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Archive of Applied Mechanics 61 (1991) 523-531 Archive of Applied Mechanics ?9 Springer-Verlag 1991 Dynamic investigations o loads on gear teeth in single gear transmission W. Nadolski, Warsaw Summary: In the paper a discrete-continuous model for the analysis of dynamic loads on gear teeth of a single gear transmission is proposed. In this model a constant equivalent mesh stiffness and ponderable shafts deformed by torsion are taken into account. In the discussion a wave method is applied which utili- zes the wave solution of the equations of motion. Numerical calculations are concentrated on the determina- tion of amplitudes for dynamic loads on gear teeth with respect to frequencies of external excitation in the first and second resonant regions. Untersuchung der dynamischen Zahnbelastung in einstufigen Getrieben Tbersicht: Zur Untersuchung der dynamischen Zahnbeanspruchung in einstufigen Getrieben wird ein )lodell aus diskreten und kontinuierlichen Massen mit einer konstanten, quivalcnten Zahneingriffssteifigkeit und massebehafteten, tordierbaren Wellen vorgeschlagcn. Die Bewegungsgleichungen werden durch einen Wellenfunktionsansatz gel5st. Numerische Berechnungen werden vorwiegend ffir die Amplituden der Zahn- belastnng bei einer nieren Erregung im Bereich der ersten und zweiten Resonanz durchgeffihrt. 1 Introduction In 1, 2 dynamic investigations were performed for the discrete-continuous model of a single gear transmission with rigid gear teeth. In the present paper a similar model is considered, how- ever the mating gears have such profiles of teeth that their equivalent stiffness can be assumed to be constant 3- 6. In the technical literature gear transmissions are mostly analyzed by means of discrete models of one and multi-degrees of freedom 7, 8. In the present paper a discrete-continuous model consists of two ponderable shafts and four rigid bodies with constant mass moments of inertia with respect to the axis of rotation. Those gear transmissions are considered where supporting bearings eliminate deformations due to bending and where the shafts are mainly torsionally deformed. The rigid body at the input is loaded by an external moment which may be arbitrary. Damping is taken into account by means of an equivalent external damping of the viscous type and an equivalent internal damping of the Voigt type. In the considerations a method utilizing the wave solution of the equations of motion is applied. This method enables the determination of dynamics loads on gear teeth, displacements, strains and velocities in steady as well as in transient states. Numerical calculations for selected parameters describing various mechanical properties of the single gear transmission are concen- trated on the determination of amplitudes of dynamic loads on gear teeth with respect to fre- quencies of the external excitation in the first and second resonant regions. 2 Assumptions and governing equations Consider the discrete-continuous model of a single gear transmission with parallel axes, Fig. 1. The ponderable shafts 1 and 2 are characterized by the shear modulus G, the polar moment of inertia Ii, the density and the length li (i = 1, 2). The mass moments of inertia of gears 4, 5 and 39* 524 Archive of Applied Mechanics 61 (199i) M(t) J, Rz. u 5 I ! () = M() + rgl() + rq;(), rg;( + 2) + rl0g;( + 2) + rlg( + 2) + n/;() + r/() (11) = rJ;() + rlJ;() + rlA() + r7() + rly(), ! rlJ2(z) -I- r2o/(z) -b r2/2(z) q- r2291(z q- 2) q- r23gl(z -q- 2) II t ! = r24q2 (z) q- rg(z) + rq(z) q- re7/(z) -q- r:s/(z ) where rl = I + AKeDz, re = A(Ke + DI), rz = AKD.e - 1, r4 =A6(K-D,), r = l + K1D1, r = K1 + Dsl, r7 = KIDI - 1, r s = K1 - Dzl, r = 1 + AK1DI, ro = A(K1 + D41 q- Cm), rll - A4Km, r12 = -ANCm, r3 = -ANKm, rl = AKD - 1, r = A(KI - DI - Cm), r - -rll , (12) r17 : -r12, rlS : -rla, rl 9 : AsK2D22 + 1, r2o = A(K2 + D51 + N2Cm), r21 : AN2K, r2 : -AsNCm, r23 : -AsNKm, r2 : AK2D2 - 1, r5 : A(K2 - D51 - NZCm), r26 : -r21, r27 : -r22, r8 : -r%. The differential equations (11) can be solved by means of the finite difference method. The functions/1,/2, g are determined from (11) for z 0 and the function gl for z : z + 2 0. The functions/i, gi are identical to zero for negative arguments, so from (11)3 it follows that g,(z2) 0 for z2 : z + 2 2. Though the functions/2(z) and gl(z + 2) are not independent, the method of finite differences enables to derive expressions for these functions in dependence on known values of appropriate functions. These expressions are given in the Appendix. 0o = 1 rad, non-dimensional ll = 12 = 1, 3 5Iumerieal results In the numerical calculations the following parameters of the single gear transmission are assumed : dimensional 11 =12 =0.25m, /5 =0.16m, J3- 1.5kgm z, (13) e = 3200m/s, =0.8.104kg/m3; K1 =0.013, K2 - 0.06627, A4 =5, A 6 =0.15, N =4/3, =r/9. The analysis includes the following non-dimensional mesh stiffness Km= 0.005 859, 0.018528, 0.05859, (in dimensions: l0 s 2/m, / 10 s N/m, 109 N/m) for the tooth length equal to 0.10 m, and the coefficient of mesh damping Cm = DI, 9-11. W. Nadolski: Dynamic investigations of loads on gear teeth in single gear transmission 527 01-NO 2 O.,-N2 10 -6 4O 30 20 10 Dil =0.01 0.05 - 0,4r . . . . . . 01-N0 0.1 0.01 /X J, o.o 0.2 0.4 0.6 p Fig. 2. Amplitude-frequency. curves of the functions 01 - NO2 and 01 - JT02 The function of the external moment M(t) can be arbitrary, i.e. irregular or regular, periodic or nonperiodie. Here it is assumed in the form M(t) = a sin (pt) (1r where a = 10 6, and p is a non-dimensional external frequency. The considerations focus on the determination of amplitudes of dynamic loads on gear teeth with respect to frequencies of external excitation in the steady states for the first and second resonant regions. The dynamic load P expressed by (2) depends on the relative displacements 01 (1, t) - N02(1, t), the relative velocities 01(1, t) - N02(1, t), and on the coefficients Kin, Urn. The effect of damping on 01 - 2V02, 01 - N02, and the effect of Km and Cm on the load P is shown in Figs. 2, 3 and 4. The amplitude-frequency curves for the relative displacements 01(1, t) -N02(1, t) (conti- nuous lines) and for the relative velocities 01(1, t) - N02(1, t) (dotted lines) presented in Fig. 2 are obtained using (11) with the parameters (13), and for the additional parameters Km =0.018528, C = 0, Dil =0,0.01,0.05,0.1 (i =3,4,5,6), (15) Di = 0, 0.01, 0.05 (i = 1, 2). It appeared that the effect of external damping on the studied functions was appreciable, but the effect of internal damping was rather insignificant. Each curve in Fig. 2 for fixed Dil corresponds to the three values of the coefficients Di2, so an effect of internal damping is not observed. Fur- ther numerical calculations are performed for coefficients of internal damping DI2 = D22 = 0.01, and for coefficients of external damping Dil = 0.05 (i = 3, 4, 5, 6). From Fig. 3 where the dia- grams of amplitudes PA of dynamic loads for the equivalent mesh stiffness Km= 0.005 859, 0.018528, 0.05859 and C = 0 are plotted it follows that the curves are regular, namely, in the first resonant region the maximum amplitude of the load increases with increase of K. The am- plitude-frequency curves for the dynamic loads shown in Figs. 4, 5, 6, 7 and 9 are obtained only for Km= 0.018528. 528 (z) + 8A(z), gl(z2) and/2(z) are expressed as follows $385 -SsS6 $3S4 -S2Se el(z) - h(z) - $1S5 -SsS4 SsS -$1S5 W. Nadolski: Dynamic investigations of loads on gear teeth in single gear transmission 531 References 1. Nadolski, W.: Application of wave method in investigations of single gear transmission. Ing. Arch. 58 (1988) 329-333 2. Nadolski, W. : Dynamic investigations of single gear transmission taking into account microcracks. Ing. Arch. 59 (1989) 362-370 3. Fronius, S. : tdber die Normung eines Berechnungsverfahrens ffir Zahnrider. Maschinenbautechnik 9 (1959) 216-221 4. Frenkel, 7. N.: Eksperimentalnoje opredelenije summarnoj deformaeii i estkosti prjamych zubjev eilindri6eskich zub6atyeh koles. Sb. ZubSatyje i ezervjaSnyje pereda6i. Magis, 1959 (in Russian) 163- 184. Frenkel, I. N. : Experimental determination of overall deformation and stiffness of gears with spur toothing (in Russian). In: Ed. Toothed wheelsand worm gears, pp. 163-184. Moscow: Magis 1959 5. Mark, W. D.: Analysis of the vibratory excitation of gear systems, I: Basic theory. J. Acoust. See. 63 (1978) 1408-1430 6. Mark, W. D.: Analysis of the vibratory excitation of gear systems, Ih Tooth