Jtam-A4.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 55, 1, pp. 253-264, Warsaw 2017 DOI: 10.15632/jtam-pl.55.1.253 MODAL ANALYSIS OF GEARBOX TRANSMISSION SYSTEM IN BUCKET WHEEL EXCAVATOR MahaKarray,NabihFeki,MohamedT.Khabou,FakherChaari,MohamedHaddar Laboratory of mechanics, Modelling and Production, National school of Engineers of Sfax, Sfax, Tunisia e-mail: fakher.chaari@gmail.com Planetary gears are widely used in modern machines as ones of the most effective forms of power transmission. In this paper, a special configuration of a gearbox composed of one stage spiral bevel gear and a two stage helical planetary gear used in a bucket wheel excavator gearbox is presented to investigate its modal properties. A lumped-parameter model is formulated to obtain equations of motion and the eigenvalue problem is solved. Themodes are presented in low-frequencyandhigh-frequencybands.Distributions ofmodal kinetic and strain energies are studied. Keywords: spiralbevel gear, helical planetarygear,bucketwheel excavator,modal properties 1. Introduction Gear transmissions such as bevel and planetary gears are widely used in transmissions of wind turbines, agricultural machinery, mining machines such as excavators and transportation such as helicopters. Spiral bevel gears BG coupled in two stages helical planetary gears can be found in gearboxes of bucket wheel excavators. The first step to investigate the dynamic behavior of such systems is the determination of natural frequencies andmode shapes. Many researchworks has been done on common parallel axis geared rotor systems dynamics, see e.g. Ozguven andHouser (1988a,b), Blankenship and Singh (1995), Velex andMaatar (1996) however few research works were dedicated to bevel gears dynamics. The existing models are mostly similar to those of parallel axis gears. Gosselin et al. (1995) proposed a general formula and applied it to analyze the load distribution and transmission error in spiral bevel gear pairs and hypoid gear pairs. Karray et al. (2013) investigated the dynamic behavior of a single stage bevel gear in the presence of local damage. Choy et al. (1991) presented vibration signature analysis for multi-stage gear transmissions which combined gear mesh dynamics and structural modal analysis in the study of transmission vibrations. For planetary gears, Kahraman (1994b) provided expressions for natural frequencies by using a rotational lumped-parameter model. Lin and Parker (1999a, 2000) showed that two- -dimensional, spur planetary gears with equally spaced and diametrically opposed planets po- ssess well-definedmodal properties.Wu and Parker (2008) proved themodal properties of spur planetary gears having elastically deformable ring gears. These vibration mode characteristics are crucial in vibration suppression strategies usingmesh phasing (Seager, 1975; Lin andParker, 2004) and eigensensitivity analysis (Lin andParker, 1999b), Guo andParker (2010) of planetary gears. Although the vibrationmodes of two-dimensional planetary gears have been studied, few studies were dedicated to those of helical planetary gears with three-dimensional motion. Ha- bib et al. (2005) determined critical frequencies for helical planetary gears and examined their sensitivity to the helix angle. Eritenel and Parker (2009) examined three-dimensional motion of helical gears and shafts. 254 M. Karray et al. Researches onmodal characteristics of multistage planetary gears are rare. Sun et al. (2014) analyzed the natural frequency and coupledmode characteristics in amulti-stage planetary gear and distinguished the dominant vibration stage by a criterion. Hammami et al. (2015) discussed the modal properties of a special configuration of two stage planetary gears mounted back-to- -back. Zhang et al. (2016) attempted to establish a translational-rotational coupled dynamic model of a two-stage closed-form planetary gear set to predict natural frequencies and vibration modes. This paperdiscusses themodal properties of a special configuration of a gearbox composed of one stage spiral bevel gear and two stage helical planetary gear used in bucket wheel excavators. A lumped-parameter model is formulated to obtain the equations of motion. 2. Dynamic model Figure 1 shows a general view of a bucket wheel excavator. The kinematic scheme of its gearbox transmission system is presented in Fig. 2. It is composed of a spiral bevel gear as the input and two stages planetary gear. Fig. 1. View of the bucket wheel excavator: (a) from the side of winning-receiving belt and (b) from the side of the transmission system (Rusinski et al., 2010) Fig. 2. Kinematic scheme of the gearbox transmission system Modal analysis of gearbox transmission system in bucket wheel excavator 255 3. Model and equation of motion Themodel of the transmission is presented in Fig. 3. Fig. 3. Lumped parameter model: (a) planetary gear, (b) bevel gear 3.1. Spiral bevel gear model The spiral BG model is divided into two rigid blocks (pinion with Z1 teeth and wheel with Z2 teeth). Each block has four degrees of freedom (three translations xi,yi,zi (i = 1,2), one rotation θ1 for the pinion, θ2 for wheel and θm for the motor). The shafts are modeled with torsional stiffness. In order to simulate the meshing, linear mesh stiffness acting along the lines of action is considered following the procedure given byKarray et al. (2013). The vector defining different degrees of freedom is qBG = {x1,y1,z1,θm,θ1,x2,y2,z2,θ2} T (3.1) 3.2. Planetary gear model For both stages of the planetary gear, a three-dimensional model is adopted. Each stage is composed of the sun gear s, ring gear r, which are coupled to each other by 3 planets P mounted on a carrier c. These elements are considered as rigid bodies. Linear springs acting on the lines of action are used to simulate themeshing stiffness (Kahraman, 1994a,c). Bearings are accounted for by linear springs. Each component has six degrees of freedom: three translations uji, vji and wji and three rotations ϕji, ψji and θji (j = c,r,s,1, . . . ,n, i = 1 for the first stage and i = 2 for the second one). These coordinates are measured with respect to a frame 256 M. Karray et al. (Oi,s1i, t1i,z1i) fixed to the carrier and rotatingwith a constant angular speedΩci. The rotations ϕji, ψji and θji are replaced by their corresponding translational gear mesh displacements as ρjix = Rbjiϕji ρjiy = Rbjiψji ρjiz = Rbjiθji j = c,r,s,1, . . . ,n i =1,2 (3.2) where Rbji is the base circle radius for the sun, ring, planet, and the radius of the circle passing through planet centers for the carrier. The system elasticity is accounted for by 6n+18 DOFs for each stage, and the planetary gear displacement vector qjPGi of each element is defined as qjPGi = [uji,vji,wji,ρjix,ρjiy,ρjiz] T j = c,r,s,1, . . . ,n i =1,2 (3.3) 3.3. Global model The objective is to obtain a unique differential system combining the BG stage and both stages planetary gear coordinates. The principle of the coupling consists in using an additional torsional stiffness joining the rotational degree of freedom of the bevel gear wheel and the sun gear of the first stage planetary gear and adding a linear spring joining the axial degrees of freedomof the samewheel and sun.The same are used to couple the axial and rotational degree of freedom of the carrier of the first stage and the sun of the second one. Introducing the following extended state variable vector composed of the bevel and two stages planetary gear displacements qG = {qBG,qPG1,qPG2} (3.4) Applying Lagrange formulation for each element allows us to obtain the equations ofmotion of the 9+2(18+6n) degrees of freedom of the global system MGq̈G +CGq̇G +[KpG +KeG(t)]qG =FG(t) (3.5) where qG, MG, CG, KpG, KeG, FG are respectively the displacement vector, mass, damping, bearing, mesh stiffness matrices and the force vector for the global system. 4. Modal analysis The characteristics of the bevel gear model are presented in Table 1 while the characteristics of the two stages planetary gear are presented in Table 2. It has a fixed ring and three planets. Table 1.Characteristics of the spiral bevel gear Parameters Pinion Wheel Number of teeth Z 27 62 Mass [kg] 300 800 Moment of inertia [kg·m2] 18 72 Axial stiffness kx1, ky2 [N/m] 1 ·10 9 2.3 ·109 Lateral stiffness ky1, kz1, kx2, kz2 [N/m] 8.8 ·10 9 1.3 ·1010 Torsional stiffness kθ1, kθ2 [Nm/rad] 1.2 ·10 4 7.4 ·104 Pressure angle α =20◦ Spiral angle β =20◦ Modal analysis of gearbox transmission system in bucket wheel excavator 257 Table 2.Characteristics of the planetary gear model Sun Ring Carrier Planet Teeth number Zs1 =21 Zr1 =150 – Zp1 =64 Zs2 =27 Zr2 =90 – Zp2 =31 Mass [kg] Ms1 =270 Mr1 =4500 Mc1 =2600 Mp1 =1200 Ms2 =446 Mr2 =1960 Mc2 =1300 Mp2 =600 J/R2bi (J/R 2 bi)1 =200 (J/R 2 bi)1 =740 (J/R 2 bi)1 =990 (J/R 2 bi)1 =592 (J/R2bi)2 =281 (J/R 2 bi)2 =387 (J/R 2 bi)2 =618 (J/R 2 bi)2 =294 I/R2bi (I/R 2 bi)1 =100 (I/R 2 bi)1 =370 (I/R 2 bi)1 =495 (I/R 2 bi)1 =296 (IR2bi)2 =140 (I/R 2 bi)2 =193 (I/R 2 bi)2 =310 (I/R 2 bi)2 =147 Gearmesh ksp1 =2.28 ·10 8, krp1 =2.6 ·10 8 stiffness [N/m] ksp2 =2.2 ·10 8, krp2 =2.3 ·10 8 Bearing stiffness kjx = kjy =10 8, kjz =10 9, j = c,s [N/m] krx = kry = krz =10 10, kxx = kyy =10 8, kzz =10 9 Torsional kjϕ = kjψ =10 9, kjθ =0, j = c,s,1, . . . ,n stiffness [N/m] krϕ = krψ = krθ =10 10 Pressure angle α =20◦ Helix angle β =20◦ Table 3.Eigenfrequencies of the system Mode type Multipli- Eigenfrequency city [m] [Hz] Bevel gear mode C f11 =56, f17 =64 T 1 f20 =67 R f81 =1237 First planetary mode R-A 1 f4 =30, f35 =165, f44 =212, f49 =220, f50 =246, f75 =649 T-T 2 f2,3 =20, f8,9 =38.7, f12,13 =58, f18,19 =66, f24,25 =84.8, f36,37 =173, f41,42 =204, f45,46 =213, f51,52 =251, f56,57 =313, f73,74 =516, f77,78 =890 Second planetary mode R-A 1 f23 =84.7, f59 =345, f60 =361, f65 =420, f68 =442, f76 =838 T-T 2 f5,6 =38.5, f14,15 =63, f27,28 =100, f31,32 =119, f33,34 =153, f47,48 =216, f61,62 =364, f63,64 =416, f66,67 =434, f69,70 =447, f71,72 =482, f79,80 =1147 Coupled modes f1 =0, f7 =38.6, f10 =46, f16 =63.7, f21 =75, f22 =79, f26 =92, f29 =111, 1 f30 =117, f38 =179, f39 =199, f40 =203, f43 =209, f53 =255, f54 =275, f55 =300, f58 =341 4.1. Natural frequencies and vibration modes The undamped eigenvalue problem derived from the equation of motion by considering only the mean stiffness matrix K is ( −ω2iMG +KG ) φi =0 (4.1) whereφi is the eigenvector and ωi is the corresponding eigenfrequency. 258 M. Karray et al. Natural frequencies and vibration modes of the system are given in Table 3. The natural modesaregroupedaccording to themultiplicity of thenatural frequencies. Several characteristics are revealed after a thorough comparison on the natural frequencies ωi andmodal vectors φi. • The first-order natural frequency is ω1 =0, and the corresponding vibration mode is the rigid body mode. It is obvious that the rigid body mode can be eliminated by removing rigid-bodymotion. • Bevel gear modes contain only modal deflection of BG components. They include four distinct natural frequencies: – two natural frequencies with combined (C) translational and rotational modal deflec- tion. An example of this mode is observed in Fig. 4. The equilibrium positions are represented by a solid black line and the deflected positions are shown by a dashed black line. Similarly, Figs. 5-7 all abide by these rules; – one natural frequency in which only rotation (R) is observed; – one in which only translation (T) is observed. Fig. 4. Combined bevel gear mode at f11 =56Hz Fig. 5. Rotational-axial mode of the first stage planetary gear at f4 =30Hz Modal analysis of gearbox transmission system in bucket wheel excavator 259 Fig. 6. Translational-tiltingmode of the second stage planetary gear at f5,6 =38.5Hz Fig. 7. Coupledmode at f7 =38.6Hz • Planetary gear modes in which there are only modal deflection of the 2 stage planetary gear components include: – twelve natural frequencies with the multiplicity m = 1. The related vibration mo- des are rotational-axial (R-A) modes in which the carriers, rings and suns rotate and translate axially, but they do not tilt or translate in-plane. The planets move identically and in phase. Figure 5 shows one rotational mode of the system; – twenty four natural frequencies with the multiplicity m = 2. The related vibration modes are translational-tilting (T-T) modes in which the carriers, rings and suns only translate in-plane and tilt but do not rotate or translate axially. In addition, the following relations between the deflections are noticed for each double mode: ui1 = vi2, vi1 = ui2 and ϕi1 = ψi2, ψi1 =−ϕi2 (i = c,r,s), where ui1, vi1, ϕi1, ψi1 are modal deflections in the first mode and ui2, vi2, ϕi2, ψi2 are modal deflections in the second mode; – the planets exhibit sequentially phased motion. Figure 6 shows one rotational mode of the system; 260 M. Karray et al. – the planet modes exist only if the number of planet N > 3 and have themultiplicity m = N−1. In both stages of theplanetary gear, there are only three planets (N =3). So, only the previous classes of modes appear when solving the eigenvalue problem. • Modal properties of the two-stage planetary gears are analogous to those of simple, single- stage planetary gears. Features of rotational and translational modes are identical. • Thecoupledmodewhich includes seventeendistinctnatural frequencies includesmovement of the different stages. Figure 7 illustrates the vibration modes of the system. 4.2. Analysis of the distribution of modal kinetic strain energies Computation of the modal strain energy and the modal kinetic energy distributions gives information on bodies brought to critical speeds (which excite the natural frequencies) in terms of dominant motion and deformation. The total modal strain energy can be written as the sum of strain energies of rotation and translation from each component of the system Epφ = 1 2 φ T i Kφi = ∑ Epφk + ∑ Epφkω +Epφpw + ∑ Epφji + ∑ Epφjiω + ∑ (Epφsin +Epφrin) (4.2) where Epφk and Epφkω are the strain energies of the bearing stiffness in the rotational and translational motion of the pinion andwheel (k = p,w), respectively. Epφpw is the strain energy of the pinion-wheel meshing. Epφji and Epφjiω are the strain energies of the bearing stiffness in the rotational and translational motion of the carriers, suns, rings and planets (j = c,r,s,1,2,3) in both stages (i =1 for the first stage and i =2 for the second stage). Epφsin and Epφrin are the strain energies of the sun i-planets and ring i-planets meshing in both stages. The modal kinetic energy can also be written as the sum of the kinetic energies of rotation and translation from each component of the system Ecφ = 1 2 ω2iφ T i Mφi = ∑ Ecφk + ∑ Ecφkω + ∑ Ecφji + ∑ Ecφjiω (4.3) where Ecφk and Ecφkω are the kinetic energies of the bearing stiffness in the rotational and translational motion of the pinion and wheel (k = p,w). Ecφji and Ecφjiω are the kinetic energies in the rotational and translational motion of the suns, carriers, rings and planets (j = s,c,r,t,1,2,3) in the first stage and the second one (i =1,2). Figure 8 shows the distribution of modal kinetic energies in low frequencies. In the X-axis, the contribution of each degree of freedom in the total modal strain energy is represented. Details are given in Table 4. Figure 9 shows the distribution ofmodal strain energies in low frequencies, where the X-axis is defined in Table 5. Table 6 presents a resume of themodal dominantmotion and the dominant strain energy in each low-frequency mode. Figure 10 shows the distribution of modal kinetic and strain energies in high frequencies. The X-axis is defined in Table 4. Modal analysis of gearbox transmission system in bucket wheel excavator 261 Fig. 8. Modal kinetic energies in low-frequencies Table 4.The X-axis location of kinetic energies 1-3 Translation of pinion 40-42 Translation of planet 3 of 1st stage 4 Rotation of bearingmotor 43-45 Rotation of planet 3 of 1st stage 5 Rotation of pinion 46-48 Translation of carrier of 2nd stage 6-8 Translation of wheel 49-51 Rotation of carrier of 2nd stage 9 Rotation of wheel 52-54 Translation of ring of 2nd stage 10-12 Translation of carrier of 1st stage 55-57 Rotation of ring of 2nd stage 13-15 Rotation of carrier of 1st stage 58-60 Translation of sun of 2nd stage 16-18 Translation of ring of 1st stage 61-63 Rotation of sun of 2nd stage 19-21 Rotation of ring of 1st stage 64-66 Translation of planet 1 of 2nd stage 22-24 Translation of sun of 1st stage 67-69 Rotation of planet 1 of 2nd stage 25-27 Rotation of sun of 1st stage 70-72 Translation of planet 2 of 2nd stage 28-30 Translation of planet 1 of 1st stage 73-75 Rotation of planet 2 of 2nd stage 31-33 Rotation of planet 1 of 1st stage 76-78 Translation of planet 3 of 2nd stage 34-36 Translation of planet 2 of 1st stage 79-81 Rotation of planet 3 of 2nd stage 37-39 Rotation of planet 2 of 1st stage Fig. 9. Strain energy in low-frequencies 262 M. Karray et al. Table 5.The X-axis location of strain energies 1-3,5 Bearing of pinion 52-57 Bearing of ring of 2nd stage 4 Bearing of motor 58-63 Bearing of sun of 2nd stage 6-9 Bearing of wheel 64-69 Bearing of planet 1 of 2nd stage 10-15 Bearing of carrier of 1st stage 70-75 Bearing of planet 2 of 2nd stage 16-21 Bearing of ring of 1st stage 76-81 Bearing of planet 3 of 2nd stage 22-27 Bearing of sun of 1st stage 82 Meshing pinion-wheel 28-33 Bearing of planet 1 of 1st stage 83-85 Meshing ring-planets of 1st stage 34-39 Bearing of planet 2 of 1st stage 86-88 Meshing sun-planets of 1st stage 40-45 Bearing of planet 3 of 1st stage 89-91 Meshing ring-planets of 2nd stage 46-51 Bearing of carrier of 2nd stage 92-94 Meshing sun-planets of 2nd stage Table 6.Dominant motion and dominant strain energy in low-frequencies No. of Frequencies Modal dominant movement Dominant strain energy mode [Hz] 2,3 20 (T-T) Transl. of planet 3 of 1st stage Bearing of carrier of 1st stage 4 30 (R-A) Transl. of planet 3 of 1st stage Meshing sun-planets of 1st stage 5,6 38.5 (T-T) Transl. of planet 3 of 2nd stage Bearing of carrier of 2nd stage 7 38.6 (C) Transl. of planet 3 of 2nd stage Meshing ring-planets of 2nd stage 8,9 38.7 (T-T) Transl. of all planets of 1st stage Bearing of sun of 1st stage 10 46 (C) Transl. of all planets of 1st stage Bearing of carrier of 1st stage 11 56 (B) Transl. of wheel Bearing of wheel 12,13 58 (T-T) Transl. of planet 2 of 1st stage Bearing of carrier of 1st stage 14,15 63 (T-T) Transl. of planet 3 of 2nd stage Bearing of sun of 2nd stage 16 63.7 (C) Transl. of planet 3 of 2nd stage Bearing of carrier of 2nd stage Meshing sun-planets of 2nd stage Bearing of pinion Fig. 10. Kinetic and strain energy in high-frequencies Table 7 presents a resume of themodal dominantmotion and the dominant strain energy in each high-frequencymode. Modal analysis of gearbox transmission system in bucket wheel excavator 263 Table 7.Dominant motion and dominant strain energy in high-frequencies No. of Frequencies Modal dominant movement Dominant strain energy mode [Hz] 77,78 890 (T-T) Rotation of ring of 1st stage Bearing of ring of 1st stage 79,80 1147 (T-T) Rotation of ring of 2nd stage Bearing of ring of 2nd stage 81 1237 (B) Rotation of bearingmotor Rotation of pinion Meshing pinion-wheel 5. Conclusion This paper investigatesmodal properties of a special configuration of a gearbox composed of one stage spiral bevel gear coupled in a two stage helical planetary gear. A lumped-parametermodel is formulated to obtain the equations ofmotion. Thehelical planetary gear system is represented by a three-dimensional lumped-parameter model with six degrees of freedom per gear and the shaft body supported by bearings. Solution of the eigenvalue problem allowed recovering modal characteristics of the transmission. It hasbeen found that natural frequencies canbedivided into three main mode classes: coupled modes, bevel gear modes and planetary gear modes. The last class of modes includes two types: the first one is a rotational-axial mode in which the central components rotate andmove axially butdonot tilt or translatewith identicalmodaldeflection of the planets; there are also 12 rotational-axialmodeswithdistinct natural frequencies, the second one is a translational-tilting mode in which the central members tilt and translate in-plane but do not rotate or move axially; there are 12 pairs of degenerate translational-tilting modes with the natural frequency multiplicity two. 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