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Technical Memorandum 104434 AIAA912019
Technical Report 90C032
Analytical and Experimental Study of Vibrations in a Gear Transmission
F.K. Choy and YF. Ruan The University ofAkron Akron, Ohio
and
J.J. Zakrajsek, F.B. Oswald and J.J. Coy Lewis Research Center Cleveland, Ohio
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Prepared for the 27th Joint Propulsion Conference cosponsored by the AIAA, SAE, ASME, and ASEE Sacramento, California, June 2427, 1991
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ANALYTICAL AND EXPERIMENTAL STUDY OF JIBRhTIONS IN A GEAR TRANSMISSION F.K. Choy, Y.F. Ruan Department of Mechanical Engineering The University of Akron Akron, Ohio 44325 J.J. Zakrajsek, F.B. Oswald, and J.J. Coy National Aeronautics and Space Administration Lewis Research Center Cleveland, Ohio 44135
ABSTRACT This paper presents an analytical simulation of the dynamics of a gear transmission system compared to experimental results from a gear noise test rig at the NASA Lewis Research Center. The analytical procedure developed couples the dynamic behaviors of the rotorbearinggear system with the response of the gearbox structure. The modal synthesis method is used in solving the overall dynamics of the system. Locally each rotorgear stage is modeled as an individual rotorbearing system using the matrix transfer technique. The dynamics of each individual rotor are coupled with other rotor stages through the nonlinear gear mesh forces and with the gearbox structure through bearing support systems. The modal characteristics of the gearbox structure are evaluated using the finite element procedure. A variable time stepping integration scheme is used to calculate the overall time transient behavior of the system in modal coordinates. The global dynamic behavior of the system is expressed in a generalized coordinate system. Transient and steady state vibrations of the gearbox system are presented in the time and frequency domains. The vibration characteristics of a simple single mesh gear noise test rig is modeled. The numerical simulations are compared to experimental data measured under typical operating conditions. The comparison of system natural frequencies, peak vibration amplitudes, and gear mesh frequencies are generally in good agreement.

NOMENCLATURE A i(t)
A t(t)
modal function of the ith mode in xdirection
modal function of the i th mode in 8direction
Bi(t)
modal function of the ith mode in ydirection
[C bx] C][C bz (CT ] ICxx],
FBx FBy
gearbox damping matrices torsional damping matrix
C[y]CYX], [C
]
bearing direct and crosscoupling damping matrices
bearing excitation forces
F r(t) FGx(t),F y(t)
gear mesh torque gear mesh force in x and ydirections
FT(t) F (t),F (t)
external excitation moment external excitation forces gryoscopicangular acceleration matrix gyroscopicangular rotation matrix identity matrix rotational mass moment of inertia matrix average stiffness matrix gearbox stiffness matrix compensation matrices in x and ydirection shaft stiffness matrix torsional stiffness matrix gear mesh stiffness between ith and kth rotor bearing direct and crosscoupling stiffness matrix massinertia matrix of rotor massinertia matrix of gearbox the number of mod3s used to define each motion radius of gear in the ith rotor gear generated torque generalized motion in x and ydirections gearbox motion in x, y and zdirections gearbox motion at bearing supports
th gear displacements in x and ydirections of the i
(GA]
(GV
(I] [J] (K]
[Kbx [Kby](Kbz] Kdx Kdy
Ks [K ] Ktik [Kx×],[Ky][Ky×],(K yyI [M] IMb] m Rcl TF X,Y XbYb Zb Xbe Ybs Xci, Yc
rotor XF YF
gear forces in x and ydirections motion of rotor at bearing support angle of tooth mesh between kth and ith rotor rotational displacement of the ith rotor at the gear rotational displacement vector of the ith rotor
X ' Y
aki
Oci {01 i
2
2 (A ],(At2]
I
lateral and torsional eigenvalue diagonal matrices
[f
~
lateral and torsional orthonormal eigenvector matrices of the kth rotor
jth
0kJl
orthonormal mode of kth rotor at 1t h station
INTRODUCTION The study of gearbox vibration is an important area of research among the engineering community. With increased speed and torque requirements and reduced allowable vibration levels, a critical need exists to develop an accurate analytical simulation of the dynamics of gear transmission systems. Recently, a number of analytical modeling and simulation procedures have been developed to predict the dynamics of multistage gear transmission systems. Some experimental work has also been performed to improve the understanding of gear stress and vibration. There exits a need to correlate analytical and experimental efforts so that the analytical methods can be refined and verified. There is a wealth of literature on vibration analysis and simulation of gear transmission systems. August (1986) simulated the vibration characteristics of a threeplanet transmission system. Pike (1987) and later Choy (1988b) used the gear relationships introduced by Cornell (1981) to study the dynamics of coupled gear systems. Mitchell (1987) applied and matrix transfer method to simulate gear vibrations. Ozguven and Houser (1988) and Kahraman (1990) used a finite element model to predict the dynamics of a multigear mesh system. Choy (1991) applied a modal synthesis technique in conjunction with finite element and the matrix transfer methods in both time and frequency domains to calculate the transient rotor and casing motions. There are few studies correlating analytical predictions and experimental results. Some experimental correlation work by Lim (1990) compared analytical predictions of vibration of a gear housing and mounting with measurements. This paper reviews the development of a global dynamic model of a gear train system, and presents results of a comparison made between the analytical predictions of the model and experimental results from a gear test rig. A combined approach using the modal synthesis and finite element method for analyzing the dynamics of gear systems was developed. The method includes dynamic coupling between the housing and the gearrotor system. The single mesh gear noise test rig at NASA Lewis was used as an example in this analysis. The gearbox modal characteristics were evaluated by solving a finite element model. These modal characteristics were compared to those measured in the experimental study to insure accurate modeling of the experimental rotorgearhousing system. Transient vibrat.ions of the rotorgear stages and the gearbox housing were analytically evaluated by the modal method. A numerical FFT (Fast Fourier Transform) algorithm was used to examine vibration in the frequency domain. Frequency spectra of the predicted gearbox vibration were compared to those measured in the experimental study. The following discussion and conclusions are drawn from these comparisons between the experimental results and analytical predictions.
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ANALYTICAL PROCEDURE The equations of motion for a single mesh multimass rotorbearinggear shaft system, with the coupling effects of gearbox vibrations and the rotor inertiagyroscopic effects, can be written in matrix form for the ith rotor (Choy 1987, 1989) for the XZ plane as:
[M]{R}i + [Gifj+ [C~x],{A

Ab~i + [C~y]j{j'

±b}j +
[GA]i(Y~i + [KXX + K
I)
(1)

[KXx]{XbS)
+
[Kxy]i(y  Ybs}i = {Fx(t)}i + {F Gx (t
)}i
and in the YZ plane as:
[M1{fli

[G~]j{;0i +.
[Cy.,] 1{A


±tb~j
+
[Cyy],{k

~
[GA]{ii+ [xyy
+ K
(2)
X (Y}1i
[Kyy](YbsB}
+
[K.it
Ybi
(Fy(t)
}i
+
( FGYMt)
}
Here Fx and Fy are force excitations from the effects of mass imbalance and shaft residual bow in the X and Ydirections. F and F are the X GX and Y gear mesh forces induced from gear teeth interactions. Te bearing forces are evaluated from the relative motion between the rotor {X}, {Y} and the gearbox {Xb}, {Yb} at the bearing locations (Choy 1987). The massinertia and gyroscopic effects are incorporated in the mass matrix [M] and the gyroscopic matrices [Gv] and [GA]. The coupled torsional equations of motion for the single gearrotor system can be written as: Pi1+ [CT]i {B})1+[T
=8J(FT
(t)
}
{F~t (t)
}1
(3)
In Eq. (3), {F T(t)} represents the externally applied torque and {F (t)} represents the gear mesh induced moment. Note that Eqs. (1) to (3) repeat for each single gearrotor system. The gear mesh forces couple the force equations of each gearrotor system to each other as well as the torsional equations to the lateral equations (Choy, 1989; Cornell, 1981; David, 1987 and 1988). In addition, there are equations of motion for the gearbox which couple the various gearshaftsystems through the bearing supports. The gearbox equations can be written as: Xequation
[Mb]{(Rk}
+ [Cb.]{(±k} + [Cx,] (Ab k + [Cxy]~ t. )S + [Kb.]tXb}
(4)
[KX](Xb
 X. )+
[Kxy]{Yb  Y.}
0
Yequation
4
[Mb] t~b}) + [Cby] t%
) + [Cy.] (b

A
. +
Y9 }t+
 Y. + []Xb
[Kby](Yb}
(5)
[Kt]{Yb
X

0
and Zequation
[Mb](t2b}) + [Cb.
b+[bz]{b (9Zb
F bz(t)
(6)
where Fbz(t) is an excitation function due to external forces in the axial direction. Since the bearing is assumed to be uncoupled in the Zdirection, Eq. (6) can be solved independently without considering shaft motion. The torsional and lateral vibrations of a single individual rotor and the dynamic relationships between each gearrotorsystem are coupled through the nonlinear interactions in the gear mesh. The gear mesh stiffness varies in a periodic nonlinear pattern with each tooth pass engagement period (August, 1986; Cornell, 1981; Savage, 1986) and can be represented by a high order polynomial (Cornell, 1981; Boyd, 1987). For the coordinate system as shown Fig. 1, the following gear mesh coupling equations are established by equating forces and moments (Choy, 1989). For the kth gear of the system, summing forces in the Xdirection results in:
FGxk
En
i1, i0k
K
RB
tki 1
RB
cck
+(X
c
 X
k
ICOS a
ki
+
(y
ci

Yksin
J,
a iCos a(7)
ij
k
ici
Summing forces in the Ydirection results in: n F yk =
E
i=l, i k
KtkiRci~ci  RBCkk
+ (XciXck)cOs Oki
+
i
 Yck)sin
ak] sin aki
1
(8)
Summing moments in the Zdirection results in:
F
FGtk
~~
i=1, itk
Rci
Ki
~

Rk[ B oc)
c
ck
+
(Xc ck)CO
XcosiR+
alki
Y:iYk~
Y~i
9
where
n
is the number of gears in the system.
Using the modal expansion approach (Choy, 1987, 1988a, 1989, and 1990), the motion of the system can be expressed as:
5
M
M
inl
inl
{X
i~}{b
M
b(bi
(Y) 
Bi(Oi}rfyb)
i=I. i=l
B bj
(b{yi
(10)
=
inl
Ati{Zb} A8i (01
inl
DbiD(Oz1}
where
m
is the number of modes used to define each motion.
Using modal expansion and the orthogonality conditions, (Choy 1989, 1990) with the bearing forces due to the base motion expressed on the right hand side of the equations the modal equations of motion for the gearrotor system (Choy 1989) can be written as: XZ equation
+, } [,.J[Cxy][tJl + + A[o 1] (A) +, } [tf [Cx,,][t,](A (B) [,,y[Gv[,t]( },[,,
+ [A2](A)
f[Kd.
0(+][{A}
[.
xy][t] (B
=
[f{Fx(t).
Fx.(t)
+ F.x(t)
}
which can also be expanded into modal parameters as where
FBx(t)
and
= [[Ctb..
]{Ab)
+
[C.Y []{6b)+ [K.][b] {Ab} b }
+ [K'][s"]by
}
(1)
YZ equation
(a) [Gv][I,(A}  [,,
+ [A
where
FBy(t)
+ [
xT]Cy. + OJ{, c ][[ ][ ,, [(A)

[fGA][fl {A}
(13)
B)
+
[of([KdyJ0I(B}
+
T [fl [K xJ[O{ A) = [OJT {Fy(t) + Fy,(t) + F,(t
}
(14)
[cy][, bX]{A)
+ [cyyj][,byl{}+
[Ky][,xl](A}+
[Kyy1][,by](1
and the 8equation can be expressed as
t)~ + [t[CT][t](t+
rt)
6
=
.T(t(t)
+
FGt(t)
}(5
The gear mesh forces and moments can also be expressed in the modal form, for the kt gear with the gear location at the 1t node, as:
($IFGX
141 G.Okj
Ktk tR 0 Re tk ICi ci k c
+
(Xci Xck)cos
a ki
+ (Yci

YCk) sin ai]COS ak
J
(16)
10
(F Gy
J=1
O k3Jl
i=1
~k(17)
K tki[R~ioC i 
Rckc
+(x,,,

X,,4cos a
+ (Yci

Yck) sin akil]sin
aki
(01
{F~t}
E
ij
Rc
~Ktk
[Rcic
0
0 Rck~k )
+ t
x
xCkcos ak
(18)
+ (Yi
Y ck) sin akill
where k is the gearshaftsystem number, j the station location of the gear mesh.
is
the mode number,
and
1
is
A set of modal equations of motion can also be written for the gearbox (Choy 1989, 1990) as:
[IA}
[b]A,,} + f.
[&']{AD
[$XfFB,.t) L

[§}
(J[c~x][O]{A}[cx~][tJ{b} +
L(19)
bi
+ Similarly, the Yequation can be written as
[Kxx ]1IA}) + [Kxy][jIB}
0
['IAb}) + [CbyJ{t
b ) + [A2]tBb)
[tbyr{F.Y(t) I

[Obyf{([cy][0]{tA})
+
+ C][]tA
(20)
=Y 0
[Kyx][IA})e +
7
and the Zequation as
[I] 'b}
I + Cb.]b}
+
[A21{Db) = [bzTjFbz(t)}
(21)
The procedure involves the solution of the coupled modal equations of motion between the (gear) rotors and the gearbox structure. The coupling effects of the gear mesh and the bearing supports are also expressed in modal coordinates such that the global equations are solved simultaneously in modal form. A set of initial conditions for both displacement and velocity of the global system are calculated from the steady state conditions at the rotorbearing systems and zero vibration at the gearbox. The modal accelerations A, B, At , Ab, Bb, and Db of the system can be evaluated (Eqs. (11), (13), (15), and (19) to (21)). A variable time stepping integration scheme (the NewmarkBeta Method) is used to integrate the modal acceleration to evaluate the modal velocities and displacements at the next time step. A regular time interval of 200 points per shaft revolution is used in this study except for a refined region of smaller steps at the transition from single to multiple tooth contact. The modal acceleration, velocity, and displacement calculated from the transient integration scheme are transformed back into the generalized coordinates (Eq. (10)). The nonlinear bearing forces and gear mesh forces can be evaluated from the velocity and displacement differentials between the rotors and the gearbox structure.
EXPERIMENTAL STUDY
The gear noise rig (Fig. 2) was used to measure the vibration, dynamic load, and noise of a geared transmission. The rig features a simple gearbox (Fig. 3) containing a pair of parallel axis gears supported by rolling element bearings. A 150kW (200hp) variablespeed electric motor powers the rig at one end, and an eddycurrent dynamometer applies powerabsorbing torque at the other end. The test gear parameters are shown in Table I. Two phases of experiments were performed on the gearbox; (1) static modal analysis of the gearbox and (2) dynamic vibration measurements during operation. Modal parameters, such as system natural frequenceis and their corresponding mode shapes, were obtained through transfer function measurements using a two channel dynamic signal analyzer and modal analysis software. For this experiment, 116 nodal points (with 30 of freedom each) were selected on the aearbox housing. Vibration data was collected from accelerometers placed on the gearbox at a few of the node points used in the modal survey. Waterfall plots of frequency spectra from node 21 on the gearbox top and from node 40 on the gearbox end are presented in Figs. 4 and 5. The modal frequencies (from Figs. 6 and 7) and the gear mesh frequency are shown on figures 4 and 5.
DISCUSSION
The experimentally obtained modes, shown in Figs. 6 and 7, represent the major vibration modes of the gear noise rig in the 0 to 3 kHz region. Although these modal frequencies are only a small part of the total modes of
8
the system, they represent a major part of the total global vibration of the system. In order to produce a compatible analytical simulation of the test apparatus, a similar set of modes were generated using a finite element model of the gearbox structure. Out of a total of 25 modes existing in the analytical model in the 0 to 3 kHz frequency region, the eight dominant modes were used to represent the gearbox dynamic characteristics. These analytically simulated modes are shown in Figs. 8 and 9. As shown in Table II, the natural frequencies of the simulated modes are within 5 percent of the measured modes. The threedimensional analytical mode shapes are very similar to the experimental modes shapes (Figs. 6 and 7). The correlation in the results between the analytical model and the experimental measurements confirms the accuracy of the dynamic representation of the test gearbox using only a limited amount of modes. In the experimental vibration spectra of Fig. 4 (from the gearbox top), there are prominent peaks at the gear mesh frequency on the traces on 1500 rpm (700 Hz), 3750 rpm (1750 Hz), and 5500 rpm (2567 Hz). These frequencies are near natural frequencies found in the modal survey (658, 1762, and 2536 Hz, respectively). At the highest speed (5750 rpm), the higher modes (2536, 2722, and 2962 Hz) are excited by the gear mesh vibration and its sidebands. Similar behavior is shown in the vibration spectra from the side of the gearbox (Fig. 5). In both of these figures, the amplitude of the peaks increases with speed as the mass imbalance force increases with the square of the rotational speed. The vibration response of the gearbox was simulated by the analytical method presented in section II. The gearbox vibration modes were calculated by finite element methods. The gearrotor system response was calculated with a discrete rotor model using the matrix transfer technique. The gearrotor system was coupled to the gearbox in modal coordinates to solve for the transient vibration response of the global system. The vibration response was then transformed into the frequency domain through a fast Fourier transformation routine. The analytical vibration spectra shown in Figs. 10 and 11 simulate the experimental results of Figs. 4 and 5. Both the experimental spectra (Figs. 4 and 5) and the analytical spectra (Figs. 4 and 5) show excitation at the natural frequencies at shaft speeds of 1500, 3500, and 5500 rpm. At the higher shaft speed (5500 rpm), several modes are excited by the shaft frequency and its sidebands. Small differences between the analytical and experimental spectra may be due to (1) small errors (<5 percent) in the calculated modes, (2) nonuniformaties in the gearbox not present in the modal, (3) errors in measurement of rotational speeds and vibration amplitudes, and (4) limitations in modeling the bearings as a radial stiffness element only.
CONCLUSIONS Analytical and experimental studies system were performed. Results from the correlation with experimentally obtained basic conclusions from this study can be of a single stage gear transmission analytical simulations show good data from the test gearbox. Some summarized as follows:
1. A modal synthesis approach can be used to simulate the dynamics of a gear transmission system. 9
2. The accurate gearbox model can be developed by correlating the modal characteristics from experimental study with those from analytical simulations. 3. The proper choice of modes used in the modal synthesis will reduce the number of modes required in the analysis, without sacrificing accuracy in the numerical solution.
REFERENCES August, R. and Kasuba, R., "Torsional Vibrations and Dynamic Loads in a Basis Planetary Gear System," Journal of Vibration, Acoustics, Stress, and Reliability in Design, Vol. 108, No. 3, July 1986, pp. 348353. Boyd, L.S. and Pike, J.A., "Epicyclic Gear Dynamics," AIAA Journal, Vol. 27, No. 5, May 1989, pp. 603609. Choy, F.K. and Li, W.H., "Frequency Component and Modal Synthesis Analysis of Large RotorBearing Systems with Base Motion Induced Excitations," Journal of the Franklin Institute, Vol. 323, No. 2, 1987, pp. 145168. Choy, F.K., Padovan, J., and Li, W.H., "Rub in High Performance Turbomachinery: Modeling; Solution Methodology; and Signature Analysis," Mechancial Systems and Signal Processing, Vol. 2, No. 2, Apr. 1988, pp. 113133. Choy, F.K., Townsend, D.P., and Oswald, F.B., "Dynamic Analysis of MultimeshGear Helicopter Transmissions," NASA TP2789, 1988. Choy, F.K., Tu, Y.K., Savage, M., and Townsend, D.P., "Vibration Signature Analysis of Multistage Gear Transmission," 1989 International Power Transmission and Gearing Conference, 5th, Vol. 1, ASME, New York, 1989, pp. 383390. (Also, NASA TM101442). Choy, F.K., Tu, Y.K., Zakrajsek, J.J., and Townsend, D.P., "Dynamics of Multistage Gear Transmission with Effects of Gearbox Vibrations," Proceedings of the 1990 CSME Mechanical Engineering Forum, Vol. 2, Canadian Society for Mechanical Engineering, 1990, pp. 265270. (Also, NASA TM103109). Cornell, R.W., "Compliance and Stress Sensitivity of Spur Gear Teeth," of Mechancial Design, Vol. 103, No. 2, Apr. 1981, pp. 447.459. David, J.W. and Park, N., "The Vib~aticn Problem in Gear Coupled Rotor Systems," llth ASME VibratiUns and Noise Conference, Boston, MA, Sept. 29, 1987. David, J.W., Mitchell, L.r , and Daws, J.W., "Using Transfer Matrices for Parametric System Forced Response," Journal of Vibration, Acoustics, Stress and Reliability in Design, Vol. 109, No. 4, Oct. 1987, pp. 356360. Kahraman, A., Ozguven, H.N., Houser, D.R., and Zakrajsek, J.J., "Dynamic Analysis of Geared Rotors by Finite Element," NASA TM102349, AVSCOMTM89C006, 1990. Journal
10
Lim, T.C., Signh, R., and Zakrajsek, J.J., "Modal Analysis of Gear Housing and Mounts," 7th International Modal Analysis Conference, Vol. 2, Society for Experimental Mechanics, Bethel, CT, 1990, pp. 10721078. (Also, NASA TM101445). Mitchell, L.D., and David, J.W., "Proposed Solution Methodology for the Dynamically Coupled Nonlinear Geared Rotor Mechanics Equations," Journal of Vibration, Acoustics, Stress, and Reliability in Design, Vol. 107, No. 1, Jan. 1985, pp. 112116. Ozguven, H.N. and Houser, D.R., "Mathematical Models Used in Gear Dynamics  A Review," Journal of Sound and Vibration, Vol. 121, No. 3, Mar. 22, 1988, pp. 383411.
11
TABLE I.

TEST GEAR PARAMETERS
Gear type....................Standard involute, fulldepth tooth Number teeth.....................................28 Module, mm (diametrial pitch in.'I).......................3.174(8) Face width, mm (in.)..........................6.35(0.25) Pressure angle, deg.................................20 Theoretical contact ratio...........................1.64 Driver modification amount, mm (in.)...................023t0.0009) Driven modification amount, mmn (in.).................0.025(0.0010) Driver modification start, deg............................24 Driven modification start, deg............................24 Toothroot radius, mm (in.).........................1.35(0.053) Gear quality.............................AGMA class 13 Nominal (100 percent) torque, Nm(in.lb) ............. 71.77(635.25)
TABLE II.

COMPARISON OF EXPERIMENTAL
MEASURED AND ANALYTICAL MODELED NATURAL
____________FREQUENCIES
Experimental, Hz 658 1049 1710 2000 2276 2536 2722 2962
Analytical, Hz 658 1006 1762 2051 2336 2536 2752 3012
Difference, percent 0 4.1 3.0 2.6 2.6 0 1.1 1.7
0k
=rr+ 0

ith stage Y
kth stage Figure 1Geometry of simulation of gear force. 12
DynamometerSoft coupling increaser Drive motor
Test
gearbox
Figure 2. Picture of gear noise rig.
Node 21 Z 
Node 40Y (on back side)

7
Figure 3  Test gear box.
13
Modes 1 Speed, rpm 6000
'
2
3 4 5 678 Mesh frequency
'
5500
5000 4500 E 4000
.2 3500 . 3000 2500 2000

1500 0 1 Mesh frequency J
L
2
,
3 Frequency, 4 kHz 5 6
Figure 4. Experimental vibration frequency spectrum at node 21 (note: The measured modal frequencies are shown at top of figure).
Modes 1 Speed, rpml 6000 5500 E
2
2
34 5 678
I I I If ehrequency
5000 4500 4000 3500 3000 2500 2000 0 ,1 Mesh frequency I
6
0
E < 1 >
1500
2
3 4 Frequency, kHz
5
6
Figure 5. Experimental vibration frequency spectrum at node 40 (note: The measured modal frequencies are shown at top of figure).
14
y
y
Mode: 1 Frequency, 658.37 Hz
Mode: 2 Frequency, 1048.56 Hz
z
z
Y Mode: 3 Frequency, 1709.77 Hz Mode: 4 Frequency, 1999.95 Hz
Y
Figure 6. Gear box experimental mode shapes.
15
z xy Mode: 5 Frequency, 2275.69 Hz Mode: 6 Frequency. 2535.77 Hz xy
z
x y Mode: 7 Frequency, 2722.16 Hz Mode: 8 Frequency, 2961.71 Hz Figure 7. Goor box oxporimontal modo shapos
16
x Mode: 1 Frequency  658 Mode: 2 Frequency  1006
x
Mode:3 Frequency  1762 Figure 8.
Mode:4 Frequency  2051 Gear box analytical mode shapes.
17
y Mode: 5 Frequency  2336 Mode: 6 Frequency  2536
y
'C
x
Mode:7 Frequency = 2752
Mode:8 Frequency  3012 Figure 9. Gear box analytical mode shapes.
19
Mesh frequency Modes 1 Speed, rpm 2 3 4 5 6 7 8
E 2 4500
. E 3500
0
2500
1500
50 100 150 0 Mesh frequency 1 JFrequency,
J.ii
I
I
200
L
250 kHz
I
300
I
350
400
I
Figure 10. Analytical vibration frequency spectrum at node 21 (note: The prediced modal frequencies are shown at top of figure).
Modes 1
2
3
4
5
6
7
7
8
Speed, rpm
Mesh frequency
5500
E5
2500

,'

,___
.__._______._
.6 2500
1500
0 50 Mesh frequency
100
150
200 Frequency, kHz
250
300
I
350
I
400
I
Figure 11 . Analytical vibration frequency spectrum at node 40 (note: The prediced modal frequencies are shown at top of figure).
19
NASA
1. Report No.
Nao4 Awonaulic and Spune Ad&Wnistam' NASA TM 104434
Report Documentation Page
2. Government Accession No. 3. Recipient's Catalog No.
AVSCOM TR 90C032 AIAA9120191
4. Title and Subtitle 5. Report Date
Analytical and Experimental Study of Vibrations in a Gear Transmission
6. Performing Organization Code
7. Author(s)
8. Performing Organization Report No.
F.K. Choy, Y.F. Ruan, J.J. Zakrajsek, F.B. Oswald, and J.J. Coy
E6144
10. 9. Perfrming Organization Name and Address
Work Unit No.
5056356
1L16221 A47A
11. Contract or Grant No.
NASA Lewis Research Center Cleveland, Ohio 44135  3191 and
and Propulsion Directorate U.S. Army Aviation Systems Command
Cleveland, Ohio 44135  3191
12. Sponsoring Agency Name and Address
13. Type of Report and Period Covered Technical Memorandum
14. Sponsoring Agency Code
National Aeronautics and Space Administration
Washington, D.C. 205460001 and U.S. Army Aviation Systems Command St. Louis, Mo. 63120 1798
15. Supplementary Notes
Prepared for the 27th Joint Propulsion Conference cosponsored by the AIAA, SAE, ASME, and ASEE, Sacramento, California, June 2427, 1991. F.K. Choy and Y.F. Ruan, Department of Mechanical Engineering, The University of Akron, Akron, Ohio 44325. J.J. Zakrajsek, F.B. Oswald, and J.J. Coy, NASA Lewis Research Center. Responsible person, J.J. Zakrajsek, (216) 433 3968.
16. Abstract
This paper presents an analytical simulation of the dynamics of a gear transmission system compared to experimental results from a gear noise test rig at the NASA Lewis Research Center. The analytical procedure developed couples the dynamic behaviors of the rotorbearinggear system with the response of the gearbox structure. The modal synthesis method is iied in solving the overall dynamics of the system. Locally each rotorgear stage is modeled as an individual rotorbearing system using the matrix transfer technique. The dynamics of each individual rotor are coupled with other rotor stages through the nonlinear gear mesh forces and with the gearbox structure through bearing support systems. The modal characteristics of the gearbox structure are evaluated using the finite element procedure. A variable time stepping integration scheme is used to calculate the overall time transient behavior of the system in modal coordinates. The global dynamic behavior of the system is expressed in a generalized coordinate system. Transient and steady state vibrations of the gearbox system are presented in the time and frequency domains. The vibration characteristics of a simple single mesh gear noise test rig is modeled. The numerical simulations are compared to experimental data measured under typical operating conditions. The comparison of system natural frequencies, peak vibration amplitudes, and gear mesh frequencies are generally in good agreement.
17. Key Words (Suggested by Author(s))
18. Distribution Statement
Gears Vibration Vibration effects
Unclassified  Unlimited Subject Category 37
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Unclassified NASA FORM 142 OCT 06
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20
A03
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