Gear Noise and Vibration Episode 2 Part 3 doc

180 Chapter 10 rubber auxiliary mass main mass support stiffnesses I \ original response ampl. response with tuned damped absorber frequency Fig 10.7 Tuned damped vibration absorber response. Although it is possible in theory to use steel springs and oil damping, this is rare due to sealing and tuning problems. The device needs careftil tuning to the correct frequency and is, in general, only worthwhile if the auxiliary mass can be about 10% of the effective mass of the resonance and the original dynamic amplification factor (Q) of the resonance was greater than 8. The absorber can then reduce the Q factor to below 4. Improvements 181 Untuned (Lanchester) dampers which use only mass and viscous damping will work over a range of frequencies but require greater mass and give much less damping so they are little used except for torsional engine vibrations which occur over a wide range of frequencies as speed varies. 10.8 Production control options When trouble strikes and the customer's installation cannot be altered there is a tendency to panic and to halve all drawing tolerances on principle, to make sure that all the gears are being made "better." This is, of course, no help if it is a faulty gear design (or installation) and is very expensive to achieve. On the assumption that development has investigated permissible loaded T.E. and found that it must be kept below, say, 4 um at once-per-tooth, there are several options available. The first is the obvious one to run a model and to see how tolerant the design is to errors of profile, helix and pitch. This should give a good idea of the sensitivity of the design which could decide how tightly manufacturing tolerances should be specified. If these tolerances are not economically sensible then the choices are: (a) alter the design to make it less sensitive (if possible); (b) greatly reduce tolerances; or (c) manufacture scrap. Option (b), though often used, is usually far too expensive. Option (c), deliberately catering for a percentage of scrap, is guaranteed to produce acute hysteria with production directors and accountants. However, it is surprisingly often the most economic solution and is politically permissible provided that the small percentage of noisy boxes are not allowed to go to the customer. This means 100% T.E. checking on the production line. This suggestion of 100% T.E. production checking seems expensive but may actually save money because some of the earlier checks on profile and pitch can be reduced or eliminated since detailed faults or changes will be picked up by the T.E. check. There is also a large hidden bonus, due to the statistics of the process, provided that a pair of mating gears are checked as a pair, not separately against "master" gears which these days may well be little more accurate than the gears they are meant to be testing. If gears are checked individually for a total error band of 4 um in the mesh then each gear must individually be within +/- 2 um to ensure that any pair are within 4 um. This could well generate scrap rates of the order of 10% on wheel and pinion. Testing together will greatly reduce the scrap rate, as indicated in Fig. 10.8 since, of the "scrap" pinions, most of those "negative" will encounter wheels which are not too large and will mate satisfactorily. 182 Chapter 10 pinion scrap pass max permitted difference is T pass scrap C size wheel scrap pass pass scrap A production - T/2 limits + T/2 size Fig 10.8 Combination of tolerance limits with gear pair testing showing how the number of failed gears is greatly reduced. A wheel of size A (which must be scrapped if tested separately) will mate perfectly with a pinion of size B, and with any pinion of a size less than C, covering about 75% of the pinions manufactured. This effect can easily reduce scrap rates by a factor of four with corresponding savings. The cost of T.E. checking is relatively low. The standard commercial checker can cost up to $300,000 (£200,000), much the same as a profile, helix or pitch checker but the testing is very fast (it can easily be < 1 minute) so throughput is high, reducing costs. Alternatively, a dedicated check rig can be set up for a standard component such as a back axle. The cost of the mechanics, encoders and electronics is then of the order of $30,000 (£20,000) since all the high precision slides and variable settings of the general purpose equipment are not needed. There is one hazard which sometimes causes puzzlement when gear design is improved and that is the oddity that the statistical scatter on the final noise levels is increased. A poor and rather noisy design might give a Improvements 183 measured noise level variation of ± 2 dB. When the design is improved, the variation can easily rise to ± 5 dB so the customer may complain about greater inconsistency in the gear noise and assume that quality control has deteriorated. The reason for this is that the variations in T.E. are mainly due to manufacturing so they will stay roughly constant at, say, ± 2 um. A poor design might give a fairly regular "design" T.E. of 8 um so ±2 um gives 6 to 10 um, a range of roughly 4 dB. Improvement of the average T.E. to 4 um, still subject to ± 2 urn variation gives a range of 2 to 6 um or a total range of 10 dB. This manufacturing range cannot be reduced by the improved design so the customer has to be educated. It is difficult to convince a customer that the better the basic design, the larger the statistical variation will appear to be. The ultimate case is when the design is good enough to occasionally (accidentally/miraculously) give zero T.E. and the dB range (at a given frequency) is then infinite, regardless of how quiet the average gear pair is. References 1. Fahy, FJ. Sound and Structural Vibration. Academic Press, London, 1993. 2. Maag Gear Handbook, Maag, Zurich, 1990 (in English), section 5.271. 3. DIN 3963, Tolerances for cylindrical gear teeth, (in English), DIN standards, Beuth Verlag GmbH, Berlin 30. 4. Smith, J.D., 'Gear Transmission Error Accuracy with Small Rotary Encoders,' Proc. Inst. Mech. Eng., Vol. 201, No. C2, 1987, pp 133- 135. 5. Den Hartog, J.P., 'Mechanical Vibrations.', Dover, New York, 1985, Section 3.3. [...]... - k34 (s3-s4) - D34 (V3-V4) = F45 r4 15 A5 + D5 V5 + k5 (s5) + = F45 r5 Divide throughout by base circle radii to get "linear" equations and take rl=r2 [Il/r 22] (Al.r2) + [Dl/r 22] (VI r2) + [D 12/ r 22] (Vlr2-V2r2) + [k !2/ r 22] (slr2-s2r2+te) - Q/r2 [I2/r 22] (A2.r2) + [D2/r 22] (V2 r2) + [D 12/ r 22] (V2r2-Vlr2) + [k !2/ r 22] (s2r2-slr2-te) = - F 23 [I3/r 32 ] (A3.r3) + [D3/r 32 ] (V3 r3) + [D34/r 32 ] (V3r3-V4r3)... te 23 and te45 are due to meshes Input Q, inertia 1, shaft, input gear 2, lay gear 3, shaft, differential pinion 4, differential wheel 5, half shaft, earth Motion II Al = Q - Dl VI -k !2 (sl-s2 +te !2) - D 12 (V1-V2) rearranges to 11 Al + Dl VI + k !2 (sl-s2+te !2) + D 12 (V1-V2) = Q and similarly 12 A2 + D2 V2 - k !2 (sl-s2+te !2) - D 12 (V1-V2) = - F 23 r2 13 A3 + D3 V3 + k34 (s3-s4) + D34 (V3-V4) = - F 23 r3... 1.5*V(4) 0.7*V (3) 0]; if X (2) +X (3) +te 23 > 0; % drive flank +ve force F 23 = 2e8*(X (2) +X (3) +te 23) + 3e2*(V (2) +V (3) ); elseif X (2) +X (3) -te23r+bl 1 < 0; % overrun flank -ve force F 23 - 2e8*(X (2) +X (3) -te23r+bll) + 3e2*(V (2) +V (3) ); else F 23 = 0; % in backlash end if X(4)+X(5)+te45 < 0; % drive flank F45 = -3e8*(X(4)+X(5)+te45) - 3e2*(V(4)+V(5)); elseif X(4)+X(5)+te45r - b !2 > 0; % overrun flank F45 = -3e8*(X(4)+X(5)+te45r-bl2)... [D34/r 32 ] (V3r3-V4r3) + [k34/r 32 ] (s3r3-s4r3) = - F 23 [I4/r 42] (A4.r4) + [D4/r 42] (V4 r4) + [D34/r 42] (V4r4-V3r4) + [k34/r 42] (s4r4-s3r4) = + F45 Lightly Loaded Gears 197 [I5/r 52] (A5.r5) + [D5/r 52] (V5 r5) + [k5/r 52] (s5r5) = + F45 [M] [A] - -[Dabs] [V] - [Drel][V] + [Drel][Vtr] - [Krel][X]+ [Krel][Xtr] = [F] Tooth forces F 23 = K 23 [s2 r2 + s3 r3 + te 23] + D 23 [V2 r2 + V3 r3] F45 - - K45 [s4 r4 + s5... zeros (2, Z); % setup final results for n = 1 :Z; % +++++++++++++++++ start time step loop te !2= 5e-5*sin(tors*n); % due to 2/ rev torsionals ~ 100 micron te 23= 2e-6*sin(toothl*n); % TE 4 ^m p-p te23r=2e-6*sin(toothl*n + 3) ;% reverse about m lag te45=2e-6*sin(tooth2*n);te45r=2e-6*sin(tooth2*n + 3) ;%TE +ve for +ve metal Xtr = [(X (2) -tel2) (X(l)+tel2) 1.5*X(4) 0.67*X (3) 0]; % includes coupling Vtr = [V (2) V(l)... masses M = [ 6 .3 0. 63 1.0 1 .2 5 ]; % pi*0.045(4th)*0. 02* 7840/ (2* 0.04sq) kg Dabs = [ 20 0 100 100 100 100]; % start low damping freq order 30 Hz Drel - [30 0 30 0 30 0 30 0 0];% rel shaft damping, 1 -2 3- 4 freq order 400 ,30 Hz K = [8e6 8e6 2e6 4.5e6 Ie6 ]; % shaft stiffiiesses l -2, 3- 4,5-earth/r(sq) % turned into equiv linear stiffiiesses at teeth % T/lrbsq = 81e9*pi*0.01(4th) /2* 0 Ix0.04(sq) for 1 -2 torsional... equivalent Q/rb bll = 3e-5 ; b !2 = 4e-5 ; % 30 micron backlash rev = input('Input revs/sec '); % Angle is rev x teeth/rev x 2pi x time % set input rev to rev/s then tors is 2* rev *2* pi rad/s 198 Chapter 11 % 1st tooth is 29 *rev *2* pi rad/s 2nd is 17*rev *2* pi rad/s tors= 12. 6*rev*tint; tooth 1= 1 82* rev*tint; tooth2= 107*rev*tint; A =[0 0 0 0 0];V=[0 0 0 0 0];X= [3. 1e-4 2. 9e-4 -2. 9e-4 -1.2e-4 1.2e-4];% initial... fine details of the mesh contacts because the impacts are extremely short and high force so the contact will be right across the full facewidth and so a constant stiflhess assumption is reasonable 20 00 1500 1000 - 500 -500 - -1 000 116 118 120 122 124 126 128 time in milliseconds 130 1 32 134 Fig 11.8 First mesh tooth forces at 36 00 rpm as modelled on computer ... - b !2 > 0; % overrun flank F45 = -3e8*(X(4)+X(5)+te45r-bl2) - 3e2*(V(4)+V(5)); else F45 = 0; % in backlash end F = [CF -F 23 -F 23 F45 F45]; % ext and tooth forces A = (F - Dabs.*V - Drel.*V + Drel.*Vtr -K.*X + K.*Xtr)./M; % acelerations V = V + tint*A ; X = X + tint*V; seq(:,n) = (X1); % stores displacements for plot force(l,n) = F 23 ; force (2, n) = F45 ; % mesh forces end % +++++++++++++-+++++++ end... where, to economise on the heavy and expensive variable part of the drive, the power is split Part goes directly through gears to one member of a planetary gearbox and part is taken through the variable drive section which only has to deal with about one third of the power The powers are then added in the planetary gear to give the drive to the wheels At zero output speed the gears are essentially running . (V2r2-Vlr2) + [k !2/ r2 2 ] (s2r2-slr2-te) = - F 23 [I3/r3 2 ] (A3.r3) + [D3/r3 2 ] (V3 r3) + [D34/r3 2 ] (V3r3-V4r3) + [k34/r3 2 ] (s3r3-s4r3) = - F 23 [I4/r4 2 ] (A4.r4) + [D4/r4 2 ] . equations and take rl=r2 [Il/r2 2 ] (Al.r2) + [Dl/r2 2 ] (VI r2) + [D 12/ r2 2 ] (Vlr2-V2r2) + [k !2/ r2 2 ] (slr2-s2r2+te) - Q/r2 [I2/r2 2 ] (A2.r2) + [D2/r2 2 ] (V2 r2) + [D 12/ r2 2 ] . -k !2 (sl-s2 +te !2) - D 12 (V1-V2) rearranges to 11 Al + Dl VI + k !2 (sl-s2+te !2) + D 12 (V1-V2) = Q and similarly 12 A2 + D2 V2 - k !2 (sl-s2+te !2) - D 12 (V1-V2) = - F 23 r2 13