Gear Noise and Vibration Episode 2 Part 3 doc

182 Chapter 10pinion scrap pass max permitted difference is T pass scrap wheel scrap pass pass scrap A production - T/2 limits + T/2 size Fig 10.8 Combination of tolerance limits with ge

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

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

Lightly Loaded Gears

11.1 Measurement problems

The first hint that a gear drive may be "lightly loaded" usually comes when vibration or noise measurements do not make sense Amplitudes vary for no apparent reason, frequencies appear which bear no relation to tooth frequency or the "phantom" frequency (from the gear manufacturing machine) and, most characteristic of all, the vibration levels are extremely dependent on load levels

The standard response of taking a test run and doing an FFT analysis just produces even more confusion as the signal gives roughly equal amplitudes

at all frequencies and appears to be trying to approximate to white noise There may be stronger components near tooth frequency and harmonics but there is a high background continuous spectrum right through the range Even worse, there may be significant peaks at half tooth frequency and half phantom frequency or at other subharmonics of the obvious frequencies, or at curious ratios such as two-thirds of the tooth meshing frequency

Since all the rules of linear vibration are being broken, the obvious deduction is that the vibration is non-linear and that application of intelligence rather than mathematics may be required Since all frequency analysis is based

on the assumption of linearity, it is hardly surprising that non-linear systems cause trouble since most vibration engineers have been brainwashed (at university) into carrying out an FFT before they start thinking

The first question usually asked is "what do you mean by lightly loaded?" This is best answered by saying that when the angular accelerations

of the system multiplied by the effective moment of inertia exceed the steady load torque, which is trying to keep the teeth together, then the teeth will start losing contact since the dynamic component is greater than the mean torque level

This can occur when the angular accelerations (due to T.E or torsional vibration) are high, the moment of inertia is high or the load torque is low This is analogous to driving very fast over a bumpy road when (above a critical speed) a lightly loaded trailer will start leaving the ground

185

186 Chapter 11

A

/\ \

w v

/\ /\ A >V /\

\/

/\

_ A /x A rx y^\

—^"^y \y \ / x / '

one revolution

Fig 11.1 Vibration on successive revolutions of gear.

Lightly Loaded Gears 187

The first essential with a non-linear (or linear) system is to look at the raw vibration (or noise) signal on the oscilloscope, preferably synchronised to once per rev With recorded traces the same effect can be obtained by displaying perhaps 10 revs in succession staggered down the page like a waterfall plot as in Fig 11.1 As always it is very worthwhile having a I/rev probe to give an exact synchronising signal

11.2 Effects and identification

As mentioned previously, humans are good at averaging viewed signals on an oscilloscope or the same effect comes from time averaging the signal so the regular part of the pattern can be seen In many cases a human is better than a computer for seeing what is happening

In one engine test in an anechoic chamber, at idling, the timing train

was extremely noisy and FFT analysis of the output from a microphone gave apparently pure white noise with no individual frequency peaks, much to the puzzlement of the team of development engineers The installation was so elaborate (and extremely expensive) that a request for a look at the original time signal caused dismay because it was not available However, after an hour's hard work the relevant signal was located and brought out to a simple oscilloscope together with a I/rev pulse Once the signal had been synchronised on the display, no explanatory words were needed and the dominating sound was of heads being banged against walls The time signal was as sketched in Fig 11.2

A ~

\J V w time

one revolution

Fig 11.2 Time trace of vibration synchronised to once per rev

188 Chapter 11

The time signal not only showed clearly what was happening in this case but showed exactly where in the revolution the large engine torsionals were acting to bring the timing gear teeth back into contact impulsively The fundamental frequency, 2/rev, about 25 Hz, was too low to be picked up powerfully by microphone or accelerometer so it was solely the high harmonics (with much modulation) that dominated the measurements As far as frequency analysis is concerned there is no difference between amplitude distributions for white noise and for isolated short impulses (see section 9.3)

Both distributions contain equal amplitudes at all frequencies and the only difference is in the phase synchronisation at the pulse

More commonly, the torsional excitation is due to the T.E so there is

a likelihood of an impulsive vibration at about 1/tooth frequency, varying in amplitude and period The mechanism (Fig 11.3) is similar to bouncing a ball

on a tennis racket or driving over a very bumpy road at high speed A short and rather violent impact is followed by a "flight" out of contact until the load torque (or gravity) brings the teeth back into contact after about one cycle of T.E excitation It is perfectly possible to bounce powerfully enough to land 2

or 3 cycles later and we then have the "subharmonic" phenomenon of an excitation at 1/tooth giving an irregular vibration at once per 2 teeth or once per 3 teeth It is difficult for the bounce to maintain consistent time and this gives a very irregular variation in bounce height

It may seem strange that an excitation as small as T.E can give trouble, but feeding in a few typical figures shows what is involved A T.E of

± 5 um (0.2 mil) at a I/tooth frequency of 1000 Hz corresponds to an acceleration of 5 E-6*(6283)2 which is roughly 200 m/s2 or 20 g

bouncing response

ampl

input vibration (T.E.)

Fig 11.3 Impulsive bouncing response to roughly sinusoidal input.

Lightly Loaded Gears 189

A pinion of mass 20 kg will have an effective linear mass J/r2 at pitch radius of about 10 kg so to keep the teeth in contact requires a load of about 2000 N (450 Ibf) which at O.lm radius is 200 N m (150 Ib ft) This is easily achieved in a normal loaded gearbox but, in a machine such as a printing machine, 20 g acceleration on a printing roll with an effective mass of

500 kg would require 10 tons tooth load, and the load due to printing is at least

an order lower than this, so it is difficult to keep teeth in contact

Testing with portable high speed T.E equipment on a printing machine will show the manufacturing gear errors repeating consistently at low speeds but as the speed rises the observed T.E becomes erratic and the drive can be seen bouncing out of contact for long periods

From an understanding of the basic mechanism it is soon clear that varying the load on the system will have a major effect on the vibration and the quickest and most telling test for non-linearity is to vary the load This may mean temporarily braking the driven component to increase the torque despite the power waste involved Major changes in vibration immediately indicate non-linearity whereas minor (<30%) changes suggest a linear system Curiously, both increasing and decreasing the load may make the system better If the vibration becomes worse, then usually the alternative will improve it

11.3 Simple predictions

As with all problems it helps to have a simple model of what is happening to see what the effects of varying the parameters are likely to be The methods using a full computer time-marching approach as described in chapter 5 are necessary if we wish to detail the effects of misalignment, profile, crowning, etc., in a multi-degree of freedom system Simple systems can be looked at rather quickly by making some very basic assumptions

The simplest possible model is the single degree of freedom system shown in Fig 11.4 The response of this system will have the shape shown in Fig 11.5 The torsional moment of inertia has been turned into an equivalent

"linear" mass Due to the non-linearity, any original narrow resonance widens

as the resonance bends to the left at high amplitude

Contact will be lost initially when F = myo , where y is the vibration

of the mass The response above this frequency is generally unstable and erratic but we can make some estimates for the condition of maximum amplitude just before the downward jump

We make the assumption that there are no energy losses during the

"flight" so that the initial "upward" velocity is the same as the final

"downward" velocity

190 Chapter 11

input vibration

Fig 11.4 Simple model of non-linear system.

downward jump as frequency decreases

upward jump as frequency increases

frequency

Fig 11.5 Response of "bouncing" system as frequency varies.

Taking the coefficient of restitution at the short impact as e and the

"landing" velocity as V then, as the maximum upward velocity of the "base"

is hco (where h is the amplitude of vibration of the base), the relative velocity after impact must be e times the relative velocity before impact:

(V - hco) = e (V + hco)