Gear Noise and Vibration Episode 2 Part 5 pps

% program for finding centre distance change for contact ratio 2 % gears for TE for changeover... % take first approximation to new pressure angle phil as due to% centres moving 1 module

I a

roll

base pitch

pure involute

tip

purb involute

Fig 13.4 Sketch of T.E effects at (a) correct centre distance and (b) extended

centre distance

Fig 13.4 shows diagrammatically the difference between two pairs of teeth meshing at correct and extended centres to give the same changeover points The test requirement is then to decide what increase in the centre distance will give crossover points in exactly the same positions up the profiles

of the gears as when handing over contact under loaded conditions

The requirement is to find the exact positions up the profiles, not along the roll pressure line, where handover occurs for the original contact geometry and match these to the handover points at extended centres This is

an iterative calculation and it is simplest to use a computer routine to assist the process

% program for finding centre distance change for contact ratio 2

% gears for TE for changeover Work in terms of nominal module 1

phio = 18*2*pi/360 ; % design pressure angle 18 at contact ratio 2

nl = 32 ; % number of pinion teeth

n2 = 131 ; % number of wheel teeth

brl = nl*0.5*cos(phio); br2 = n2*0.5*cos(phio); % base radii

psil = (tan(phio) + 2*pi/nl) ; psi2 = (tan(phio) + 2*pi/n2);

% determine unwrap angles psi at changeover points assuming both

% are 1 base pitch away from pitch point

% these unwrap angles must be the same for extended test to be

% the same points on the flanks but will occur at roughly

% 0.5 base pitches away from the pitch point

% take first approximation to new pressure angle phil as due to

% centres moving 1 module apart so

phil = acos(cos(phio)*(brl+br2)/(brl+br2+cos(phio)));% new angle

% then calculate distances from pitch point to changeover points

% divided by original base pitch

rl = (brl*psil - brl*tan(phil))/(pi*cos(phio));

% new pressure angle only original base radius real

r2 = (br2*psi2 - br2*tan(phil))/(pi*cos(phio));

conratio = rl + r2;

disp('angle rl r2 contact ratio')

disp([phil rl r2 conratio]) % line 18

phi2 = input('enter new pressure angle '); % ****

rl = (brl*psil - brl*tan(phi2))/(pi*cos(phio));

r2 = (br2*psi2 - br2*tan(phi2))/(pi*cos(phio));

conratio = rl + r2;

disp(f angle rl r2 contact ratio')

disp([ phi2 rl r2 conratio ])

phi3 = input('enter new pressure angle ') ; % ****

rl = (brl*psil - brl*tan(phi3))/(pi*cos(phio));

r2 = (br2*psi2 - br2*tan(phi3))/(pi*cos(phio));

conratio = rl + r2;

dispC angle rl r2 contact ratio')

disp([phi3 rl r2 conratio]) % Iine28

% calculate increase in centre distance from original

incr = (brl +br2)*(l/cos(phi3) - l/cos(phio)); % modules

disp('centre distance increase modules')

disp( incr)

The programme assumes that the original crossover points were placed symmetrically one base pitch away from the pitch point and calculates the involute unwrapping angles to these points When the centre distance changes the only factors that remain the same are the two base radii and the two unwrap angles to the correct crossover points The approach and recess distances after the centre change will normally not be equal After the first guess at the new pressure angle only small changes are needed to adjust the angle (in radians) to give the contact ratio exactly 1

If the original design was not symmetrical about the pitch point the original design values of the unwrap angles psil and psi2 to the crossover points should be used

Gregory, R.W., Harris, S.L and Munro, R.G., 'Dynamic behaviour

of spur gears.' Proc Inst Mech Eng., Vol 178, 1963-64, Part I, pp 207-226

Leming, J C., 'High contact ratio (2+) spur gears.' SAE Gear Design, Warrendale, 1990 Ch 6

Yildirim, N., Theoretical and experimental research in high contact ratio spur gearing University of Huddersfield, 1994

Munro, R.G and Yildirim, N., 'Some measurements of static and dynamic transmission errors of spur gears.' International Gearing Conf., Univ of Newcastle upon Tyne, September 1994

Low Contact Ratio Gears

14.1 Advantages

Conventional industrial gears tend to use the standard 20° pressure angle and standard proportions and thus encounter undercutting problems when the number of pinion teeth falls below about 18 If gears are highly stressed they will normally be carburised and the standard AGMA2001 or ISO

6336 calculations will typically give a so-called "balanced" design at about 27 teeth This means that there is an equal likelihood of failure by flank pitting or

by root cracking In practice as root failure would be disastrous, it is normal to have considerably less than 27 pinion teeth to make sure that root breakage is ruled out This leads to most standard spur designs having between 18 and 25 pinion teeth and typically having a nominal contact ratio about 1.6

Alternatively we can still get involute meshing with much lower tooth numbers if we are prepared to use non-standard teeth on the pinion Tooth numbers of 13 or 11 are common on the first stages of small, high reduction gear boxes and the low tooth numbers allow larger reduction ratios The designs use increased pressure angles typically of 25° and are "corrected" so the pitch circle is no longer roughly 55% of the way up the tooth but is only about one third of the way up the tooth when meshing with a large wheel

For two equal gears meshing the practical limit is about 9 teeth and Fig 14.1 shows two such gears in mesh For pinion and large gear or the ultimate pinion and rack meshing the practical limit is down to 7 teeth Again the pressure angle is 25° and the teeth are relatively narrow at the tips The theoretical contact ratio for these gears is about 1.05 to 1.1 but this nominal value does not allow for the relatively large contact area Fig 14.2 shows the geometry for a standard design which is used on oil jacking rigs where very large loads must be taken but pinion diameters must be minimised These seven tooth gears with modules of the order of 100 mm (0.25 DP) are used with racks either 5" or 7" facewidth to lift the high loads of oil jacking platforms for use in waters up to several hundred feet deep The loads on each tooth are then of the order of 500 tonnes and dozens of meshes work in parallel Fig 14.3 gives an expanded view of the contacts near the changeover point and it can be seen that there is very little overlap when there are two pairs of teeth in contact

223

Fig 14.1 Shapes of two meshing gears with nine teeth and 25° pressure angle.

As can be seen in Fig 14.3 with the contact ratio only slightly greater than 1, contact is occurring very near the pinion tooth tip and very near to the pinion base circle

100 200 300 400 500

Fig 14.2 Seven tooth gear meshing with rack.

These highly loaded jacking gears work extremely slowly so noise is not a problem but stresses dominate the design The major advantage in using only seven teeth is that the tooth size is dictated by the load carried If the pinion were to have more teeth, not only would the pinion itself be larger and

so much more expensive, but the driving torque necessary would be increased and so the cost of each drive gearbox would be greatly increased as cost is roughly proportional to output torque Rather different considerations apply in the case of low power but high reduction ratio gearboxes Here the main advantage of low tooth numbers lies in the reduced number of reduction stages and so less components such as bearings to be bought and mounted with the attendant costs Less obvious advantages come from the more rapid reductions

in shaft speeds so that there are fewer high frequency tooth meshes to rattle and give noise and there are fewer high speed shafts so lubrication and churning losses are lower Lower tooth frequencies generally give lower noise

50 100 150 200 250

Fig 14.3 Detail of contacts for seven tooth and rack.

tip

base pitch

roll

I purfe involute

Fig 14.4 Contrast between tip relief shape for conventional design and

corresponding fast change at tip for low contact ratio design

14.2 Disadvantages

The major advantages in root strength associated with large teeth would appear to give low contact ratio gears a great advantage but in practice they are little used The main reason for this is that it is difficult to get a smooth changeover with a low contact ratio as any theoretical tip relief design must occur in a very short distance if there is to be low T.E Fig 14.4 shows Harris maps which contrast the tip reliefs for a high contact ratio mesh and a conventional mesh The changeover is very dependant on accuracy of profile generation and on having the centre distance exact

This is not important for very low speed gears where dynamics can be ignored It is also less important for very small gears since for small gears the manufacturing errors become much larger in relation to elastic deflections and pitch and profile errors become sufficiently large that they dominate the meshing As the changeover errors are large they dominate the T.E changes regardless of the nominal contact ratio so there is little noise penalty associated with using a low contact ratio

The main disadvantages from strength aspects lie in the problems at the ends of the flanks where changeover occurs as exceptionally high stresses are generated As can be seen in Fig 14.3, at the bottom of the pinion tooth the contact is very near the base circle so the radius of curvature of the involute profile is very small The standard Hertzian contact stress formulae for cylindrical contacts depend on the effective combined radius of curvature which in this case, with rack teeth, is equal to the local pinion curvature As this drops near the base circle the contact stresses rise to about double the value

at the pitch point

At the tip of the pinion teeth there is a different problem in that the radius of curvature is relatively large so the Hertzian stresses are below half those at the root but the tips of the teeth are very narrow There is high friction with very slow running gears so in one direction of rotation there can be a high force, approaching tangential in direction, attempting to shear off the tips of the teeth The shear stresses across the narrow tip combined with the local contact stresses can give failure Another problem can arise as the pinion tip is narrow, only allowing a small radius of curvature so manufacturing or positioning inaccuracies may run the contact onto the tip which, with its small radius, will give high contact stresses

14.3 Curvature Problems

The small radius of curvature of the profile at the pinion root was mentioned as a problem in stressing Our standard assumption is that the radius of curvature is equal to the length of the tangent from the base circle In

1.06

1.04

1.02

tangent from - 0 1 position

base circle 0.98L •- < - 1 - — — - —

-0.12 -0.1 -0.08 -0.06 -0.04 -0.02 0

Fig 14.5 Expanded view of involute near base circle.

the limit, if the working profile reaches down to the base circle, the length of the tangential unwrapping string becomes zero and then theoretically we have zero radius of curvature and so very high contact stresses

This does not agree with commonsense because if we look at the shape of an involute as it starts out from the base circle, it does not look like a small radius of curvature It starts out by moving almost radially outwards as can be seen in Fig 14.3 with no hint of the sharp point we would expect with zero radius of curvature Double-checking the mathematics by alternative methods still gives zero as the radius of curvature

When mathematics and common sense do not agree it is usually (invariably) the mathematics that is wrong In this case the reason for the silly answer is that near the base circle the centre of the radius of curvature (at the tangent point to the base circle) is travelling as fast in the tangential direction

as the radius is reducing The net effect is that the effective curvature is not as sharp as expected This presents problems when assessing contact stresses since the effective radius of contact is very much higher than the theoretical value

Various attempts have been made to modify the involute shape near the base circle to avoid the theoretical low radius problem but it is debatable whether there is much point in such modification when there is in reality a

higher radius than expected Fig 14.5 shows the involute shape down near the base circle drawn out accurately and shows the tangent at the point 0.1 radian unwrap angle With seven teeth this unwrap angle corresponds to only about one tenth of a base pitch As can be seen, there is no detectable reduction in curvature for the first part of the involute

For highly loaded gears such as jacking rig gears there is an additional factor that eases the local stresses It is customary to design for the rack teeth to reach the plastic state each time they are loaded The deformations involved spread the contact patch over a large area and so reduce stress levels greatly The teeth surfaces deform permanently and the width of the rack teeth increases but the rack material is relatively soft and does not fracture and the required life of the gears is a restricted number of cycles so the gears are satisfactory

14.4 Frequency gains

As mentioned previously, a standard "fix" for noise problems is to alter the number of teeth to alter the excitation frequency This has usually taken the form of increasing the number of teeth to push the tooth frequency out of a troublesome resonance region There is a stress penalty associated with finer teeth as root stresses rise and, in general, this approach will only help if the tooth frequencies are already high, say above 1 kHz In general, reducing the size of the teeth does not reduce the T.E at I/tooth so it is equally likely that noise will rise

An alternative that can be useful is when the 1/tooth is relatively low, say below 500 Hz Reducing the number of teeth will drive the frequency down to the region where human hearing becomes much less sensitive and this

is reflected in the standard A weighting used At a given sound pressure level reducing the frequency from 200 Hz to 100 Hz corresponds to a nearly 10 dB improvement on the A weighting scale

Another advantage of reducing frequency is that sound pressure levels depend on velocity of panel vibration so that if the vibration is at constant amplitude (as the T.E remains constant amplitude) the frequency reduction reduces velocity correspondingly

Speeds must be relatively low for frequency reduction to help The standard motor speed of 1450 rpm will give about 400 Hz with a 19 tooth pinion so the number of teeth needs to be reduced to about 11 to pull tooth frequency down to the order of 200 Hz If possible it is much quieter to use the traditional design of a 3 or 4 to 1 initial reduction by belt drive- then tooth frequencies are in a quiet region