K-Notes

ME Krash (Sample)1 1 x2 1 1 x2dx tan 1 x cdx cot 1 x c 1 xx 112 xx 12dx cosec 1x cdx sec 1x c1 x dx sin 1 c a a2 x211 1 x a2 x2 dx a tan a c11 x a a2 x2 dx 2a ln x a c11 x a x2 a2 dx 2a ln x a c x a c dx ln x x a1 x a1 2222dx ln x x2 a2 c22Properties of definite integrala.bbaa f x dx f t dtbab. f x dx f x dxabbcaabc. f x dx f x dx f x dxcbbaad. f x dx f a b x dx ae. f(x)dx 2 f(x)dx if f(2a-x) f(x) 00 0 if f(2a-x) f(x) 2a af. f(x)dx 2 f(x)dx if f(x) f(-x) 0 a 0 if f(x) f(x) ag. t df x dx f t ' t f t ' t dt t Kreatryx. All Rights Reserved.www.kreatryx.com

ME Krash (Sample)Theory of MachineTerminology of CAMSBase circle: Smallest circle drawn from the centre of rotation of the cam forming a part ofcam profile. Radius of the circle is called the least radius of cam.Pitch curve: Path of trace point assuming cam is fixed and follower rotates.Prime circle: Smallest circle that can be drawn from the cam centre and tangent to the pitchcurve.Pitch circle: Circle drawn from the cam centre and passes through the pitch point. Pitchpoint corresponds to maxCam profile: Surface of cam that comes in contact with the follower. While drawing camprofile, we consider that cam is stationary and follower rotates over it.Dwell: It is zero displacement or absence of rotation of the follower during the motion of thecam.Angle of Ascent a : It is the angle through which the cam turns during the time followerrise.Angle of dwell : it is the angle through which cam turns while the follower remainsstationary at the highest or lowest position.Angle of dwell d : It is the angle through which cam turns during the time the followerreturns to the initial position. For a flat face and knife edge follower, prime circle and the base circle are the same,because in these types, trace point lies on the base circle.\ Maximum lift of follower Stroke of follower 30 or max 30 , otherwise a reciprocating type of follower will jam the bearing. Kreatryx. All Rights Reserved.www.kreatryx.com

ME Krash (Sample)Force exerted by camFcos : Vertical component which lifts the followerFsin : Exerts lateral pressure on the bearing Lateral force Has to be reduced by making surface more convex and longer The minimum value of cannot be reduced from certain value Base circle diameter Pressure angle Kreatryx. All Rights Reserved.www.kreatryx.com

ME Krash (Sample)Engineering MechanicsImpulse, Momentum and CollisionsThe impulse of a constant force F is defined as the product of the force and the time t forwhich it acts.Impulse F.tThe effect of the impulse on a body can be found using below equations, where a isacceleration, u and v are initial and final velocities respectively and t is time.v u at v u atSoI F t ma t m v u change in momentumSo we can say that,Impulse of a constant force F.t change in momentum produced.Impulse is a vector quantity and has the same units as momentum, Ns or kg m/sThe impulse of a variable force can be defined by the integraltImpulse F.dt , where t is the time for which F acts.0By Newton's 2nd lawF m.a m dv / dt So impulse can also be writtenvImpulse muvvdvdt mdv mv udtuwhich for a constant massImpulse m (v-u)Impulsive force Suppose the force F is very large and acts for a very short time. During this time thedistance moved is very small and under normal analysis would be ignored. Under theseconditions, the only effect of the force can be measured is the impulse, or change inmomentum which is called an impulsive force. In theory this force should be infinitely large and the time of action infinitely small. Someapplications where the conditions are approached are collision of snooker balls, ahammer hitting a nail or the impact of a bullet on a target.Conservation of linear momentumConsider the direct collision of two spheres A and B. Kreatryx. All Rights Reserved.www.kreatryx.com

ME Krash (Sample) When the spheres collide, then by Newton's third law, the force F exerted by A on B isequal and opposite to the force exerted by B on A.The time for contact is the same for both. The impulse of A on B is thus equal andopposite to the impulse of B on A. It then follows that the change in momentum of A isequal in magnitude to the change in momentum in B, but it is in the opposite direction.The total change in momentum of the whole system is thus zero.This means that the total momentum before and after a collision is equal, or that linearmomentum is conserved. This is called the principle of conservation of linear momentumand in summary this may be stated:The total momentum of a system, in any direction, remains constant unless an externalforce acts on the system in that direction.Caution: Take proper sign convention while solving problems.Impact of inelastic bodies When two inelastic bodies collide they remain together. They show no inclination toreturn to their original shape after the collision.An example of this may be two railway carriages that collide and become coupled onimpact. Problems of this type may be solved by the principle of conservation of linearmomentum.Momentum before impact Momentum after impact(Take proper sign convention) Although momentum is conserved, it is important to realize that energy is always lost inan inelastic collision (it is converted from mechanical energy to some other form such asheat, light or sound.)Impact of elastic bodies In the last section the bodies were assumed to stay together after impact. An elastic bodyis one which tends to return to its original shape after impact. When two elastic bodiescollide, they rebound after collision. An example is the collision of two snooker balls.If the bodies are travelling along the same straight line before impact, then the collision iscalled a direct collision. This is the only type of collision considered here.Consider the two elastic spheres as shown. By the principle of conservation of linearmomentumMomentum before impact Momentum after impactm1u1 m2u2 m1 v1 m2 v2where the ui initial velocity of body i.vi final velocity of body i. Kreatryx. All Rights Reserved.www.kreatryx.com

ME Krash (Sample) When the spheres are inelastic v1 and v2 are equal as we saw in the last section. For elasticbodies v1 and v2 depend on the elastic properties of the bodies. A measure of theelasticity is the coefficient of restitution, for direct collision this is defined ase v u11 v2 u2 The values of ‘e’ in practice vary between 0 and 1. For completely inelastic collision, e 0and for completely elastic collision, e 1. In the latter case, no energy is lost in thecollision.Both the law of restitution & conservation of momentum are applicable along x and ydirections in case of oblique collision.Rolling, torque and angular momentumRolling motion:Combination of translational motion & rotational motion.In rolling motion, the centre of the object moves in a line parallel to the surface.Relation between length and angle of rotation:When the object rotates through an angle ' ' , a point at a distance R from the rotation axismoves through a distance of SS RThe arc length S is the same as the distance that the wheel translates.The linear (translational) speed VCOM of the wheel isThe angular speed of the wheel is So,dS;Vis the velocity of centre of mass.dt COMd dtdSd RdtdtVCOM RRolling motion is the combination of pure rotational motion and pure translational motion. Kreatryx. All Rights Reserved.www.kreatryx.com

ME Krash (Sample)Pure rotation pure translation Rolling motionThe velocity of a point at the top of the rolling wheel is twice that of the centre of the wheelVtop 2R 2 R 2VCOMKinetic energy of rolling:As an object rolls, the point at the very bottom, the contact point with the surface, isinstantaneously stationary.We will call this point P and we can treat rolling about this point.12K.E IP 2IP : Rotational inertia about the point PParallel axis theorem says IP ICOM MR21 2I 2 P11K.E ICOM 2 MR 2 222112K.E ICOM 2 MVCOM22K.E Kinetic energy of a rolling object comes from rotational kinetic energy and translationalkinetic energy.Forces in rolling: If a wheel rolls smoothly, there is no sliding at the contact point so there is no friction. However if there is an external force that produces an acceleration, there will be anangular acceleration . The acceleration will make the wheel want to slide at the contactpoint. Then a frictional force will be on the wheel to oppose the tendency to slide.Direction of static frictional force: If a wheel moving to the right were to accelerate, the bottom of the wheel would want tomove to the left compared to the surface. Thus static friction force is to the right. If same wheel was to slow down, the direction of the acceleration and angular accelerationwould switch and the static friction force will now be pointing towards the left. Kreatryx. All Rights Reserved.www.kreatryx.com

ME Krash (Sample)Rolling down a ramp:The direction of the static friction force is the confusing part here. It points up along theramp. If the wheel were to slide down the ramp, the friction opposing the sliding would bepointing up.fs Mgsin MaCOM I Only force on the wheel that produces torque is the frictionRfs ICOM We will need to make use of aCOM R(a is down the ramp, negative X-direction but the wheel rolls counter-clockwise, is positive) aCOMRSo we can solve for fs ICOMaCOM aCOMR2 gsin I1 COM2MRYo-YoA Yo-Yo behaves similar to the wheel rolling down a ramp.1) Instead of rolling down a ramp of angle ' ' , Yo-Yo follows an angle of 900 with horizontal.2) Yo-Yo rolls down a string on a radius R.3) Instead of friction, the tension shows up in the Yo-Yo.aCOM gI1 COM2MR 0 Kreatryx. All Rights Reserved.www.kreatryx.com

ME Krash (Sample)Machine DesignFailure TheoriesMaximum principal stress theory It is also known as Rankine’s theory.Best for brittle materials. For safe design 1 SytNorSwN 1 Maximum principal stress developed at a critical pointN Factor of safetyMaximum shear stress theory (M.S.S.T) Also known as Guest and Tresca theory.Suitable for ductile materials.It gives more safety to component (most safe design).Dimension and cost of component is more.It is not suitable under hydrostatic state of stress.For safe designAbsolute Emax S YsSor YtN2NSYt tensile yield strengthSYs Yield shear stress of material SYt SYs 2 For tri-axial state of stress: 2 2 3 3 1,, max Larger of 122 2 For biaxial state of stress: max 1 22Maximum principal strain theoryAlso known as St. Venant’s theoryFor safe design 1 Y.P orSYtN E 1 Maximum principal strain at critical point Y.P Strain corresponding to yield pointN Factor of safety Kreatryx. All Rights Reserved.www.kreatryx.com

ME Krash (Sample)E Young modulus of elasticity1 1 1 2 3 ETotal strain energy theory [T.S.E.T]Also known as Haigh’s theoryFor safe design, total strain energy per unit volume should be less than that of yield point TotalS.E1111 1 1 2 2 3 3 12 22 32 2 1 2 2 3 3 1 2222EVolume( Poisson’s ratio)2 S.E 1 S YL Volume Yield point 2E N For trivial state of stress condition S yt 1 2 3 2 1 2 2 3 3 1 N 2222For biaxial state of stress condition S yt 1 2 2 1 2 N 222Maximum distortion energy theory (M.D.E.T) Also known as Von-Mises theory.Best for ductile material.Gives economical design (Less cost).It is less safer design than those corresponding to maximum shear stress theory.For safe design Maximum distortion energy Distortion energy per unit volume Per unit volume At yield point D.E 1 2 2 2 2 3 3 1 2 Vol6E 1 D.E 1 S yt Vol 3E N Y.P2 S 222 1 2 2 3 3 1 2 NYt 2For biaxial state of stress condition S yt 12 22 1 2 N 2 Kreatryx. All Rights Reserved.www.kreatryx.com

ME Krash (Sample)Consolidated table for theories of failure t perS.No.1.Theoriesof failureMax.PrincipalstresstheoryM e or Te equationsDesign equations1 M M2 T 2 2 3 d t per32Me S ytS 1 or ut NN S ytNequations(used whennormal stress isacting in only 1direction) 1 x 2 22 x 4 xy S ysS ytShape ofsafeboundaryi.e.S yc Syt1SquareLarger theory/max.distortionenergytheory 1 2 , 2 S23 yt 2 2 N 3 1 , 2 1 2 3 Te M2 T 2 S yt 3d per16-N 2x 4 2xy0.5Hexagon-11 Rhombus 12 22 32 1 2 S yt 2 2 3 N 3 1 12--2 1 2223 3 1 2 S yt 2 N Kreatryx. All Rights Reserved.23Te M2 T24 3 d t per32EllipseSemi-majoraxisS yt 1 Semiminor axisS yt 1 EllipseSemi-majoraxis21 2x 3 2xyValid for1 2 Syt3Semiminor axis 2S3 ytBest forbrittlematerials.Used forductilematerialswhen1.Uniaxialstate ofstress2. Biaxial if 1,2 are likein nature.3.Hydrostaticstate ofstress.Used forductilematerials(excepthydrostaticstate ofstress). Givesover safeanduneconomical design.-Best forhydrostaticstate ofstress.Best forductilematerials.Gives safeandeconomicaldesign.www.kreatryx.com

ME Krash (Sample)ClutchesIt is a mechanical device which is used to engage/disengage the driven shaft to/from drivershaft without stopping the prime mover.Properties of friction lining material used in clutches:(i)(ii)(iii)(iv)(v)High coefficient of frictionHigh wear resistanceHigher conductivityLower coefficient of thermal expansion ( )Good strengthBasic calculationRi Inner radius of clutchR 0 Outer radius of clutchW Total operating forcep Pressure intensityElemental area dA 2 r drdW elemental forcedW p 2 rdrdFf .dW .p.2 rdrdFf Frictional force on elemental areadTf dFf r .p.2 r dr2dTf Torque transmitted by clutch for elemental areaTf Total torque transmitted by clutchTheories of design of clutch:(i) Uniform pressure theory (UPT)(ii) Uniform wear theory (UWT) Kreatryx. All Rights Reserved.www.kreatryx.com

ME Krash (Sample)(A) Uniform pressure theoryPressure remains constant over entire friction platep ConstantR0W p.2 r.dr p R 02 R i2 RipUPT W R 02 R i2 R0R0RiRiTf n p 2 r 2dr n R 03 R i3 22 n W 2 rdr 22 3 R 02 Ri2 R 0 R i WT n WReff where, R eff 332 R0 Ri 2 3 R 0 R i2 n number of frictional surfaces plates (n 1)(B) Uniform wear theory (UWT)Wear uniformly distributed over entire surface area of clutchp r constantwhere, r radius at any sectionR0WUWT p 2 r dr p 2 r R 0 R i RipUWT W2 r R 0 Ri Pressure intensity varies over entire plateAt r ri p pmax [Maximum pressure]r r0 p pmin [Minimum pressure]R0R0Tf n p 2 r dr n Ri2Ri R Ri n W RW 2 r 2dr n W 0eff22 r R 0 R i ; R eff R0 Ri2Note: Frictional torque as per uniform pressure theory is more than friction torque by uniformwear theory. That’s why for designing clutches, it is better to use uniform wear theorybecause clutches are used to transmit power by utilising frictional forces. Also pressure isnon-uniformly distributed when clutches are in service so UWT is used. In case of new clutches, UPT is more appropriate but in case of old clutches UWT is moreappropriate. In friction clutches, UWT should be considered.For multiplate disc clutch, n n1 n2 1where n1 discs on driving shaftn2 discs on driven shaft Kreatryx. All Rights Reserved.www.kreatryx.com

ME Krash (Sample)Industrial EngineeringSequencing Aim of sequencing is to find the order in which different number of jobs are to beproceeded on different machines, so that the idle time can be minimized and utilizationmay be optimized.Job Flow time: It is the time from some starting point until that particular job iscompleted.Make Span time (MST): It is the time from when processing begins on first job in the setuntil the last job is completed.Tardiness: It is the amount of time by which a job is delayed beyond its due date.Tardiness Job Flow time – Due DateTotal job flow timeAverage Number of jobs in the system Make span timeSequencing Rule Shortest processing time (SPT) Jobs are arranged in increasing order of theirprocessing time. Earliest due date (EDD): Jobs are arranged in increasing order of due dates. Critical Ratio Rule (CRR):Due dateCritical Ratio processing time Jobs are arranged in increasing order of critical ratio.Slack time remaining (STR):STR Due date – Processing timeJobs are arranged in increasing order of their Slack time remaining.Sequencing of N-jobs on two machines: For sequencing, Johnson’s rule is applied.Ex:Machine – 1

(c) Reversible isothermal heat rejection (d) Reversible adiabatic compression It states that of all the heat engine operating between constant source and sink temperature, none has higher efficiency than a reversible engine. The efficiency of a reversible engine is independent of the nature or the amount of the working substance undergoing the .