Gear Noise and Vibration Episode 2 Part 4 pot

200 Chapter 11 Reference 1. Romax Ltd., 67 Newgate, Newark NG24 1HD. www.romaxtech.com 12 Planetary and Split Drives 12.1 Design philosophies The conventional parallel shaft gear drive works well for most purposes and is easily the most economical method of reducing speeds and increasing torques (or vice versa). The approach starts running into problems when size and weight are critical or when wheels start to become too large for easy manufacture. If we take the torques of the order of 1 MN m (750000 Ibf ft) that are needed for 6000 kW (8000 HP) at 60 rpm we can estimate the wheel size for a 5 to 1 final reduction. The standard rule of thumb allows us about 100 N mm" 1 per mm module so assuming 20 mm module (1.25 DP) gives us a wheel face width of about 450 mm and diameter of 2.25 m. This is not a problem but if the torque increases we rapidly reach the point where sizes are too large for manufacture and satisfactory heat treatment especially as the carburised case required thickness also increases. The solution is to split the power between two pinions so that loadings per unit facewidth remain the same but the torque is doubled. The further stage in this approach is to split the power between four pinions to give roughly quadruple increase in torque without significant increase in size. This fits in well if there is a double turbine power drive which is often wanted for reliability. The design is as sketched in Fig. 12.1. Power comes in via the two pinions labelled IP, splits four ways to the four intermediate wheels (IW) which in turn drive the four final pinions which mesh with the final bull wheel. The resulting design is accessible and reasonably compact though at the expense of extra complexity in shafts and bearings. To achieve the gains desired with power splitting it is absolutely essential that equal power flows through each mesh in parallel so as there are inevitable manufacturing tolerances, eccentricities and casing distortions some form of load sharing is needed. This is usually conveniently and easily provided by having the drive shafts between intermediate wheels and final pinions acting as relatively soft torsional springs. If the accumulated position errors at a mesh sum to 100 um and we do not want the load on a given pinion to vary more than 10% the torsional shaft flexibility must allow at least 1 mm flexure under load. 201 202 Chapter 12 intermediate wheel intermediate wheel intermediate wheel right input pinion intermediate wheel Fig 12.1 Multiple path high power drive, annulus Fig 12.2 Typical planetary drive showing forces on planets. Planetary and Split Drives 203 The logical extension of the multiple path principle is the planetary gear as sketched in Fig. 12.2 where to reduce size (and weight) further the final drive pinions are moved inside the wheel which becomes an annular gear. The further asset of the planetary approach is that a single sun gear can drive all the planets and with 3 planets the reduction ratio can be as high as 10 : 1. Planetary designs give the most compact and lightest possible drives and well designed ones can be a tenth the size and weight of a conventional drive. There is a corresponding penalty in terms of complexity and restricted access to the components. High performance is again dependant on having equal load sharing but this cannot be achieved by torsion bar drives and so there are many "best" patented systems for introducing load sharing. The simplest is to allow the sun wheel to float freely in space so that any variations in meshing can be taken up by lateral movements of the sun. More commonly in high power drives especially as designed by Stoeklicht, the annulus, which is relatively thin, is designed to flex to accommodate variations. A third variant deliberately designs the planet supporting pins to be flexible to absorb any manufacturing variations. Pedantically the term "planetary gear" is used to describe all such gears whereas the more commonly used "epicyclic" is only correct for a stationary annulus and if the planet carrier is stationary it is a star gear. When a gear is used in an infinitely variable drive as a method of adding speeds then all three, sun, annulus and planet carrier are rotating. 12.2 Advantages and disadvantages The advantages of splitting the power are mentioned above in terms of reduction of weight and size and frontal area (for aeroplanes and water turbines) and the corresponding disadvantages of increased complexity and, in the case of planetary gears, poor accessibility. Additional factors can be the problems of bearing capabilities since as designs are scaled up the mesh forces and hence the bearing loads tend to rise proportional to size squared whereas the capacity of rolling bearings goes up more slowly and the permitted speeds decrease. This imposes a double crimp on design and forces designers towards the use of plain bearings with their additional complications. Splitting power delays the changeover from rolling bearings to plain bearings for the pinions and as the pinions can be spread around the wheel the wheel bearing loads can be reduced or in the case of planetary gears the loads from the planets balance for annulus and planet carrier completely. The planet gears are very inaccessible and are highly loaded so they present the most difficult problems in cooling. For high power gears it is 204 Chapter 12 normal to have the planet carrier stationary as this makes introducing the large quantities of cooling oil required much easier. There would appear to be no obvious limit to power splitting but in an external drive it is complex to arrange to have more than four pinions and even this requires two input drives. Planetary gears can have more than three planets and five are occasionally used. However load sharing is still needed and, as the system is redundant, cannot be achieved by floating the sun so either the planet pins must be flexible or the annulus must flex. There is the additional restriction that with five planets the maximum reduction (or speed increasing) ratio is limited by the geometry to slightly less than five if used as a star gear or five if an epicyclic. Design problems can arise with heavily loaded planets because with most designs it is necessary to support the outboard ends of the planet pins and the space available between the planets for support structure is very limited as can be seen in Fig. 12.3. Fig 12.3 Maximum reduction with five planets. Planetary and Split Drives 205 Care must also be taken that the planet carrier is rigid so that the outboard support members are not allowed to pivot at their base when under load. Planetary gears automatically have input and output coaxial which can be either an advantage or disadvantage according to the installation. The fact that the reaction at the fixed member, whether annulus or carrier, is purely torsional can be a great advantage for vibration isolation purposes as a very soft torsional restraint can be used to give good isolation without fear of misalignment problems. 12.3 Excitation phasing If we have three, four or five meshes running in parallel there will be the corresponding number of T.E. excitations forcing the gear system and attempting to produce vibrations to cause trouble. It is easiest to consider a particular case such as the common three planet star drive and to make the assumption that the design is conventional with the three planets spaced exactly equally and that spur gears are used. If we then look at the vibrating forces on the sun we have the three forces as shown in Fig. 12.4, spaced at 120° round the sun and inclined at the pressure angle to the tangents. Fig 12.4 Sun to planet force directions. 206 Chapter 12 The three meshes will probably have roughly the same levels of T.E. and so the same vibration excitation and will have the same phasing of the vibration relative to each pitch contact. The three pitch contacts can be phased differently according to the number of teeth on the sun wheel. If the number of teeth on the sun is divisible by three the three meshes will contact at the pitch point simultaneously and the three excitations will be in phase. This will give a strong torsional excitation to the sun but no net sideways forcing. If not, the three excitations will be phased 120° of tooth frequency apart in time and at 120° in direction so there will be no net torsional vibration excitation on the sunwheel but a vibrating force which is constant in amplitude and whose direction rotates at tooth frequency. The direction of rotation is controlled by whether the number of teeth is 1 more or 1 less than exactly divisible by 3. The same considerations apply for the three mesh contacts between the planets and the annulus. Dependent on whether the number of annulus teeth is exactly divisible by three or not we can choose to have predominantly torsional vibration or a rotating lateral vibration excitation. 50 40 i 30 20 -20; -30 ! -40 -50 -60 -40 -20 Fig 12.5 Nine-tooth gear layout showing how contact occurs at pitch points at roughly the same time. Planetary and Split Drives 207 When there are five planets there are similar choices as to whether the excitations are phased or not to give predominantly torsional vibration or lateral vibration. The choice should depend on whether the installation is more sensitive to torsional or lateral problems. Similar considerations apply for the planets where the 2 meshing excitations on a planet can either be chosen to be in phase or out of phase. The former gives tangential forcing on the planet support, the carrier, while the latter gives rotational forcing on the planet itself which being light can usually rotate easily. As the contact is on the opposite flank it is not immediately obvious whether an odd or even number of teeth is needed on the planet but an odd number of teeth will give simultaneous pitch point contact to sun and annulus and an even number will give 180° phasing and so less torsional excitation on the carrier. Fig. 12.5 shows the rather extreme case of a nine-tooth 25° pressure angle gear which is meshing on both sides as in a double rack drive or as in a planet (though it would not be normal to use less than about eighteen teeth in practice). The pressure lines are shown tangential to the base circle and it can be seen that contact (along the pressure lines) will occur at the (high) pitch points at roughly the same instant in time so there will be low net tangential forces on the planet but sideways forcing on the planet pin. The Matlab program to lay out the pinion is % profile 9 tooth 10 mm module 25 deg press angle % starting from root with radius 5 % base circle 45 cos 25 = 40.784 root centre -5, 40.784 N = 65; % no of points for each flank. xl=zeros(18*N,l); yl=zeros(18*N,l); for i = 1:15 % root circle xl(i) = -5 + 5*cos(1.4488 -(i-l)*0.1); yl(i) - 40.784 - 5*sin(1.4488 -(i-l)*0.1); end fori=16:N; % involute ra = (i-16)*0.02; xl(i)=40.784*(sin(raHi-16)*0.02*cos(ra)); yl(i)=40.784*(cos(ra)+(i-16)*0.02*sin(ra)); end for i=(N+l):2*N ; % Image in x=0 other flank x2(i) = - xl(2*N+l-i); y2(i) - yl(2*N+l-i); rot 1=0.45413; xl(i) =x2(i)*cos(rotl) +y2(i)*sin(rotl); yl(i) = -x2(i)*sin(rotl) +y2(i)*cos(rotl); 208 Chapter 12 end for th = 1:8; % rotate for other 8 teeth xl((th*2*N +l):(th+l)*2*N) = xl(l:2*N)*cos(0.69813*th)+yl(l:2*N)*sin(0.69813*th); yl((th*2*N +l):(th+l)*2*N) =- xl(l:2*N)*sin(0.69813*th)+yl(l:2*N)*cos(0.69813*th); end saveteeth9 xl yl for ang = 1:44 % plot base circle xo(ang) = 40.784*cos(ang*0.15); yo(ang) = 40.784*sin(ang*0.15); end xtl = [17.236 -17.236]; ytl - [36.963 53.037] ; % tangent xt2 = [17.236 -17.236]; yt2 = [-36.963 -53.037] ; % tangent axl = [0 0] ; ax2 = [-54 54]; % vertical axis phi = -0.05 ; % rotate gear to symmetrical position u2 = xl*cos(phi)+yl*sin(phi) ; v2 = -xl*sin(phi)+yl*cos(phi); figure plot(u2,v2, t -k',xo,yo, l -k t ,xtl,ytl,'-k',xt2,yt2, 1 -k',axl,ax2,'-k 1 ) axis([-58 58 -58 58]) axis('equal') 12.4 Excitation frequencies For simple parallel shaft gears it is easy to see what the meshing frequencies will be as they are rotational speed times the number of teeth. In a planetary gear there will be at least two and possibly three out of the sun, planet carrier and annulus rotating so the tooth meshing frequency is less obvious. The simplest case occurs with a star gear as the planets, though rotating are stationary in space. In Fig. 12.6 with S sun teeth, P planet teeth and A annulus teeth, the ratio will be A/S and as 1 rotation of the sun will involve S teeth, the frequency will be S times n where n is the input speed in rev s" 1 . This is the same as A times R where R is the output speed which will be in the opposite direction but this does not alter the meshing frequency. When the planet carrier is rotating then both the sun to planet mesh and the planet to annulus mesh are moving in space so there is not a simple relationship and we must first bring the carrier to rest. As before, with the carrier at rest the tooth frequency will be S times n where n is the input (sun) speed relative to the (stationary) carrier. On top of this we impose a whole body rotation to bring the carrier up to the actual speed and the other gears will also have this speed added but the meshing frequency will not be altered as it is controlled solely by the relative sun to carrier speed. Planetary and Split Drives 209 Rrps Fig 12.6 Sketch of planetary gear for meshing frequencies. The general relation between speeds is determined relative to a 'stationary' carrier. Then with speeds o or (G> S - co c ) / (« a - co c ) = -R co s = (1 +R)oo c - where R = Na / Ns In general, whatever the speed we take the (algebraic) difference between sun and carrier speeds and multiply by the number of teeth on the sun to get the tooth meshing frequency or the corresponding difference between carrier and annulus speeds and multiply by the number of teeth on the annulus. 12.5 T.E. testing Complications arise if the T.E. of a complete planetary or split drive is required because there are several drive paths in parallel under load. If the drive is as sketched in Fig. 12.6 and there is an error in one of the three sun-to-planet meshes, we will not necessarily detect a relative torsional movement between sun and planet. The error may be [...]... have equal amounts of in-phase and 120 ° phase so will sum to 60° phase The next sixth of a rev will be dominated by the vibration from the 2nd planet and so will be 120 ° phase Similarly the next sixth will average 2nd and 3rd planet phases and so be at 180° and the next at 24 0° as the 3rd planet dominates, while the final sixth will average between 24 0° and 360° (or 0°) and so be at 300° phase The next... transmission path between the mesh and the accelerometer The theory is given by McFadden in Ref [1] accelerometer Fig 12. 7 Diagram of five planet epicyclic 21 2 Chapter 12 50 100 150 20 0 25 0 300 350 40 0 degrees rev of carrier Fig 12. 8 Vibration observed at a stationary accelerometer There is simple explanation in the time domain as indicated in Fig 12. 8 which shows the vibration received at the accelerometer... small part of the profile Since we have to achieve two base pitches from changeover to changeover and allow for errors, etc., the nominal contact ratio must be above 2. 0 and in practice about 2. 25 minimum pure involute no load T.E for one pair \\ pair 1 pair 2 pair 3 pair 4 deflected position under full load Fig 13 .2 Geometry of tip relief and deflections for contact ratio of 2 High Contact Ratio Gears... rack 21 5 21 6 Chapter 13 Fig 13.1 shows the limiting case for standard teeth when a very large gear mates with another large gear or a rack A pressure angle of 20 ° and an addendum equal to the module gives the maximum length of approach and recess as m cosec . 5*cos(1 .44 88 -(i-l)*0.1); yl(i) - 40 .7 84 - 5*sin(1 .44 88 -(i-l)*0.1); end fori=16:N; % involute ra = (i-16)*0. 02; xl(i) =40 .7 84* (sin(raHi-16)*0. 02* cos(ra)); yl(i) =40 .7 84* (cos(ra)+(i-16)*0. 02* sin(ra)); end for . i=(N+l) :2* N ; % Image in x=0 other flank x2(i) = - xl (2* N+l-i); y2(i) - yl (2* N+l-i); rot 1=0 .45 413; xl(i) =x2(i)*cos(rotl) +y2(i)*sin(rotl); yl(i) = -x2(i)*sin(rotl) +y2(i)*cos(rotl); 20 8 . lateral vibration excitation. 50 40 i 30 20 -20 ; -30 ! -40 -50 -60 -40 -20 Fig 12. 5 Nine-tooth gear layout showing how contact occurs at pitch points at roughly the same time. Planetary and