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Journal of Advanced Mechanical Design, Systems, and

Manufacturing

Vol. 1, No. 4, 2007

541

Vibration Analysis of the Helical Gear System Using the Integrated Excitation Model*

Takayuki NISHINO** **Mazda Motor Corporation,

3-1, Shinchi, Fuchu-cho, Aki-gun, Hiroshima730-8670, Japan E-mail: [email protected]

Abstract The vibration of the helical gear system is generated by three kinds of excitation. The first cause is a displacement excitation due to the tooth surface error. The second is a parametric excitation by the periodical change of the tooth mesh stiffness. The third is a moving load on the tooth surface during the progress of mesh of the teeth. In mesh of a pair of helical gears, the composite load of the distributed load along a contact line moves its operating location from one end of face width to the other end during the process of mesh progress. This moving load causes fluctuation of bearing load that excites the housing. Therefore, it is important to treat gear mesh excitation as moving load problem. For this purpose, a tooth mesh model, in which three different types of excitations above are incorporated, is proposed. In this model, a pair of gear tooth is represented by the multiple springs and the moving load can be taken into account by the multiple mesh excitation forces that have the phase differences from each other. This model is applied to the vibration analysis of a single stage gearbox. The analytical and experimental results show that this method is accurate and effective enough for practical use.

Key words: Gear, Helical Gear, Tooth Mesh Model, Moving Load, Mesh Exciting Force, Finite Element Method, Substructure Synthesis Method

1. Introduction

The contribution of the structure-borne noise is pretty high for the gear noise(1), that is, dynamic load excites the gear pair, shafts, bearings and housing, and then the noise is radiated from the vibrating surface of the housing.

To improve this structure-borne noise, many analytical methods of the vibratory behavior of the helical gear system are reported(2)–(5). In these reports, excitation forces caused by the tooth surface error and the variation of tooth mesh stiffness are incorporated. However, helical gears have another excitation force, which was reported for the first time by Nogami et al.(6), caused by the load distributed on a contact line moving in the axial direction while the gears rotate. Fig. 1(a) shows how a contact line moves from the beginning of the mesh to the end. The operating point of the load distributed along the contact line moves in the axial direction from one end A of the face width to the other end B during mesh progress as shown in Fig. 1(b). This moving load causes the fluctuation of the bearing load. This fluctuation of the bearing load

*Received 9 July, 2007 (No. 07-0294) [DOI: 10.1299/jamdsm.1.541]

z

A BMesh

progress

Distributed loadContact line

zAB

(a) (b)

Fig. 1 The moving load

Journal of Advanced Mechanical Design,Systems, and Manufacturing

Vol. 1, No. 4, 2007

542

excites the housing of the gearbox. Therefore, in addition to the excitation force caused by the tooth error and the variation of tooth mesh stiffness, this moving load should be taken into account for the analytical study of the vibratory phenomenon in the helical gear system.

Nogami et al.(6) analyzed the influence of the moving load on the fluctuation of bearing loads by using some simplified vibratory model. However they did not incorporate the tooth surface error and the variation of tooth mesh stiffness quantitatively in their excitation model. Accordingly, any excitation model in which excitation forces due to the tooth surface error, the variation of tooth mesh stiffness and the moving load are considered seems not to have been established.

The objective of this study is to develop a procedure that is easily applicable to the response analysis of the helical gear system excited by the three kinds of excitation forces above mentioned. For this purpose, a tooth mesh model is proposed. The feature is that those three excitations with different character are incorporated in the excitation model as integrated external forces. Then the proposed method is applied to the vibration analysis of a single stage gearbox and the influence of the moving load on the vibratory behavior of the helical gearing is examined.

Nomenclature

E Mesh excitation force in the direction of tooth normal(8) Eq Divided mesh excitation forces ei (xi) Composite tooth surface error of the driving and driven gear on the contact line Li

Fd Dynamic tooth load(10) j Imaginary unit k Tooth mesh stiffness in the direction of tooth normal(8) kq Divided tooth mesh stiffness kθ Rotational tooth mesh stiffness regarding relative rotation of the contact lines Li i-th contact line Lφ1, Lφ 2 Representative length of arm for converting the rotational displacement of the

gears in the plane of action into the linear displacement along the line of action(9) Mθ Mesh excitation moment(11) Md Dynamic tooth moment(11) m Number of divided tooth springs and mesh excitation forces min ei (xi) Minimum value of ei (xi) on the contact line Li n Number of teeth that meshes simultaneously pi (ξ i) Distributed load along the contact line Li rb Radius of base cylinder xi Coordinate of the measuring point of the tooth deflection on the contact line Li βb Base helix angle ξ i Coordinate of the load operating point on the contact line Li ϕ , ϕq, ϕθ Initial phase angle of E, Eq and Mθ respectively ω Angular velocity Index i Index for contact line q Index for divided tooth spring 1, 2 Index for the driving and driven gear Character decorations ¯ Time averaged value ~ Time varying value In this report, E, Eq, ei (xi), k, and kq are defined in the tooth normal section.

2. Formulation of Tooth Mesh Models


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543

2.1 Outline of tooth mesh model In this paper, the gear body is modeled as

a portion of the shaft and the teeth pairs are modeled as linear and rotational springs. The tooth mesh model is defined as the method how to set up the tooth springs and excitation forces. Figure 2 shows the tooth mesh model proposed in this report. The tooth mesh stiffness is modeled by a set of linear springs with time-averaged spring constants qk that are arranged in equally spaced on the pitch line. A set of mesh excitation forces Eq is given as the same manner as the springs.

Springs and excitations above are set in the plane of action. Each spring is connected to the two points at the both ends of it. These points are shown separately in the figure for convenience, however they are actually located at the same position. Therefore, let us consider that each spring have infinitesimal length. Moreover, let us assume that each point is rigidly connected to the body of the driving and driven gear respectively.

2.2 Arrangement of Divided Tooth Springs Figure 3 shows how the springs are

arranged in the plane of action. The plane in the figure is the same plane of action as shown is Fig. 2. Axis x and z coincides with the line of action and the pitch line respectively. Axis y is perpendicular to the plane of action and directed upward from the drawing plane. The tooth mesh stiffness is expressed by a set of linear springs of number of m that are arranged at the points S1,···, Sq,···, Sm. These points are located at the center of intervals obtained by dividing the face width b with m. Therefore, their coordinates along the axis z are given by

),,1,/(,2/)}12({ mqmbhqhbzq ⋅⋅⋅==−+−= . (1)

The two points where each spring are connected are assumed to exist at the same location as point Sq, however the orientation of the spring is assumed to be parallel to the tooth normal.

The lines that pass through the point Sq (q = 1,⋅⋅⋅, m) and are inclined to the line of action by base helix angle βb represent the direction of forces generated by expansion and contraction of springs. Therefore, we call these lines the operating lines of springs and denote them by lq (q = 1,⋅⋅⋅, m).

2.3 Expression of Moving Load The concept for expressing the moving load is explained as follows. Figure 3(b) shows

the situation where a number of contact lines appear simultaneously on the plane of action and the distributed loads act along them. Focusing one contact line among them, the mesh starts at point S' and ends at point E. As the gears rotate, the magnitude of the resultant load of the distributed load changes. At the same time, its operating point changes along the gear axis from point A to B. We can catch this moving load as a set of spring forces by

h

z

S′

E

x

B

Aβb

Operating line of springsE′

S

y

lq

l1

lm

ξe

zξ i

S′

E

ξs

x

B

A

kq

Li+1

Li−1

k1

Pi,q

Pi,1

Pi,m

km

Contact line Li

βb E′

S

y

lq

l1

lm

Oi

S1

Sq

Sm

(a) The position of the springs

(b) The orientation of the springs⋅⋅⋅⋅

⋅⋅⋅⋅

⋅⋅⋅⋅

⋅⋅⋅⋅

⋅⋅⋅⋅

⋅⋅⋅⋅

⋅⋅⋅⋅

⋅⋅⋅⋅

pi(ξ i)

Eq　

z　

x　

Driving

kq

Driven

Plane of action　

Eq　

Eq　

z　

x　

Driving

kqkq

Driven

Plane of action　

Eq　

Fig. 2 Tooth mesh model

Fig. 3 Layout of tooth springs


Vol. 1, No. 4, 2007

544

performing the following procedure at each rotation step of the gears. (i) The first task is converting the distributed load pi (ξi ) along each contact line into the

equivalent concentrated loads Pi, q (i = 1,⋅⋅⋅, n, q = 1,⋅⋅⋅, m) that are applied at nodal points on

the contact line. Here nodal points are intersections of operating lines of springs and the contact lines.

(ii) Obtain sum of the equivalent concentrated loads that exist on each operating line of spring by

),,1(1

, mqPWn

iq qi ⋅⋅⋅=∑=

=, (2)

where Wq is the load shared by q-th spring.

2.4 Distributed Load along Contact Line The distributed load pi (ξi) along the contact line Li is obtained by solving a set of

non-linear integral equations(7)(8)

),,1()()()()(),( nixexpxKdpxK iiiicL iiiib ii i ⋅⋅⋅⋅=−=+∫ ∆ξξξ (3)

under the transmitted load condition

∑∫==

n

iL iii i dpW

1)( ξξ , (4)

where Kb (xi, ξi) is the influence function regarding bending and shearing deformation of the tooth; Kc (xi) is the contact approach between tooth flanks when pressured by unit load; ∆ is the linear displacement of the rotational delay of driven gear to driving gear in the direction of tooth normal; W is the transmitted load in the same direction.

2.5 Discretization of Distributed Load Figure 4(a) shows the same distributed load pi (ξ i ) on the contact line Li as shown in

Fig. 3(b). We discretize pi (ξi ) into nodal forces Pi,1,⋅⋅⋅, Pi,q,⋅⋅⋅, Pi, m. Here, ξ i,1 ,⋅⋅⋅, ξ i, q

,⋅⋅⋅, ξ i, m are coordinates of the nodal points. We define the segment of a contact line between two nodal points as an element. There are complete and incomplete elements among them. Complete one has nodal points at both sides and incomplete one has one nodal point at one side of it.

(i) Complete element: Each contact line has the complete elements of number of (m−1). Figure 4(b) shows the q-th complete element. Qq, Qq+1 are element nodal forces equivalent to the element load pi (ξi ), ξi ∈[ξi, q, ξi, q+1]. Applying the theorem of virtual work, we obtain

∫=

∫=+

+

++1,

,

1,,

)()(

)()(

11qiqi i

qiqi i

iiqiq

iiqiq

dNpQ

dNpQξξ

ξξ

ξξξ

ξξξ, (5)

where

δξξξ

δξξξ

/)()(

/)()(

,

1,

1 qi

qi

iiq

iiq

NN

−=

−=

+

+

and δ is the width of each element. Looping over all of the complete elements,

we obtain all element nodal forces by using Eq. (5). These are added to the global nodal forces Pi, q at the corresponding nodal points. Thus, we discretize the distributed load that acts in the interval [ξ i,1, ξ i, m].

(ii) Incomplete element: Figure 4(c) shows incomplete elements. We denote by Q1 and Qm element nodal forces that are equivalent to element load acting in the interval [ξ s, ξ i,1] and

(b) Complete element

Oi ξi

ξs ξe

Incompleteelement

ξi

)( iip ξ

Incompleteelement

(a) Distributed load along a contact line

Pi,2 =Pi,q

ξi,q ξi,q+1

Completeelements

Oi

Pi,1 Pi,3 =Pi,m

Qq+1Qq

Oi ξi

ξs ξeξi,1

Q1 Qm

ξi,m

)( iip ξ

)( iip ξ

ξi,1 ξi,2=ξi,q ξi,3=ξi,m

(c) Incomplete elements

Fig. 4 Equivalent concentrated loads


Vol. 1, No. 4, 2007

545

[ξ i, m, ξ e], here ξ s and ξ e are coordinates of both ends of the contact line Li. By considering the equilibrium of force and moment, we have

∫−−

+∫−−

=

∫−−

+∫−−

=

emi

imi

ii

is

imi

ii

emi

imi

mii

is

imi

mii

ii

iii

im

ii

iii

i

dpdpQ

dpdpQ

ξξ

ξξ

ξξ

ξξ

ξξξξξ

ξξξξξξ

ξ

ξξξξξ

ξξξξξξ

ξ

,1,,

1,1,

1,,

1,

,1,,

,1,

1,,

,

)()(

)()(1

. (6)

Now, superimposing Q1, Qm to Pi, 1, Pi, m above respectively, a set of the concentrated loads Pi, q

is obtained.

2.6 Divided tooth mesh stiffness By using the concentrated loads Pi, q

, divided tooth mesh stiffness is formulated. Figure 5 shows that the tooth mesh stiffness of a pair of teeth is represented by a set of linear spring Ki, q (q = 1,⋅⋅⋅, m). Here, each spring constant is defined as

),,1,,,1(min

,, mqni

eP

Ki

qiqi ⋅⋅⋅=⋅⋅⋅=

−=∆

. (7)

Divided tooth mesh stiffness kq is defined as a composite of spring constants Ki, q that exist on the common operating line lq. So, kq is expressed as

),,1(1

, mqKkn

iq qi ⋅⋅⋅=∑=

=. (8)

2.7 Divided mesh excitation force So far, the gears are supposed to have rigid bodies. In dynamic condition, however, gear

body should be treated as flexible. When the gears vibrate with distortions, the point where each spring is connected changes its position, because the point is supposed to be rigidly connected to the gear body as mentioned previously. Strictly speaking, the displacement of such a point differs according to the position on the line lq. This makes the formulation quite complicated. Therefore, we use the displacement of a representative point for the line lq and we choose the point Sq in Fig. 3(a) for such a point. We denote by X1, q and X2, q linear displacements of the point Sq in the direction of tooth normal when the point is rigidly connected to the driving and driven gear respectively. Consequently, denoting the relative displacement by Xq = X1, q− X2, q

, the sum of loads shared by Ki, q (i = 1,⋅⋅⋅, n) on the line lq is

expressed by qqqqq WXkF φ+−= , (9)

where Wq is the time-averaged load of Fq; φq is expressed as

qn

iiq WeK qi /min

1,∑=

=φ . (10)

As gears rotate, the variables Fq, kq, Xq and φq fluctuate around the steady state values. Separating these variables into the steady state values and fluctuating values, ignoring higher order values qq Xk ~~ and leaving only fluctuating terms, we obtain

qqqq EXkF +−= ~~ , (11)

where }~)1(~{ qqqqq WE κφφ +−= , (12)

qq kkq /~~ =κ , (13)

and (−) and (~) means steady state value (time-averaged value) and fluctuating value (time-varying value) respectively.

Equation (11) shows Eq is an excitation force and qq Xk ~ is the reaction force to it.

zξ i

S′

E

xkq

Li+1

Li−1

k1

Ki,q

Ki,m

km

Ki,1

Contact line Li

βbOperating line of springs

E′

S

y

lq

l1

lm

⋅⋅⋅⋅

⋅⋅⋅⋅

⋅⋅⋅⋅

⋅⋅⋅⋅

Fig. 5 Divided tooth mesh stiffness


Vol. 1, No. 4, 2007

546

Therefore, we can consider Eq as a mesh excitation force. In other words, the forced vibratory system of a helical gearing is considered that a pair of gears connected by the springs with time-averaged spring constant qk (q = 1,⋅⋅⋅, m) is excited by the excitation forces Eq (q = 1,⋅⋅⋅, m) as shown in Fig. 2. Moreover, applying a set of excitation force Eq (q = 1,⋅⋅⋅,

m) as shown in Fig. 2, we can express the excitation caused by the moving load in addition to the excitations caused by the tooth surface error and the variation of tooth mesh stiffness.

2.8 Non-Linearity of Tooth Mesh Stiffness to the Transmitted Load Figure 6 shows how the deflection δ of the teeth increases as the transmitted load W

increases. In the gear pair with tooth surface error, the tooth contact area considerably changes according to amount of the load. As the result, tooth mesh stiffness shows non-linearity to the load. Therefore, the non-linearity of the tooth mesh stiffness should be considered. Since we deal with the small displacement around the equilibrium position in the vibratory analysis of the gears, however, we only modify qk and qE as follows.

∑⋅==

n

iq qiK

kkk

10, (14)

}~)1(~{0

qqqkkWE qq κφφ +−= (15)

where 0k is the tooth mesh stiffness when the non-linearity isn't taken into account; k is the tangential tooth mesh stiffness at the point P as shown in Fig. 6.

3. Experiment

The acceleration of the gearbox housing is measured to verify the effectiveness of the tooth mesh model described in the previous chapter.

3.1 Experimental Apparatus Figure 7 shows the testing unit and coordinate

system. Axis x is the axial direction, axis y is the vertical direction and axis z is the horizontal direction. The gearbox is bolted to the base through rectangular steel blocks of 4 cm. Figure 8 shows the internal structure of the gearbox. The housing is the welded structure made of steel plate of thickness 6 mm. And the housing is welded to the steel sole plate of thickness 20 mm. The top plate is bolted to the gearbox trough rectangular bar of 10 mm (M6×16 bolts). Bearing housings are bolted to the gearbox (M8×6 bolts). The driving and driven shafts of diameter 24 mm are supported by taper roller bearings HR33205J. The preload of the bearings are adjusted by the thickness of the shims inserted between outer races and stopper . The values of the preload are 3500 N on the driving shaft and 2400 N on the driven shaft. Shaft fit to inner ring is tight fitted of approximately 20 µm, and bearing housings to outer ring are clearance fitted of about 15 to 30

Fixed to base apparatus

zx

y

P2Y

P3X

P1Z

Steel blocks

123

325

155

271

304

20

t=6 mm

t=6 mm

φ24

33

44

55

33

22

11

4455

33

k0= δ

k= dδdW

Pk= dδ

dWP

Transmitted load W (N)

Def

lect

ion

of to

oth δ

(µm

)

k0tan−1 1

k0tan−1 1

0

0

P

W

δ

0 20000

10Wktan−1 1

ktan−1 1

Fig. 6 Non-linearity of tooth mesh stiffness

Fig. 7 Outer view of gearbox

Fig. 8 Internal structure of gearbox


Vol. 1, No. 4, 2007

547

µm. The gears are fixed to the shafts by ETP-bushes (Miki Pulley type ETP-38).

3.2 The Gear Pairs A pair of helical gears is used for the

experiment. The gear data is shown in Table 1. Their face widths are 35mm and are comparatively wide for their diameters. The material is SCr420. After case-hardened (HRC60), tooth surfaces are finished by grinding. The tooth surface modifications of convex type are given for both gears. Their composite tooth surface topography is shown in Fig. 9.

3.3 Measurement of Vibration of Gearbox Figure 10 shows the power absorbing

gear-testing rig. DC motor of 150 kW power is used for driving and DC dynamo of 150

kW power is used for power absorbing. The gear unit is connected to the intermediate shafts by flexible couplings (Tokai Rubber type CGM5505) to isolate the vibratory system from the driving and absorbing units.

The torque value on the driving shaft is 93.1 Nm. The rotating speed of the driving shaft is swept from 300 rpm to 2000 rpm with variation rate of 15rpm/s. Three piezoelectric accelerometers are fixed on the gearbox housing at the positions shown in Fig. 7, and off-plane accelerations are measured at each position. We denote these accelerations by P1Z, P2Y and P3X.

4. Simulation

The FE model for the testing unit previously mentioned is generated, and the accelerations of the gearbox housing are simulated. The analytical results are compared with the experimental results. Thus, the validity of the tooth mesh model is examined.

4.1 FE Model of the Structure Figure 11 shows the FE model of the

whole system. The gearbox housing is modeled by thin-shell elements of 2nd order except for the bearing housings and the rectangular bars that fix top plate. These thick- walled parts are modeled by brick elements of 2nd order. Coincidence nodes connect the shell and brick elements. The bolts are modeled by beam

100

50

0

Mod

ifica

tion,

µm

Profile, mm

Face

wid

th, m

m

-17.

5

17.5

-7.3 7.3

22 33 44 8877665511

① DC motor ② Flexible coupling ③ Tested Gearbox④ Universal joint ⑤ Flywheel ⑥ Torque&Revolution pick-up⑦ Reduction gear ⑧ DC generator

22

Driver / Driven2.25

48 / 5014.5 deg26.75 deg

RH / LH35 / 35

123Center distance (mm)

ModuleNumber of teethPressure angle

Helix angleDirection of helixFace width (mm)

Fig. 12 FE-model of gear train

Fig. 11 FE-model of gearbox

Fig. 10 Power absorbing test rig

Fig. 9 Tooth surface geometry

Table 1 Gear data

B1

B3

B2

B4

Driving

Driven

Inner race

Coupling


Vol. 1, No. 4, 2007

548

elements. The bolts and circumferential area at bolt holes of top plate and bearing housings are connected by rigid elements.

Figure 12 shows the FE model of the gear train(12). The shafts are modeled by beam elements of 2nd order. Inner races of the bearings are modeled by brick elements of 2nd order and connected to the shafts by rigid elements. The gear bodies are modeled by tetrahedral elements and teeth are modeled by beam elements. The gear bodies and the shafts are connected to each other by 6 DOF springs that represents the stiffness of the ETP bushes. The flexible couplings are modeled by lumped mass and 6 DOF springs. The bearings are modeled by 5DOF springs except for the rotation about the axis of the shaft. These springs are used for connecting the shafts to the bearing housings.

The values of the spring constants of ETP bushes, flexible couplings and bearings are shown in the Ref. (12).

4.2 FE Model of Tooth Mesh The divided tooth mesh stiffness and

the divided excitation force are calculated by the methods previously described. Here number of division is two (m = 2) and torque value is 93.1Nm. The results are shows in Figs. 13 and 14. Figure 13 shows the excitation force. We can observe that two excitation forces

1E and 2E have initial phase difference from each other. This phase difference causes an excitation moment described later on. Figure 14 shows the tooth mesh stiffness 1k and 2k .

The teeth mesh area is modeled as shown in Fig. 15. The spring 1k and 2k are aligned in the direction of tooth normal and connect nodal points N1-N2 and N3-N4 respectively. Though N1 and N2 are separately shown in the figure for convenience, they are actually located at the same positions. This is the same thing about N3 and N4. Moreover, they are located on the pitch lines at the distance of 4/b± from the center of the face width. Then N1-N2 and N3-N4 are connected to the root circle of the driving and driven gear respectively by rigid elements. Thus, excitation force 1E , 2E applied at N1-N2 and N3-N4 can be transmitted to the gear bodies.

4.3 Method of Response Analysis I-Deas ms9 (FE Analysis Code) is used for the response analysis. The contact area of

the base of the gearbox housing to the steel blocks are completely constrained, and then eigenvalue analysis is conducted by Lanczos method. Using this result, the accelerations of the gearbox housing are calculated by mode superposition method at the frequency

0 1 2-100

0

100E1

Composite (=E)E2

Non-dimensional line of action x/tes

Mes

h ex

cita

tion

forc

e (N

)

0 1 20

200

400

k1

Composite (=k)

k2

Non-dimensional line of action x/tesTo

oth

stiff

ness

(×10

6N/m

)

N4N3

N1 N2

k2

Constraint elementsk1

bb 4b 4b 4b 4β b

E1Plane of action

Pitch line

E1

E2

E2

Fig. 14 Variation of tooth mesh stiffness

Fig. 15 Modeling of tooth mesh

Fig. 13 Mesh excitation forces


Vol. 1, No. 4, 2007

549

intervals of 5Hz. Modal damping ratio ζ=0.025, which is the average value of modal damping ratios obtained by shaker test, is applied to all modes(12).

4.4 Analytical and Experimental Results Two types of excitation forces are

used for the response analysis as follows. (i) [Explanatory note] FEM (Phase):

Two divided mesh excitation forces E1, E2 with initial phase difference shown in Fig. 13.

(ii) [Explanatory note] FEM (No Phase): Both of E1 and E2 are equal to (E1 + E2) /2.

The latter is the similar to the conventional methods that many researchers adopt(2)-(5); two excitation forces have no phase differences to each other in the latter method.

Figure 16 shows the acceleration of gearbox housing obtained by analysis and experiment. The analytical results [FEM (Phase)] using mesh excitation forces with initial phase difference show good correlation with the experimental results. It should be emphasized that resonance peaks marked by that appear around 1200Hz in experimental results of P1Z and P3X also appear in the analytical results. In contrary to this, the resonance peaks are not observed at all in the analytical results [FEM (No Phase)]. Therefore, we can conclude that the tooth mesh model proposed in this report is effective.

5. Effect of the Moving Load

We examine the effect of the moving load on the vibratory response. For this purpose, we use the tooth mesh model shown in Fig. 17. This model is approximately equivalent to the tooth mesh model shown in Fig. 2(11). In this model, the tooth mesh stiffness is modeled by a linear spring k and a rotational spring θk instead of kq (q = 1,⋅⋅⋅, m). In addition, a mesh excitation force E and a mesh excitation moment Mθ are given instead of Eq (q = 1,⋅⋅⋅, m). Here, k is the tooth mesh stiffness that is sum of divided tooth mesh stiffness defined by

∑==

m

qqkk

1. (16)

θk is the rotational tooth mesh stiffness regarding relative rotation of the contact line, and defined by

∑ ⋅==

m

qbqq zkk

1

22 cos βθ . (17)

E is the mesh excitation force that is the sum of divided mesh excitation forces defined by

P1ZP1Z

P2YP2Y

P3XP3X

FEM (Phase)FEM (No phase)Experiment


0 1000500 1500Frequency (Hz)





P1Z

P2Y

P3X

Acc

eler

atio

n (m

/s2 )

5

10

15

0A

ccel

erat

ion

(m/s2 )

5

10

15

0A

ccel

erat

ion

(m/s2 )

5

10

15

0

Acc

eler

atio

n (m

/s2 )

5

10

15

0

Acc

eler

atio

n (m

/s2 )

5

10

15

0

Acc

eler

atio

n (m

/s2 )

5

10

15

0

z　

x　

DrivingDriven

Plane of action　

EMθ

kθ

kP1

P2

E

Mθ

z　

x　

DrivingDriven

Plane of action　

EMθ

kθkθ

kkP1

P2

E

Mθ

Fig. 17 Modeling of tooth mesh

Fig. 16 Analytical and measured gearbox accelerations


Vol. 1, No. 4, 2007

550

∑==

m

qqEE

1. (18)

θM is the mesh excitation moment that acts in the plane of action. Because the mesh excitation forces Eq (q = 1,⋅⋅⋅, m) have the initial phases that are different from each other, they cause a moment. This moment exerts in the plane of action and is given by

∑ ⋅==

m

qbq qq zjEM

1cos]exp[ βϕθ . (19)

From above definitions, the excitation by E and Mθ is equivalent to the excitation by Eq (q =

1,⋅⋅⋅, m). Here we should keep in mind the followings: E is the excitation force caused by tooth surface error and periodical change of the tooth mesh stiffness as proved in the Ref. (11); Mθ is an additional moment caused by the moving load. In other words, Mθ is the excitation by the moving load.

Using the values ( k = 2.49×108N/m, θk = 1.54×104N/m, E = 36.4N and Mθ =

0.66Nm), the acceleration P3X is calculated based upon the model shown in Fig. 17. The result is shown in Fig. 18. The broken line represents the acceleration excited by E; the thin solid line represents the acceleration excited by Mθ ; the thick solid line represents the total acceleration. The result shows that the acceleration excited by Mθ has major contribution in the frequency rage of 1000 – 1500Hz. Especially, we should note that the resonance peak is excited by Mθ . Therefore, remembering that Mθ is the excitation by the moving load, we can conclude that the effect of the moving load on the vibratory response is significant. But this conclusion depends on the structure of system and the range of tooth mesh frequency.

Next, eigenvalue analysis is conducted to examine why such a big resonance peak appears. Figure 19 shows the mode shape of the gear train at the resonance peak (1210Hz). The mode shape shows the coupled mode between 2nd bending modes of the driving shaft and that of the driven shaft. This mode has the node at the gear mesh position and only tilting vibration is possible there. Therefore, this mode is not induced by E, but induced by Mθ .

Finally, we examine why such a strongly coupled vibration between 2nd bending modes of the shafts appears. For this purpose, it is very useful to use the excitation model by the dynamic tooth load Fd and the dynamic tooth moment Md

(11) when the tooth mesh is de-coupled in Fig. 17. Here, Fd and Md are given by

))((1]exp[

21 nnnd GGjk

jEF c +++=

ωϕ , (20)

))((1]exp[

21 θθθθ

θθ

ωϕ

GGjkjMM cd

+++= , (21)

where 1nG and 2nG are driving point compliances when the point P1 and P2 are excited by the unit force in the direction of tooth normal; 1θG and 2θG are driving point compliances regarding tilt when the point P1 and P2 are excited by the unit moment that acts in the plane of action; cn (250 Ns/m) and cθ (0.5 Nms/rad) are the coefficients of damping between the

P3XP3X

P3X

0 1000500 1500Frequency (Hz)

Acc

eler

atio

n (m

/s2 )

5

10

15

0

Acc

eler

atio

n (m

/s2 )

5

10

15

0

Vibration excited by EVibration excited by MθTotal vibration

xy

z

Fig. 19 Mode shape at 1210Hz

Fig. 18 Contribution of E and Mθ on the acceleration of the housing


Vol. 1, No. 4, 2007

551

teeth regarding linear and rotational displacement respectively. Figure 20 and 21 show the bode-diagrams of 21 nn GG + and

21 θθ GG + respectively. The horizontal lines in the figures shows Hn )/1( k= and Hθ

)/1( θk= that are the tooth mesh compliance regarding linear and rotational displacement respectively. Using these data, dF and dM are calculated. The obtained force transmissibility EFd / and moment transmissibility θMM d / are shown in Fig. 22. We observe several resonance peaks in the transmissibility. The generation mechanism of these resonances is clarified in the Ref. (10). For example, the major resonance peaks appear in dynamic tooth load when 21 nn GG + has the intersection with nH at its downhill side from the peak (right-side of the peak). This is the same thing about the dynamic tooth moment. According to this mechanism, we presume that the dynamic tooth load should be small value in the frequency range below 1600Hz, because the relation nnn HGG >>+ 21 is valid as shown in Fig. 20. Actually, EFd / is far below one )1/( <<EFd as shown in Fig. 22, and the excitation force is damped enough. In other words, the dynamic lateral stiffness of the shafts used in this study is much softer than tooth mesh stiffness. This makes the coupling of the vibrations in the direction of tooth normal weak. On the other hand, the downhill side of 21 θθ GG + has an intersection with θH at the point Q shown in Fig. 21. As the result, θMM d / is much amplified and resonance peak appears at this frequency. This makes the tilting vibration much coupled. In other words, the dynamic tilting stiffness of the shafts is small because of small diameters of the shafts; in addition to this, the rotational tooth mesh stiffness is comparatively large because of the wide face width. This makes it easier for the moving load to excite tilting vibration of the gears. Even if the stiffness of the shafts is high, the moving load should not be disregarded if the tooth mesh frequency is high, because the bending modes of higher order exist in the frequency range.

6. Conclusion

In this report, an integrated excitation model of the helical gear system is presented. This model incorporates the excitation caused by moving load as well as the excitation force caused by the tooth error and the variation of tooth mesh stiffness. This model is applied to the vibration analysis of a single stage gearbox. We can summarize our

0 1000500 1500Frequency (Hz)

0

3

2

1

Tran

smis

sibi

lity Md /Mθ

Fd /E

0 1000500 1500Frequency (Hz)

0

3

2

1

Tran

smis

sibi

lity Md /Mθ

Fd /E

0 1000500 1500Frequency (Hz)

10−6

10−3

10−4

10−5

Com

plia

nce

(rad

/Nm

)

−180Phas

e 0

Gθ1Gθ2Gθ1+Gθ2

Gθ1Gθ2Gθ1+Gθ2

Hθ

Q

0 1000500 1500Frequency (Hz)

0

10−9

10−5

10−6

10−7

10−8

Com

plia

nce

(m/N

)

−180Phas

e

10−9

10−5

10−6

10−7

10−8

Com

plia

nce

(m/N

)

−180Phas

e

Gn1Gn2Gn1+Gn2

Gn1Gn2Gn1+Gn2

Hn

Fig. 20 Bode-diagram of Gn1+Gn2

Fig. 21 Bode-diagram of Gθ1+Gθ2

Fig. 22 Force and moment transmissibility


Vol. 1, No. 4, 2007

552

observations in analytical and experimental results as follows. (1) The analytical and experiment results show that the presented model is appropriate

and effective for practical use. (2) If we neglect the excitation by the moving load, some vibrations never appears in

the analytical results. Therefore, the moving load should not be disregarded in the study of the vibratory phenomenon in the helical gearing.

(3) The main effect of the moving load is causing the mesh excitation moment. This moment excites the bending modes of the shafts. As the result, the tilting vibrations of the gears appear.

References

(1) Saiki, E., Matsunaga, T., Structure-borne and Air-borne Noise in a Gear Box, Proc. of 14th. JSME Symposium, No.780-2 (1978), pp. 49-54 (in Japanese).

(2) Houjoh, H., Umezawa, K. and Matsumura, S., Vibration Analysis for a Pair of Helical Gears Mounted on Elastic Shafts, ASME Power Transmission and Gearing Conference, DE-Vol.88 (1996), pp. 501-508.

(3) Wang, S., Houjoh, H., Matsumura, S. and Umezawa, K., Investigation of the Dynamic Behavior of a Helical Gear System, Transactions of JSME, Series C, Vol.64, No. 617 (1998), pp. 324-329.

(4) Chung, J., Steyer, G., Gear Noise Reduction through Transmission Error Control and Gear Blank Dynamic Tuning, Transaction of SAE, 1999-01-1766 (1999), pp. 1-9.

(5) Miyauchi, Y., Fujii, K., Nishino, T., Hatamura, K. and Kurisu, T., Introduction of Gear Noise Reduction Ring by Mechanism Analysis Including FEM Dynamic Tuning, Transaction of SAE, 2001-01-0865 (2001), pp. 1-9.

(6) Nogami, M., Tanaka, N., Nakamura, Y., Matsunaga, T., Noise and Vibration in Power Transmission System (7th Report, The Relationship between Fluctuation of Bearing Load and Noise in a Pair of Helical Gears), Proc. of 22th. JSME Symposium, No.850-3 (1985), pp. 233-238 (in Japanese).

(7) Kubo, A. and Umezawa, K., On the Power Transmitting Characteristics of Helical Gears with Manufacturing and Alignment Errors, Transactions of JSME, Vol.43, No. 371 (1977), pp. 2771-2779 (in Japanese).

(8) Nishino, T., Vibratory Response in Helical Gear System (1st Report, Analysis of Mesh Excitation Force), Transactions of JSME, Series C, Vol.64, No. 623 (1998), pp. 2688-2694.

(9) Nishino, T., Vibratory Response in Helical Gear System (2nd Report, Equations of Forced Vibration in Multi Degree of Freedom System), Transactions of JSME, Series C, Vol.64, No. 623 (1998), pp. 2695-2701.

(10) Nishino, T., Vibratory Response in Helical Gear System (3rd Report, Dynamic Tooth Load and its Control), Transactions of JSME, Series C, Vol.65, No. 631 (1999), pp. 1132-1139.

(11) Nishino, T., Vibratory Response in Helical Gear System (4th Report, Mesh Excitation Model Taking Moving Load into Account), Transactions of JSME, Series C, Vol.67, No. 664 (2001), pp. 3952-3960.

(12) Nishino, T., Vibration of Helical Gear System Including a Gearbox (Study on Analysis Method), Transactions of JSME, Series C, Vol.69, No. 679 (2003), pp. 743-751.

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