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Theory of Machines

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Theory of Machines THEORY OF MACHINES For MECHANICAL ENGINEERING THEORY OF MACHINES & VIBRATIONS SYLLABUS Theory of Machines: Displacement, velocity and acceleration analysis of plane mechanisms; dynamic analysis of linkages; cams; gears and gear trains; flywheels and governors; balancing of reciprocating and rotating masses; gyroscope. Vibrations: Free and forced vibration of single degree of freedom systems, effect of damping; vibration isolation; resonance; critical speeds of shafts. ANALYSIS OF GATE PAPERS Exam Year 1 Mark Ques. 2 Mark Ques. Total 2003 6 - 15 2004 8 - 18 2005 6 - 14 2006 9 - 21 2007 1 6 13 2008 1 3 7 2009 2 4 10 2010 5 3 11 2011 1 3 7 2012 2 1 4 2013 3 2 7 2014 Set-1 2 3 8 2014 Set-2 2 3 8 2014 Set-3 2 4 10 2014 Set-4 2 3 8 2015 Set-1 1 2 5 2015 Set-2 2 2 6 2015 Set-3 3 3 9 2016 Set-1 2 3 8 2016 Set-2 1 2 5 2016 Set-3 3 3 9 2017 Set-1 1 3 7 2017 Set-2 2 4 10 2018 Set-1 2 3 8 2018 Set-2 2 1 4 © Copyright Reserved by Gateflix.in No part of this material should be copied or reproduced without permission CONTENTS Topics Page No 1. MECHANICS 1.1 Introduction 01 1.2 Kinematic chain 05 1.3 3-D Space Mechanism 07 1.4 Bull Engine / Pendulum Pump 12 1.5 Basic Instantaneous centers in the mechanism 15 1.6 Theorem of Angular Velocities 16 1.7 Mechanical Advantage of the mechanism 22 2. GEARS 2.1 Introduction 23 2.2 Classification of Gears 23 2.3 Analysis of Involutes Gears 30 2.4 Methods to prevent Interference 32 2.5 Different Types of Involutes Systems 34 3. GEAR TRAIN 3.1 Introduction 37 3.2 Components 37 3.3 Simple Gear Train 37 3.4 Compound Gear Train 37 3.5 Reverted Gear Train 38 3.6 Planetary Gear Trains 42 3.7 Kinetic analysis of single slider crank mechanism 43 4. FLYWHEEL 4.1 Introduction 48 4.2 Turning moment diagram of single cylinder double-acting stream engine 48 4.3 Role of Flywheel 48 4.4 Natural vibrations 54 © Copyright Reserved by Gateflix.in No part of this material should be copied or reproduced without permission 4.5 Vibrations 55 4.6 Methods of initial disturbances 55 4.7 Springs in combination 56 4.8 Energy Method 56 4.9 Transverse vibrations of the beam 62 5. VIBRATIONS 5.1 Introduction 65 5.2 Logarithmic Decrement(S) 65 5.3 Death of the Shaft 69 6. CAMS & FOLLOWERS 6.1 Introduction 71 6.2 Classification of Cams 71 6.3 Classification of Followers 72 6.4 Application of Cams 74 6.5 Terminology of Cam & Follower 74 6.6 Derivatives of Follower Motion 75 6.7 Motions of Followers 77 7. GOVERNORS 7.1 Introduction 80 7.2 Classification of Governors 80 7.3 Gravity Controlled Centrifugal Governors 81 7.4 Proell Governor 83 7.5 Spring Controlled Centrifugal Governors 83 7.6 Governor Effort and Power 85 7.7 Characteristics of Governors 86 7.8 Controlling Force and Stability of Spring Controlled Governors 86 7.9 Insensitiveness in the Governors 87 8. GATE QUESTIONS 89 9. ASSIGNMENT 160 © Copyright Reserved by Gateflix.in No part of this material should be copied or reproduced without permission 1 MECHANICS 1.1. INTRODUCTION : 1.1.1 KINEMATIC LINK OR ELEMENT The subject Theory of Machines may be Every part of a machine which is having the defined as that branch of Engineering- relative motion w.r.t some other part is science, which deals with the study of known as kinematic link or element and for relative motion between the various parts a body to be a link it should be a resistant of a machine, and forces which act on them. body. The knowledge of this subject is very essential for an engineer in designing the various parts of a machine. Sub-divisions of Theory of Machines The Theory of Machines may be sub- divided into the following four branches : 1. Kinematics. It is that branch of Theory of Machines which deals with the relative motion between the various parts of the machines. 2. 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. Note: For a link the body should be a 3. Kinetics. It is that branch of Theory of resistant body so that it is capable of Machines which deals with the inertia transmitting Motion from one element to forces which arise from the combined another. effect of the mass and motion of the machine parts. 1.1.2 TYPES OF LINK 4. Statics. It is that branch of Theory of Machines which deals with the forces 1. Rigid Link A rigid link is one which and their effects while the machine does not undergo any deformation parts are at rest. The mass of the parts while transmitting motion. Strictly is assumed to be negligible. speaking, rigid links do not exist. However, as the deformation of a ds m connecting rod, crank etc. of a V s reciprocating steam engine is not dt appreciable, they can be considered as dv d rigid links a2 m F(ext) mv dt s dt Links in which the deformations are da negligible (Microscpoic). j m 3 Eg :- Connection Link. dt s V=Velocity 2. Flexible Link : A flexible link is one A=Acceleration which is partly deformed in a manner J=Impact not to affect the transmission of motion. © Copyright Reserved by Gateflix.in No part of this material should be copied or reproduced without permission For example, belts, ropes, chains and When the motion between a pair is limited wires are flexible links and transmit to a definite direction irrespective of the tensile forces only direction of force applied, then the motion Links in which deformations are is said to be a completely constrained there but they are in the permissible motion. For example, the piston & cylinder limit. (in a steam engine) form a pair and the Eg :- Belt Drives, Rope Drives. motion of the piston is limited to a definite direction Fluid Link A fluid link is one which is Completely Constrained Motion formed by having a fluid in a receptacle For One Input → One output (Without the and the motion is transmitted through help of surrounding) the fluid by pressure or compression only, as in the case of hydraulic presses, jacks and brakes. Sometimes fluid behaves like a link because the power is transmitted due to the Pressure of the fluid. Eg. Hydraulic brake, hydraulic ram, hydraulic lift. Bernoulli Principle P1A1=P2A2 P1=Pressure on Left side of tank P2= Presuure on right side of tank A1=Area of left side of tank The motion of a square bar in a square A2=Area on right side of tank hole, as shown in, and the motion of a shaft with collars at each end in a circular hole, as shown in, are also examples of completely constrained motion 2. Successfully Constrained Motion NormalForce Thrust Pressure = = When the motion between the elements, Area Area forming a pair, is such that the constrained Kinematic pair: The two links or elements motion is not completed by itself, but by of a machine, when in contact with each some other means, then the motion is said other, are said to form a pair. If the relative to be successfully constrained motion. motion between them is completely or Consider a shaft in a foot-step bearing as successfully constrained (i.e. in a definite shown in Fig. direction), the pair is known as kinematic The shaft may rotate in a bearing or it may pair. move upwards. This is a case of incompletely constrained motion. But if the 1.1.3 TYPES OF CONSTRAINED MOTIONS load is placed on the shaft to prevent axial 1. Completely Constrained Motion. © Copyright Reserved by Gateflix.in No part of this material should be copied or reproduced without permission upward movement of the shaft, then the another link, the pair is known as turning motion of the pair is said to be successfully pair A shaft withcollars at both ends fitted constrained motion. The motion of an I.C. into a circular hole, the crankshaft in a engine valve (these are kept on their seat journal bearing in an engine, lathe spindle by a spring) and the piston reciprocating supported in head stock, cycle wheels inside an engine cylinder are also the turning over their axles etc. are the examples of successfully constrained motion. examples of a turning pair. A turning pair Motion With the help of Surrounding. also has a completely constrained motion. Sliding Pair (Prismatic Pair) When the two elements of a pair are connected in such a way that one can only slide relative to the other, the pair is known as a sliding pair. The piston and cylinder, cross-head and guides of a reciprocating steam engine, Footstep Bearing ram and its guides in shaper, tail stock on 3. Incompletely Constrained Motion the lathe bed etc. are the examples of a When the motion between a pair can take sliding pair. place in more than one direction, then the motion is called an incompletely constrained motion. The change in the direction of impressed force may alter the direction of relative motion between the pair. e.g. A circular bar or shaft in a circular hole Rolling Pair When the two elements of a For one input → two or more output.
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