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Weight and rated characteristics of machines: positive displacement pumps, motors and gear sets
Mechanism and Machine Theory Vol. 18, No. I, pp. 1~, 1983 Printed in Great Britain.
(g~94-114X/83/010001-06503.00[0 ,~ 1983 Pergamon Press
WEIGHT AND RATED CHARACTERISTICS OF MACHINES: POSITIVE DISPLACEMENT PUMPS, MOTORS AND GEAR SETS GIUSEPPE RICCI Assistant of Applied Mechanics at the Politecnico di Torini, Istituto de Meccanica Applicata elle Macchine, C. so Duca DegliAbruzzi, 24-10129, Torino, Italy
(Received[or publication 26 July 1982) Abstract--Resuminga previous study[l] samples of positive displacementhydraulic machines and gear reduction units are analyzed, in order to find out correlations between their weight and rated characteristics.The particular tendencies of the positive displacement hydraulic machines resulting from the former work are confirmed and specified in this one. As far as the gear reduction units are concerned, correlations between the variables "weight/rated torque" WIT and "rated power" P are pointed out; they are well definedfor singlefields of output rotation speed. Some hypotheses about the interpretation of those correlations are advanced and further developments stated.
i. INTRODUCTION
the motor/machine sets. Unlike the machines previously studied, both the input and the output are now purely mechanical (torques applied to rotating shafts), if we neglect the thermal output corresponding to the power losses due to friction. A comparison with the respective specific weight properties will be of some interest. As far as we know all the machines considered are for land and sea application.
In a former study/l/we wrote about the results of the analysis of a sample of machines (456 units), which showed the existence of a correlation between the weight W and the rated characteristics of such machines, i.e. power P, rotation speed n and therefore torque T =
P/(2~rnl60). The studied sample included machines of the most various types (turbo- and reciprocating-machines; electric, hydraulic, pneumatic machines, etc.) and sizes (from less than I kW up to more than 100 MW). All of them were of simple type, in which, i.e. mechanical power entered or left through a rotating shaft and no reduction gear unit or other device converted the power mechanical factors (torque and angular velocity). One of the results of the study was that the behaviour of the positive displacement hydraulic machines (subsample A, 60 units) appeared to be clearly distinct from that of the remaining types of machines (sub-sample B, 396 units). For the latter only one correlation was in fact to be seen, linear in a bilogarithmic diagram, between the variables WIT and P, whereas the points corresponding to the former occupied a separated area in that diagram and did not show any tendency to gather around a line. But the size of the sub-sample A was too small to allow either a sufficient confidence in the generality of the result or a further analysis aiming at clearing the causes of its singular behaviour. The study of that group then set aside, is now reconsidered on the basis of a larger sample (370 units). We will refer to such new sample as sample A and call sample B what was previously called sub-sample B. In addition, the research has been extended, without change of method or point of view, to gear reduction units as components of general use in the formation of
2. POSITIVE DISPLACEMENT HYDRAULIC MACHINES
The general characteristics of the chosen sample are shown in Table 1. On the basis of the mainly empiric considerations already mentioned in the previous study, the weight~rated torque WIT and rated power P have been calculated for each machine. Points having co-ordinates WIT and P corresponding to all sample units were plotted on a bilogarithmic diagram. In it the resulting cloud of points occupies an area beneath the one occupied by the remaining types of machines (sample B of the previous study) and shows no tendency to gather around a line whatever. The results of the previous study are thus confirmed on a larger statistical basis. Attempts have been made to resolve the cloud of points by distinguishing with particular graphical symbols the machines belonging to different ranges of rotation speed, maximum continuous pressure and theoretical displacement. The results of such attempts have not been successful. The points corresponding to the different ranges of rotation speed mix up in an unorderly way; the same happens to the different ranges of max continuous pressure and theoretical displacement. On the contrary if we distinguish graphically the points according to the different types of machines, the cloud shows a distinct stratification, as we can see in Figs. 1 and 2.t Therefore, a correlation between the variables WIT and P is definable only within the limits of a single type of machine, unlike what had been observed for the machines of the group B.
tThe machines of the group A3 have not been represented in order to clarify the diagrams. MMT Vol. 18, No. I--A
I
2
G. RICCl
Table 1. Positive displacement hydraulic machines h
SOUFCeS - - 7
r!
max cont speed n ( r p m ',,
m max. COR("
~-
machines
reciprocating pumps
A~ for general appliances oleodynamic A~ pumps and motors
I~ / ~
power p
ro
(~ w ':
max con1 pressure p , ,bar i:
!
21 =28600 1952=78
9
125=8500 i
70=500
2 =-4700
i970-78 12-4
i200 3600 -
70 i75
8a390
%70+74- 4 4
~60 ~60 I 30!
3+42
170=-800
o5~85
250=-1450
ToTA 137o1 03÷,7
53+8500
"
i
90=260 42=35 42+500
1 [
0 6=.117
1970=77
4
4
47-1000
1970+77
2
2
3-II000
1970+80
4
1
1952+80 i 2 6
7
015 + 2 8 6 0 0
102
r
W
b - -
I I
T
3
3+158 03=73
1100~
vane gear
A3 -merit pumps
=o,~ o •=
3=309
piston
5oii
~b ~_
7--436
1 +1176
various posihve d~splace-
dale
53+1500 i
1701
0rbilal
theoretical d~spiacement D (cm]/rev
lo
-
~ ~
[
. recipr, pumps for general appliances (A~) F,~ vane pumps and motors 7~, o-rbital motors fk/42)
~
"
• ~
^ ,~ _
..
•
,~ ~A~ q °
.............0.
•
~
•
"
o~O
..
^ ~, g
".
I
i
• : 0 .~•o . . o _*~0 ~OooO _ _ •
71
[
.'oO_-O u
•
I~.
, i
0.1 0.1
10
102
10 3
10 4
(kW) Fig. 1.
Some layers have a clearly rectilinear trend (reciprocating pumps for general appliances, gear pumps and motors), while others would be better represented by a parabolic curve.t In order to simplify calculations we have adopted lines of regression of the variable WIT on the variable P being straight on the bilogarithmic diagram. Such lines (al, a2, a3) are represented in Fig. 5, together with the line b corresponding to the machines of the group B and with the lines cl, c2, ca corresponding to the gear reduction units, to be considered below. Lines a~, a2, t13 have still a negative slope, generally more evident than that of line b. The upper limit of the power range (100 to 1000 kW), is clearly lower than that of the machines of group B (640 MW); at the moment we do not know if this is a consequence of our lack of tAs for orbital motors a correlation is definable only by extending to this group the behaviour of the other groups with some forcing.
documentation or if it depends on inconveniencies or difficulties inherent in the construction of large units of this type of machine. 3. GEAR REDUCTION UNITS
The general characteristics of the chosen sample are shown in Table 2. It does not include epicyclic reduction gears, as the data available were too few. The variable WIT has still been associated with each unit, T being the torque acting on the output shaft or slow shaft, i.e. the higher of the two input and output torques now acting on the machine. We did not consider the variable WIP, being a less synthetic index of the specific weight of a machine, as seen in the former study. Some additional remarks are now to be made about the rated torque T. The diagram of Fig. 3 shows the rated performances of a gear reduction unit as they can be seen in the catalogues of the manufacturing companies. Unlike the previously considered machines, here we do
Weight and rated characteristics of machines
W
10 2
b y
T
1o pistonpumpsand molorsq,^ ,1
I
b
gear pumpsand motors ;t~2) I
10
.-o
° oO o°°oO°o
•
o 00
o .I~o
"" •
.
-I~
,~
0~
O.
000 0
• ~ o _%o~'•o ~ 0 0 0 0 0 < 0 0 0 2 0 • • •
o&
i l ~ ' ~ O ~
i
0 ^0~_
I
• ••~2 • ••211R'o °lll,,°.O~o • 7"•
0.1
i i li Z i ~ i
I0
0.1
"
102
IO~
P IkW)
104
Fig. 2.
Table 2. Gear reduction units Sources
transmission
C1
ratio
n& oF units
class n2 _< 10 rprn
C2 10
spur lworm I tot, 37 38 75 57 57 114
input speed
i:n~ 1"12
n, (r pro)
1 5 + 4900
60 ÷ 1800
2.24
+ 120
60 ÷ 3(300
output
output
speed
P (kW)
n2 (rpm) 0.15+10 11÷100
!i
power
• 0.007+430
date
12# 1960+79 6
10.019÷ 1210 1960+7£
o== ~8 3
6
I
i
C3 I
83
n2 > 100 rpm
24
107
1.5+ 20 15 + 4900
total
~//
~X~'~x
P
':ikw'I
<,v/
~Y
f ,/ [ 4~ | / ?L
/..~
worm gear unit nominal ratings transm ratio i : 5 weight W=17 daN i
0
100
260
0
500
I000
360 n2
1500 n~ rp.m.
Fig. 3.
not have a single rated working condition, but several conditions corresponding to the different values of the rotation speed at which the specific reducer may be employed. To represent the above quoted characteristics we thought it necessary and correct to introduce into the
400 + 3000 60 + 3000
103+
1200
~.~5-3~00
-0.208 + 50953 1961:-80 ~0 067 +50953
9
sample to be studied three rated operating conditions for each unit. They are shown in Fig. 3 with the symbols L, M, H, corresponding respectively to low, medium and high speed operating conditions, as if they were three distinct units. So in the sample C (Table 2) each actual machine is represented by three units each having a single rated operating condition. The rated power P is that available on the output shaft, in conditions of continuous duty with service factor equal to unity (shock-free operation of 10-20h r/day, few starts/hr with starting torque of about two times the running torque). Points having co-ordinates WIT and P corresponding to all sample units were plotted, as previously, on a bilogarithmic diagram. The resulting cloud of points has stretched about a fairly constant value of W/T, approximately equal to 1 N/Nm. By distinguishing graphically between the points belonging to different ranges of output rotation speed n2, the cloud of points is seen to show a clear stratification, Fig. 4. In order to clarify the diagram only the points belonging to the classes cl(n2~<10rpm) and c3(n2> 100 rpm) have been represented in it; the intermediate
‘3. RICCI _.d_
__
-.
nz ~1OOrp.m “,Ilorpm
1
10
lo2
lo4
lo3
iCzi ‘C,
lo5
P (kW) Fig. 4.
class ~~(10< n2 s 100rpm), however, shows a similar shape and lies between the two contiguous classes. As one can see, the layers show a rectilinear parallel trend, with a slight negative slope; they follow one another upwards in order of increasing rotation speeds. In the same Fig. 4, line b corresponds to the machines of the previously studied group B. By distinguishing graphically the points belonging to different ranges of transmission ratio i, some stratification can be noticed in the whole cloud of points, analogous to that mentioned above but decidely less definite. From Fig. 4 it can be seen that the variables W/T and P that are pertinent to the gear reduction units belonging to a single range of output rotation speed n2 are connected by a sole correlation. The lines (c,, cz, c,) of regression corresponding to the three ranges of n, con-
1
sidered are shown in Fig. 5, together with those that apply to the samples A and B. 4. CONCLUSIONS
The anomalous behaviour of positive displacement hydraulic machines (sample A) compared with the remaining types of machines (sample B) in the former study [l] has been confirmed and specified by the larger sample of the present study. Correlations between the variables W/T and P have been proved also for gear reduction units (sample C). As it can be observed from Fig. 5, the weight per unit of torque of the latter ones is lower than that of the machines of the group B (electrical machines and fluid machines, except for positive displacement hydraulic machines) and shows a slightly descending trend when transmitted power increases. Indeed, if we leave the
L __c,
lo-‘_
Fig. 5
Weight and rated characteristics of machines 5. As already mentioned at the end of par. 2, at present we cannot exclude the possibility that the limit may be due to faults, or shortcomings in the documentation. But if it were not so, and if the value WIT constant relative to the gear reduction units were to be regarded as a construction limit imposed by the strength conditions, then the above limit would indicate that some difficulty exists in reconciling, for larger power, the requirements of the fluid circuit with those of the strength design of such machines. Similar difficulties should then be expected also from the machines of the groups A1, A3 and (for much larger powers) from the group B. Of course, the above statements wait for confirmation through a general theoretical explanation of the behaviour that has been pointed out. On the basis of the diagrams of Fig. 5 it seems possible to develop some interesting considerations about the weight of the assemblies motor-gear reduction unitworking machine. We intend to come back to this subject in a further study. As to the use of the diagrams of Fig. 5 for the plain prediction of the weight of machines it does not seem to need special remarks. The accuracy allowed by the transverse size of the cloud of points is not very high. As a matter of curiosity we can, however, say that it has been high enough to show, while gathering and elaborating the statistical data, misprints in the catalogues of the manufacturing companies and also to reveal the presence (non-declared and non-apparent) of speed reducers incorporated in some machines. Diagrams and remarks about the specific weight of positive displacement hydraulic machines are found in Geyer and Treseder[2]; they have been examined and discussed in [1]. Diagrams and remarks about the weight of gear reduction units are found in Niemann[3] and Broersma[4]. As far as point of view and range of examined machines are concerned, their treatment sensibly differs from our study and a comparison with the respective results does not appear to be meaningful at present. A dimensional analysis of the factors influencing the weight of the gear reduction units is put forward by Broersma. It will be taken into account in the further developments of this study.
distinction in classes of n2 out of consideration, the ratio WIT is for this group of machines approximately constant. The two above-mentioned facts allow us to put forward the following considerations with regard to the interpretation of the observed behaviours: (1) As far as the gear reduction units are concerned, they are (quasi-) purely mechanical machinesf and their weight is then mainly determined by the mechanical strength conditions. According to what we saw in the first study [Ref. 1, par. 2], the above-mentioned conditions make the ratio WIT fairly constant. The slight dependence on the rotation speed (and therefore on the transmitted power) may be connected with the conditions imposed by dynamic, thermal and fatigue stresses. (2) As with the machines of the group B, the higher values of the ratio WIT and the descending trend of it when power increases, seem to be connected, proceeding by exclusion, with the material flow (fluid machines) or to the energy flux (electrical machines) related to the non-mechanical terminal of the machines [see Ref. 1, Fig. 1]. (3) The positive displacement hydraulic machines, like gear reduction units, show values of the ratio WIT lower than the machines of the group B, under the same power. On the other hand they show, like the latter group, a pronounced descending trend of the ratio WIT when power increases. From a qualitative point of view such an intermediate behaviour is quite comprehensible, because although the positive displacement hydraulic machines are in fact fluid machines, like most of the machines of the group B, they (especially hydrodynamic pumps and motors) may also be considered without excessive forcing as mechanical power transformers, similar in that respect to the gear reduction units. Their particular characteristic is to have a member (oil or another hydraulic fluid) which has not its own shape and therefore must be contained and guided; for the rest (incompressibility, statical behaviour, internal stress magnitude) such a member does not differ substantially from other mechanical machine members, e.g. flexible members such as chains and wire ropes. (4) We found no hydrodynamic pumps and motors whose power exceeded the point (area) of intersection of the lines a2 with the lines c,, c2, c~ as can be seen in Fig.
REFERENCES
l. G. Ricci, Mechanism and Machine Theory, 17, 99-106 (1981). 2. H. M. Geyer, Trans. AIEE, Part II, 71, 118-126 (1952). 3. G. Niemann, Maschinenelemente-Zweiter Band: Getriebe, Springer-Verlag, Berlin, pp. 6-13 (1960). 4. G. Broersma, Marine Gears, De Technische Uitgeverij. H. STAM, N.V., Harlem, the Netherlands, pp. 171-177 (1961).
fPower enters and mainly leaves through rotating shafts. Heat due to friction losses may nevertheless play a non-negligible part.
MASSE UND NENNDATEN HYDRAULISCHER VOLUMETRISCHER
PUMPEN UND MOTOREN UND UNTERSETZUNGSGETRIEBF
G. Ricci Kurzfassun£ hydraulischen ziehungen Die
-
Im A n s c h l u 6
zwischen
sue d e r
an e i n e
volumetrischen Masse
fruheren
frdhere
Maschinen
und N e n n d a t e n
Arbeit
Analyse
und e i n e s dieser
hervorgehobenen
(Ref.
1) w i r d
ein
statisches
UntersetzungsgetrXebes
Maschine
Muster
untersucht,
der
um Be-
zu f i n d e n .
Sondertendenzen
der
hydraulischen
volumetri-
6
G. RICCI
schen Maschinen werden b e s t ~ t i g t Bei dem U n t e r s e t z u n g s g e t r i e b e
und n~her b e s t i m m t .
ergeben s i c h K o r r e l a t i o n e n
drehmoment und der V a r i a b i e n N e n n i e i s t u n g , winkelgeschwindigkeit
klar
auf w e i t e r e
zwischen der V a r x a b l e n Masse/Nenn-
f o r beschrQnkte B e r e l c h e der A u s t r i t t -
abzeichneten.
Es werden Hypothesen b e z 0 g l i c h und es w i r d
die sich
der Deutung der herausgefundenen K o r r e i a t i o n e n
E n t w i c k i u n g e n der A r b e i t
hingewiesen.
aufgestellt,
(g~94-114X/83/010001-06503.00[0 ,~ 1983 Pergamon Press
WEIGHT AND RATED CHARACTERISTICS OF MACHINES: POSITIVE DISPLACEMENT PUMPS, MOTORS AND GEAR SETS GIUSEPPE RICCI Assistant of Applied Mechanics at the Politecnico di Torini, Istituto de Meccanica Applicata elle Macchine, C. so Duca DegliAbruzzi, 24-10129, Torino, Italy
(Received[or publication 26 July 1982) Abstract--Resuminga previous study[l] samples of positive displacementhydraulic machines and gear reduction units are analyzed, in order to find out correlations between their weight and rated characteristics.The particular tendencies of the positive displacement hydraulic machines resulting from the former work are confirmed and specified in this one. As far as the gear reduction units are concerned, correlations between the variables "weight/rated torque" WIT and "rated power" P are pointed out; they are well definedfor singlefields of output rotation speed. Some hypotheses about the interpretation of those correlations are advanced and further developments stated.
i. INTRODUCTION
the motor/machine sets. Unlike the machines previously studied, both the input and the output are now purely mechanical (torques applied to rotating shafts), if we neglect the thermal output corresponding to the power losses due to friction. A comparison with the respective specific weight properties will be of some interest. As far as we know all the machines considered are for land and sea application.
In a former study/l/we wrote about the results of the analysis of a sample of machines (456 units), which showed the existence of a correlation between the weight W and the rated characteristics of such machines, i.e. power P, rotation speed n and therefore torque T =
P/(2~rnl60). The studied sample included machines of the most various types (turbo- and reciprocating-machines; electric, hydraulic, pneumatic machines, etc.) and sizes (from less than I kW up to more than 100 MW). All of them were of simple type, in which, i.e. mechanical power entered or left through a rotating shaft and no reduction gear unit or other device converted the power mechanical factors (torque and angular velocity). One of the results of the study was that the behaviour of the positive displacement hydraulic machines (subsample A, 60 units) appeared to be clearly distinct from that of the remaining types of machines (sub-sample B, 396 units). For the latter only one correlation was in fact to be seen, linear in a bilogarithmic diagram, between the variables WIT and P, whereas the points corresponding to the former occupied a separated area in that diagram and did not show any tendency to gather around a line. But the size of the sub-sample A was too small to allow either a sufficient confidence in the generality of the result or a further analysis aiming at clearing the causes of its singular behaviour. The study of that group then set aside, is now reconsidered on the basis of a larger sample (370 units). We will refer to such new sample as sample A and call sample B what was previously called sub-sample B. In addition, the research has been extended, without change of method or point of view, to gear reduction units as components of general use in the formation of
2. POSITIVE DISPLACEMENT HYDRAULIC MACHINES
The general characteristics of the chosen sample are shown in Table 1. On the basis of the mainly empiric considerations already mentioned in the previous study, the weight~rated torque WIT and rated power P have been calculated for each machine. Points having co-ordinates WIT and P corresponding to all sample units were plotted on a bilogarithmic diagram. In it the resulting cloud of points occupies an area beneath the one occupied by the remaining types of machines (sample B of the previous study) and shows no tendency to gather around a line whatever. The results of the previous study are thus confirmed on a larger statistical basis. Attempts have been made to resolve the cloud of points by distinguishing with particular graphical symbols the machines belonging to different ranges of rotation speed, maximum continuous pressure and theoretical displacement. The results of such attempts have not been successful. The points corresponding to the different ranges of rotation speed mix up in an unorderly way; the same happens to the different ranges of max continuous pressure and theoretical displacement. On the contrary if we distinguish graphically the points according to the different types of machines, the cloud shows a distinct stratification, as we can see in Figs. 1 and 2.t Therefore, a correlation between the variables WIT and P is definable only within the limits of a single type of machine, unlike what had been observed for the machines of the group B.
tThe machines of the group A3 have not been represented in order to clarify the diagrams. MMT Vol. 18, No. I--A
I
2
G. RICCl
Table 1. Positive displacement hydraulic machines h
SOUFCeS - - 7
r!
max cont speed n ( r p m ',,
m max. COR("
~-
machines
reciprocating pumps
A~ for general appliances oleodynamic A~ pumps and motors
I~ / ~
power p
ro
(~ w ':
max con1 pressure p , ,bar i:
!
21 =28600 1952=78
9
125=8500 i
70=500
2 =-4700
i970-78 12-4
i200 3600 -
70 i75
8a390
%70+74- 4 4
~60 ~60 I 30!
3+42
170=-800
o5~85
250=-1450
ToTA 137o1 03÷,7
53+8500
"
i
90=260 42=35 42+500
1 [
0 6=.117
1970=77
4
4
47-1000
1970+77
2
2
3-II000
1970+80
4
1
1952+80 i 2 6
7
015 + 2 8 6 0 0
102
r
W
b - -
I I
T
3
3+158 03=73
1100~
vane gear
A3 -merit pumps
=o,~ o •=
3=309
piston
5oii
~b ~_
7--436
1 +1176
various posihve d~splace-
dale
53+1500 i
1701
0rbilal
theoretical d~spiacement D (cm]/rev
lo
-
~ ~
[
. recipr, pumps for general appliances (A~) F,~ vane pumps and motors 7~, o-rbital motors fk/42)
~
"
• ~
^ ,~ _
..
•
,~ ~A~ q °
.............0.
•
~
•
"
o~O
..
^ ~, g
".
I
i
• : 0 .~•o . . o _*~0 ~OooO _ _ •
71
[
.'oO_-O u
•
I~.
, i
0.1 0.1
10
102
10 3
10 4
(kW) Fig. 1.
Some layers have a clearly rectilinear trend (reciprocating pumps for general appliances, gear pumps and motors), while others would be better represented by a parabolic curve.t In order to simplify calculations we have adopted lines of regression of the variable WIT on the variable P being straight on the bilogarithmic diagram. Such lines (al, a2, a3) are represented in Fig. 5, together with the line b corresponding to the machines of the group B and with the lines cl, c2, ca corresponding to the gear reduction units, to be considered below. Lines a~, a2, t13 have still a negative slope, generally more evident than that of line b. The upper limit of the power range (100 to 1000 kW), is clearly lower than that of the machines of group B (640 MW); at the moment we do not know if this is a consequence of our lack of tAs for orbital motors a correlation is definable only by extending to this group the behaviour of the other groups with some forcing.
documentation or if it depends on inconveniencies or difficulties inherent in the construction of large units of this type of machine. 3. GEAR REDUCTION UNITS
The general characteristics of the chosen sample are shown in Table 2. It does not include epicyclic reduction gears, as the data available were too few. The variable WIT has still been associated with each unit, T being the torque acting on the output shaft or slow shaft, i.e. the higher of the two input and output torques now acting on the machine. We did not consider the variable WIP, being a less synthetic index of the specific weight of a machine, as seen in the former study. Some additional remarks are now to be made about the rated torque T. The diagram of Fig. 3 shows the rated performances of a gear reduction unit as they can be seen in the catalogues of the manufacturing companies. Unlike the previously considered machines, here we do
Weight and rated characteristics of machines
W
10 2
b y
T
1o pistonpumpsand molorsq,^ ,1
I
b
gear pumpsand motors ;t~2) I
10
.-o
° oO o°°oO°o
•
o 00
o .I~o
"" •
.
-I~
,~
0~
O.
000 0
• ~ o _%o~'•o ~ 0 0 0 0 0 < 0 0 0 2 0 • • •
o&
i l ~ ' ~ O ~
i
0 ^0~_
I
• ••~2 • ••211R'o °lll,,°.O~o • 7"•
0.1
i i li Z i ~ i
I0
0.1
"
102
IO~
P IkW)
104
Fig. 2.
Table 2. Gear reduction units Sources
transmission
C1
ratio
n& oF units
class n2 _< 10 rprn
C2 10
spur lworm I tot, 37 38 75 57 57 114
input speed
i:n~ 1"12
n, (r pro)
1 5 + 4900
60 ÷ 1800
2.24
+ 120
60 ÷ 3(300
output
output
speed
P (kW)
n2 (rpm) 0.15+10 11÷100
!i
power
• 0.007+430
date
12# 1960+79 6
10.019÷ 1210 1960+7£
o== ~8 3
6
I
i
C3 I
83
n2 > 100 rpm
24
107
1.5+ 20 15 + 4900
total
~//
~X~'~x
P
':ikw'I
<,v/
~Y
f ,/ [ 4~ | / ?L
/..~
worm gear unit nominal ratings transm ratio i : 5 weight W=17 daN i
0
100
260
0
500
I000
360 n2
1500 n~ rp.m.
Fig. 3.
not have a single rated working condition, but several conditions corresponding to the different values of the rotation speed at which the specific reducer may be employed. To represent the above quoted characteristics we thought it necessary and correct to introduce into the
400 + 3000 60 + 3000
103+
1200
~.~5-3~00
-0.208 + 50953 1961:-80 ~0 067 +50953
9
sample to be studied three rated operating conditions for each unit. They are shown in Fig. 3 with the symbols L, M, H, corresponding respectively to low, medium and high speed operating conditions, as if they were three distinct units. So in the sample C (Table 2) each actual machine is represented by three units each having a single rated operating condition. The rated power P is that available on the output shaft, in conditions of continuous duty with service factor equal to unity (shock-free operation of 10-20h r/day, few starts/hr with starting torque of about two times the running torque). Points having co-ordinates WIT and P corresponding to all sample units were plotted, as previously, on a bilogarithmic diagram. The resulting cloud of points has stretched about a fairly constant value of W/T, approximately equal to 1 N/Nm. By distinguishing graphically between the points belonging to different ranges of output rotation speed n2, the cloud of points is seen to show a clear stratification, Fig. 4. In order to clarify the diagram only the points belonging to the classes cl(n2~<10rpm) and c3(n2> 100 rpm) have been represented in it; the intermediate
‘3. RICCI _.d_
__
-.
nz ~1OOrp.m “,Ilorpm
1
10
lo2
lo4
lo3
iCzi ‘C,
lo5
P (kW) Fig. 4.
class ~~(10< n2 s 100rpm), however, shows a similar shape and lies between the two contiguous classes. As one can see, the layers show a rectilinear parallel trend, with a slight negative slope; they follow one another upwards in order of increasing rotation speeds. In the same Fig. 4, line b corresponds to the machines of the previously studied group B. By distinguishing graphically the points belonging to different ranges of transmission ratio i, some stratification can be noticed in the whole cloud of points, analogous to that mentioned above but decidely less definite. From Fig. 4 it can be seen that the variables W/T and P that are pertinent to the gear reduction units belonging to a single range of output rotation speed n2 are connected by a sole correlation. The lines (c,, cz, c,) of regression corresponding to the three ranges of n, con-
1
sidered are shown in Fig. 5, together with those that apply to the samples A and B. 4. CONCLUSIONS
The anomalous behaviour of positive displacement hydraulic machines (sample A) compared with the remaining types of machines (sample B) in the former study [l] has been confirmed and specified by the larger sample of the present study. Correlations between the variables W/T and P have been proved also for gear reduction units (sample C). As it can be observed from Fig. 5, the weight per unit of torque of the latter ones is lower than that of the machines of the group B (electrical machines and fluid machines, except for positive displacement hydraulic machines) and shows a slightly descending trend when transmitted power increases. Indeed, if we leave the
L __c,
lo-‘_
Fig. 5
Weight and rated characteristics of machines 5. As already mentioned at the end of par. 2, at present we cannot exclude the possibility that the limit may be due to faults, or shortcomings in the documentation. But if it were not so, and if the value WIT constant relative to the gear reduction units were to be regarded as a construction limit imposed by the strength conditions, then the above limit would indicate that some difficulty exists in reconciling, for larger power, the requirements of the fluid circuit with those of the strength design of such machines. Similar difficulties should then be expected also from the machines of the groups A1, A3 and (for much larger powers) from the group B. Of course, the above statements wait for confirmation through a general theoretical explanation of the behaviour that has been pointed out. On the basis of the diagrams of Fig. 5 it seems possible to develop some interesting considerations about the weight of the assemblies motor-gear reduction unitworking machine. We intend to come back to this subject in a further study. As to the use of the diagrams of Fig. 5 for the plain prediction of the weight of machines it does not seem to need special remarks. The accuracy allowed by the transverse size of the cloud of points is not very high. As a matter of curiosity we can, however, say that it has been high enough to show, while gathering and elaborating the statistical data, misprints in the catalogues of the manufacturing companies and also to reveal the presence (non-declared and non-apparent) of speed reducers incorporated in some machines. Diagrams and remarks about the specific weight of positive displacement hydraulic machines are found in Geyer and Treseder[2]; they have been examined and discussed in [1]. Diagrams and remarks about the weight of gear reduction units are found in Niemann[3] and Broersma[4]. As far as point of view and range of examined machines are concerned, their treatment sensibly differs from our study and a comparison with the respective results does not appear to be meaningful at present. A dimensional analysis of the factors influencing the weight of the gear reduction units is put forward by Broersma. It will be taken into account in the further developments of this study.
distinction in classes of n2 out of consideration, the ratio WIT is for this group of machines approximately constant. The two above-mentioned facts allow us to put forward the following considerations with regard to the interpretation of the observed behaviours: (1) As far as the gear reduction units are concerned, they are (quasi-) purely mechanical machinesf and their weight is then mainly determined by the mechanical strength conditions. According to what we saw in the first study [Ref. 1, par. 2], the above-mentioned conditions make the ratio WIT fairly constant. The slight dependence on the rotation speed (and therefore on the transmitted power) may be connected with the conditions imposed by dynamic, thermal and fatigue stresses. (2) As with the machines of the group B, the higher values of the ratio WIT and the descending trend of it when power increases, seem to be connected, proceeding by exclusion, with the material flow (fluid machines) or to the energy flux (electrical machines) related to the non-mechanical terminal of the machines [see Ref. 1, Fig. 1]. (3) The positive displacement hydraulic machines, like gear reduction units, show values of the ratio WIT lower than the machines of the group B, under the same power. On the other hand they show, like the latter group, a pronounced descending trend of the ratio WIT when power increases. From a qualitative point of view such an intermediate behaviour is quite comprehensible, because although the positive displacement hydraulic machines are in fact fluid machines, like most of the machines of the group B, they (especially hydrodynamic pumps and motors) may also be considered without excessive forcing as mechanical power transformers, similar in that respect to the gear reduction units. Their particular characteristic is to have a member (oil or another hydraulic fluid) which has not its own shape and therefore must be contained and guided; for the rest (incompressibility, statical behaviour, internal stress magnitude) such a member does not differ substantially from other mechanical machine members, e.g. flexible members such as chains and wire ropes. (4) We found no hydrodynamic pumps and motors whose power exceeded the point (area) of intersection of the lines a2 with the lines c,, c2, c~ as can be seen in Fig.
REFERENCES
l. G. Ricci, Mechanism and Machine Theory, 17, 99-106 (1981). 2. H. M. Geyer, Trans. AIEE, Part II, 71, 118-126 (1952). 3. G. Niemann, Maschinenelemente-Zweiter Band: Getriebe, Springer-Verlag, Berlin, pp. 6-13 (1960). 4. G. Broersma, Marine Gears, De Technische Uitgeverij. H. STAM, N.V., Harlem, the Netherlands, pp. 171-177 (1961).
fPower enters and mainly leaves through rotating shafts. Heat due to friction losses may nevertheless play a non-negligible part.
MASSE UND NENNDATEN HYDRAULISCHER VOLUMETRISCHER
PUMPEN UND MOTOREN UND UNTERSETZUNGSGETRIEBF
G. Ricci Kurzfassun£ hydraulischen ziehungen Die
-
Im A n s c h l u 6
zwischen
sue d e r
an e i n e
volumetrischen Masse
fruheren
frdhere
Maschinen
und N e n n d a t e n
Arbeit
Analyse
und e i n e s dieser
hervorgehobenen
(Ref.
1) w i r d
ein
statisches
UntersetzungsgetrXebes
Maschine
Muster
untersucht,
der
um Be-
zu f i n d e n .
Sondertendenzen
der
hydraulischen
volumetri-
6
G. RICCI
schen Maschinen werden b e s t ~ t i g t Bei dem U n t e r s e t z u n g s g e t r i e b e
und n~her b e s t i m m t .
ergeben s i c h K o r r e l a t i o n e n
drehmoment und der V a r i a b i e n N e n n i e i s t u n g , winkelgeschwindigkeit
klar
auf w e i t e r e
zwischen der V a r x a b l e n Masse/Nenn-
f o r beschrQnkte B e r e l c h e der A u s t r i t t -
abzeichneten.
Es werden Hypothesen b e z 0 g l i c h und es w i r d
die sich
der Deutung der herausgefundenen K o r r e i a t i o n e n
E n t w i c k i u n g e n der A r b e i t
hingewiesen.
aufgestellt,