Shafts & Axles: State of Stress, Engineering & Checking PDF

Gépelemek 1. SHAFTS & AXLES STATE OF STRESS, ENGINEERING & CHECKING Authors: Dr. Kerényi György Molnár László, Dr. Marosfalvi János, Dr. Horák Péter, & Dr. Baka Ernő Tengelyek | GÉPELEMEK 1. előadás 1 Shafts or axles Gépelemek 1. Tengelyek | GÉPELEMEK 1. előadás 2 Shafts or axles Gépeleme...

Gépelemek 1. SHAFTS & AXLES STATE OF STRESS, ENGINEERING & CHECKING Authors: Dr. Kerényi György Molnár László, Dr. Marosfalvi János, Dr. Horák Péter, & Dr. Baka Ernő Tengelyek | GÉPELEMEK 1. előadás 1 Shafts or axles Gépelemek 1. Tengelyek | GÉPELEMEK 1. előadás 2 Shafts or axles Gépelemek 1. Tengelyek | GÉPELEMEK 1. előadás 3 Shafts, axles types, grouping Gépelemek 1. A shaft is a rotating machine element, usually circular in cross-section, which is used to transmit power from one part to another, or from a machine that produces power to a machine that absorbs power. • Axles • Non-rotating element (wheels, clutches, on axleseg. automotive) • Rotating-carrying element (rotates together with on-mounted parts) • Shafts • The task of the connecting shafts is to carry the rotating structural elements as well as to transmit torque. The torque is produced by gear, worm, sprocket, pulley, rope pulley, etc. Tengelyek | GÉPELEMEK 1. előadás 4 Axles & shafts in general Gépelemek 1. Shaft Axle It is a rotating member. It is a non-rotating member. Its primary function is to transmit power. Its primary function is to provide support to parts like a wheel, pulley, drum, etc. It does not transmit power. It is subject to primary bending and torque. It is subject to bending moment primarily due to transverse load. Its design is more complex than that of the axle. Its design is simpler than that of the shaft. The cross-section of the shaft is usually circular The cross-section of the axle may be different because it gives minimum vibration and stress. such as rectangular, square, circular, etc. They are used for electric motors, I.C. Engine shaft, gear, etc. They are used in cars, trucks, railway buggies, etc. They may or may not be under normal load. It is fitted into the housing by means of bearings. The shaft is basically divided into two types, the The axle is basically divided into three types, the first is the transmission shaft and the second is first is the front axle, the second is the rear axle, the machine shaft. and the third is the stub axle. Tengelyek | GÉPELEMEK 1. előadás 5 Grouping Gépelemek 1. Stationary connecting axle (wagon bearing arrangement) Bicycle axle Tengelyek | GÉPELEMEK 1. előadás 6 Grouping Gépelemek 1. Wheel pressure Wheel pressure Automotive beam axles Tengelyek | GÉPELEMEK 1. előadás 7 Grouping Gépelemek 1. Rotating shafts (train bearing arrangement) Tengelyek | GÉPELEMEK 1. előadás 8 Grouping Gépelemek 1. Connecting shafts (gearboxes) Tengelyek | GÉPELEMEK 1. előadás 9 Grouping Gépelemek 1. Special connecting shaft 4-stroke internal combuston engine Crankshaft Tengelyek | GÉPELEMEK 1. előadás 10 Grouping Gépelemek 1. Gearbox shafts: The cross-section of the connecting shafts is a circle or ring. (tubular) Their typical use is bending and twisting. The type of fatigue stress is rotary bending, and general swinging twisting may also occur. Tengelyek | GÉPELEMEK 1. előadás 11 Shafts, axles & engineering in motion Gépelemek 1. The largest and most powerful series of steam locomotives ever built. Produced from 1941 to 1944 by the American Locomotive Company of Schenectady, N.Y. The Big Boy locomotives had an articulated design; the frame of the front engine was hinge-connected to the rear engine under a single boiler. The wheel arrangement was designated 4-8-8-4—i.e., a set of 4 pilot wheels led a set of 8 coupled driving wheels, which were compounded by another set of 8 coupled drivers, with 4 trailing wheels. LEGO Ideas Tengelyek | GÉPELEMEK 1. előadás 12 Shafts, axles & engineering in motion Gépelemek 1. Tengelyek | GÉPELEMEK 1. előadás 13 Materials of shafts, axles Gépelemek 1. The material of the shafts is most often steel, less often cast steel or cast iron. Under normal operating conditions, the material of the shafts is general purpose structural steel (EN 10025). For places with greater stress, the shafts are made of unalloyed or alloyed steel (EN 10083). The surface of the shaft can be made wear-resistant by using hardened steel (EN 10084). Shafts are almost, without exception, subjected to fatigue stress, so it is reasonable to use technological processes that increase the service life: hardening, flame hardening, induction hardening, nitriding, rolling, shot peening, hard chrome plating, etc. Tengelyek | GÉPELEMEK 1. előadás 14 Pre-planning of axles Gépelemek 1. Main use of them is bending: Mh  = Mh bending moment causes the normal stress: h K Where K in case of solid axle & tubular section D 3 K= 32 ( D 4 − d 4 ) 2 K=  64 D For bending the most advantageous design is (varying cross-sectional beam) Tengelyek | GÉPELEMEK 1. előadás 15 State of stress in axles Gépelemek 1. Carrying axles. Their cross-section is usually round, in special cases I-section or closedsection beam. Their typical use is bending. The nature of the fatiguing stress is pulsating. (Shearing stress also appears, but since it is much smaller than bending, it is usually neglected.) They are in steady state. M [Nm] M [Nm] Tengelyek | GÉPELEMEK 1. előadás 16 Engineering & design of shafts Gépelemek 1. Main use of them is rotary bending & twisting: Their design is the same as the design of axles... Pre-planning: determining the characteristic size (diameter or wall thickness) of the shaft from the allowable stress on the material of the shaft and the stress resulting from bending, assuming a static load; Defining the design: of the shaft in accordance with the installation and operating conditions. Control: to deformation; to fatigue; to critical speed, to reduced static use Tengelyek | GÉPELEMEK 1. előadás 17 State of stress in axles & shafts Gépelemek 1. Rotating shafts: Their cross-section is almost without exception a circle or ring. (tubular) The nature of the fatiguing stress is rotary folding bending (which means tensile and compressive stress is generated in the extreme fiber of the crosssection). M [Nm] M [Nm] Tengelyek | GÉPELEMEK 1. előadás 18 Extreme fiber in beams Gépelemek 1. Tengelyek | GÉPELEMEK 1. előadás 19 Engineering & design of shafts Gépelemek 1. Tengelyek | GÉPELEMEK 1. előadás 20 Engineering & design of shafts Gépelemek 1. Pre-planning for staic twisting:  cs = Mcs twisting torque causes Tau state of stress: where Kp at solid shaft M cs Kp at tubular section D 3 Kp = 16 ( D 4 − d 4 ) 2 K=  32 D   where D is the diameter of solid shaft, and outer diameter of the tube, and d is the innner diameter of tube.. cs meg ...and comparison with: where τmeg a is the allowable stress value for material of the shaft. Tengelyek | GÉPELEMEK 1. előadás 21 Engineering & design of shafts Gépelemek 1. Checking for complex static stress: From left figure: an external point load acting on the shaft in a general direction (e.g. in the case of a helical gear) has three components: radial (Fr), circumferential (Fk), and axial (Fa) 1. Determination of the reaction forces occurring in the supports based on the principle of superposition. Tengelyek | GÉPELEMEK 1. előadás 22 Engineering & design of shafts Gépelemek 1. 2. Determination of the bending stress of the shaft, based on the principle of superposition: the torque diagrams are shown in two mutually perpendicular planes. 3. Determination of the resultant of the bending moment: Mh = ( M hr + M ha ) + M 2hk 2 Tengelyek | GÉPELEMEK 1. előadás 23 Engineering & design of shafts Gépelemek 1. 4. Determination of the normal stress (σh) arising in the dangerous crosssection (possibly also taking into account the stress from the axial load). (The location of the dangerous cross-section is where the greatest tension arises, which does not necessarily coincide with the location of the biggest moment.) 5. Determination of the torque and shear stress (τcs) resulting from the circumferential force Fk. Tengelyek | GÉPELEMEK 1. előadás 24 Engineering & design of shafts Gépelemek 1. 6. /a. In case of tough shaft materials, the reduced stress is calculated according to the Huber – Mises – Hencky theory:  red =  h2 + 3 cs2 6/b. In case of cast iron shafts, the reduced stress is calculated according to the relationship valid for brittle materials: 3 8  red =  h + 5  h2 + 4 cs2 8 7. Comparison of reduced and allowable stress: where  red meg Re H = z and z = 1,5 . . 2,0 is the safety factor.  red   red meg Tengelyek | GÉPELEMEK 1. előadás 25 Engineering & design of shafts Gépelemek 1. Checking for deformation: In some cases, the extent of flexible bending and twisting of shafts can be important from operation point of view. E.g.: - in case of gear connections, bending of the shaft results an incorrect meshing; - in the case of electric motors, pump or turbine impellers, the deformation of the rotor affects the size of the radial gap; - also affects the operation and service life of shaft seals and bearings There are no general regulations for the allowable elastic deformation of the shafts. In mechanical engineering, the permissible value of the usual elastic deflection or sagging is: where l a is the distance between bearings... f  0,00035l Tengelyek | GÉPELEMEK 1. előadás 26 Engineering & design of shafts Gépelemek 1. Another condition is to limit the elastic deformations: the angular rotation of the cross-sections under the bearings.   0, 05 which does not effect the regular operation of the bearings... The below specification for the degree of twisting: []  0,005l[mm] where [] is the twisted angle in degrees, and l is the lenght of shaft in [mm]. The theories and methods learned in elasticity studies can be used in here to determine elastic deformations! Tengelyek | GÉPELEMEK 1. előadás 27 Engineering & design of shafts Gépelemek 1. Checking for critical speed: Due to the flexibility, rotors are oscillating systems which are excited by centrifugal forces and/or external force effects and torque fluctuations. If the oscillation frequency of the excitation effect is the same as the natural frequency of the shaft, the phenomenon of resonance occurs. The speed which equals with the natural frequency is called the critical speed. The operating speed must not be in the vicinity of the critical speed, because the strong vibrations which occur can damage the equipment and even lead to the breakage of individual structural elements. Excitations & deformations which can occur from: • bending swing and/or • torsion swing as well as the corresponding critical speeds. Tengelyek | GÉPELEMEK 1. előadás 28 Engineering & design of shafts Gépelemek 1. Checking for bending swings W hen a shaft is turning, eccentricity causes a centrifugal force deflection, which is resisted by the shaft’s flexural rigidity. As long as deflections are small, no harm is done. m The center of mass S of the disc mounted on the shaft is located at e distance from the axis of rotation. If the shaft rotates with an angular velocity ω, the elastic deflection of the center line of the shaft due to the centrifugal force is y. Balance of elastic retrieving force and centrifugal force : e sy = m( y + e) 2 S y where s bending stifness of shaft, m mass of disc. ...from the equation: m 2 y= e 2 s − m Tengelyek | GÉPELEMEK 1. előadás 29 Engineering & design of shafts Gépelemek 1. If denominator would be zero, s − m = 0 then y =  deflection can be... (this phenomenom can not be possible because the shaft breaks before infinity!!!) The angular velocity at which the elastic deformation is infinite called the critical angular (velocity) speed. 2 k = s m The dynamic behavior of the shaft is characterized by the extent to which, this eccentricity before the rotation, increased due to the rotation...so y + e ...how big is the magnification factor.  m 2 e e+e y + e s − m 2 = = = ...substituted & arranged: e e Tengelyek | GÉPELEMEK 1. 1  1 −   k    előadás 30 2 Resonance trasmittibility Gépelemek 1. The change in the absolute value of the magnification factor versus /k gives the resonance curve. overtuned range undertuned range /k < 1.41 is the so-called overtuned range, the range /k > 1.41 is the socalled undertuned range. In the range /k > 1.41, the shaft automatically adjusts to the central position and runs more calmly than in the overtuned range. Tengelyek | GÉPELEMEK 1. előadás 31 Rotor experiment Gépelemek 1. Tengelyek | GÉPELEMEK 1. előadás 32 Stiffness of shafts (deflections) Gépelemek 1. The bending stiffness s can be determined from the weight force of the disc (mounted on the shaft) and the elastic deflection corresponding to the weight force. arrangement formula ha a = b G s= f The most frequent cases are summarized in the left table for prismatic beams. Tengelyek | GÉPELEMEK 1. előadás 33 Stiffness of a stepped shaft (one-mass oscillating system) Gépelemek 1. F=1N d i s1 s2 ... Mi si .. s n-1 sn M i+1 Way of calculation: 1. A unit force 1 [N] is applied at the location of the disk. 2. We draw the bending moment diagram. Tengelyek | GÉPELEMEK 1. előadás 34 Stiffness of a stepped shaft (one-mass oscillating system) Gépelemek 1. 3. We divide the shaft into sections as shown in the previous drawing. 4. Then calculate the value of... ( 1 n xi c= M i2 + M i2+1 + M i M i +1  3E i =1 I i ) expression, where xi lenght of given section, M i , M i+1 bending moment at the certain section boundary, Ii moment of inertia od the section, E Elastic modulus of the shaft. 5. In the end the stiffness of the shaft is: 1 s= c Tengelyek | GÉPELEMEK 1. előadás 35 Two-mass oscillating system Gépelemek 1. Swinging of a two mass system The first critical speed of a shaft with more than one concentated masses (as depicted above) can be calculated with the Dunkerley-approximation method. Tengelyek | GÉPELEMEK 1. előadás 36 Engineering & design of shafts Gépelemek 1. Torsion swinging: The natural frequency of torsional oscillations in case of a single-mass oscillating system can be calculated as follows: st tk =  where st is the torsional stiffness of the shaft, and  is the torsional moment of inertia of an impeller (mounted on the shaft). In the case of two rotating masses, a reduced moment of inertia must be formed to use the above relationship: 1 1 1 = +   1 2 As well as... 1 2 = 1 +  2 Tengelyek | GÉPELEMEK 1. előadás 37 Balancing of rotors Gépelemek 1. Static imbalance Dynamic imbalance Tengelyek | GÉPELEMEK 1. előadás 38 Recommended residual imbalance (VDI Verein Deutscher Ingenieure) Gépelemek 1. Rotor quality class Type of the rotor assembly Velocity of centre of mass [mm/s] G40 Car wheels, rims, passenger and truck crankshafts. 40 G16 Propeller shafts, cardan shafts, rotating parts of agricultural machines 16 G6,3 Centrifuge drums, fans, pump impellers, flywheels, 6,3 electric machine fittings. G2,5 Gas and steam turbines, turbo generators, machine tools, small electric motors. 2,5 Gl Driving tape recorders, CD players, grinding machines, special small electric motors. 1 G0,4 Precision machine tools, fine grinding machines, spinners. 0,4 Tengelyek | GÉPELEMEK 1. előadás 39