## Short Note On Machine Design

Published in: Mechanical | Science
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Short Note On Machine Design With Formulas.

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Yielding occurs when the state of stress is at the boundary of the rectangle. Maximum Principal Strain Theory (St. Venant's theory): If El and €2 are maximum and minimum principal strains corresponding to 01 and 02, in the 01 V02 02 2 VOI limiting case: Eel = 01 Eh 02 > 01 Boundary of a yield surface in Maximum Strain Energy Theory is given below 02 01=oo+V02
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Maximum Shear Stress Theory (Tresca Theory): At the tensile yield point 03 = O and thus maximum shear stress is oy/2. Yield surface corresponding to maximum shear stress theory in biaxial stress situation is given below . Maximum strain energy theory ( Beltrami's theory): Failure would occur when the total strain energy absorbed at a point per unit volume exceeds the strain energy absorbed per unit volume at the tensile yield point. (OIEI + + + 02 + 03 2 —21) (0102 + 0203 + 0301)
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0102 Above equation results in Elliptical yield surface which can be viewed as: 02 E(l+v) Distortion energy theory (Von Mises yield criterion): Yielding would occur when total distortion energy absorbed per unit volume due to applied loads exceeds the distortion energy absorbed per unit volume at the tensile yield point. Total strain energy ET and strain energy for volume change Ev can be given as: +02€2 + and E —cave av Ed = ET- Ev 01 +02 +03 At the tensile yield point, 01 = oy , 02 = 03 = 0 which gives, 6E Y —0102 —0203 —0361 The failure criterion is thus obtained by equating Ed and Edy , which gives (01 —02 +(02 —03 + 0 01) 2 = 202 In a 2-D situation if 03 = O, so the equation reduces to,
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01 o 02 o This is an equation of ellipse and yield equation is an ellipse. This theory is widely accepted for ductile materials 450 0.577 Cotter and Knuckle Joints A cotter joint is a temporary fastening and is used to connect rigidly two co-axial road or bars which are subjected to axial tensile or compressive forces. Socket and Spigot Cotter Joints In a socket and spigot cotter joint, one end of the rods is provided with a socket type of end as shown in figure and the other end of the rod is inserted into a socket. The end of the rod which goes into a socket is also called spigot. Failures in Socket and Spigot Cotter Joints
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Failure Cases Failure of the rod in tension Failure of spigot in tension across the weakest section Failure of the rod or cotter in crushing Failure of the socket in tension across the slot Failure of cotter in shear Failure of the socket collar in crushing Failure of rod end in shear Failure of spigot collar in crushing Failure of the spigot collar •n shearing Failures in Sleeve and Cotter Joints Failure Cases Failure of the rod in tension Failure of rod in tension across the weakest section Failure of the rod or cotter in crushing Failure of sleeve in tension across the slot Failure of cotter in shear Failure of rod end in shear Failure of sleeve end in shearing Tensile Force (d2)2 dzxt ct = d. x txct P (dl)2 (d2)21— (dl P = 2b x tx T P 2(d4 — d2)cx T P = 2ax d: x P (d: (d2)21 c, Tensile Force P 4 (d2)2 d: x t q 4 d:xtxct P P=2bxtxr 2a x (12 x r P 2(dl — d2)cxr P d2)t ct €12) t
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o o o Endurance limit Stress concentration Surface characteristics -filigu.e s Endurance limit IJigb cycle føcigve 1 life 1 1 1 1 1 1 1 106 Riveted joints: • There are two basic components of riveted joints: o Rivets o Two or more plates. • The popular materials for the rivets are: Steel, Brass, Aluminium & Copper as per the requirement of the application for fluid tight joints the steel rivets are used Welded Joints It is a permanent joint. When the two parts are joined by heating to a suitable temperature with or without application of pressure. Welding Processes Fusion Welding Thermit Welding Gas Welding Electric Arc Welding Forge Welding Types of Welded Joints:
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Lap Joint or Fillet Joint In lap joint, overlapping the plate and welding the edge of the plates takes place in welding process. The strength of different types of fillet joint can be given according to their welding process as Shear strength in parallel fillet weld, or P— 0.707 hi where, P = Tensile force on the plates h = Leg of the weld I = Length of the weld = Permissible shear stress For double parallel fillet weld, 0.707h/r P = 1.414 h/r Strength of Transverse Fillet Weld o p = Throat area Allowable tensile stress = 0.707 s x 1 o For double transverse fillet joint P = 1.414 hiat Special Cases of Fillet Welded Joint Circular Fillet Weld Subjected to Torsion o Shear stress where, T = Torque acting or rod h = size of weld t = Throat thickness :rtd2 ear or 2.S3T IZhd2
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Circular Fillet Weld Subjected to Bending Moment Bending stress: xtd2 (mgt) Long Fillet Weld Subjected to Torsion t12 Shear stress: o Butt Joint Strength of Butt Joint For single V— butt joint, For double V-butt joint, Eccentric Loaded Welded Joints 5.66 if xhd2 4.242T P = (ti + t2)lxot When the shear and bending stresses are simultaneously present in a joint. Maximum normal stress Maximum shear stress Direct or primary shear stress 2 1 2 1 2
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Strength of Bolted Joint • Maximum tensile stress in the bolt where, dc = Core diameter Torque Requirement for Bolt Tightening P: cly where, Pi = Pretension in bolt, drn - 0.9 d d = Nominal diameter For ISO metric screw thread = 30 For ISO metric a = 250 Load Throat area 1.414 4 4 2tl g sec 8 — tm Q 'I—g sec 9 tan Q Eccentric Load on Bracket with Circular Base If there are n number of bolts, then load in a bolt 2wL (R r cos a) n (2R2+ r: ) In above case when n = 4
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Maximum load in bolt where, cos a = -1 Factor of Safety (FOS) in Bolted Joints It is defined as the ratio of failure stress to allowable stress. Failure strss FOS = Allowable stress For ductile material, For brittle material, where, Syt = Yield strength of component material w. L (R— a cos a) 2 (2R2+ r2) FOS= FOS= sut Sut = Ultimate tensile stress of components material o = Allowable stress. Stress concentration Factor It is defined as the ratio of highest value of actual stress near discontinuity to nominal stress obtained by elementary equations for minimum cross-section. It is denoted by kt. Highest value of actual strss near discontinuitv Nomin al stress obtained by equations or k: where, 00, To - - Nominal stresses
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The magnitude of stress concentration factor depends upon the geometry of the component. In this case, 1+2(a/b) a Design against fluctuating load where, a = Semi-axis of ellipse perpendicular to the direction of load b = Semi-axis of ellipse parallel to the direction of load If b = 0 then, hole is like as very sharp crack then, kt=oo If a = b then, hole becomes a circular hole then, = 1+2=3 Fluctuating Load It is defined as the load, of which magnitude and direction both changes with respect to time. o O Time Time (a) Fluctuating load o max (b) Repeated load cm = Mass strss, = Stress amplitude Time
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Mass stress and stress amplitude c min C For repeated stress, For reversed stress, and max max max Gears Gear can be defined as the mechanical element used for transmitted power and rotary motion from one shaft to another by means of progressive engagement of projections called teeth. Classification of Gears Spur Gear Helical Gear Bevel Gear Worm Gear Spur Gear In spur gears, teeth are cut parallel to axis of the gear. Circular pitch Rd Diametrical pitch Module
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Torque transmitted by gear 60 x IOS (k-rr) Aft = Dynamic load or incremental dynamic load 21v(ceb+R) 21 V —9 Where, v = Pitch line velocity c = Deformation factor b = Face width of tooth Pt = Tangential force due to rated torque. e = Sum of errors between two meshing teeth • Estimation of module based on beam strength 60 x IOS (hv)c: (f:) z ncv m 3 Where, = Service factor, = Velocity factor fs = Factor of safety, n = Speed (rpm) • Estimation of module based on wear strength 60 x IOS (kw)c: (f:) b z-pnpcv Helical Gear • The teeth of helical gear cut in the form of helix or an angle on the pitch cylinder.
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cos V Where, Pn = Normal diametrical pitch P = Transverse diametrical pitch W = Helix angle mn = m cos q.) mn = Normal module m = transverse module Axial pitch tan V Pitch circular diameter cos V Tooth proportions O Addendum ha = mn o Dedendum hf= 1.25 mn o Clearance C = 0.25 mn Addendum circle diameter da = d + 2ha or + 2mn COS V Dedendum circle diameter + 2.5mn cosv Component of tooth forces
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tan ,a,: cos V Pa = Pt tan V Beam strength of helical gear Where, m = Module, 0b = Permissible bending stress y = Lawis form factor • Dynamic load or incremental dynamic load Pd 21v (ceb cos-v+ P:) cos V 21v+ (ceb cos2 V + P:) Where, e = Sum of errors, C = Deformation factor Wear strength of helical gear Herringbone Gear bQ cos- V In order to avoid an axial thrust on the shaft and the bearings, the double helical gears or Herringbone gears are used. Bevel Gears Use to transmit power between two intersecting shafts. High speed high power transmission. Classification of Bevel Gear • Mitre Gear: When two bevel gears are mounted on shafts that are intersecting at right angle. Crown Gear: In pair of bevel gear, when one of the gear has a pitch angle of 900. Internal Bevel Gear: When the teeth of bevel gear are cut on the inside of the pitch. Skew Bevel Gear: Mounted on non-parallel and non-intersecting shafts. It constant of straight teeth.
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Hypoid Gear: Similar to skew bevel gear, non-parallel and non-intersecting shafts. It consists of curved teeth. Zerol Gear: Sprial bevel gear with zero spiral angle. Force Gear: Consists of a spur or helical pinion meshing with a conjugate gear or disk form. Beam strength of bevel gear 1-2 Where, bevel factor. Wear strength of bevel gears K = 0.16 Where, K = Material constant, Bearing —mbqr | —2- 0.75 b Q DpK cos 7 BEV - 100 A bearing is a mechanical element that permits relative motion between two components or parts, such as the shaft and housing, with minimum friction. Plain Bearings (Sliding Contact Bearings) A plain bearing is any bearing that works by sliding action, with or without lubricant. This group encompasses essentially all types other than rolling-element bearings.l Journal or Sleeve Bearings • These are cylindrical or ring-shaped bearings designed to carry radial loads. • The simplest and most widely used types of sleeve bearings are cast-bronze and porous- bronze (powdered-metal) cylindrical bearings. Thrust Bearings • This type of bearing differs from a sleeve bearing in that loads are supported axially rather than radially which is shown in the following figure. Thin, disk like thrust bearings are called thrust washers. Bearing Materials Babbitts
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• Two-directional, groove-race bearings consist of two stationary races, one rotating race, and two ball complements. Roller Bearing (Rolling Contact Bearings) • The principal types of roller bearings are cylindrical, needle, tapered, and spherical. • They have higher load capacities than ball bearings of the same size and are widely used in heavy-duty, moderate-speed applications.. Cylindrical Bearings Cylindrical roller bearings have high radial capacity and provide accurate guidance to the rollers. Their low friction permits operation at high speed, and thrust loads of some magnitude can be carried through the flange-roller end contacts. Needle Bearings Needle bearings are roller bearings with rollers that have high length-to-diameter ratios. Compared with other roller bearings, needle bearings have much smaller rollers for a given bore size. Loose-needle bearings are simply a full complement of needles in the annular space between two hardened machine components, which form the bearing raceways. They provide an effective and inexpensive bearing assembly with moderate speed capability, but they are sensitive to misalignment. Caged assemblies are simply a roller complement with a retainer, placed between two hardened machine elements that act as raceways. Their speed capability is about 3 times higher than that of loose-needle bearings, but the smaller complement of needles reduces load capacity for the caged assemblies. Thrust bearings are caged bearings with rollers assembled like the spokes of a wheel in a wafer like retainer. Tapered Bearings • Tapered roller bearings are widely used in roll-neck applications in rolling mills, transmissions, gear reducers, geared shafting, steering mechanisms, and machine-tool spindles. Where speeds are low, grease lubrication suffices, but high speeds demand oil lubrication, and very high speeds demand special lubricating arrangements. Spherical Bearings Spherical roller bearings offer an unequaled combination of high load capacity, high tolerance to shock loads, and self-aligning ability, but they are speed-limited. Single-row bearings are the most widely used tapered roller bearings. They have a high radial capacity and a thrust capacity about 60 percent of radial capacity. Two-row bearings can replace two single-row bearings mounted back-to-back or face-to- face when the required capacity exceeds that of a single-row bearing. Brake A brake is a device by means of which artificial frictional resistance is applied to a moving machine member, in order to retard or shop the motion of a machine.
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• The most commonly brakes use friction to convert kinetic energy into beat, though other methods of energy conversion may be employed. Single Block or Shoe Brake It consists of a block or shoe which is passed against the rim of revolving brake wheel drum. The block is made of a softer material than the rim of the wheel. If the angle of contact is less than 600 then, it may be assumed that normal pressure or force between the block and the wheel is uniform. 20 Wheel A schematic diagram of shoe brake Case l: When the line of action of tangential braking force passes through the fulcrum O of the lever. If wheel is rotating in clockwise direction then, Free Body Diagram (FBD) of wheel and block is Wheel is rotating anti-clockwise direction If wheel is rotating in anticlockwise direction then, FBD of wheel and block is Wheel is rotating anti-clockwise direction
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Braking force PPI Braking torque gPlr When wheel is rotating in anticlockwise direction then, the braking torque is same as above gP1r Case Il: When the line of acting of the tangential braking force (h) passes through a distance a below the fulcrum O. Then, there are two cases: For Clockwise: Braking force Braking torque For Anti Clockwise: Braking force 1 O a PPI x+ ga gPlr x + ga RN FBD of block API x— ga
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Braking torque x— ga (as r) FBD of block Case Ill: When the line of action of tangential braking force (h) passes through a distance 'a' above the fulcrum O. For clockwise, Braking force Braking torque For Anti- Clockwise Braking force API x— ga gP1r x— ga FBD of block FBD of block
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API 1+ ga Braking torque "Pir x— ga : Fl o, RN FBD of block When the frictional force helps to apply the brakes then, such type of brakes are said to self- energizing brakes. When P is negative or equal to zero then, these are known as self-locking brakes. Simple Band Brake A band brake consists of a flexible band of leather, one or more ropes, or steel lined with friction material, which embraces a part of the circumference of the drum is called simple band brake. We know, (9 = 3600 9') = g (3600 9') or 2.3102 — Braking force on the drum = (Tl — T2) Braking torque on the drum (TB) = (Tl - When wheel rotates in the clockwise direction and taking moment about fulcrum O b
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For anticlockwise rotation of the drum PI = T2b b where, b = Perpendicular distance from O to the line of action Tl or T2 I = Length of the lever from the fulcrum Tl = ot wt w = Width of the band t = Thickness of the band ot = Permissible stress in the band. Clutch A clutch is a mechanical device that provides for the transmission of power (and therefore usually motion) from one component (the driving member) to another (the driven member) when engaged, but can be disengaged. Friction Clutch • The friction clutch is used to transmit power of shafts and machines which must be started and stopped frequently. Friction surfaces of a clutch remain in contact to each other by applying an axial thrust or load w. Considering Uniform Pressure The uniform pressure p can be evaluated as. x(h2— r?) Total frictional torque given in this case, 2 3 where, Rm = Mean radius of friction surfaces 2 3 mw RE
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Considering Uniform Wear • Total frictional torque acting on clutch where Mean radius of friction surfaces gwRm 2 In uniform wear theory, Maximum pressure acts at the inner radius and minimum pressure acts at the outer radius. Average pressure on the friction surfaces Pav z(h2— r?) Multiple Disc Clutch • Number of pairs of contact surfaces -1 n = ni + r12 where, m = Number of discs on the driving shaft = Number of discs on the driven shaft • Total frictional torque acting on the frictional surface where, 2 3 1 [in case of uniform pressure] 2 [in case of uniform wear] where, rl and are outer and inner radii of the friction plates.
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Cone Clutch In cone clutch, driver is keyed to the driving shaft by a sunk key and has an inside conical surface or face which exactly fits into the outside conical surface of the driven. Total torque on the clutch, T gw cosec 1 (for uniform pressure) €— gw cosec h+r ) (for uniform wear) a = Semi angle of cone or face angle of the cone w w sin a b r2 Cone clutch where, w = Axial load or thrust Axial force required for engaging the clutch, = wn(sin a + g cos a) Axial force required to disengaged the clutch Wd = wn (g cos a — sin a) If face width b and mean radius of cone clutch is Rm. Then, 2
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, T = 2ng.P%Rmb 2rRm bsin Q 2mVT 60 • Power transmitted by clutch Centrifugal Clutch Centrifugal force acting on each shoe at running speed pc = m w2r Where, 60 Friction force acting on each shoe = g(Pc —PJ The direction of force is perpendicular to the radius of the rim pulley. Frictional torque on each shoe = x R • Total torque transmitted = Number of shoes g(Pc— Ps)R = - Ps)R Arc = Angle (in radian) x Radiua I = OR Where, area of contact = 1b Force exerted on each shoe = plb pc — Ibp Where, I = Contact length of the shoe b = Width of the shoe p = Pressure intensity on shoe q = Angle made by shoe at the centre of spider in radian R = Contact radius of shoe = inside radius of the rim of the pulley

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