valve ports are made with literally zero overlap, there would be some lag in operation, since the first movement of the piston valve would not be sufficient to open the ports This lag increases the probability of unstable operation 26.6 Transfer Function The transfer function is an expression showing the relation between output and the input to each unit or block of a control system Mathematically, Transfer function = θo / θi where θo = Output signal of the block of a system, and θi = Input signal to the block of a system Chapter 26 : Automatic Control l 1053 Thus, the output from an element may be obtained by multiplying the input signal with the transfer function Note : From the transfer function of the individual blocks, the equation of motion of system can be formulated 26.7 Ov erall Transfer Function Overall In the previous article, we have discussed the transfer function of a block A control system actually consists of several such blocks which are connected in series The overall transfer function of the series is the product of the individual transfer function Consider a block diagram of any control system represented by the three blocks as shown in Fig 26.2 Fig 26.2 Overall transfer function Thus, if F1 (D), F2 (D), F3 (D) are individual transfer functions of three blocks in series, then the overall transfer function of the system is given as θo θ1 θ2 θo = × × = F1 ( D) × F2 ( D) × F3 ( D) = KG ( D) θi θi θ1 θ2 where K = Constant representing the overall amplification or gain, and G(D) = Some function of the operator D Note: The above equation is only true if there is no interaction between the blocks, that is the output from one block is not affected by its connection to the subsequent blocks 26.8 Transfer Function ffor or a System with viscous Damped Output Consider a shaft, which is used to position a load (which may be pulley or gear) as shown in Fig 26.3 The movement of the load is resisted by a viscous damping torque Fig 26.3 Transfer function for a system with viscous damped output Let θi = Input signal to the shaft, θo = Output signal of the shaft, q = Stiffness of the shaft, I = Moment of Inertia of the load, and Td = Viscous damping torque per unit angular velocity 1054 l Theory of Machines After some time t, Twist in the shaft = θi − θo ∴ Torque transmitted to the load = q ( θi − θo ) We also know that damping torque d θo = Td ω0 = Td dt (∵ ω0 = d θo / dt ) Material being moved via-belt conveyor Note : This picture is given as additional information and is not a direct example of the current chapter According to Newton’s Second law, the equation of motion of the system is given by d θo I dt or = q (θi − θo ) − Td d θo dt (i) d θo dθ I = q θi − q θo − Td o dt dt Replacing d / dt by D in above equation, we get I ( D θo ) = q θi − q θo − Td ( Dθo ) I ( D θo ) + Td ( Dθo ) + q θo = q θi or T q q D θo + d ( D θo ) + (θo ) = (θi ) I I I T D θo + d ( D θo ) + (ωn )2 θo = (ωn )2 θ I q I Also we know that viscous damping torque per unit angular velocity, Td = I ξωn or Td / I = ξωn where where ωn = Natural frequency of the shaft = ξ = Damping factor or damping ratio (ii) Chapter 26 : Automatic Control l 1055 The equation (ii) may now be written as D2 θ + ξωn ( Dθo ) + (ωn ) θo = (ωn )2 θi [D2 + 2ξωn D + (ωn )2 ] θo = (ωn )2 θi or ∴ Transfer function = = (ωn )2 θo = θi D + ξωn D + (ωn )2 T D + ξT D + 2 T = Time constant = 1/ ωn where Note: The time constant (T) may also be obtained by dividing the periodic time (td) of the undamped natural oscillations of the system by 2π Mathematically, 2π td 2π 1 ∵ td = ω = × = n 2π ωn 2π ωn Example 26.1 The motion of a pointer over a scale is resisted by a viscous damping torque of magnitude 0.6 N-m at an angular velocity of rad / s The pointer, of negligible inertia, is mounted on the end of a relatively flexible shaft of stiffness 1.2 N-m / rad, and this shaft is driven through a to reduction gear box Determine its overall transfer function If the input shaft to the gear box is suddenly rotated through completed revolution, determine the time taken by the pointer to reach a position within percent of its final value Solution Given: Td = 0.6/1 = 0.6 N-ms/rad; q = 1.2 N-m/rad The control system along with its block diagram is shown in Fig 26.4 (a) and (b) respectively T= Overall transfer function Since the inertia of the pointer is negligible, therefore the torque generated by the twisting of the shaft has only to overcome the damping torque Therefore q (θ1 − θo ) = Td (d θo / dt ) θ1 = Output from the gear box where ∴ (∵ d / dt = D ) ( q + Td D ) θo = q θ1 or ∴ where q θ1 − q θo = Td ( Dθo ) Fig 26.4 θo 1 q = = = θ1 q + Td D + (Td / q) D + T D T = Time constant = Td / q = 0.6/1.2 = 0.5s .(i) 1056 l Theory of Machines Substituting this value in equation (i), we get θ0 = θ1 + 0.5D We know that overall transfer function for the control system is θo θ1 θ2 = × = × Ans θi θi θ1 (1+ 0.5D) ∵ θ1 / θi = (Given) Aircraft engine is being assembled Note : This picture is given as additional information and is not a direct example of the current chapter Time taken by the pointer Let t = Time taken by the pointer Since the input shaft to the gear box is rotated through complete revolution, therefore θi = 2π, a constant We know that transfer function for the control system is θo θ = × or (1 + 0.5D) θo = i θi (1 + 0.5 D) ∴ dθ 0.5 o dt θi + θo = Substituting θi = 2π in the above equation, we get 2π π dθ = 0.5 o + θo = dt or dθ π 0.5 o = − θo dt Separating the variables, we get dθ = dt π − θo (∵ D ≡ d / dt) Chapter 26 : Automatic Control l 1057 Integrating the above equation, we get π − loge − θo = t + constant (ii) Applying initial conditions to the above equation i.e when t = 0, θo = , we get π constant = − log e 2 Substituting the value of constant in equation (i), π π − loge − θo = t − log e 2 2 or π π log e − θo = −2 t + log e 2 2 ∴ π π − θo = e −2t × 2 π / − θo = e−2t π/ or π (1 − e −2t ) (iii) The curve depicted by above equation is shown in Fig 26.5 and is known as simple exponential time delay curve θo = i.e Fig 26.5 The output θo will be within percent of its final value when θ0 = 0.99(π / 2) Substituting this value in equation (iii), we get ( π π 0.99 = − e −2t 2 ) 0.99 = − e −2t or e −2t = 0.01 ∴ t = loge 100 = 4.6 or t = 2.3s Ans 1058 l Theory of Machines 26.9 Transfer Function of a Har tnell Go ver nor artnell Gov ernor Consider a Hartnell governor* as shown in Fig 26.6 (a) The various forces acting on the governor are shown in Fig 26.6 (b) Let m = Mass of the ball M = mass of the sleeve, r = Radius of rotation of the governor in mid position, ∆r = Change in radius of rotation, ω = Angular speed of rotation in mid position, ∆ω = Change in angular speed of rotation, (a) Hartnell governor (b) Forces acting on a Hartnell governor Fig 26.6 x = Length of the vertical or ball arm of the lever, y =Length of the horizontal or sleeve arm of the lever, h = compression of spring with balls in vertical position, h ′ = Displacement of the sleeve, s = Stiffness of the spring, c = Damping coefficient i.e damping force per unit velocity, and ξ = Damping factor * Bucket conveyor Note : This picture is given as additional information and is not a direct example of the current chapter For details on Hartnell governor, refer chapter 18, Art 18.8 Chapter 26 : Automatic Control l 1059 The various forces acting on the governor at the given position are as follows : Centrifugal force due to ball mass, Fc = m(r + ∆r ) (ω + ∆ω)2 x = m r + ( h′) (ω + ∆ω) y x d h′ y dt Inertia force of the balls, Fim = m Inertia force of the sleeve mass, d h′ FiM = M dt Damping force, dx Fd = c dt Fs = s(h + h′) Spring force, It is assumed that the load on the sleeve, weight of the balls and the friction force are negligible as compared to the inertia forces Now, taking moments about the fulcrum O, considering only one half of the governor, x m r + h′ (ω + ∆ω)2 x = m × y d h′ x d h′ x+ ×M y y dt dt dh′ + ×c y + × s( h + h′) y dt Neglecting the product of small terms, we get x mr ω2 x + m × × h′ω2 x + 2mr ω(∆ω) x y d h′ mx d h′ dh′ + × M y + ×c y + × s y (h + h′) = y dt dt dt (i) Also, we know that at equilibrium position, ×s h y Now the equation (i) may be written as mr ω2 x = x mx 2 1 × s h y + m × × h′ω2 x + 2mr ω(∆ω) x = ( D h′) + My ( D2 h′) + c y ( Dh′) y y 2 + sy ( h + h′) (∵ d / dt = D) 1060 or l Theory of Machines mx 1 mx 1 + My D2 h′ + × cy Dh′ + sy − × D2 h′ = 2mr ω(∆ω) x y y 2 2 Multiplying the above equation throughout by 2y, we get (2mx + My ) D h′ + (c y )dh′ + (sy − 2mx ω2 )h′ = 4mr ω (∆ω) x y cy sy − 2mx2 ω2 + (2mx2 + My ) D2 + 2mx + My 2mx + My h′ = 4m r ω (∆ω) xy or cy sy − 2mx ω2 4mr ω (∆ω) xy ×D+ D + h′ = 2 2 2mx + My 2mx + My 2mx + My or D2 + 2ξωn D + (ωn2 )h′ = ∴ where h′ = 4mr ω (∆ω) xy 2mx2 + My2 4mr ω(∆ω) xy 2mx + My 2 × D + 2ξωn D + (ωn ) 2 cy 2 ξ ωn = 2mx + My 2 ξ = Damping factor, and ωn = Natural frequency = sy − mx ω2 mx + My Thus, transfer function for the Hartnell governor, = Output signal Displacement of sleeve ( h′) = Input signal Change in speed (∆ω) = 4m r ω xy 2mx + My 2 × D + ξωn D + (ωn ) 2 26.10 Open-Loop Transfer Function Fig 26.7 Open loop control system Fig 26.8 Simplified open loop control system The open loop transfer function is defined as the overall transfer function of the forward path elements Consider an open loop control system consisting of several elements having individual transfer function such F1(D), F2(D), F3 (D) as shown in Fig 26 Thus Chapter 26 : Automatic Control Open loop transfer function = l 1061 θo θ1 θ2 θo = × × θi θi θ1 θ2 = F1 ( D ) × F2 ( D ) × F3 ( D ) = KG (D ) The simplified block diagram of open loop transfer function is shown in Fig 26.8 26.11 Closed - Loop Transfer Function The closed loop transfer function is defined as the overall transfer function of the entire control system Consider a closed loop transfer function consisting of several elements as shown in Fig 26.9 Fig 26.9 Closed-loop transfer function Now, for the forward path element, we know that θo θo = = K G ( D) θ1 θi − θo K G ( D ) = F1 ( D) × F2 ( D) × F3 ( D) where On rearranging, we get θo = K G ( D)θi − K G ( D )θo [1 + K G ( D )] θo = K G ( D ) θi or ∴ θo Open loop TF K G(D) = = θi + K G (D ) 1+ Open loop TF The above expression shows the transfer function for the closed-loop control system Thus the block diagram may be further simplified as shown in Fig 26.10, where the entire system is represented by a single block Fig 26.10 Simplified closed-loop system EXERCISES Define the following terms: (a) Response (b) Process control (c) Regulator (d) Transducer What you understand by open-loop and closed loop control system? Explain with an example Discuss the importance of block diagrams in control systems 1062 l Theory of Machines Draw the block diagrams for the following control systems: (a) A simple carburettor, (b) A thermostatically controlled electric furnace What is a transfer function ? OBJECTIVE TYPE QUESTIONS The device used to keep the variables at a constant desired value is called a (a) process controlled (b) regulator (c) deviation sensor (d) amplifier The transfer function of a to reduction gear box is (a) (b) (c) 1/4 (d) 1/2 A simple Bourdon tube pressure gauge is a (a) closed-loop control system (b) open-loop control system (c) manually operated system (d) none of the above The overall transfer function of three blocks connected in series is (a) F1( D ) × F2 ( D) F3 ( D ) (c) F1( D ) × F2 ( D) × F3 ( D) (b) F1 ( D ) × F3 ( D ) F2 ( D ) (d) F ( D ) × F ( D ) × F ( D) where F1 (D), F2 (D) and F3 (D) are the individual transfer functions of the three blocks The transfer function for a closed-loop control system is K G ( D) (a) + K G ( D ) (b) K G( D)[1 + KG( D)] + K G ( D) KG ( D ) (d) K G ( D) − K G (D ) (c) ANSWERS (b) (c) (b) (c) (a) GO To FIRST [...]... this, we find it quite convenient to use some standard abbreviations We shall use : m for metre or metres km for kilometre or kilometres kg for kilogram or kilograms t for tonne or tonnes s for second or seconds min for minute or minutes N-m for newton × metres (e.g work done ) kN-m for kilonewton × metres rev for revolution or revolutions rad for radian or radians * In certain countries, comma is still... laws, these are also justified as the results, so obtained, agree with the actual observations These three laws of motion are as follows: 1 Newton s First Law of Motion It states, “Every body continues in its state of rest or of uniform motion in a straight line, unless acted upon by some external force.” This is also known as Law of Inertia The inertia is that property of a matter, by virtue of which... Indian Standards (BIS) previously known as Indian Standards Institution (ISI) has been created for this purpose We have already discussed that the fundamental units in M.K .S and S. I units for length, mass and time is metre, kilogram and second respectively But in actual practice, it is not necessary to express all lengths in metres, all masses in kilograms and all times in seconds We shall, sometimes, use... Weights and Measures recommended only the fundamental and derived units of S. I units But it did not elaborate the rules for the usage of the units Later on many scientists and engineers held a number of meetings for the style and usage of S. I units Some of the decisions of the meetings are as follows : 1 For numbers having five or more digits, the digits should be placed in groups of three separated by spaces*... Mass and Weight Sometimes much confu-sion and misunderstanding is created, while using the various systems of units in the measurements of force and mass This happens because of the lack of clear understanding of the difference between the mass and the weight The following definitions of mass and weight should be clearly understood : The above picture shows space shuttle 1 Mass It is the amount of. .. circular path of radius r We have seen in Art 2.19 that the centripetal acceleration, ac = v 2 /r = ω2 .r and Force = Mass × Acceleration ∴ Centripetal force = Mass × Centripetal acceleration or Fc = m.v2 /r = m.ω2 .r 28 l Theory of Machines This force acts radially inwards and is essential for circular motion We have discussed above that the centripetal force acts radially inwards According to Newton 's Third... we shall discuss the kinetics of motion, i.e the motion which takes into consideration the forces or other factors, e.g mass or weight of the bodies The force and motion is governed by the three laws of motion 3.2 Newton s Laws of Motion Newton has formulated three laws of motion, which are the basic postulates or assumptions on which the whole system of kinetics is based Like other scientific laws,... given forces, is known as a resultant force The forces P,Q ,R etc are called component forces The process of finding out the resultant force of the given component forces, is known as composition of forces A resultant force may be found out analytically, graphically or by the following three laws: 1 Parallelogram law of forces It states, “If two forces acting simultaneously on a particle be represented... defined as the shortest distance (at 0°C) between the two parallel lines, engraved upon the polished surface of a platinum-iridium bar, kept at the International Bureau of Weights and Measures at Sevres near Paris 1.11 Kilogram The international kilogram may be defined as the mass of the platinum-iridium cylinder, which is also kept at the International Bureau of Weights and Measures at Sevres near Paris... mean solar day 1.13 Presentation of Units and their Values The frequent changes in the present day life are facilitated by an international body known as International Standard Organisation (ISO) which makes recommendations regarding international standard procedures The implementation of ISO recommendations, in a country, is assisted by its organisation appointed for the purpose In India, Bureau of