Page 52 - Automotive Engineering

Measurement of torque, power, speed and fuel consumption CHAPTER 2.1

This calls for a reduction in the torsional stiffness in If in place of an eddy current dynamometer we were
the ratio: to employ a d.c. machine, the inertia I b would be of the
2
2 order of 1 kg m , four times greater.
1500 This has two adverse effects:
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1. Service factor, from Table 2.1a-4 increased from 4.8
to 5.8;
i.e. to 3364 Nm/rad.
The combined torsional stiffness of several elements 2. The denominator in eq. (7) is reduced from
in series is given by: (1þ 0.25/0.30) ¼ 1.83 to (1 þ 0.25/1.0) ¼ 1.25,
corresponding to an increase in the vibratory torque
1 1 1 1
¼ þ þ þ / (14) for a given exciting torque of nearly 50 per cent.
C C 1 C 2 C 3
This is a general rule: the greater the inertia of the dy-
This equation indicates that the desired stiffness namometer the more severe the torsional stresses generated
could be achieved by the use of two flexible couplings by a given exciting torque.
each of stiffness 7480 Nm/rad. A manufacturer’s cata- An application of eq. (1) shows that for the same
logue shows a multi-bush coupling having the following critical frequency the combined stiffness must be in-
characteristics: creased from 3364 Nm/rad to 5400 Nm/rad. We can
meet this requirement by changing the bushes from
Maximum torque 814 N m (adequate)
Shore Hardness 50/55 to Shore Hardness 60/65, in-
Rated torque 170 N m creasing the dynamic torsional stiffness of each coupling
Maximum continuous from 8400 Nm/rad to 14000 Nm/rad (in general, the
vibratory torque 136 Nm usual range of hardness numbers, from 50/55 to 75/80,
Shore (IHRD) hardness 50/55
corresponds to a stiffness range of about 3:1, a useful
Dynamic torsional stiffness 8400 Nm/rad degree of flexibility for the designer).
Substituting this value in eq. (14) indicates a combined Eq. (1) shows that with this revised coupling stiffness
stiffness of 3800 Nm/rad. Substituting in eq. (1) gives n c changes from 1573 cycles/min to 1614 cycles/min,
n c ¼ 1573, corresponding to an engine speed of 786 rev/ and this should be acceptable. The oscillatory torque
min, which is acceptable. generated at the critical speed is increased by the two
It remains to check on the probable amplitude of any factors mentioned above, but reduced to some extent
torsional oscillation at the critical speed. Under no-load by the lower dynamic magnifier for the harder rubber,
conditions, the imep of the engine is likely to be in the M ¼ 8.6 against M ¼ 10.5. As before, prolonged running
region of 2 bar, indicating, from eq. (5a), a mean turning at the critical speed should be avoided.
moment M mean ¼ 8Nm. For completeness, we should check the whirling speed
From Table 2.1a-1, p factor ¼ 1.91, giving T ex ¼ 15 from eq. (11). The mass of the shaft per unit length is:
Nm per cylinder: W s ¼ 9.80 kg/m.
X s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
T ex ¼ 4 15 ¼ 60 N m
9
30p 200 10 p 0:04 4
N w ¼ ¼ 19 100 r:p:m:
Table 2.1a-3 indicates a dynamic magnifier M ¼ 10.5, 0:50 2 64 9:80
the combined dynamic magnifier from eq. (13) ¼ 7.4.
The corresponding value of the vibratory torque, from The mass of the shaft þ half couplings is found to be
eq. (7), is then: 12 kg and the combined radial stiffness 33.6 MN/m.
From eq. (12a):
60 7:4 r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
T v ¼ ¼ 242 N m 6
ð1 þ 0:25=0:30Þ 30 33:6 10
N t ¼ ¼ 16 000 r:p:m:
p 12
This is in fact outside the coupling continuous rating
of 136 Nm, but multiple bush couplings are tolerant of then from eq. (12b), whirling speed ¼ 12 300 rev/min,
brief periods of quite severe overload and this solution which is satisfactory.
should be acceptable provided the engine is run fairly Note, however, that, if shaft length were increased
quickly through the critical speed. An alternative would from 500 to 750 mm, whirling speed would be reduced
be to choose a coupling of similar stiffness using SBR to about 7300 rev/min, which is barely acceptable. This
bushes of 60/65 hardness. Table 2.1a-3 shows that the is a common problem, usually dealt with by the use of
dynamic magnifier is reduced from 10.5 to 2.7, with tubular shafts, which have much greater transverse
a corresponding reduction in T v . stiffness for a given mass.

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