The paper presents a method of optimizing the geometry of a cycloid positive displacement pump using genetic algorithm. This allowed for increasing its delivery and efficiency and reduce pulsation

Lempa, Paweł

Lisowski, Edward

Portal Czasopism Naukowych (E-Journals)

* MSc. Eng. Paweł Lempa, Prof. Eng. Edward Lisowski, Institute of Applied Informatics, Faculty 
of Mechanical Engineering, Cracow University of Technology.
PAWEŁ LEMPA*, EDWARD LISOWSKI*
GENETIC ALGORITHM IN OPTIMIZATION 
OF CYCLOID PUMP
ALGORYTM GENETYCZNY W OPTYMALIZACJI 
POMPY CYKLOIDALNEJ
A b s t r a c t
The paper presents a method of optimizing the geometry of a cycloid positive displacement 
pump using genetic algorithm. This allowed for increasing its delivery and efficiency and 
reduce pulsation.
Keywords: optimization positive displacement pump genetic algorithms
S t r e s z c z e n i e
W artykule przedstawiono metodę optymalizacji geometrii pompy wyporowej cykloidal-
nej z wykorzystaniem algorytmu genetycznego. Pozwoliło to na zwiększenie jej wydajności 
i sprawności oraz obniżenie pulsacji.
Słowa kluczowe: optymalizacja pompa wyporowa algorytm genetyczny 
TECHNICAL TRANSACTIONS
MECHANICS
1-M/2013
CZASOPISMO TECHNICZNE
MECHANIKA
230
1. Introduction
The paper presents the task of optimizing the displacement gear pump with three teeth. 
Due to the small number of teeth is obtained discharge and suction chamber with large 
volume. This allows on the one hand, to obtain a high delivery in a unitary pump with small 
dimensions, on the other hand a low sensitivity of the pump parameters for the pumped 
medium and the contamination contained in pump. Selection of geometric parameters 
is essential for pump parameters. To help select solution optimization task can be formulated. 
The paper presents own software that using a genetic algorithm [1,2,6,7] which by API 
interface is connected to the CAD system [8,9], and can automatically generate 3D models 
and perform multi-criteria optimization for gear pump [5].
2. Cycloid gear wheel with three teeth
Gear wheels of cycloidal profile are designed using whole cycloid curves arcs and have 
the form as on figure 1.
The tooth profile is created by rolling in a circle with a diameter db of a circle of radius 
r
h
, on the inner side, resulting in that a point M which lies on that circle creates an arc 
Fig. 1. Gear wheel with cycloidal profile
Rys. 1. Koło zębate o profilu cykloidalnym
231
of ordinary hypocycloid. From the outside circle of radius ρe rolls creating an arc of ordinary 
Hypocycloid. However, at the point P, which is located on the diameter db of the two arcs 
[3, 10]. Ratio of base circle db and radius ρ rolling circle must be:
 
d
zb
ρ
= 4  (1)
Length of the pitch circle is:
 m = 4r (2)
Pitch circle divides tooth on the head with a height ha and foot with a height hf, according 
to the relationship:
 ha = hf =2r (3)
Formulas for determining the diameter of addendum circle da and the diameter of root 
circle df have form:
 da = d + 2ha = 4rz + 4r = 4r (z + 1) (4)
 df = d – 2hf = 4rz – 4r = 4r (z – 1) (5)
3. Optimization of pump parameters
Optimization of pumps can be carried out taking into account the different objective 
functions. In the case of positive displacement pumps it may be getting the biggest pump 
delivery or the total efficiency, minimalize pulsation or cost of manufacture product, getting 
higher operating pressure, etc. The most common approach known in the literature rely on 
formulating one objective with a fixed range of decision variables [4, 11], which facilitates 
the solution of the problem. However, it appears that higher effects are obtainable using 
a multiobjective optimization [7, 11]. As the objectives of optimizing for the analyzed pump 
in this article were selected: maximum unitary delivery, maximum pump delivery,  minimum 
of pulsation and the total efficiency. Mathematical relations for these functions can be 
determined from the following relations [3]:
Geometric dependencies of the positive displacement pump
Geometrical relations are as the following formulas:
– Unitary delivery:
 Q bm z zu = + − −
2
1
2 2
8
2 1 2 2[ ( ) ]  (6)
where:
Qu – unitary delivery,
b – gearwheel width,
m – module,
z – number of teeth.
232
– Theoretical delivery[3]:
 Q bm z zt = + − −
pi 2 2 2
4
2 1 2 1[ ( ) ]  (7)
where:
Qt – theoretical delivery.
– Pulsation:
 δ
pi
pi
pi
=
+




 − − +






2 2
2
2
2
1
2
1
2
2
2 2 2
2
z
R
R
z
z z z
R
z
R
R
z
sin
sin
 (8)
where:
d – pulsation,
R – tooth shape factor
The efficiency of the pump
Parameters related to the operation of the pump are: working pressure, fluid parameters 
and above all, the total pump efficiency, which is defined as the product of the volumetric, 
mechanical and hydraulic efficiency [3, 4]. Total efficiency hc is defined as the quotient of 
output power Pout to the power input Pin or can be calculated as the product of the volumetric 
efficiency hv and hydraulic-mechanical efficiency hhm.
 η η ηc
out
in
v hm
P
P
= =  (9)
Volumetric efficiency is defined as the quotient of real delivery Qr to a theoretical:
 ηv
r
t
t
t t
Q
Q
Q Q
Q
Q
Q
= =
−
= −
∆ ∆1  (10)
where:
DQ – volumetric losses.
 DQ = Qm + QV (11)
where:
Qu – losses associated with viscosity of the liquid,
QV – losses associated with the density of the liquid.
 Q c p qu
d
= µ piµ2
 (12)
where:
md – dynamic viscosity of liquid,
cm – loss coefficient associated with the viscosity, depending on the unitary delivery.
 Q c p qrς ς
=
2 23  (13)
233
where:
ς – density of the liquid,
cr – loss coefficient associated with the density, depending on the type and size of the 
slots, and the unitary delivery. 
Substituting relations (11), (12), (13) for (10) receives a formula which defines the 
volumetric efficiency:
 η
piµ ςµv d
rc
p
n
c
n
p q= − − −1
2
1 2 13  (14)
Hydraulic-mechanical efficiency is defined as the quotient of the theoretical moment Mt 
on the pump shaft to the actual moment Mr:
 ηhm
t
r
t
t
M
M
M
M M
= =
+ ∆
 (15)
where:
DM – a moment of loss.
 M = Mv + MV + Mp (16)
where:
Mv – moment of losses associated with the resistances caused by the friction of viscous 
liquids,
MV – moment of losses associated with the density of the working medium,
Mp – moment of losses associated with the mechanical losses dependent on the 
working pressure of the pump.
These moments can be calculated from the following formulas [3]:
 Mv = cvmdnq (17)
 M c n qς ς pi
=
2
53
4
 (18)
 M c pqp p= 2pi
 (19)
where:
cv – proportionality coefficient,
cV – ensity coefficient,
cp – pressure coefficient.
Substituting dependeces (17), (18), (19) and (16) for (15)receives a formula defining 
the hydraulic-mechanical efficiency:
 η
pi
µ ς
pi
hm
v
d
p pc
n
p
c n q c
=
+ + +
1
1 2
2
2
23
 (20)
234
Substituting the formula for volumetric efficiency (14) and model of the hydraulic- 
-mechanical efficiency (20) we get the following formula on the total efficiency:
 η
piµ ς
pi
µ ς
µ
ς
c
d
r
v
d
p
c p
n
c
n
p q
c n
p
c n
p
q c
=
− − −
+ + +
1
2
1 2 1
1 2
2
3
2
23
 (21)
The objective function
On basis of formulas (6), (7), (8), (21) constituting mathematical model of the pump was 
built four objective functions . The following notation were prepared: f1 – function of the 
unitary delivery (22), f2 – function of the theoretical delivery (23), f3 – function of the pulsation, 
for it is used inverse function (24) (for compatibility with other functions, i.e. maximum f
3
 
means the minimum pulsation), f
4
 – function of the total efficiency (25).
 f1 = f1 (q (b, rw, rt)) (22)
 f2 = f2 (Qt (x, y, z, b)) (23)
 f
3
 = f
3
 (d (z, y, z, b)) (24)
 f
4
 = f
4
 (hc (Qew, p, md, V)) (25)
For the defined objective function multi-criteria optimization problem can be written 
as [11]:
 f = max (w1 · f1 + w2 · f2 + w3 (–1) f3 + w4 f4) (26)
4. The simulation studies
Calculations were carried out for various variants of the pump design.
To carry out the calculations a hydraulic oil with the following specifications was selected: 
dynamic viscosity of liquid u
k
=41 mm2/s, liquid density ς = 840 kg/m3, and the number of 
teeth equal to 3.
Decision variables xi:
– height tooth coefficient y x
1min
 < x1 < x1max
– gearwheel width b x
2min
 < x2 < x2max
– the pumping pressure p x
3min
 < x
3
 < x
3max
The range of values of the decision variables are shown in Table 1.
T a b l e  1
The Range of the decision variables
xi xi min xi max units
×1 0.6 1.2 –
×2 30 50 mm
×3 1.2 2.5 MPa
235
For optimization using a genetic algorithm the following parameters were selected:
– The type of representation: floating point
– Crossover: 2-point, with a probability: 0.9
– The probability of mutation: 0.1
– The size of population: 20
– The number of algorithm iterations: 2000
The greatest effect of the conducted optimization was reduction of the pump pulsation 
by 47.2%, total efficiency improved by 9.2%, unitary delivery improved by 2.9% 
and theoretical delivery improved by 2.8%. This effect is presented in Figure 2.
5. Conclusions
This paper presents the problem of multi-criteria optimization displacement pump design 
using a genetic algorithm. To optimize pump was used known in the literature mathematical 
models describing the basic properties of the pump, such as unitary delivery, pump delivery, 
pulsation and the total efficiency. As the object of the investigations the standard model of 
the positive displacement pump used in industry was selected. On pump was conducted 
simulation studies looking for better solutions to reduce pulsation and increase unitary 
delivery, pump delivery and total efficiency. A software Creo (Pro/Engineer) was used for the 
purpose of construction of the gear geometry. Own software with the use genetic algorithm 
was developed for the optimization. The developed software proved to be an effective tool in 
the search for better solutions, which improved pump parameters in particular significantly 
reduce the pulsation.
Fig. 2. Gear wheels after optimization
Rys. 2. Koła zębate po optymalizacji
236
R e f e r e n c e s
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of Electrical Engineers, London 1997.
[2] Goldberg D., Algorytmy  genetyczne  i  ich  zastosowanie, Wydawnictwa Naukowo-Techniczne, 
Warszawa 2003.
[3] Kollek W., Pompy  zębate  –  konstrukcja  i  eksploatacja, Zakład Narodowy im. Ossolińskich, 
Wrocław 1996.
[4] Kollek W., Palczak E., Optymalizacja elementów układów hydraulicznych, Zakład Narodowy 
im. Ossolińskich, Wrocław 1994.
[5] Lisowski E., Automatyzacja  i  integracja  zadań  projektowania.  Z  przykładami  dla  systemu  
Pro/Engineer Wildfire, Kraków 2007.
[6] Michalewicz Z., Algorytmy genetyczne + struktury danych = programy ewolucyjne, Wydawnict-
wa Naukowo-Techniczne, Warszawa 2003.
[7] Pieczara J., Algorytmy  genetyczne  w mechanice  konstrukcji, Uczelniane Wydawnictwo Nau-
kowo-Dydaktyczne AGH, Kraków 2004.
[8] Parametric Technology Corporation, Getting Started with Creo® Parametric TOOLKI 2.0, 2012.
[9] Parametric Technology Corporation, Creo® Parametric TOOLKIT 2.0 Release Notes, 2012.
[10] Stryczek J., Koła zębate maszyn hydraulicznych, Oficyna Wydawnicza Politechniki Wrocławskiej, 
Wrocław 2007.
[11] Tarnowski W., Podstawy  projektowania  technicznego, Wydawnictwa Naukowo-Techniczne, 
Warszawa 1997.


Genetic algorithm in optimization of cycloid pump

http://www.ejournals.eu/sj/index.php/Cz/article/download/1817/1838

Genetic algorithm in optimization of cycloid pump

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