Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 46, 4, pp. 879-896, Warsaw 2008 CHAOTIC VIBRATIONS IN GEAR MESH SYSTEMS Jan Łuczko Cracow University of Technology, Institute of Applied Mechanics, Cracow, Poland e-mail: jluczko@mech.pk.edu.pl The paper is concerned with qualitative analysis of a non-linear model describing vibrations of a gear mesh system. The influence of selected parameters on the character and level of vibrations is studied. The possibility of excitation of quasi-periodic or chaotic oscillations for some regions of the parameters has been shown. Different types of vi- bration are illustrated by plots of time histories, phase portraits, stro- boscopic portraits and bifurcation diagrams. Key words: non-linear vibration, chaos, gear 1. Introduction Dynamicmodels of gear mesh systems aremost often described by non-linear parametrical equations, i.e. differential equations with time-periodic coeffi- cients (Müller, 1986; Dyk et al., 1994; Raghothama and Narayanan, 1999; Theodossiades and Natsiavas, 2001; Litak and Friswell, 2003). The variability of coefficients results in variation of themeshing stiffness inducedmainly by a change of the number of gear teeth pairs, which are simultaneously in contact (meshing of one or two pairs of teeth). The nonlinearity of equations is the result of backlashes in the transmission gear. Impacts of teeth are the effect of backlashes, and characteristics of impacts forces are strongly non-linear. As a result of these impact forces, it is possible, under some conditions, for periodic, quasi-periodic or chaotic vibration to occur in the system. Main sources of vibrations are kinematic excitations caused bymanufactu- ring errors andwear. Frequencies of such excitations depend on the product of rotational speed and the number of teeth. Non-linear characteristics of the dri- vingmoment, dependent on rotation angles and angular speed are sometimes taken into account. 880 J. Łuczko In the literature, a simplified one degree-of-freedom model is usually used where the torsional stiffnesses of shafts as well as transverse vibrations are neglected. In this classical model, the gear mesh is modelled as a pair of rigid disks connected by a spring-damper set along the line of action. The bac- klash function is usually used to represent gear clearances and an external displacement excitation is also applied at the gear mesh interface to represent manufacturing errors, intentional modifications of teeth or wear profile. The most of existing models of gear transmission systems treat the gear errors as a deterministic input. Another one-stage gearbox model was presented by Müller (1986). This model attracted a lot of attention inPoland.Müller’smodel is a two-parameter (stiffness and damping) one in which the inertia of two gear wheels is reduced to onemass. It was discussed, for example, in the paper byDyk et al. (1994). Motion of themass is equivalent to relative motion of two gear wheels. Vibra- tion is caused by relative motion of springs (having different lengths) which are in contact with the mass. In the paper byDyk et al. (1994), a comparison between the results of the analysis by the classical and Müller’s model was presented. The comparison showed that chaotic vibration canbeobserved in the casewhen theparameters of the model differ from those existing in a real construction. Raghothama and Narayanan (1999) investigated coupled transverse and torsional vibrations of a geared rotor-bearing systemusing the classicalmodel. Periodicmotions were obtained by the incremental harmonic balancemethod. The path following procedure using the arc length continuation techniquewas then used to trace bifurcation diagrams. A similar approach to the analysis of periodic vibrations was presented by Theodossiades andNatsiavas (2001)who studied the influence of gearmeshing frequency on the steady state response curve. Somemodels additionally include effects of friction between gear teeth. For example, in the paper by Vaishya and Singh (2001), the influence of sliding friction and viscous damping on unstable regions was discussed. The phenomenon of chaotic vibrations in gear transmission systems has been relatively early detected. Usually, strange attractors are shown only for suitably chosen values of gearbox parameters. In some papers, bifurcation diagramswere calculated to determine ranges of chaotic vibrations (e.g., Li et al., 2001). Taking into account the torsional stiffness, one needs to study a three degrees-of-freedommodel.LitakandFriswell (2003) showeddiagramsobtained by numerical integration, which illustrate the influence of the shaft stiffness as well as the magnitude of excitation on the character of motion. Chaotic vibrations in gear mesh systems 881 The coefficient of restitution has been also used to describe the impacts of teeth. In the paper by De Souza et al. (2004) in order to illustrate the influence of this coefficient on the type of vibration, the authors determined the bifurcation diagram as well as they calculated the Lyapunov exponents. Recently, the interest towards developing a better understanding of gear vibration has been renewed. This interest has been clearly reflected in some new studies which have dealt with a relatively simple model with rigid sup- ports.Analysing thismodel by the harmonic balancemethod (e.g., Shen et al., 2006) as well as the method of numerical integration (e.g., Gill-Jeong, 2007), the authors investigated the influence of selected parameters on frequency- response curves. A similar model, in whichmanufacturing errors were treated stochastically, was investigated by Bonori and Pellicano (2007). In the present paper, a four-degree-of-freedom model with the backlash and time varying stiffness is used to describe vibration of a one-stage gear- box. The influence of significant parameters on the character of vibration and the quality criteria are presented. Two-parameter diagrams are obtained to provide a better description of dynamic behaviour of the system. The stu- dy is done numerically using methods of numerical integration and spectrum analysis (Łuczko, 2006). 2. The model of the system A pair of gears is modelled (Fig.1) using two disks coupled by a non-linear mesh stiffness (parameters c, ∆) and a linear mesh damping (coefficient k). It is assumed that the resilient elements of the supports are described by the Voigt-Kelvin model with damping coefficients k1 and k2 and stiffness coefficients c1 and c2. Motion of the system is described by rotational angles ϕ1 and ϕ2 anddisplacements u1 and u2 of the centers of the disks.Themodel takes into account the influence of torques M1 and M2 on the driving and driven shaft, respectively. Using the laws ofmomentumand angularmomentum, transverse-torsional motion of the system can be described by the following system of four second- order differential equations I1ϕ̈1+R1S =M1(ϕ̇1,ϕ1, t) I2ϕ̈2−R2S =−M2(ϕ̇2,ϕ2, t) m1ü1+k1u̇1+ c1u1+S=0 m2ü2+k2u̇2+ c2u2−S=0 (2.1) In equations (2.1), the parameters: m1, m2 and I1, I2 are, respectively, the gear masses and the mass moments of inertia of the gears about their 882 J. Łuczko Fig. 1. A gear pair: (a) model of the system, (b) characteristics c(t) and F(u) axis of symmetry. The base radii of the gears are denoted by R1 and R2. The meshing force S is normal to the involute profile of the gear tooth. The following relationships for the torques M1 and M2 are used in the model M1(ϕ̇1,ϕ1, t)=M10− b1ϕ̇1 M2(ϕ̇2,ϕ2, t)=M20− b2ϕ̇2 (2.2) where b1 and b2 are the damping coefficients in the journal bearings. The meshing force S depends on the relative displacement of gear teeth, which describes the so-called dynamic transmission error u=R1ϕ1−R2ϕ2+u1−u2−e(t) (2.3) In equation (2.3), the so-called static transmission error e(t) takes into account the effects of gear faults such as wear of the tooth face, mount er- ror, tooth spall, etc. This error depends on the rotational angles ϕ1 and ϕ2. However, in this case, the influence of torsional vibrations on the rotational speed is usually neglected and the static error e(t) depends directly on time. Introducing the notation ωz = n1ω1 = n2ω2 for the fundamental frequency (gear meshing frequency), the static transmission error can be expressed in the form of a Fourier series e(t)= ∑ j ej cos(jωzt−θj) (2.4) where the integers n1 and n2 stand for the number of teeth of each gear. Chaotic vibrations in gear mesh systems 883 Themeshing force S is given by the following formula S= ku̇+ c(t)F(u) (2.5) The gear backlash nonlinearity F(u) ismodelled as a piecewise linear function F(u)=        u−∆ for u>∆ 0 for |u| ¬∆ u+∆ for u<−∆ (2.6) where 2∆ is the backlash. The stiffness c(t) depends on the number and position of the gear teeth pairs which are in contact. It is a periodic function of the relative angular position of the gears. The function c(t) has a similar form as equation (2.5) discussed earlier c(t)= ∑ j cj cos(jωzt−αj) (2.7) or it takes a characteristic shown in Fig.1b. This somewhat idealized charac- teristic takes into account the change of stiffness only due to the change of the number of teeth which are in contact. The periods T1 and T2, in which one or two pairs of teeth are in contact, can be expressed by the profile con- tact ratio α in the following way: T1 = (α−1)T and T2 = (2−α)T , where T =2π/ωz. The analysis will be done in a dimensionless form, and non-dimensional quantities are used where the displacements are referred to the parameter ∆ and thenon-dimensional time τ =ω0t is connectedwith the circular frequency ω0 = √ c0 m (2.8) where m= I1I2 I1R 2 2+ I2R 2 1 (2.9) is the equivalentmass representing the total inertia of a gear pair. Introducing new variables (n=1,2) xn = rnϕn = Rn ∆ ϕn yn = 1 ∆ un (2.10) and following notations 884 J. Łuczko Jn = In mR2n µn = mn m qn = Mn0 c0Rn∆ εj = ej ∆ ζn = kn 2mnω0 ζ = k 2mω0 βn = bn 2mω0R 2 n κn = cn c0 (2.11) the differential equations assume the following dimensionless form J1 d2x1 dτ2 +2β1 dx1 dτ +2ζ dz dτ +χ(τ)f(z) = q1 J2 d2x2 dτ2 +2β2 dx2 dτ −2ζ dz dτ −χ(τ)f(z) = q2 (2.12) µ1 d2y1 dτ2 +2µ1ζ1 dy1 dτ +κ1y1+2ζ dz dτ +χ(τ)f(z)= 0 µ2 d2y2 dτ2 +2µ2ζ2 dy2 dτ +κ2y2−2ζ dz dτ −χ(τ)f(z)= 0 Here f(z)=        z−1 for z > 1 z+1 for |z| ¬ 1 z+1 for z <−1 (2.13) where z=x1−x2+y1−y2−ε(t) (2.14) and ε(t)= ∑ j εj cos(jωτ −θj) (2.15) For the characteristic shown in Fig.1b, the function χ(τ) in the interval (0,2π/ω), where ω=ωz/ω0, has the following form χ(τ)= { 1+χ1 for 0¬ τ < τ1 1−χ2 for τ1 ¬ τ < τ1+ τ2 (2.16) In the case of a fixed support (κ1 →∞, κ2 →∞, y1 = y2 =0), and using an additional assumption that β1 =β2 =0, vibrations of the gearmechanism can be described by the following differential equation d2z dτ2 +2ζ0 dz dτ +χ0(τ)f(z) = q0− d2ε dτ2 (2.17) Chaotic vibrations in gear mesh systems 885 where χ0(τ)= J1+J2 J1J2 χ(τ) ζ0 = J1+J2 J1J2 ζ (2.18) q0 = q1 J1 + q2 J2 3. Analysis of torsional vibrations Some results of qualitative analysis will be described below. The results have been obtained using methods of numerical integration and the Fast Fourier Transform.More details about the use of spectrumanalysis for determination of the character of vibration were discussed in Łuczko (2006). In thediscussionof the results, the criterion index VRMS is used, definedas the velocity rms value. Moreover, it has also been assumed that J1 = J2 =2 and q1 = q2, so that the following relationships take place: χ0(τ) = χ(τ), ζ0 = ζ and q0 = q1 = q2.The following set values of parameters havebeenused in thenumerical calculations: ζ =0.025,β1 =β2 =0.01,χ1 =χ2 =χ0 =0.25, θ1 =0 and α=1.5 (then T1 =T2 =T/2). We begin the study of equations (2.12) by analysing torsional vibra- tions, neglecting the transverse displacements u1 and u2. In this case, mo- tion is described by the first two equations of (2.12) or equation (2.17) for β1 =β2 =0. Fig. 2. Influence of the excitation frequency and amplitude (χ0 =0.25, q0 =0.1, θ1 =0, ζ =0.025) on: (a) vibration zones, (b) efficiency index Figure 2 illustrates the influence of the frequency ω and amplitude ε1 of the first harmonic of the excitation on the character of vibration as well 886 J. Łuczko as on the rms value of the velocity (efficiency index). In the lower frequency range (for ω < 2), apart from periodic vibrations, interesting sub-harmonic (most often 2T-periodic) and chaotic vibrations can be observed. The regions of chaotic vibrations have irregular shapes (Fig.2a) and the changes of the efficiency index (Fig.2b) in these regions are also irregular. Additionally, the level of vibration (measured by the introduced index) is usually somewhat higher, but it is not a general rule. The raised level of vibration is also observed in the regions of parametrical resonance. These regions begin in the neighbourhoods of points ω=2,ω=1, ω = 1/2 and ω = 1/4, which is typical for parametric vibration. The widest are the first two regions. In these regions too, for a large enough amplitude ε1, chaotic vibrations are induced very often. For the determination of solutions to equations (2.12), the following initial conditions were used: x2(0) = x ′ 1(0) = x ′ 2(0) = 0, x1(0) = 1. The condition x1(0) = 1 means that at the time t=0, the contact takes place between the meshing teeth and the force acting on the gear tooth causes revolution in the direction consistent with the assumed direction of the angular velocity of the driven shaft. For other initial conditions, the results are similar, however sometimes, especially for a smaller excitation amplitude (in the vicinity of theboundaryof the regions)more solutions are observed.Amoreexact analysis of the influence of the initial conditions on the character of vibration requires determination of the basins of attraction. In the case of a system with four degrees-of-freedom, one needs to ana- lyse the influence of four initial conditions. This is arduous calculation and, additionally, the results are difficult to illustrate graphically in that case. Because the influence of the parameters β1 and β2 (damping in the be- arings) on the shape of the determined regions is insignificant, it is possible to find the basins of attraction by analysing the simpler system described by equation (2.17). The results of analysis of the influence of initial conditions on the solution to equation (2.17) are shown in Fig.3. For smaller values of the frequency, two different types of vibrations get mostly excited (Figs.3a,b). For larger va- lues of ω, the number of solutions grows drastically (Fig.3c). For instance, in the case of ω = 2.3 and ε1 = 0.05, one can show the existence, apart from 1T-periodic vibration (Fig.4a) and 3T-periodic (Fig.4b), two types of 4T-periodic vibration (Figs.4c,d). Additionally, for a small change in the pa- rameters ω and ε1, 5T-periodic vibrations appear in the system. Chaotic vibrations in gear mesh systems 887 Fig. 3. Basins of attraction (χ=0.25, q0 =0.1, ζ =0.025): (a) ω=0.8, ε1 =0.1, (b) ω=1.1, ε1 =0.05, (c) ω=2.3, ε1 =0.05 Fig. 4. Phase trajectories and time histories of the velocity (ω=2.3, ε1 =0.05): (a) 1T-periodic, (b) 2T-periodic, (c), (e) 4T-periodic, (d), (f) 4T-periodic The frequency ω and amplitude ε1 of the kinematic excitation as well as undimensional torque q0 have the biggest influence on the character of vi- bration. The influence of parametric excitation (χ1,χ2,τ1,τ2) is considerably smaller. Very similar results are obtained, by replacing the function χ(τ), defi- nedby formula (2.16), with a sine functionhavinga suitably chosen amplitude. The parameters χ1 and χ2 mainly influence the velocity rms value. Figure 5 illustrates, in the same format as in Fig.2, the effect of ω and q0 (static load torque). Chaotic vibrations are excited for small values of q0 (for q0 < 0.15). With an increase in q0, the range of chaotic vibrations strongly decreases (Fig.5a), however the range of 2T-periodic vibration simultaneously increases. Other types of sub-harmonic vibration are mainly observed on the boundary of zones of different vibration types. 888 J. Łuczko Fig. 5. Effect of ω and q0 (χ0 =0.25, ε1 =0.1, ζ =0.025): (a) vibration zones, (b) quality index In the regions of periodic vibration, the changes of quality index are re- gular (Fig.5b). With an increase in q0, in some frequency ranges, the level of vibrations slightly increased. The quality index achieves small values for higher values of ω (for ω> 2), independently of q0. Fig. 6. Bifurcation diagram – influence of the parameter q0 (ε1 =0.1, ω=1) The influence of the parameter q0 is also illustrated in the bifurcation diagram shown in Fig.6. A cursory observation of the diagram shows that for q0 < 0.0508 only chaotic vibrations are excited. However, analysing the diagram in more detail, it is possible to detect at least two very narrow ran- ges of periodic vibration (somewhat brighter trails in the neighbourhoods q0 = 0.0165 and q0 = 0.025). For q0 = 0.0164, 4T-periodic vibration ta- kes place, whereas for q0 =0.01645 and q0 =0.0165 the period of oscillations is respectively equal to 8T and 12T . In the range 0.02485 < q0 < 0.025, 3T-periodic vibration is excited, which next for q0 = 0.0255 gives place to 6T-periodic vibration. For q0 = 0.047, one can also observe 6T-periodic vi- brations. Interesting is the fact that in the next ranges of chaotic vibrations lying between the ranges of sub-harmonic vibrations, the shape of the fractal under- goes a qualitative change (Fig.7). For larger values of q0 (0.067 0.2. In the first range of the parameter ε1, quasi-periodic solutions are created from periodic solutions – the stroboscopic portrait is a single closed curve (Fig.13a). In the second range, for ε1 > 0.2, 2T-periodic solutions chan- ge into quasi-periodic, and two closed curves make the stroboscopic portrait (Fig.13b). In general, quasi-periodic vibrations have somewhat smaller amplitudes in relation to chaotic vibration,which ispartly visible on thebifurcationdiagram. It shouldbepointed out that in the discussed ranges of the parameter ?1 other types of vibrations are also possible, though probability of them to be excited is considerably smaller than in the remaining ranges. Earlier, Raghothama andNarayanan (1999) detected quasi-periodic vibra- tion by analysing a model of a gearbox with flexible supports. Chaotic vibrations in gear mesh systems 893 Fig. 13. Phase trajectories and stroboscopic portraits (κ=0.5, q0 =0.1, ω=1.05): (a) ε1 =0.04, (b) ε1 =0.205 Analogous conclusions relating to the character of vibration can be drawn from the analysis of diagrams shown in Fig.14, which illustrate the influence of the nondimensional moment q0. In the ranges of quasi-periodic vibrations, the stroboscopic points are usually distributed more regularly – this remark concerns the interval (0.13,0.17) in Fig.14a aswell as the interval (0.16,0.18) in Fig.14b. Fig. 14. Bifurcation diagram – influence of the parameter q0 (ε1 =0.1, ω=1.05): (a) κ=0.5, ω=0.9, (b) κ=1, ω=1 Both diagrams can be comparedwith that earlier shown inFig.6.One can easily observe that the zone of chaotic vibrationwith regard to transversemo- tion becomeswider. It should be noted that the range of changes of q0 is twice as large as now than that inFig.6. The character of chaotic and quasi-periodic motion is similar. This is confirmed by the similarity of stroboscopic portraits obtained for chaotic (Fig.15a) and quasi-periodic oscillations (Fig.15b). 894 J. Łuczko Fig. 15. Stroboscopic portraits (κ=0.5, ω=0.9, ε1 =0.1): (a) q0 =0.128, (b) q0 =0.135 5. Conclusions Detailed conclusions have been successively drawn in the discussion of the re- sults of numerical calculations. The most important conclusions can be sum- marized as follows: • The analysis of the effect of parameters on the character of vibration and the introduced quality index enable one to determine ranges of the parameters, for which vibration amplitudes are sufficiently small. Hi- gher levels of vibration are observed in the low frequency range (for ω< 2), mainly in regions of parametric resonance (in the neighbourho- odof points ω=1/4,ω=1/2,ω=1and ω=2). In these regions, apart from the rotational speed, the parameters of parametric excitations play an important role. • The investigation of the character of motion is interesting mainly from the cognitive point of view. The parameters of the kinematic excitation aswell as the static load torque have the decisive influence on the type of oscillations.However, since a raised level of oscillation isusually observed in chaotic regions, it is possible to use the results of qualitative analysis for selection of parameters of the gear transmission system. • For small values of the nondimensional static load, the solutions to dif- ferential equations strongly depend on the initial conditions. We can have different types of subharmonic and chaotic vibrations, considerably differing in the amplitude of oscillation. • The taking into account the stiffnesses of the gearbox supports results in growth of the ranges of chaotic oscillations. Moreover, a new type of oscillation – quasi-periodic vibration – gets excited in the system. Chaotic vibrations in gear mesh systems 895 • By analysing the model of the system, it is possible to observe diffe- rent scenarios of generation of chaotic oscillations. They can come into existence both from sub-harmonic oscillations as well as quasi-periodic vibrations. References 1. Bonori G., Pellicano F., 2007, Non-smooth dynamics of spur gears with manufacturing errors, Journal of Sound and Vibration, 306, 271-283 2. De Souza S.L.T., Caldas I.L., VianaR.L., Balthazar J.M., 2004, Sud- den changes in chaotic attractors and transient basins in a model for rattling in gearboxes,Chaos, Solitons and Fractals, 21, 763-772 3. Dyk J., KrupaA.,Osiński J., 1994,Analysis of chaos in systemswith gears, Mechanika Teoretyczna i Stosowana, 32, 3, 549-563 4. Gill-Jeong C., 2007, Nonlinear behavior analysis of spur gear pairs with a one-way clutch, Journal of Sound and Vibration, 304, 18-30 5. Li Y., Zheng L., Mi L., 2001, Bifurcation and chaotic motion of a piecewi- se linear vibration system in gear-pairs, International Journal of Gearing and Transmissions, 3, 126-133 6. Litak G., Friswell M.I., 2003, Vibration in gear systems, Chaos, Solitons and Fractals, 16, 795-800 7. Łuczko J., 2006,Numerical methods of determining the regions of subharmo- nic and chaotic vibrations,Czasopismo Techniczne, 11-M, 39-62 [in Polish] 8. Müller L., 1986,Tooth Gears: Dynamics, WNT, Warszawa [in Polish] 9. RaghothamaA.,NarayananS., 1999,Bifurcationandchaos in gearedrotor bearing systemby incrementalharmonicbalancemethod,Journal of Sound and Vibration, 226, 3, 469-492 10. Shen Y., Yang S., Liu X., 2006, Nonlinear dynamics of a spur gear pair with time-varying stiffness andbacklashbasedon incrementalharmonicbalance method, International Journal of Mechanical Sciences, 48, 1256-1263 11. Theodossiades S., Natsiavas S., 2001, Periodic and chaotic dynamics of motor-driven gear-pair systems with backlash, Chaos, Solitons and Fractals, 12, 2427-2440 12. Vaishya M., Singh R., 2001, Analysis of periodically varying gear mesh sys- temswithCoulomb friction using Floquet theory, Journal of Sound and Vibra- tion, 243, 3, 525-545 896 J. Łuczko Drgania chaotyczne w przekładniach zębatych Streszczenie W pracy przeprowadzono analizę jakościowąmodelu jednostopniowej przekładni zębatej.Uwzględnionowpływzmiany sztywności zazębienia, luzumiędzyzębnegooraz wymuszeń kinematycznych. Do analizy wykorzystano procedury numerycznego cał- kowania skojarzone z algorytmami szybkiej transformaty Fouriera. Zbadano wpływ parametrów układu na charakter drgań oraz wprowadzone wskaźniki jakości. Wy- kazano możliwość generowania się w badanym układzie drgań chaotycznych, prawie okresowych i podharmonicznych. Manuscript received March 19, 2008; accepted for print July 3, 2008