Article
citation information:
Maláková, S. Teeth deformation of
noncircular gears. Scientific Journal of
Silesian University of Technology. Series Transport. 2021, 110, 105114. ISSN: 02093324. DOI: https://doi.org/10.20858/sjsutst.2021.110.9.
Silvia MALÁKOVÁ[1]
TEETH
DEFORMATION OF NONCIRCULAR GEARS
Summary. In practice, gear units
whose transmission number is not constant during one revolution are used. Such
gears include the proposed elliptical transmission. Its application can be
(missing a word?) and the automotive industry. The gears set consists of a pair
of identical elliptical gears. The transmission ratio of the designed
elliptical gear is not constant. The basic kinematic characteristics of this
transmission are described in this work. The deformation in contact point of
non  circular gears is determined by the finite element method in this paper.
The results are compared with the deformation of the teeth of the spur gears.
Keywords: noncircular gears, teeth deformation, FEM,
spur gear
1. INTRODUCTION
Gears are a
logical continuation of the invention of the wheel. That gears are called the
biggest invention after the wheel is not unthinkable. More so, we cannot live a
normal day in our lives without gears: without gears no production, no energy
and no transport [4].
By the year 100 v.C., the Greeks used metal gears with
cylindrical teeth in complex computing equipment and astronomical calendars. We
know this by the discovery of the Antikythera Mechanism (Fig. 1), the oldest
known gear machine. This ancient mechanism was found in 1900, in a wreck off
the coast of the Greek island of Antikythera. It contains more than 30 gears
for complicated astronomical calculations of time. Was this the first computer?
In later centuries, the gear became one of the most important parts of modern
technology, incorporated into almost all mechanisms, machines and vehicles.
Fig.
1. Remains of the oldest gear machine (Antikythera Mechanism) [2]
"Standard"
gearboxes are commonly used in practice. They are characterised by a constant
gear ratio. This means that when the drive gear rotates, the driven gear also
rotates evenly, so the gear ratio has to be constant during one revolution. The
teeth of these "standard" gears have the same shape on one gear and
the teeth profile is symmetrical or asymmetric in rare cases.
Special
gears, including noncircular gears, are being increasingly used. The idea of
noncircular gearboxes comes from the forerunners of engineering thinking [11].
In Richard of Wallingford’s Horologium [23], we find the first known
example of a noncircular wheel, which dates back to the 14th century. The
tradition of using noncircular gears in clockwork has been followed by other
clockmakers. For instance, Lorenzo Della Volpaja (14461512) made
–towards the end of the 15th century, an astronomical planetary clock,
for the City of Florence [1]. Noncircular gears from Leonardo da Vinci's
sketches are also known [24].
Noncircular
gears are among gears with a variable transmission ratio. Examples of the use
of noncircular gears in practice include, for example, mechanical presses, for
optimisation of work cycle kinematics [14,26]; forging machines, for optimising
the work cycle parameters; high torque hydraulic engines for bulkhead drives;
textile industry machines, for improving machine kinematics resulting in the
process optimisation and application in oval gear flowmeters. There is noncircular transmission usually used to
provide a periodically variable ratio.
In this paper,
the kinematic properties for the proposed elliptical gear set are presented. It
is a model of a nonstandard eccentric elliptical transmission with a
continuously changing gear ratio for specific parameters. The deformation
in contact point of non – circular gears is determined by the finite
element method in this paper. The results are compared with the deformation of
the teeth of the spur gears.
One should not
forget about the possibility of diagnosing damages with the use of
vibroacoustic diagnostics [2,3,5,6,12,17].
2. Investigated elliptical gears set
To sponsor the work of the
private sector has been created using the CAD model of gear with variable
transmission in the range u = 0.5 to 2.0, with the
number of teeth z_{1} = z_{2}
= 24 and gearing module m_{n}
=
Fig.
2. Designed elliptical
gears set
In pursuit of kinematic ratios on the
proposed gearings, we assume from the right mesh conditions. Kinematic
conditions were processed for the gear 1 (the centre of rotation at point O_{1})
and the gear 2 (with the centre of rotation at point O_{2}). The two
gears are shown in a kinematic dependence one graph (on the horizontal
axis of the wheel teeth first). The angular velocity on the drive wheel gear
and the driven wheel gear is constant for standard spur gears. For designed
elliptical gearing with variable transmission ratio, the angular velocity of
the driven wheel is not constant but is changed according to the
continuous changing of the gear ratio. This is shown in Fig. 4, the angular
velocity is constant for the drive elliptical wheel (ω_{1} = 100 s^{1})
and the driven elliptical wheel is not constant (ω_{2i}).


Fig.
3. Changing the transmission
ratio 
Fig.
4. The angular velocity 
Real
of load gear teeth with variable gear ratio is not constant. By way of
illustration is given unit input torque (driven) spur gear M_{k1} = 100Nm.
Fig. 5 shows the course of torque M_{k1}
on the input gear and torque M_{k2}
on the output (driven) gear (M_{k2i}
= M_{k1}.u_{i}), where u
is transmission ratio. In Fig. 6 are the value of changing tangential tooth
load the driver and driven gear F_{01} = F_{02} (F_{01}=M_{k1}/r_{1i}), where r_{1i}
is the radius of mesh points for the driver gear wheel and the radial force F_{r1} = F_{r2} (Fr_{1} = F_{01}.tgα)
and the resultant force acting on the side of the tooth F_{N1} = F_{N2}
(F_{N1} = F_{01 }/ cosα), where α is
a pressure angle is 20°.


Fig.
5. The course of torque 
Fig.
6. The course of force 
3. Teeth deformation of designed noncircular
gear set
Knowledge
of the deformation properties of gearing is very important. Consider first the
one tooth (Fig. 7a). Action of the resulting normal force F is deformed tooth. This is shown in the figure by a thin
line. The resultant deformation in the direction of action of the normal force d_{i} (i = 1,2
 index that distinguishes whether it is a tooth of the piniondrive wheel or
driven wheel tooth) consists of a deflection bending, shear, deformation
in the area of constraint and the touch deformation [7,15,20].
It is
necessary to determine the deformation meshing teeth, that is, deformation of
teeth pair that you can imagine and illustrate two ways (Fig.7b,c). Figure 7b
shows a pair of teeth that in nonloaded aspect are contact at point X on the line of contact t_{B}. The
profiles meshing teeth are deformed after loading. The deformed teeth profiles
intersect the line of contact at points X_{1}
and X_{2}. We can then
determine the total deformation of the pair of teeth as the sum of the deformation
of both teeth d_{1} and d_{2}. Figure 8c
shows a pair of teeth that in nonloaded aspect are contact at point X´ on the line of contact t_{B}. The
deformed teeth profiles intersect the line of contact at points X_{1} and X_{2}, because one of wheel gears is fixated (restrained).
The total deformation of one pair of meshing teeth determined by equation (1).
(1)
a)
b) c)
Fig.
7. a) The deformation of the tooth, b)c) deformation of a one pair of spur gear
teeth [21]
Due to the complex shape of the teeth, the
theoretical determination of tooth deformation is difficult. A lot of work is
devoted to this issue. The initial assumption is usually a strongly idealised
idea of the linear dependence of the tooth deflection on the load and the tooth
is considered to be a beam stressed by bending [8,13]. The question of the
exact determination of the deformation of the teeth remains unresolved at the
required level; therefore, the experimentally determined values are
usually used.
Recently, with the
everevolving computational technology, which performs extensive calculations,
we increasingly come across modern numerical methods for solving a wide range
of gear problems in the available literature. These methods include the finite
element method [25]. This method belongs to the numerical methods of
mathematics, it is widely used for solving problems of elasticity and strength,
dynamics of flexible bodies, heat transfer and many other engineering problems
[18]. In this paper, the deformation of the designed gearing is solved using
the finite element method (FEM).
When solving the requirements for the accuracy of
the calculation of technical problems using the finite element method, the
basic problem of solving the deformation analysis of the gearing of the
examined gear lies in the choice of the solved area of the gear wheel [10,27].
It is not expedient to calculate the gears as a whole by the finite
element method, but only as a part of the gear [19,22].
Since it is a gear set with a timevarying transmission ratio, the transmission ratio of which
ranges from 0.5 to 1 to 2, the deformation of those teeth engaged when the transmission ratio 0.5 is
examined; 1 and 2. The assumed segments of the investigated elliptical gear are
shown in Fig. 8, assuming that, similarly to "standard" spur gears
with straight teeth, it is not expedient to calculate the deformation of teeth
using FEM on the whole wheel model, but only parts  segment.
Fig. 8. Drive wheel segments
To determine the deformation of gearing under
load, it is necessary to know the apportionment of the load on each gearing
pairs with two pairs meshing. In the beginning, let us consider with the
simplest, ideal load apportionment when on two pairs meshing is the load
divided by half for each couple of teeth in the meshing (Fig. 9). Points A, B,
C, D, E are the characteristic points of the meshing for spur gears.
Fig. 9. Load distribution on the line of contact
To determine the computer model for the studies of deformation of the
teeth using FEM, it was necessary to determine material constants, define the
type of the finite element, and to select appropriate boundary conditions
(geometry and power).
The course
of the deformation of the toothing in the designed elliptical eccentrically
mounted gear set, if during the whole engagement we observe the tooth of the
drive wheel, which together with the engaging tooth reaches the transmission
number 1, is shown in Figure 10, (the force
F_{N}=1000[N]). In this
figure, is the progress of the overall deformation of teeth solved by FEM for
the spur gears, in the teeth, which in the meshing reaching gear ratio 1 and
for the division of load by Fig. 9. Deformation of pairs of teeth over the
meshing along the length of meshing line changes. Maximum value of deformation
shall, in this case, be the endpoints lonely meshing (if we consider the
imagepair) and the minimum value shall also meshing in two pairs of endpoints
lonely meshing. The points B and D, which is the solitary meshing points, leads
to a step change deformation teeth and it will input the next couple teeth to
meshing.
Fig. 10. Course of tooth deformation for elliptical gear set
To determine the resulting deformation of the teeth,
it is necessary to determine the deformation of individual pairs. Figure
11 shows the progress of the overall deformation of teeth solved by FEM for the
spur gears with number of teeth z_{1,2}=24,
the module of teeth m=3,75[mm], the
force F_{N}=1000[N] and width
of gearing b_{1,2=}10[mm],
which in the meshing reaching gear ratio 1 and for the ideal division of load.
Deformation of pairs of teeth over the meshing along the length of meshing
line changes. Maximum value of deformation shall,
in this case, be the endpoints lonely meshing (if we consider the imagepair)
and the minimum value shall meshing in two pairs of endpoints lonely meshing as
well. The points B and D, which is the solitary meshing points, leads to a step
change deformation teeth and it will input the next couple of teeth to meshing.
Fig. 11. Course of tooth deformation for spur gear set
The difference during the deformation in the designed gearing and the
standard ring spur gear (Fig. 12) results from the smooth change of the gear
number, thus, the change of the force acting on the individual teeth, the teeth
being loaded with different resulting normal forces. Another difference results
from the different length of the engagement line and the related engagement
duration factor, which acquires smaller values for the designed elliptical
gears than for comparable spur gears.
Fig. 12. Deformation comparison
4. CONCLUSION
The gearing with
changing transmission gear ratio is used in the practice, even though the
"standard" gearing with constant transmission gear ratio is usually
used. In practice, we also encounter noncircular gears. In this paper, an
elliptical gear set designed for specific parameters is presented. It is the
gearing with variable transmission.
The angular velocity
on the drive wheel gear and the driven wheel gear is constant for standard spur
gears. For designed elliptical gearing with variable transmission ratio, the
angular velocity of the driven wheel is not constant but is changed according
to the continuous changing of the gear ratio. Real of load gear teeth with
variable gear ratio is not constant.
Deformation of pairs
of teeth over the meshing along the length of meshing line changes. Maximum
value of deformation shall, in this case, be the endpoints lonely meshing (if
we consider the imagepair) and the minimum value shall also be meshing in two
pairs of endpoints lonely meshing.
The difference during the deformation in the designed gearing and the
standard ring spur gear results from the smooth change of the gear number,
thus, the change of the force acting on the individual teeth, the teeth being
loaded with different resulting normal forces. The deformation of the
designed elliptical eccentrically mounted gear differs from the stiffness of
the standard spur gearing.
Acknowledgement
This paper was
written within the framework of Grant project “VEGA 1/0290/18 – Development of new methods of determination
of strain and stress fields in mechanical system elements by optical methods of
experimental mechanics” and “KEGA 041TUKE4/2017– Implementation of new technologies specified
for solution of questions concerning emissions of vehicles and transformation
of them into the educational process in order to improve quality of
education” and “VEGA 1/0528/20  Solution of new elements for
mechanical system tuning”.
References
1.
Addomine M., et al. 2018.
„A landmark in the history of noncircular gears design:
The mechanical masterpiece of Dondi’s astrarium“. Mechanism and Machine Theory 122:
219232.
2.
Berlato Francesco, Gianluca
D’elia, Mattia Battarra, Giorgio Dalpiaz. 2020. „Condition
monitoring indicators for pitting detection in planetary gear units“. Diagnostyka 21(1): 310.
3.
Chen Dingke, Changbin Mao, Bin
Qin. 2020. „Study on transformer fault diagnosis technology of VMD local
signal denoising based on kurtosis  approximate entropy“. Diagnostyka 21(1): 8187.
4.
Chen Y.Z., H. Huang. 2016.
„A variableratio line gear mechanism“. Mechanism and Machine Theory 98: 151163.
5.
Czech Piotr. 2013. „Diagnosing a car
engine fuel injectors' damage”. Communications
in Computer and Information Science 395: 243250.
DOI: https://doi.org/10.1007/9783642416477_30. Springer, Berlin,
Heidelberg. ISBN: 9783642416460; 9783642416477. ISSN: 18650929. In:
Mikulski Jerzy (eds), Activities of
transport telematics, 13th International Conference on Transport Systems
Telematics, Katowice Ustron, Poland, October 2326, 2013.
6.
Czech Piotr. 2013. „Intelligent
Approach to Valve Clearance Diagnostic in Cars”. Communications in Computer and Information Science 395: 384391.
DOI: https://doi.org/10.1007/9783642416477_47. Springer, Berlin,
Heidelberg. ISBN: 9783642416460; 9783642416477. ISSN: 18650929. In:
Mikulski Jerzy (eds), Activities of
transport telematics, 13th International Conference on Transport Systems
Telematics, Katowice Ustron, Poland, October 2326, 2013.
7.
Fattahi A.M., M.Gh. Khosroshah.
2017. „Three Dimensional Stress Analysis of a Helical Gear Drive
with Finite Element Method“. Mechanika
23(5): 630638.
8.
Figlus Tomasz.
2019. „A Method for Diagnosing Gearboxes of Means of Transport Using
MultiStage Filtering and Entropy”. Entropy
21(5): 113. DOI: 10.3390/e21050441.
9.
Freeth T., et al. 2006. „Decoding
the ancient Greek astronomical calculator known as the Antikythera
Mechanism“. Nature 444(7119):
58791. DOI: 10.1038/nature05357.
10.
Grega Robert, et
al. 2017. „Failure analysis of driveshaft of truck body caused by
vibrations”. Engineering Failure
Analysis 79: 208215. ISSN:
13506307.
11.
Kapelevich Alexander. 2000. „Geometry
and design of involute spur gears with asymmetric teeth”. Mechanism and Machine Theory 35(1):
117130.
12.
Kluczyk Marcin, Andrzej
Grządziela. 2020. „Detection of changes in the opening pressure of
marine engine injectors using vibration methods“. Nase More 67(1): 18. ISSN: 04696255. DOI: 10.17818/NM/2020/1.1.
13.
Kuľka J. et al. 2018.
“Failure analysis of the foundry crane to increase its working
parameters”. Engineering Failure
Analysis 88: 2534.
14.
Murčinková Z.,
K. Vasilko. 2017. “The proposal how to make the basic machining
technologies – turning, milling, planing  more productive.” Manufacturing Technology 17(2): 261266.
ISSN: 12132489.
15.
Neusser Zdenek, Tomas Vampola,
Michael Valasek. 2017. „Analytical gear mesh model using 3D gear
geometry“. Mechanika 23(3):
425431.
16.
Pacana Jacek, et al. 2015. „Improvement of the gear production process by
automating their strength calculations”. Acta Mechanica Slovaca 19(4): 2225. ISSN: 13352393.
17.
Pawlik Pawel. 2019. „The
diagnostic method of rolling bearing in planetary gearbox operating at variable
load“. Diagnostyka 20(3):
6977.
18.
Randall R.B., D.W. Kelly. 1998.
„Modelling of spur gear mesh stiffness and static transmission
error“. Proceeding of the
Institution of Mechanical Engine 1: 112.
19.
Rincon Femandez, Fernando Viadero, 2013.
„A model for the study of meshing stiffness in spur gear
transmissions”. Mechanism and
Machine Theory 61: 3058.
20.
Šalamoun Č., I. Suchý. 1990.
Čelní a šroubová
soukolí s evolventním ozubením. SNTL, Praha, p. 466.
[In Czech: Spur and helical gears with
involute gearing].
21.
Shimanovsky Alexandr, et al. 2016.
„Simulation of spur gear for their nonparallel axes”. Acta
Mechanica Slovaca 20(1): 2832.
DOI: 10.21496/ams.2016.005.
22.
Tong S.H., D.C.H. Yang. 1998.
„Generation of identical noncircular pitch curves”. Journal of Mechanical Design 120:
337341.
23.
Turner J.A. 1975. „The
tragical history of giovanni de dondi“. J. History Astron 6:
126131.
24.
Walter Isaacson. 2017. Leonardo da Vinci. Simon & Schuster. ISBN: 1474466767. 599 p.
25.
Wojnar Grzegorz, Michał
Juzek. 2018. „The impact of nonparallelism of toothed gear shafts axes
and method of gear fixing on gearbox components vibrations numerical“. Acta
Mechanica et Automatica 12(2): 165171.
26.
Zarebski
Igor, Tadeusz Salacinski. 2008. „Designing of noncirculas gears“. The Archive
of Mechanical Engineering LV(3): 275292.
27.
Zhang Xin, Shouwen Fan. 2016.
„Synthesis of the steepest rotation pitch curve design for noncircular
gear”. Mechanism and Machine Theory
102: 1635. DOI: 10.1016/j.mechmachtheory.2016.03.020.
Received 14.07.2020; accepted in revised form 03.11.2020
Scientific
Journal of Silesian University of Technology. Series Transport is licensed
under a Creative Commons Attribution 4.0 International License
[1]
Faculty of Mechanical Engineering, Technical University of Košice,
Letná 9, 042 00 Košice, Slovakia. Email: silvia.malakova@tuke.sk.
ORCID: https://orcid.org/0000000316606333