A non-linear mathematical model for dynamic analysis of spur gears including shaft and bearing dynamics

Journal of Sound and Vibration (1991) 145(2), 239-260

A NON-LINEAR ANALYSIS

MATHEMATICAL

OF

SPUR

GEARS

BEARING

MODEL

INCLUDING

FOR

DYNAMIC

SHAFT

AND

DYNAMICS

H. N. ~ZG~JVEN Mechanical Engineering Department, Middle East Technical University, Ankara, Turkey (Received 29 November 1989, and in jinal form 15 May 1990)

A six-degree-of-freedom non-linear semi-definite model with time varying mesh stiffness has been developed for the dynamic analysis of spur gears. The model includes a spur gear pair, two shafts, two inertias representing load and prime mover, and bearings. As the shaft and bearing dynamics have also been considered in the model, the effect of lateral-torsional vibration coupling on the dynamics of gears can be studied. In the non-linear model developed several factors such as time varying mesh stiffness and damping, separation of teeth, backlash, single- and double-sided impacts, various gear et-rots and profile modifications have been considered. The dynamic response to internal excitation has been calculated by using the “static transmission error method” developed. The software prepared (DYTEM) employs the digital simulation technique for the solution, and is capable of calculating dynamic tooth and mesh forces, dynamic factors for pinion and gear, dynamic transmission error, dynamic bearing forces and torsions of shafts. Numerical examples are given in order to demonstrate the effect of shaft and bearing dynamics on gear dynamics.

1. INTRODUCTION The first vibratory models for gear dynamics were given in the 1950s [ 1,2], although the concern with gear loads goes back to the eighteenth century. Experimental studies and

more reliable models on the dynamic behavior of gears have been reported in the past 30 years. Even though there are several mathematical models developed for gear dynamics, there are only a few extensive experimental studies that can be used to verify these models. Attia [3] presented a set of experimental results giving the dynamic loads in spur gears. Munro [4] used a lightly damped test rig to measure the dynamic transmission of a spur gear pair at different speeds. Kubo [5] has measured dynamic tooth stresses in spur gears at a wide range of speeds. Mathematical models developed for the dynamic analysis of gears range from simple single-degree-of-freedom linear systems to non-linear rotor dynamics models. Recent researches in this field include the studies of Kubo et al., [6,7], Kasuba and Evans [!I], Wang and Cheng [9, lo], Bahgat ef al. [ll], Umezawa et al. [ 121, Kiiciikay [ 131, Lin et al. [ 14,151, &giiven and Houser [ 161, Kahraman et al. [ 171, and Kahraman and Singh [18]. An extensive review of the literature on dynamic modeling of gears has been given by Ozgiiven and Houser [19]. It has been observed that dynamic models suggested for gear systems may vary considerably, and yet it may still be possible to obtain similar predictions by using completely different models for certain systems. However, this depends on the relative dynamic properties of the systems. For instance, when the relative values of torsional and flexural stifhresses of shafts and of the mesh stiffness are such that the mesh mode can be uncoupled from the other vibrational modes, it may be sufficient to use a 239 0022-460X/91/050239+22

$03.00/O

@ 1991 Academic

Press Limited

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H. NE~Z,~T~ZC~IJVEN

single-degree-of-freedom model for obtaining accurate predictions. Use of a multi-degreeof-freedom model which includes the coupling between the torsional and transverse vibrations will not increase the accuracy of the predictions significantly under such circumstances. When the relative values of the dynamic properties cause a strong coupling among the mesh mode and other modes, however, a single-degree-of-freedom model, no matter how elaborate it is, cannot be useful for accurate predictions. While the degree of freedom of the model constructed is important, a more important feature of the model is the effects included, such as non-linearity of the elements, excitation due to gear errors, time variation of mesh stiffness and damping. The purpose of the dynamic simulation plays a very important role in constructing a suitable model. The major concern in dynamic modeling of gears varies considerably from dynamic tooth stresses to stability, and from noise radiation to rotor whirl. Therefore it is not possible to construct the “best” model unless the purpose of the analysis and the relative dynamic properties of the gear system are known. Although several multi-degree-of-freedom models have been developed for gear dynamics, only torsional vibrations are considered in most of them. David and Mitchell [20] have pointed out the importance of dynamic coupling in geared rotor systems by studying the effect of linear coupling terms. David [21,22] also studied dynamic coupling in non-linear geared rotors. His model consists of a single geared rotor excited at mesh frequency. Iida et al. [23-251 have investigated coupled torsional-lateral vibration of geared shafts and concluded that, when at least one of the shafts has a soft support, there will be torsional-lateral vibration coupling. They have studied the effect of some parameters on the dynamic coupling and the critical speeds. They used linear models in their studies. Iwatsubo et al. have first assumed constant mesh stiffness [26] and then included the periodic variation of mesh stiffness [27] in their rotor dynamics models in which two mutually perpendicular displacements of the gear and pinion have been considered, as well as two torsional degrees of freedom. Bearing, load and prime mover dynamics and backlash have not been considered in their models. In the eight-degree-of-freedom model of a helical gear pair suggested by Kiiciikay [28] parametric excitation due to mesh stiffness variation has been considered and a perturbation method has been used for the steady state dynamic response computations. The effects of backlash and load dependent contact ratio have been studied. The load and prime mover dynamics were not considered in Kiiciikay’s model, and only harmonic excitatons were assumed. In the finite element model of Neriya et al. [29,30] constant mesh stiffness has been assumed and a linear dynamic analysis has been made to determine the coupled torsional-lateral response to mass unbalance and geometric eccentricity in the gears. In a recent study, Kahraman et al. [17] have suggested a finite element model for a geared rotor system on flexible bearings. The model is an extension of the rotor-disk-bearing model of &giiven and Ozkan [31], in which several effects such as the rotary inertia of shaft elements, axial loading on shafts, shear deformations, flexibility and damping of bearings, material damping of shafts were included. Even though a constant mesh stiffness was used in this recent finite element model of geared shafts, the excitation effect of mesh stiffness variation was included as an internal excitation in the form of loaded static transmission error, as recommended by dzgiiven and Houser [ 161 for complicated models. The purpose of this paper is to develop a multi-degree-of-freedom non-linear model for a spur gear pair that can be used to study the effect of lateral-torsional vibration coupling on dynamic tooth forces and dynamic transmission error, by retaining all other features of a previously developed [ 161 single-degree-of-freedom non-linear model (namely, variation of mesh stiffness and damping, backlash, separation of teeth in mesh, back side collision that may follow tooth separation, excitation due to gear errors and

NON-LINEAR

GEAR

DYNAMICS

241

profile modifications). In the new model, torsional compliances of shafts, transverse bearing compliances and dampings, material dampings of shafts, and load and prime mover inertias are also considered. The model has six degrees of freedom (four rotations and two translations). In the formulation, the STE (Static Transmission Error) method developed by &gilven and Houser [16] is employed, and thus the dynamic transmission error of a gear system is calculated from theoretically calculated loaded static transmission error (by using STEPSpur Gear Transmission Error Program-[32]) or from experimentally measured loaded static transmission error. The computer program prepared, DYTEM, is very similar in function to the program DYTE prepared in the previous study [ 161 where a single-degreeof-freedom model was used. The program can be used to predict the dynamic mesh and tooth forces, dynamic factors for pinion and gear, dynamic transmission error, torsion of shafts, and dynamic bearing forces in high-speed gearing. The dynamic factors have been defined either as the ratio of the maximum dynamic root stress to the maximum static root stress in a gear tooth, or as the ratio of the maximum dynamic force to the maximum static force on a gear tooth. Although the computer program DYTEM can calculate the dynamic factors in either way, in this paper the term “dynamic factor” refers to the first definition. The approximate method suggested for the single-degree-of-freedom non-linear model of a gear pair in the previous work [ 161 has also been applied to the six-degree-of-freedom model developed in this study. In this approach a constant mesh stiffness is used while the excitation effect of the mesh stiffness variation is included in the analysis through the loaded static transmission error input. In program DYTEM either the STE method or the approximate method can be used for the dynamic simulation. 2. FORMULATION 2.1. COUPLING

OF MODES

In most of the gear pair systems the coupling between the torsional vibration mode controlled by mesh stiffness, and the other vibrational modes is weak. Therefore a mathematical model representing only the torsional vibrations of a gear pair which is controlled by mesh stitIness (see Figure 1) may yield quite accurate results in most practical cases, as demonstrated in a previous work [ 161. It was concluded in this previous work that increasing the degree of freedom of the model by including the compliances and inertias of other system elements might not yield a considerable improvement in the

Cm

e(t) -

p--1: ---b

a.

I2

Figure 1. Two d.o.f. semi-definite

dynamic

model for a gear pair.

242

H.

NEVZAT

~ZG~~VEN

dynamic stress and transmission error predictions when the modes are almost uncoupled. When the torsional mode which is controlled by mesh stiffness is coupled with other vibration modes, however, the compliances responsible from these modes are to be included in the dynamic analysis. In such problems, increasing the degree of freedom of the model by including compliances of other elements is more important than elaborating the single-degree-of-freedom model with some “secondary effects” such as tooth friction. The model considered here consists of two gears on two shafts which are connected to a load and a prime mover. The model includes four inertias, two of them representing the load and the prime mover, and the other two representing the gears. The torsional compliances of shafts, and the transverse compliances of bearings combined with those of shafts are included in the model. Both bearing and shaft dampings are also considered in the model. The transverse vibrations of gears are considered along the line of action. With this model, the response including modulations due to transverse and torsional vibrations stemming from bearing and shaft compliances can be calculated. Thus, the coupling between the torsional and transverse vibrations of gear-shaft-bearing systems can be investigated, and the effect of bearing compliances and dampings on gear dynamics can be studied. The time variation of mesh stiffness and damping, and non-linearities due to backlash and tooth separation, are also considered in the model. 2.2.

SIX-DEGREE-OF-FREEDOM

NON-LINEAR

MODEL

The six-degree-of-freedom non-linear model suggested in this study is shown in Figure 2. It has four angular rotations (of prime mover, pinion, gear and load), and two translations (of pinion and gear) along the line of action.

Figure 2. The six d.o.f. model of a gear system suggested in this study.

NON-LINEAR

GEAR

243

DYNAMICS

The effects which are included in the mathematical model and thus considered in the dynamic analysis are as follows: (1) time varying mesh stifEness and mesh damping; (2) torsional compliances of pinion and gear shafts; (3) material damping in shafts (linear viscous); (4) bearing compliances and dampings (linear viscous); (5) transverse compliantes of shafts; (6) inertia of prime mover and load; (7) drive and load torques; (8) separation of teeth in mesh; (9) backlash; (10) back side collision that may follow tooth separation; (11) gear errors (pitch, profile and runout errors), and profile modifications. The mesh stiffness calculated by program STEP [32] is used in the analysis. In the computation of mesh stiffness, shear deformations, bending, base rotation and Hertzian deflections are taken into consideration. Dynamic analysis is made in the plane of gears, and any out-of-plane motion is neglected. As the excitation is basically along the line of action, transverse vibration of gears in the direction perpendicular to the line of action is neglected. Another simplifying assumption made is to neglect the off-line of action and the friction between the teeth in contact. When the approximate method suggested in reference [ 161 is used, a further assumption of constant mesh stiffness and mesh damping will be made. However, the parametric excitation effect of time varying mesh stiffness will be included into the analysis, as discussed in detail in reference [16]. 2.3.

APPLICATION

OF

THE

MULTI-DEGREE-OF-FREEDOM

STATIC

TRANSMISSION

ERROR

METHOD

TO

THE

MODEL

The governing equations of motion for the model depicted in Figure 2 can be written as follows (a list of symbols is given in the Appendix): Z,di,+c,,(i,-~,)+k,,(e,-e,)=

TD,

Z,Bi,+c,,(e,-8,)+k,,(e,-e,)=-WR,, (1,X)

I,& + c,,( 62 - d, ) + kJ 192- OL) = WRz , m& + c,Y, + k,y, = w,

Here W( = W(t))

ZL~~+c,~(~L-~*)+k,Z(eL-ez)=-TL, m& + czjz + k2y, = - W.

(3,4) (596)

is the dynamic mesh force given by

W=k,(8,R,-eZR2+y2-y,)-k,,ea-kbeb+c,(~,R,-~,R,+);,-)i,)-c,e,-c~PC,

(7)

and a dot-denotes differentation with respect to time. In the above formulation, the transverse stiffness of the shafts at the pinion and gear locations can also be considered in the analysis by taking k, and k2 as the equivalent stiffness values representing both bearing and shaft stithiesses. This may be important when the bearings are not located close to gears so that the transverse compliances of the shafts will contribute to the total transverse compliances. By introducing the dynamic transmission error x, which is defined as x=R,8,-R*8*,

(8)

equation (7) can be rewritten as W=k,(x-ty,-y,)-k,e,-kbeb+c,(i+~,-j,)-c,P,-cb~b.

The dynamic transmission x=f!+(y,-y*)+ m

(9)

error, x, can be written from equation (9) as

ke, + kbeb c, -k(~+l:?_i’ ,)+cai~k+cbpb. km m m

(10)

244

H. NEVZATG~CXJVEN

By introducing mesh compliance C( = l/k,,,), and decomposing C and the dynamic mesh load W respectively into their average components C and W, and their fluctuating components SC and 6 W, the first term in equation (10) can be written as W/k,=

WC=(W,+SW)(~+K)=

W,C+&W+SWSC.

(11)

Substitution of equation (11) into equation (10) and the use of the definition of loaded static transmission error, x,, which is given by x, = WO/k, + (k,e, + kbeb)/ k,,

(12)

yield x=csw+x,+(y,-y*)-c,c(3i.+4;*-j,)+c(c0~~+c&).

(13)

Then, by following an approach similar to the one used in reference [16], the dynamic mesh load W can be obtained as W= w,+~(l/~)/(1+SC/~)}(e,R,-e,R,-x,+y,-y,)+c,(~,R,-~,R,+);,-jl,). (14) The time varying mesh damping can approximately damping ratio 5 as

be expressed in terms of the mesh

c, = 25( k,m,)“‘,

(15)

where m, is the equivalent mass representing

pinion and gear inertias, defined as

m, = ZIZz/(ZIR:+ Z,R:). The definitions given above for are exact for a model consisting the six-degree-of-freedom model of shafts in torsional vibrations material damping of shafts, as

(16)

the equivalent mass m, and the mesh damping ratio 6 of two gears only, and can be used approximately for introduced. Similarly, the viscous damping coefficients can be calculated from damping ratio &, representing

ct, =25&Z,,)“*,

c,, = 25&ZeJ”‘,

(17918)

where the equivalent inertias Ze, and Z,, are given by Z,, = ZDZ,l(ZD+ Z,),

L,=LZ*/(k+Z*).

(19920)

Then, equations (l)-(6) and (14) provide the governing differential equations of motion, with the damping coefficients ct,, ch and c,,, being given by equations (15)-(20). The only input to the system is the loaded static transmission error which includes both gear error information and a part of mesh stiffness variation information. The above equations are valid if there is no tooth separation or back side collision. When tooth separation occurs between two gears, it is necessary to check whether or not there is back side collision, If there is no back side collision while there is tooth separation, the dynamic load W will be zero, as there will be instantaneously no contact between two gears. If there is back side collision, however, W can be calculated approximately from w= w,+{(l/~)/(l+6C/~)}(e,R,-e,R,-x,+y2-y,+b)+c,(B,R,-8*R,+)i2-~,)r (21) where b is the backlash.

NON-LINEAR APPLICATION

2.4.

OF THE

GEAR

APPROXIMATE

245

DYNAMICS

METHOD

the approximate method suggested in reference [ 161 is used for the multi-degreeof-freedom model, in which SC is taken to be zero, the equations of motion will be given by equations (l)-(6), where W is expressed as When

w= W,+{(e,R,-e,R,-x,+y~-y,)/~}+~~(8,R,-~,R,+jl,-jl,),

(22)

in which i;, = 25( m,/ C)“*.

(23)

Although a constant mesh stiffness is used in this method, the excitation effect of the time varying mesh stiffness will be indirectly included in the analysis through the loaded static transmission error, which is taken as a displacement input. However, the change in the natural frequency of the system due to changing from single tooth pair to double tooth pair contact will be ignored in this approximate analysis. The tooth separation and back side collision possibilities are checked in this method as well, and when separation or back side collision occurs, the dynamic mesh load W is taken, respectively, as zero or w= w,+{(e,~,-e,~,-x,+y2-y,fb)/~}+~~(d,R,-~2RZ+)iZ-);,).

(24)

3. COMPUTATIONAL PROCEDURE The solutions of the differential equations were obtained by numerical integration. The fifth order Runge-Kutta with adaptive step size control has been employed after writing each second order differential equation in the form of two first order differential equations. Thus 12 non-linear first order differential equations were obtained. These equations were written such that each equation contains the time derivative of only one variable. Then it is possible to express them as ii

=_L(zl~

z2,

i =

* * * 3 z*2)v

1,2,. . . ,12,

(25)

where zi represents a state variable, which is either a displacement or velocity. For instance, when there is no separation or back side collision, the equations in terms of state variables will be as follows: i, = z2,

i2=(1/zD)[~D-cl,(Z*-Z~)-k,,(z,-z,)l,

&=z4,

(26-28)

i4=(1/z,)r-C,,(Z~-Z2)-~r,(Z3-Z,)-R,Wl, ~6=(1/z*)[-C,~(Z~-Z*)--kr,(Z5-z7)-~2Wl,

z5=i%, i, = z*,

(29)

~8=(~/zL)[-~L-c,~z(~8-z~)-~,~(~7-z~)1,

%o=(I/m,)(-c,z,o-k,z,+

W),

&,=z12,

&=zlO,

%2=(l/m2)(--c,z,2-~2z,,-

(30,3I) (32-34) w.

(35-37) Here w= wo+[1/(1+SC/~)][(1/~)(R,z,-R,z,-x,+z,,-z~)l+C~(R,Z~-R2Z~+Z*2-Z,O) (38) when the static transmission z1 = eD,

error method is used. Also here

z3 = e,,

Z5=e2,

z7 = eL,

z9=y1,

Zll =y2.

(39)

246

H. NEVZAT

6ZGtiVEN

In the solution each mesh cycle was divided into 50 points. Although fewer points can be used in the analysis, it has been observed that 50 points per mesh cycle can provide good accuracy in the numerical solution. The loaded static transmission error calculated in STEP [32] is employed as an input in the dynamic analysis. After finding the time step, At, corresponding to the travel between two successive contact points at a given rotational speed, the zi are calculated by using the initial values of zi (i = 1,2,. . . , 12), and then numerical integration is performed to find the new values of zi. In obtaining zi, the dynamic mesh force W is calculated. Negative mesh force calculated from equation (38) indicates the separation of gears. Then, a back side collision is to be checked by comparing dynamic transmission error by the backlash. Depending on whether there is back side collision or just a separation, a different equation (equation (21) or W = 0) is used in the calculation of the actual dynamic mesh force. If a back side collision is detected, then the dynamic mesh force that will be calculated from equation (21) will have a negative sign. Therefore, a negative mesh force in output indicates a back side collision. As the initial values of zi, three different sets of values may seem logical: (a) all zero values; (b) initial displacements corresponding to the static displacements due to the nominal torque, initial angular velocities corresponding to the nominal operating speed, and zero translational velocities; (c) initial displacements corresponding to the static displacements due to the nominal torque, and zero values for initial velocities. All of these sets may have physical meanings under different conditions. The first set corresponds to a gear pair at rest without any load on it. Therefore, starting with all zero initial values would correspond to the dynamic analysis of a gear system which is initially at rest and unloaded, and so the analysis would give the transient response. This option is provided in the computer program prepared. However, due to the relatively low shaft stiffnesses, the first two natural frequencies of the system will be much lower than the natural frequency governed by mesh stiffness and, therefore, the free oscillations at these lower natural frequencies (excited by the torque applied suddenly at t = 0) will have longer periods compared to the mesh period. As a result of free oscillations with a very large period and a very low damping, it will require a considerable time for the system to reach steady state. Hence, if the steady state vibrations of the system due to the loaded static transmission error are required, this option will not be helpful. Although in terms of real time it does not take long to reach steady state, computational time may very easily reach very high values. However, this analysis provides very valuable information about the transient behavior of the system when the external torques are suddenly applied. The second set of initial values is useful in reaching the steady state solution very quickly, and is used in similar studies [14,15]. However, it has been observed that when this set of initial values is used, due to high angular velocities some numerical problems may arise, especially when the analysis is made for several mesh cycles. It has been numerically observed, for instance, that after calculating correct results for several mesh cycles, incorrect values can be obtained due to very high angular velocities. Furthermore, the computational time increases extremely when this set of initial conditions are used, due to numerical integration difficulties arising, again, from very high angular velocities. It is obvious that the main part of a high angular velocity is due to the rigid body rotation of the system. Therefore the third set of initial values, where zero initial angular velocities are assumed, has been used. Thus, the rigid body rotation of the system is eliminated and only the vibration of the system is considered. This is analogous to using an equivalent single-degree-of-freedom model for a two-degree-of-freedom model of a gear pair without shaft and bearing dynamics. The third set of initial values was successfully used in the numerical solution in finding the steady state response. It was also noted that neither the

NON-LINEAR

GEAR

247

DYNAMICS

accuracy nor the computational time changes when, instead of static values, zero values are used for the initial values of the transverse displacements. The computer program prepared has two options for the response computations: either the first set of initial values is used and the transient response is obtained or the third set is used and the steady state response is calculated. When the transient response is calculated, a few complete rotations of the pinion may need to be analyzed in order to gain an understanding of the general dynamic behavior of the system. When the steady state response is required, the third set of initial values is used, and thus the excitation of the system due to suddenly applied torque is avoided. It is observed that after a few mesh cycles are analyzed the steady state is reached in this option. Although the steady state angular velocities are not our concern in the dynamic analysis, if one is concerned with these values then the nominal speed of the system is to be added to the angular velocities found by the computer program. Similarly, the angular displacements do not include the rigid body rotation, which is given by the product of nominal speed and the time elapsed. It is important to note that in either option there is no real iteration, although the word “convergence” is used for steady state response computations. When steady state response is required, angular deflections corresponding to the nominal torque applied are taken as the initial values at the beginning of the integration routine. But after completing the analysis for one mesh cycle, the calculated values of Zi are not compared with the assumed initial values for a sufficiently small difference. Instead, it is continued with the computational process, which is equivalent to taking the calculated values as the new initial values. The response in one mesh cycle may not be exactly repeated in the consequent mesh cycles, due to the effect of the shaft-bearing dynamics. In other words, the response of the system in two successive mesh cycles may be almost the same, but the whole response may show a very slight fluctuation in time due to shaft and bearing compliances which may cause amplitude modulations. Therefore, the word “convergence” is not to be treated rigorously in the analysis presented here. The process of calculating and plotting the response for all mesh cycles analyzed shows whether or not a “convergence” is obtained (see Figure 3). Any amplitude modulation can also be detected in such plots. In some previous studies in which multi-degree-of-freedom models have been analyzed by numerical solution, iteration has been made by comparing the calculated values of the displace-

1600

600

250

300

350

400

Contact point Figure

3. Convergence

to steady

state; mesh load at 11 000 rpm.

248

H. NEVZAT ~ZGUVEN

ments and speeds after one period of mesh with the assumed initial values, and the averages of the initial and calculated values have been used as new initial values unless the differences between them were sufficiently small. However, it is believed that such a procedure forces the solution approach a steady state which may be a fictitious one in disregard of any possible modulation. For instance, when the steady state response is composed of a cyclic response at the mesh period superposed on a harmonic function with a larger period and a small amplitude, the later component of the response cannot be obtained if an iterative solution technique is used. Whether or not a gear system has such a response is another question, the answer to which depends on several factors such as the relation between the Fourier components of the loaded static transmission error and the lower natural frequencies of the system due to shaft compliances. If the shaft and bearing compliances were not included in the model, then amplitude modulation would not occur at all. A typical case where this effect is very clear is a transient response of a system in which the cyclic response in one mesh period shows fluctuation in a larger period (see Figure 4). An iterative solution technique cannot yield the correct solution in such cases. The amplitude variation due to fluctuations in larger periods, however, is observed to be very small in several example runs made for steady state response analyses.

Contactpoint Figure 4. Transient response; mesh load at 6000 rpm.

It has been observed that for systems where the non-linearity causes a jump in the frequency response the numerical results may converge to different values around resonance region when different but logical initial conditions are used. This is because of having two steady state solutions over certain frequency regions, when the non-linearity is high enough to cause a jump phenomenon. This problem has been noted by Comparin and Singh [33] in their comprehensive study on the non-linear frequency response characteristics of an impact pair. When the backlash and other non-linearities cause a

NON-LINEAR

GEAR

DYNAMICS

249

jump in the frequency response of a gear system, the numerical integration results around the jump frequency are to be used cautiously. This is the only region where the computational results are sensitive to the initial conditions chosen. Current research in this field is directed toward obtaining both of the steady state solutions by using different initial conditions and by modifying the numerical integration algorithm.

4. COMPUTER PROGRAM DYTEM In this section the Dynamic Transmission Error Program with a Multi-degree-offreedom model (DYTEM) is described, the options available in the program are given, and input and output of the program are explained. Program DYTEM is quite similar to program DYTE developed in a previous work [16], as far as the dynamic response quantities calculated are concerned. While a semidefinite two-degree-of-freedom model is used in DYTE by assuming uncoupled vibrational modes, torsional-lateral vibration coupling is considered in the six-degree-of-freedom model of DYTEM by including shaft and bearing dynamics. The following dynamic response quantities are calculated in program DYTEM: (a) dynamic mesh loads; (b) dynamic tooth forces (shared loads); (c) dynamic factors based on stress for pinion and gear; (d) dynamic factors based on tooth forces; (e) dynamic transmission error; (f) torsion of shafts; (g) dynamic bearing forces; (h) transverse vibration response at bearings. Unlike in DYTE, the dynamic bearing forces are calculated from the bearing properties (stiffness and damping) and the response of bearings. Therefore, bearing forces are not the same as dynamic mesh load. Especially for elastic bearings, a considerable difference can be observed between mesh load and bearing forces, as will be discussed later. All such computations can be made for either transient or steady state response: that is, DYTEM can be used for either transient response or steady state response analysis. Although the main object in developing this program was the determination of the steady state response, it is also possible to obtain the transient response as well by just simply choosing this option in the program. In transient response computations the external torques Tb and TL are taken as step inputs. Due to the large difference between the periods of the natural oscillations stemming from the mesh compliance and from the torsional compliances of the shafts, in most practical cases, several mesh cycles may need to be analyzed in order to obtain general information about the transient response of the system. The input data required by DYTEM is given in two parts: (1) data provided by program STEP [32]; (2) data supplied by the user (interactively or in a data file). The second part of the input is divided into seven data groups which can be provided interactively or can be read from an input file: (a) program control and geometry data; (b) dynamic data-mass and inertia; (c) dynamic data-stiffness; (d) dynamic datadamping; (e) dynamic data-loading; (f) kinematic data; (g) gear error data. There are two options for the method that will be used in the analysis: (a) the static transmission error (STE) method; (b) the approximate method. Although the theory for the dynamic analysis by either method allows for variable external torques, in both programs constant drive and load torques are assumed. Still, variable external torque may be included in the analysis approximately by using nominal values of TL and T,, but the loaded static transmission error calculated in STEP for variable torque. Yet, the accuracy of such an analysis is to be investigated. Further information on program DYTEM can be found in reference [34].

250

H. NEVZAT 6ZGih’EN 5.

NUMERICAL

APPLICATIONS

AND DISCUSSION

OF RESULTS

In this section some selected results from several runs made with DYTE and DYTEM are presented. The purpose is not to make a complete parametric study, but to demonstrate the effects of bearing stiffness and damping on the dynamic response of gear systems. Since DYTE and DYTEM are general purpose programs, they can be used to make the dynamic analysis of a given system, and therefore the study of the effects of some system parameters on gear dynamics can be made on a certain system defined. As an example case, the experimental set-up of Kubo [5] is chosen. The same system has been used in demonstrating the accuracy of DYTE [16]. The properties of the gear pair used in the following analyses are given in Table 1. TABLE 1 Properties of the gear pair used in computational examples

Tooth profile Number of teeth Module Pressure angle Face width Backlash Center distance

involute 25125 4mm 20 degrees 15 mm O-1 mm 100 mm

Contact ratio Damping ratio Outside diameter Base diameter Tooth diameter Tooth thickness

1.56 0.10 108 mm 94 mm 90 mm 6.233 mm

Further data about the system were not given in the literature. They can only be estimated from the picture of the set-up. However, it is believed that it is not important to know the actual data, since the aim here is not to make a comparison with experimental results but just to have an example problem with some realistic values. Some of the values estimated for the system parameters are kept the same in each run, while some of them were varied in order to see the effect of them on the dynamic response. The numerical values used for the former group of parameters are as follows: I, = I2 = IL = 0.0102 lb in s2; ID = 0.051 lb in s’; M, = MZ = 5.36 x 10m3lb s2/in; TD = T,_= 954.8 lb in; 5, = 0.005. The data which were not kept the same in each run are the torsional stiffness of shafts (k,, and k,J, and the stiffnesses and dampings of bearings (k,, kZ, cl and cZ). The values of these parameters for the five systems analyzed (Systems A, B, C, D and E) are given in Table 2. The natural frequencies calculated by using the average mesh stiffness are also shown in this table. The torsional stiffness values in System A were taken as the minimum possible values that can be estimated from approximate geometry of Kubo’s system, whereas the bearing stiffness and damping values were chosen rather high. 5.1. SINGLE-DEGREE-OF-FREEDOM MODEL VERSUS MULTI-DEGREE-OF-FREEDOM MODEL to verify that the effects The purpose of the first numerical example is computationally of the transverse vibrations and the torsional vibrations due to shaft compliances upon the gear dynamics may be negligible under certain conditions: that is, under certain circumstances the single-degree-of-freedom model of DYTE can be successfully used instead of the multi-degree-of-freedom model of DYTEM for an accurate dynamic analysis. For instance, when the torsional stiffness of the shafts are much smaller than the equivalent torsional stiffness corresponding to tooth mesh, the torsional modes will be lightly coupled. Furthermore, if the bearing stiffnesses are so high that bearings can be treated as rigid, the results obtained by the single-degree-of-freedom model DYTE

NON-LINEAR

GEAR

TABLE

251

DYNAMICS 2

Bearing and shaft properties of systems A, B, C, D and E k,(= 4

c,(= 4

k

kll

(lb in/rad)

(lb/in)

System

“II

(lb in/rad)

(lb s/in)

(rad/s)

2.0 x 1o’O

1.7x

lo4

3.0 x lo4

2000

907,2179,31 525, w4, o5 very high

2.0 x 1o’O

1.7x

lo6

3.0 x lo6

2000

8400,20 734, 35 765, w4, wg very high

1.477 x 10’

1.7x

lo4

3.0 x lo4

50

907, 2179, 27 890 52 496, 59 356

1.477 x lo6

1.7 x lo4

3.0 x lo4

50

906, 2178, 12 898 16 601, 40 674

1.477 x 10’

1.7 x 10”

3.0 x lo4

500

907, 2179, 27 890 52 496, 59 356

are not expected to be different from those that can be obtained by DYTEM. System A satisfies these conditions. The comparison of the maximum dynamic factors calculated by DYTE and DYTEM at different speeds for System A shows that they are very close to each other with an exception at the speed where sudden jump occurs, as can be seen in Figure 5. This deviation is because of the slightly shifted resonance frequency in the multi-degree-of-freedom model due to the coupling between the vibrational modes. As will be discussed later, when the torsional stiffnesses of shafts are increased, the resonance frequency governed by mesh stiffness will increase as well. Therefore, when there is a very light coupling between the vibrational modes, the speed at which sudden jump occurs in dynamic factors will be slightly shifted.

2.01.8 6

1.61.4 1.2-

-_ 2.0N k 1.5 -

1.0 0.4

1 0.6

I

I

I

I

0.8

1.0

1.2

1.4

1.6

Pinion speed (rpm) X 10S4

Figure 5. Comparison of the results of DYTE and DYTEM-System (b) gear.

A. Dynamic factors for (a) pinion and

252

H. NEVZAT~ZGWEN

The good agreement between the computer simulation results of DYTE and DYTEM for System A explains why the dynamic factors calculated by DYTE were observed [16] to be in good agreement with the experimental results of Kubo, although only a singledegree-of-freedom model is used in DYTE.

5.2. EFFECT OF BEARING AND SHAFT PROPERTIES ON GEAR DYNAMICS As system A is a typical system where the effects of bearing and shaft dynamics on gear dynamics are negligible, it is expected that the bearing force be almost equal to the mesh force at any instant. The comparison of the dynamic mesh and bearing forces in System A at 11 000 rpm are shown in Figure 6. It is clearly seen from this figure that, when almost rigid bearings are used, the mesh force can be taken as the bearing force. Increasing the torsional stiffnesses of the shafts so that two torsional modes will be coupled to a certain extent, does not cause a considerable difference between dynamic mesh and bearing forces. For instance, when the torsional stiffnesses of the shafts are increased 100 times (System B) the bearing and mesh forces at 13 000 rpm will be as shown in Figure 7. However, when compliant bearings are used (System C) there will be a considerable difference between the mesh forces and the bearing forces, as shown in Figure 8. The values used in System C are reasonable for a roller bearing and correspond to ki/k, = 10.

1

-500 0

10

I 20

I 30 Contact

Figure 6. Dynamic

1 40

50

point

mesh and bearing forces at 11000 rpm-System

A.

The effect of bearing compliance and damping on the dynamic factors at different speeds can be seen from Figures 9-13. In Figures 9-12 dynamic factors calculated for System A are compared with those found for Systems C and D for a range of frequency. System D is the same as System C with 10 times smaller bearing stiffness so that ki/k,,, = 1. In System C, the stiffnesses of bearings are such that dynamic coupling between transverse and torsional vibrations lowers the third natural frequency which is governed by tooth mesh. Therefore, the peaks shift to lower values, since they occur at speeds corresponding to the third natural frequency and its subharmonic (Figures 9 and 10). When the bearing stiffnesses are reduced further (System D), the coupling between

NON-LINEAR 2000

I-

15oc

l-

toot

)-

T

GEAR

253

DYNAMICS

1

z Z 8 k LL 5oc

)-

CI-

-5oc

)._

I

I

0

10

20

I

30 Contact

Figure

7. Dynamic

mesh and bearing

I

40

0

point

forces

at 13 000 rpm-System

B.

torsional and transverse vibrations becomes such that the resonances mainly governed by bearing compliances drop below the torsional resonance governed by mesh compliance. The torsional resonance controlled by mesh stiffness is also increased, and therefore, the major peak in System D is observed at a higher frequency (compared to System A), as can be seen in Figures 11 and 12. It is also observed in Figures 9-12 that lowering the bearing stitInesses causes a decrease in the values of dynamic factors, as noted in a previous study [17]. However, lowering bearing stiffnesses may not always decrease the natural frequency governed by the tooth mesh, unlike as observed in reference [17].

0

10

20

30 Contact

Figure

8. Dynamic

mesh and bearing

40

point

forces

at 10 000 rpm-System

C.

254

H. NEvzAT

System

~~ZGCJVEN

A-

System C

7 .; z 0”

1.6 -

1.4 -

1.2-

1.01 0.4

u

,u. 0.6

0.8

I

I

1.0

1.2

,\ 1.4

5

Pinion speed (rpm) X lo-’

Figure

9. Effect of bearing

compliance

on dynamic

factor;

dynamic

factor

for pinion.

In order to see to what extend bearing damping may affect the dynamic factors, System C has been analyzed with 10 times increased damping values (System E). The comparison of the dynamic factors calculated for System E with those calculated for System C is shown in Figure 13. Finally, dynamic factors at different pinion speeds are obtained for System B (which has rigid bearings like System A, but has shaft stiffnesses 100 times larger than those of System A). The comparison of the dynamic factors calculated for System A and B is shown in Figures 14 and 15. The coupling between the torsional mode governed by tooth mesh and those governed by the shafts causes the former mode shift to higher frequencies, as can be seen from Figures 14 and 15. One of the two lower peaks around 7000-8000 rpm in Figure 14 corresponds to the torsional natural frequency governed by shaft (at

!.4-

r.2 -

:.o -

,6-

0.6

0.6

1.0

1.2

14

Pinion speed (rpm) X 10s4

Figure

10. Effect of bearing

compliance

on dynamic

factor,

dynamic

factor

for gear.

NON-LINEAR

GEAR

255

DYNAMICS

2.2

2.0

System A.

I---

2

H

System Cl

.u 1.6 E : D

\

I

1.4

\ 0.4

0.6

0.6

1-o

1.2

1.4

18

16

Pinion speed (rpm) x 10e4

Figure

11. Effect of bearing

compliance

on dynamic

20 734 rad/s), and the other to the subharmonic mesh (35 765 rad/s).

factor;

dynamic

factor

for pinion.

of the natural frequency

due to tooth

6. CLOSURE

The mathematical models and the methods developed in this study are proved to be successful in the dynamic analysis of spur gears. It can be concluded from the study of the numerical results presented here that the single-degree-of-freedom non-linear model of DYTE, in which all the compliances except those of gear teeth are excluded, may be adequate for an accurate dynamic analysis in some gear systems. If the torsional compliantes of the gear carrying shafts are much higher than that of the tooth mesh and, furthermore, if the system is carried on very stiff bearings, then the gear pair’s torsional

2 .4-

I

-r

I

r

2 .2 System

A-

I

I

2 ,oz 51 e .: @

,6-

’.6

Syste,m D

\

-

I

,

0”

I

1 -4-

1 .21

0.6

0.8

1.0

1.2

1.4

1.6

l-6

Pinion speed (rpm) X toe4

Figure

12. Effect of bearing

compliance

on dynamic

factor;

dynamic

factor

for gear.

256

H.

NEVZAT

6ZGijVEN

1.9. l-8 1.7I.6 -

1.6-

1.55

1,41.3 1.2 I.1 l.O0.9 0.8

I.0

I.2

0.81 0.8

1.4

Pinion speed kpm) Figure

13. Effect of bearing damping

on dynamic

1.2

I.0

l-4

X 10S4

factor for (a) pinion and (b) gear.

vibration mode controlled by mesh stiffness will be very lightly coupled with other vibrational modes. For the dynamic analysis of such systems, use of the single-degree-offreedom model of DYTE, rather than the six-degree-of-freedom model of DYTEM, will not make much difference as far as the accuracy is concerned: this is because the vibrational mode governed by mesh stiffness can then be taken to be uncoupled without loss of accuracy. This has been demonstrated by successfully simulating Kubo’s system by using DYTE [ 161. It is also numerically verified in this study that, under such conditions, using DYTEM instead of DYTE for the dynamic analysis will not change the simulation results. Consequently, it can be said that when the mesh mode is uncoupled from the other modes, there is no need to use a multi-degree-of-freedom model for an accurate dynamic analysis. However, extreme care is to be taken to be sure that the mesh mode can be uncoupled from the other modes before using a single-degree-of-freedom model. In several previous studies, the accuracy of a single-degree-of-freedom model has been

2.0

-

1.8 5 e0 1.6.u E g I.4

-

0” I.2

-

1.0 0.81 0.4

Figure

14.

Effect of torsional

I 0.6

1

I

I

I.2 0.8 1.0 Pinion spectd (rpm) X 10m4

shaft compliance

on dynamic

1

1.4

factor; dynamic

I

I.6

factor for pinion.

NON-LINEAR 2.4

I

GEAR

I

257

DYNAMICS

I

I

2.2 2.0 h

1.8-

$ _z 1.6z 0” 1,41.2 l,O0.8 0.4

1 0.6

I

1

I

I

0.8

1.0

1.2

1.4

Pinion speed

1.6

(rpm) x 10m4

Figure 15. Effect of torsional shaft compliance on dynamic factor; dynamic factor for gear.

demonstrated by comparing predicted responses by experimental results. The test rigs used in these studies have been designed such that the shaft and bearing dynamics would not affect the gear dynamics notably. Since this point has not always been clearly stated in the literature, it may easily lead to the incorrect conclusion that a certain single-degreeof-freedom model is always successful in gear dynamic simulations. When the torsional mode governed by mesh stiffness is coupled with the torsional and transverse vibration modes governed by shaft and bearing compliances, the coupling between these modes cannot be neglected. Then the dynamic response will be strongly affected by the compliantes and damping of bearing and shafts, as can be seen from the results presented here. For such systems it is required to use a multi-degree-of-freedom model including torsionallateral vibration coupling. It also has been observed in this study that when stiff bearings are used the bearing forces will be almost the same as the mesh forces, as expected. Then for stiff bearings the mesh forces calculated with the single-degree-of-freedom model of DYTE can easily be taken as bearing forces. When compliant bearings are used, however, there may be a considerable difference between the dynamic bearing forces and mesh forces. Then it is necessary to include the bearing dynamics in the model in order to obtain correct bearing forces, since they are significantly affected by bearing stithresses and dampings. The model developed and the program prepared in this study can be used to predict the dynamic forces and dynamic responses in gear systems, including modulations due to transverse vibrations of bearings and torsional vibrations of shafts, which cannot be obtained by any single-degree-of-freedom model. It is also possible to simulate the transient response due to suddenly applied external torque by including the effect of bearing and shaft dynamics, which is again not possible using a single-degree-of-freedom model. The advantage of the model suggested here over the finite element model developed in an earlier study [17], on the other hand, is that the model developed here includes several non-linear effects as well as an accurate treatment of the gear errors and profile modifications, which cannot be considered in the finite element model. Therefore, it can be concluded that the model developed in this study combines the advantages of making a non-linear dynamic analysis with including shaft and bearing dynamics and thus considering torsional-lateral vibration coupling in gear dynamics. However, the numerical results obtained around the mesh resonance are to be used with caution when the

258

H. NEVZAT ijZGuVEN

non-linearity is high enough to cause a jump in the frequency response, since the numerical simulation yields only one of the solutions when there are two steady state solutions. Current research is directed toward modifying the numerical simulation algorithm so that both of the steady state solutions can be obtained.

ACKNOWLEDGMENTS The author would like to thank the sponsors of the Gear Dynamics and Gear Noise Research Laboratory (The Ohio State University), its director D. R. Houser, and NASA Lewis Research Center for supporting this research.

REFERENCES 1. W. A. TUPLIN 1950 Proceedings of the Institution of Mechanical Engineers 16, 162-167. Gear tooth stresses at high speed. 2. W. A. TUPLIN 1953 Machine Design 25, 203-211. Dynamic loads on gear teeth. 3. A. Y. ATTJA 1959 Journal of Engineering for Industry, Transactions of the American Society of Mechanical Engineers 81, l-9. Dynamic loading of spur gear teeth. 4. R. G. MUNRO 1962 Ph.D. Dissertation, University of Cambridge. The dynamic behaviour of spur gears. 5. A. KUBO, K. YAMADA, T. AIDA and S. SATO 1972 Transactions of the Japan Society of Mechanical Engineers 38, 2692-2715. Research on ultra speed gear devices (Reports l-3). 6. A. KUBO and S. KIYONO 1980 Bulletin of the Japan Society of Mechanical Engineers 23, 1536-1543. Vibration excitation of cylindrical involute gears due to tooth form error. 7. A. KUBO, S. KIYONO and M. FUJINO 1986, Bulletin of the Japan Society of Mechanical Engineers 29, 4424-4429. On analysis and prediction of machine vibration caused by gear meshing. 8. R. KASUBA and J. W. EVANS 1981 Journal of Mechanical Design, Transactions of The American Society of Mechanical Engineers 103, 398-409. An extended model for determining dynamic loads in spur gearing. 9. K. L. WANG and H. S. CHENG 1981 Journal of Mechanical Design, Transactions of The American Society of Mechanical Engineers 103, 177-187. A numerical solution to the dynamic load, film thickness and surface temperatures of spur gears, part 1: analysis. 10. K. L. WANG and H. S. CHENG 1981 Journal of Mechanical Design, Transactions of The American Society of Mechanical Engineers 103, 188-194. A numerical solution to the dynamic load, film thickness, and surface temperatures in spur gears, part 2: results. 11. B. M. BAHGAT, M. 0. M. OSMAN and T. S. SANKAR 1983 JournalofMechanisms, Transmissions, and Automation

12.

13. 14.

15.

16. 17.

in Design, Transactions of the American Society of Mechanical Engineers 105,

302-309. On the spur gear dynamic tooth load under consideration of system elasticity and tooth involute profile. K. UMEZAWA, T. SATO and K. KOHNO 1984 Bulletin of the Japan Society of Mechanical Engineers 27, 569-575. Influence of gear errors on rotational vibration of power transmission spur gears; (1) pressure angle error and normal pitch error. F. F~C~KAY 1987 Dynamik der Zahnradgetriebe. Berlin: Springer-Verlag. H. H. LIN, R. L. HUSTON and J. J. COY 1988 Journal of Mechanisms, Transmissions, and Automation in Design 110, 221-225. On dynamic loads in parallel shaft transmissions: part I-modelling and analysis. H. H. LIN, R. L. HUSTON and J. J. COY 1988 Journal of Mechanisms, Transmissions, and Automation in Design 110, 226-229. On dynamic loads in parallel shaft transmissions: part II-parameter study. H. N. OZG~VEN and D. R. HOUSER 1988 Journal of Sound and Vibration 125,71-83. Dynamic analysis of high speed gears by using loaded static transmission error. A. KAHRAMAN, H. N. OZG~VEN, D. R. HOUSER and J. ZAKRAJSEK 1989 Z+oceedings of

Fifth International Power Transmission and Gearing Conference, the American Society of Mechanical Engineers, 375-382. Dynamic analysis of geared rotors by finite elements. 18. A. KAHRAMAN and R. SINGH 1990 Journal of Sound and Vibration 142, 49-75. Non-linear

dynamics of a spur gear pair.

NON-LINEAR

GEAR DYNAMICS

259

19. H. N. &GWEN and D. R. HOUSER 1988 Journal of Sound and Vibration 121, 383-411. Mathematical models used in gear dynamics-a review. 20. J. W. DAVID and L. D. MITCHELL 1986 Journal of Vibration, Acoustics, Stress, and Reliability in Design, Transactions of the American Society of Mechanical Engineers 108, 171-176. Linear dynamic coupling in geared rotor systems. 21. J. W. DAVID 1984 Ph.D. Thesis, Virginia Polytechnic and State Uniuersity. Analytical investigation of dynamic coupling in nonlinear geared rotor systems. 22. J. W. DAVID and N. G. PARK 1987 Rotating Machinery Dynamics, the American Society of Mechanical Engineers, DE-2, 297-303. The vibration problem in gear coupled rotor systems. 23. H. IIDA, A. TAMURA, K. KIKUCH and H. AGATA 1980 Bulletin of the Japan Society of Mechanical Engineers 23,2111-2117. Coupled torsional-flexural vibration of a shaft in a geared system of rotors (1st report). 24. H. IIDA, A. TAMURA and M. OONISHI 1985 Bulletin of the Japan Society of Mechanical Engineers 28, 2694-2698. Coupled dynamic characteristics of a counter shaft in a gear train system. 25. H. IIDA, A. TAMURA and H. YAMAMOTO 1986 Bulletin of the Japan Society of Mechanical Engineers 29, 1811-1816. Dynamic characteristics of a gear train system with softly supported shafts. 26. T. IWATSUBO, S. ARII and R. KAWAI 1984 Bulletin of the Japan Society of Mechanical Engineers 27, 271-277. Coupled

lateral-torsional method).

vibration

of rotor system trained by gears (1. Analysis

by transfer-matrix 27. T. IWATSUBO, S. ARII and R. KAWAI 1984 Proceedings of the Third International Conference on Vibrations in Rotating Machinery, Institution of Mechanical Engineers, 59-66. The coupled lateral torsional vibration of a geared rotor system. 28. F. K~~c~~KAY 1984 Proceedings of the Third International Conference on Vibrations in Rotating Machinery, Institution of Mechanical Engineers, 81-90. Dynamic behaviour of high speed gears. 29. S. V. NERIYA, R. B. BHAT and T. S. SANKAR 1985 American Society of Mechanical Engineers, Paper 85-DET-124. Vibration of a geared train of rotors with torsional-flexural coupling. 30. S. V. NERIYA, R. B. BHAT and T. S. SANKAR 1985 The Shock and Vibration Bulletin 55, 13-25. Coupl$ torsional-flexural v.ibration of a geared system using finite element analysis. 31. H. N. OZG~~VEN and Z. L. OZKAN 1984 Journal of Vibration, Acoustics, Stress and Reliability in Design, Transactions of the American Society of Mechanical Engineers 106, 72-79. Whirl speeds and unbalance response of multibearing rotors using finite elements. 32. M. TAVAKOLI and D. R. HOUSER 1986 Journal of Mechanisms, Power Transmissions, and Automation in Design, Transactions of the American Society of Mechanical Engineers loS, 86-94. Optimum profile modification for the minimization of static transmission errors of spur gears. 33. R. J. COMPARIN and R. SINGH 1989 Journal of Sound and Vibration 134,259-290. Nonlinear frequency response characteristics of an impact pair. 34. H. N. OZG~VEN 1988 Technical Report, Gear Dynamics and Gear Noise Research Laboratory, Department of Mechanical Engineering, The Ohio State University. Development of a mathematical model and a computer program for dynamic analysis of gears including shaft and bearing dynamics.

APPENDIX: b C C cir

c2

%,

ch

backlash mesh compliance time average of the mesh compliance viscous damping coefficients of pinion and gear bearings, respectively viscous damping coefficients of the first and second meshing tooth pairs, respectively

CIPl Cl, 9 Cl2 eo,

ID,

eh

Z,,k

k,,

k2

k,,

kh

km

LIST OF SYMBOL

9 L

mesh damping viscous damping coefficients of pinion and gear shafts respectively displacement excitation representing the relative gear errors of the first (a) and second (b) meshing tooth pairs, respectively mass moment of inertia of drive, pinion, gear and load, respectively pinion and gear bearing stiflnesses, respectively stiffnesses of the first and second meshing tooth pairs, respectively time varying mesh stiffness

I

260

H. NEVZAT

k,, k,z ml, m2 4, TD, X XS

R2 TL

ijZGi.h’EN

torsional stiffnesses of pinion and gear shafts, respectively masses of pinion and gear, respectively base circle radii of pinion and gear, respectively drive and load torques, respectively dynamic transmission error static transmission error translational displacements of pinion and gear, respectively, along the line of action dynamic mesh load static mesh load damping ratio angular displacements of drive, pinion, gear and load, respectively