Investigation of characteristics of gear meshing noise under three-axis driving planetary gear set.
Planetary gear sets are widely used in hybrid or electric vehicles. Although they have many advantages, they have been known to develop noise and vibration problems. There have been many reports dealing with the problems' characteristics for two-axis driving, but very few for three-axis driving. This paper reports three-axis driving performed on a developed driving stand, with driving torque measured by a high response torque sensor on each driving axis. Under these conditions it was found that the meshing frequency occurred at two kinds of frequency harmonics.
CITATION: Nakagawa, M., Abbes, M., Hirogaki, T., and Aoyama, E., "Investigation of Characteristics of Gear Meshing Noise under Three-Axis Driving Planetary Gear Set," SAE Int. J. Passeng. Cars - Mech. Syst. 9(1):2016.
Hybrid and electric vehicles have recently been attracting increasing attention due to global warming and the drying up of fossil fuels . Both vehicle types use planetary gear sets; hybrid vehicles use them as the power distribution system and electric vehicles as a high reduction system. However, planetary gear sets are complex devices and consequently their dynamic characteristics are not fully understood.
Many reports have cited the problem of planet gear displacement occurring in two-axis driving. Parker et al. noted the structured vibration modes of general compound planetary gear systems , used analytical and finite element models to ascertain the nonlinear dynamics of planetary gears , and conducted planetary gear modal vibration experiments to identify the correlation against lumped-parameter and finite element models . Velex et al. reported on the dynamic simulation of eccentricity errors in planetary gears  and on a dynamic model for studying the effects of planet position errors in planetary gears . Torque sharing of planet gears based on FEM  has also been reported. However, only a few reports have dealt with dynamic characteristic in three-axis driving.
In this paper, we report three-axis driving performed on a developed driving stand, with driving torque measured by a high response torque sensor on each driving axis. In particular, we focus on characteristics of gear meshing frequency components due to planet gear set driving.
We observed that the meshing frequency under three- axis driving occurred at two kinds of frequency harmonics, caused by the meshing relationship between ring and planet gear and between planet and sun gear. Comparing these results, we constructed a method of calculating their frequencies on the basis of instantaneous rotation center theory. Moreover, the meshing frequency component of the ring gear was found to be pure sound noise due to its high inertia and rotational stability, and that of the sun gear was found to be mixed sound noise with side bands due to its low inertia and rotational instability.
PLANETARY GEAR SET
A planetary gear set is shown in Fig.1. It has four basic components: ring gear, sun gear, planet gears, and a carrier that, connects the planet gears. [Suffixes.sub.s, c, r, p] correspond to the sun gear, carrier, ring gear, and planet gears. The rotational speed of each respective shaft is defined [N.sub.s], [N.sub.c], [N.sub.r] and [N.sub.p]. The input can be given through three components excluding the planet gears. A planetary gear set can provide a multi gear ratio by changing the input and output axis and deal with multi-inputs or multi-outputs at the same time. However, planetary gear sets have a complicated structure and consequently their dynamic characteristics are not fully understood.
Planetary gear sets are used for either a three-axis or a two-axis driving. Three axis driving is general driving condition of planet gear set. In three-axis driving, the sets deal with 1 input-2 outputs or 2 inputs-1 output; combining power from engine and motor to a vehicle, or distribute engine power to a vehicle and a generator. Three-axis driving can provide variable gear ratio by controlling multi inputs. Furthermore, there is a rigid rotation mode in which all components except the planet gears rotate at the same speed in the same direction like a rigid body ([N.sub.r]=[N.sub.c]=[N.sub.s], [N.sub.p]=0).0n the other hand, two-axis driving condition has one fixed component and it is same as ordinal gear set; it provides single gear ratio, single planet gear set can provide multi different gear ratio by changing its in and output. In two-axis driving, there are three fixed conditions. The ring gear-fix ([N.sub.r]=0) condition is called the planet mode, the carrier-fix ([N.sub.c]=0) condition is called the star mode, and the sun gear-fix ([N.sub.s]=0) condition is called the solar mode. Three-axis driving is considered to be a combination of these three fixed conditions.
In this paper, 2 inputs from the ring gear and carrier and 1 output to the sun gear are taken into consideration.
USAGE OF PLANETARY GEAR SET
Planetary gear sets are widely used in stepped automatic transmissions with their advantages like multi-gear ratio, compactness, and in and out on the same axis. A single planetary gear set ordinarily provides a two-gear ratio in a stepped automatic transmission: Another ratio is used for reverse rotation. Three planetary gear sets are used in 5-speed and 6-speed transmissions. In every transmission type, planetary gear sets are used for two-axis driving. Recently, the number of automatic transmission speeds has been increasing from an efficiency aspect, in order to use the high efficiency area of internal - combustion engines, or from a friction aspect to enable down speeding. ZF has released the 9-speed transmission 9HP, which is used in Range rover .
Series-Parallel Hybrid System
Toyota uses planetary gear sets as the core of series-parallel hybrid system to combine engine power and motor power to achieve higher efficiency. In such system, planetary gear sets are used in a very complex way, not only for two-axis driving to transmit engine power to the car as with an ordinary automatic transmission, or to transmit braking energy to the generator to make electricity (called kinetic energy recovery brake), but also for three-axis driving functions like transmitting combined power from the engine and motor (called power mode), or distributing engine power to the car and generator (called charge mode). A 5-planet planetary gear set is widely used in such system. Control of the planetary gear set is the key for achieving higher efficiency. At the same time, however, the noise generated by planetary gear set has emerged as a problem due to the quietness of hybrid vehicles.
Differential Gear in Formula1
A planetary gear set is used as the differential gear in Formula1 due to its compactness, light weight and high torque capability. This is a very special case and a clutch is required to lock it up as a limited slip differential (LSD). The ring gear is the input axis and the carrier and sun gear are the output axis for the left and right wheels, respectively. A 4-planet planetary gear set is widely used due to its balance of torque capability and weight. Its rigid rotation is used when the vehicle is running on the straight or if there is no difference between the right and left wheels, and it is used in three-axis driving during the cornering or when there is difference between the right and left wheels.
PLANETARY GEAR SET THEORY
Relation of Component Speeds
The speed of each component is determined by synthesis of two kinds of rotation modes, i.e., star mode and rigid rotation mode, as below.
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)
In these equations, N is determined as a positive value and a negative value means opposite rotation direction. Eq. (1) is for star mode and Eq. (2) for rigid rotation mode. The relation of rotation speed can be calculated by solving these equations with the undecided coefficient method.
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)
As shown in Eq. (3), the rotation speed of planetary gear N is determined by the difference in rotation speeds between any two other components and the gear ratio of the planetary gear set. From this equation, the rotation condition can be divided into four states as shown in Table 1.
In Table 1, the ring gear rotation direction is defined as a positive value, i.e., described as +. When the input ratio is [N.sub.c]/[N.sub.r]=0, it is in star mode, and when it is [N.sub.c]/[N.sub.r]=1, it is in rigid rotation mode. When it is [N.sub.c]/[N.sub.r] =[Z.sub.r]/([Z.sub.r] +[Z.sub.s]), it is in solar mode and no output exists although there are two inputs. In this case, its [Z.sub.r]/([Z.sub.r] +[Z.sub.s]) value can be controlled by the gear ratio.
According to Table 1, meshing forces around the planet gear can be described as in Fig.2.
A simple lumped parameter model (Appendix 1) is used in Fig.2. The rotation center O is on the XY plane, the rotation center of the planet gears is [O.sub.p], its tangential direction is [X.sub.p], and its normal direction is
[Y.sub.p]. The pitch radius R is defined as in Eq. (4) below, with module m and the number of teeth Z.
The meshing force F on the meshing point is defined as in Eq. (5) below, with torque T.
In the figure, F is nominal meshing force, i.e., the force tangential to the pitch circle. The former suffix is the element that is given the meshing force and the latter suffix is the one that gives the meshing force. The respective forces from the ring gear, sun gear and carrier to the planet gears are [F.sub.pr], [F.sub.ps], and [F.sub.pc]. In each, transitional force should be balanced during constant driving.
Instantaneous Rotation Center
In the three-axis driving, the planet gears not only rotate but also revolve at the same time. The velocity distribution of the planet gears' rotation is shown in Fig. 3. In this paper, we define tangential speed as a positive value for the CCW (Counter Clockwise) of planet gear. In the figure, [V.sub.sp] is the velocity at the sun gear-side and [V.sub.rp] is that at the ring gear-side of the planet gears turning at the speed of [N.sub.p]. The ratio of these two is given using Eq. (6).
[V.sub.sp]: [V.sub.rP]=-1:1 (6)
This shows that the velocity at the two sides is the same but the direction is opposite.
Furthermore, the revolution speed varies according to the distance from the rotation center O, as shown in Fig. 4. The revolution of the planet gears is equal to the rotation of the carrier.
In the figure, [V.sub.sc] is the velocity at the sun gear-side and [V.sub.rc] is that at the ring gear-side of the carrier turning at the speed of [N.sub.c]. The ratio of these two is given using Eq. (7).
[V.sub.sc]: [V.sub.rc] = [R.sub.s]: [R.sub.r] = [Z.sub.s]: [Z.sub.r] (7)
This shows that [V.sub.rc] is faster than [V.sub.sc] because of the difference in the pitch radius or the number of teeth.
When a planet gear rotates and revolves at the same time, the velocity distribution is like that shown in Fig. 5, or the combination of that shown in Figs. 3 and 4.
The ratio of [V.sub.s] and [V.sub.I], respectively the sun gear-side and ring gear-side velocity of the planet gear, is given using Eq. (8).
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)
Instantaneous rotation center [O.sub.p]' is the point at which its velocity is zero on the [Y.sub.p] axis. The planet gear rotates around [O.sub.p]' instantaneously.
Then, if the difference in tangential speeds between the planet gear and the other gears, or ring gear and sun gear, is taken into consideration,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)
Eq. (9) and (10) show that the tangential speed at the meshing point is the same and that proves nonslip at the meshing point.
Distance R' between O and instantaneous rotation center [O.sub.p]' is defined with [V.sub.[GGAMMA]] and [V.sub.s] in Eq. (11).
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)
The ratio [alpha], i.e., that of R' to the actual rotation center of the planet gear is calculated in Eq. (12).
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)
Here, [R.sub.p] and [R.sub.s] are the pitch radius of the planet gear and sun gear, respectively. The [alpha] values calculated with the input speed ratio of the ring gear and carrier are shown in Fig.6.
There is an asymptote in Fig.6 and when the speed ratio is close to it, a goes to infinity as shown in Eq. (13).
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)
When the input ratio is at the exact ratio of the asymptote, the velocity distribution of planet gear is calculated as below.
From Eq. (3) [N.sub.s] can be described as below.
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)
When [N.sub.c]/[N.sub.[GAMMA]] =2[Z.sub.[GAMMA]][N.sub.[GAMMA]]/([Z.sub.[GAMMA]] +[Z.sub.s]),
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)
Furthermore, [N.sub.p] becomes as below.
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)
Then, [V.sub.s] and [V.sub.[GAMMA]] are calculated as below.
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)
In accordance with the tangential speed being defined as a positive value, [V.sub.s] and [V.sub.[GAMMA]] become the same value and the same direction. This means the planet gear translates instantaneously at that input ratio as in Fig.7.
When the speed ratio goes to infinity, [alpha] approaches 1 as shown in Eq. (19).
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)
When a planetary gear set is in star mode or its speed ratio is equal to zero, radius ratio [alpha] is equal to 1. This means the instantaneous rotation center corresponds to the actual rotation center of the planet gear. When it is in lumped rotation mode or the speed ratio is equal to 1, radius ratio [alpha] is equal to 0. This means the instantaneous rotation center corresponds to the rotation center of the sun and ring gears, or that of the planetary gear set itself. As shown in Eq. (13), when the speed ratio is the exact amount of the asymptote, the instantaneous rotation center goes to infinity. In this case the planet gear translates successively.
Meshing frequency is the frequency generated by meshing or meshing times per second. The meshing frequency of a planetary gear set is determined by the meshing times of the planet gears. As shown in Eq. (8), planet gears has two different tangential speeds due to their rotation and revolution, i.e., the ring gear side tangential speed and sun gear side tangential speeds are different from each other. As a result, the meshing distance per unit time is different and the number of meshings is different due to the nonslip nature of the gears. This means the meshing times are different between the ring gear side and sun gear side. Meshing frequency can be calculated from velocity with unit exchanging. Meshing frequency fz [Hz] = [teeth/s] can be written by velocity [mm/s], circumference per tooth 2[pi][R.sub.p]/[Z.sub.p] [mm/tooth] as in the following Eq. (20).
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)
Then by substituting Eq. (3) into Eq. (20), each meshing frequency or ring gear-side meshing frequency and sun gear-side meshing frequency can be calculated as below.
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (21)
If we substitute [N.sub.p] from Eq. (3) as follows,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (22)
The ring gear-side meshing frequency becomes the meshing frequency of the ring gear itself.
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (23)
If we substitute [N.sub.p] from E. (3) as follows,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (24)
The sun gear-side meshing frequency becomes the meshing frequency of the sun gear itself.
This shows that two kinds of meshing occur in a planetary gear set with three axis driving. Each meshing frequency is determined simply by the rotation speed of the ring gear or sun gear.
EXPERIMENTAL RESULTS AND DISCUSSION
The prototype equipment we used to estimate three axis driving is shown in Fig. 8. The torque of each axis was measured by using (a) a UTM-3Nm (Uni pulse) torque meter and the rotation pulse was measured by using an EE-SX870-2M (Omron) photo micro sensor. The torque meter specifications were 25,000 [min.sup.-1] maximum speed, [+ or -]3 Nm maximum torque, 1/10000 resolution and 200 Hz cutoff frequency. A radio type codeless accelerometer ASH-A-500 (Tateyama Denshi) called an FM telemeter was put on the carrier to record acceleration and analyze it by FFT Two types of planetary gear sets, a 3-planet (P3) set and a 6-planet (P6) set, were used for this experiment. The gear geometry factors are shown in Table 2.
The input was given through both the ring gear and the carrier ([N.sub.r], [N.sub.c]= 200~800 [min.sup.-1]) and the output through sun gear [N.sub.s]. The brake torque was input on the sun gear axis ([T.sub.s]= 0.05 Nm).
Balance Around Planet Gear
Fig.9 shows an example of raw data from the experiment.
The driving conditions for the P6 set were input [N.sub.r]: 800, [N.sub.c]: 600 [[min.sup.-1]] (condition c in Fig.2), [T.sub.s]: 0.05 Nm and output [N.sub.s]: 200 [min.sup.-1].
There were three torque and three pulse data items for each axis. Meshing forces were calculated from torque and rotation speeds were calculated from pulse.
Transitional force and rotational torque are theoretically balanced when there is steady rotation. The transitional force or force in the [X.sub.p] direction in Fig.10 is balanced as in Eq. (25).
[F.sub.pr]= [F.sub.ps]+[F.sub.pc] (25)
In our experimental results, [F.sub.pr]=4.5, [F.sub.pc]=3.3, [F.sub.ps]=0.9 [N], and the transitional forces were almost balanced. However, the rotational torques around the rotation center of the planet gears Op were not balanced because [F.sub.pr][not equal to][F.sub.ps]. Accordingly, the rotational torques around Op' were considered. Under these experimental conditions, the rotational torque was almost balanced around the instantaneous rotation center.
These results confirmed that the transitional forces and rotational torques are balanced when there is a steady rotation.
Fig.11 shows an example of the FFT analysis results we obtained with data from the FM telemeter. This telemeter can only measure acceleration in the tangential direction or acceleration generated by meshing. For the P6 set, the driving conditions for the results shown in the figure were input [N.sub.r]: 800, [N.sub.c]: 600 [[min.sup.-1]] (condition c in Fig.2), [T.sub.s]: 0.05 Nm and output [N.sub.s]: 200 [min.sup.-1].
Fig.12 shows another example of the FFT analysis results obtained with data from the FM telemeter. For the P3 set, the driving conditions for the results shown in the figure were input [N.sub.r]: 800, [N.sub.c]: 600 [[min.sup.-1]] (condition c in Fig.2), [T.sub.s]: 0.05 Nm and output [N.sub.s]: 200 [min.sup.-1].
The sharp peak at 10 Hz was caused by carrier rotation because the FM telemeter was on the carrier and, as shown in the driving conditions, [N.sub.c] =600 [min.sup.-1] or 10 Hz. There were also many other peaks caused by natural frequencies. The natural frequencies of the P6 set are 117, 130, 152, 311, 352, 548, 588, 665 and 691 Hz, while those for the P3set, are 131, 170, 256, 322, 380, 503, 625 and 680 Hz. These were determined through a hammering test and through calculation with a 9 degrees of freedom vibration model.
Two clear peaks can be seen in both Fig.12 and 13. The right side peak ([fz.sub.r]=800 Hz) is the meshing frequency of the ring gear as in Eq. (20). The peak is quite high due to the gear's stable rotation and its high inertia. The left side peak ([fz.sub.s] =100 Hz) is the meshing frequency of the sun gear as in Eq. (21). The peak is low and has few sidebands due to the gear's unstable rotation and its low inertia. This can be explained by Eq. (26).
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (26)
Here, I is the inertia of the planet gear set components. For the P3 set, [I.sub.s]=8.0*[10.sup.-4] and [I.sub.r]=2.97*[10.sup.-2] [kg*[m.sup.2]], and the maximum torque differences are [DELTA] [T.sub.s]=0.56 and [DELTA][T.sub.r]=0.74 [Nm]. These results show that, although the maximum torque differences are of the same order of the magnitude, the inertia differs by two orders of magnitude.
We investigated gear meshing in planetary gear sets for three-axis driving with a proto-type of the driving equipment. The following results were obtained;
* Although planet gear transmits internal forces, ring gear-side meshing force and that of sun gear-side aren't equal.
* It was demonstrated that transitional force and torque around the instantaneous rotation center are balanced.
* Two kinds of rotational vibrations at different meshing frequencies were found due to meshing of (a) the planet gear and sun gear teeth and (b) the planet gear and ring gear teeth.
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Masao Nakagawa and Mohamed Ali Ben Abbes
Doshisha university Graduate School of Science and Engineeri
Toshiki Hirogaki and Eiichi Aoyama
Doshisha university faculty of science and engineering
Doshisha university graduate school of science and engineering
MODEL OF PLANETARY GEAR SET
The geometry of an involute gear results in its having a pitch circle and a base circle. The meshing force can be described as shown in Fig. 13. The ISO defines meshing force as the force on the common tangential line of the pitch circle. It is called nominal tangential force [F.sub.wt] and is vertical to the axis of the gear. On the other hand, meshing occurs on the common tangential line of the base circle (called the line of action). The meshing force on that line is defined as [F.sub.bt], which corresponds to the actual meshing force and is vertical to the tooth surface.
The relation between [F.sub.wt] and [F.sub.bt] is as in Ew. (27).
[F.sub.wt] =[F.sub.bt]cos [alpha] (27)
Here, [alpha] is the normal (20[degrees]) pressure angle. There is radical force [F.sub.r] as shown in Fig. 11, but it means nothing in terms of engineering and [F.sub.wt] corresponds to the axial force.
The radii of the pitch circle and base circle are respectively defined as in Eq. (28) and (29).
d = mz (28)
[d.sub.b] = mz cos [alpha] (29)
Two kinds of planetary gear set models are conceivable, i.e., a model with a pitch circle and one with a base circle. The model of the former is shown in Fig. 14. Nominal meshing force can be taken into consideration in this model; it is sufficient to consider the balance around the planet gears. In this paper we refer to the model as a simple lumped parameter model.
Furthermore, the model with a base circle is shown in Fig. 15. The characteristic of this model is that the base circle of the planet gear is outside that of ring gear. This can be proved as follows. The center of the planet gear's base circle is at the same point as the center of the planet gear's pitch circle. The distance [R.sub.pc] between the center O and the center of the planet gear's pitch circle can be calculated as below with pitch radius.
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Then, the distance [R.sub.bp]' between the center O and the outside of the planet gear's base circle is as below.
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (31)
On the other hand, the radius of the ring gear's base circle is defined as in Eq. (32).
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (32)
The difference between [R.sub.bp]' and [R.sub.br] is as in Eq. (33)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (33)
This equation shows that the planet gear's base circle is always outside that of the ring gear. This model corresponds to actual meshing and it can be used to consider vibration due to meshing. The model can be described as a vibration model like that shown in Fig.16.
Table 1. Range of speed [N.sub.c]/[N.sub.r] a <0< b< [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] <c <1< d [N.sub.r] + + + + + + + [N.sub.c] - 0 + + + + + [N.sub.p] + + + + + 0 - [N.sub.s] - - - 0 + + + Table 2. Gear geometry Sun Planet Ring Carrier Module [mm] 1 --- Number of teeth 30 15 60 --- Normal pressure 20 --- angle Helix angle 0 --- Tooth depth 2.25 --- coefficient Tooth width [mm] 6 10 10 --- Pitch diameter [mm] 30 15 60 --- Base diameter [mm] 28.2 14.1 56.4 --- Mass [kg] 0.09 0.01 0.28 0.72 Material S45C Heat treatment None
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|Author:||Nakagawa, Masao; Abbes, Mohamed Ali Ben; Hirogaki, Toshiki; Aoyama, Eiichi|
|Publication:||SAE International Journal of Passenger Cars - Mechanical Systems|
|Date:||Apr 1, 2016|
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