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An efficient finite element solution for gear dynamics
This article has been downloaded from IOPscience. Please scroll down to see the full text article. 2010 IOP Conf. Ser.: Mater. Sci. Eng. 10 012150 (http://iopscience.iop.org/1757-899X/10/1/012150) View the table of contents for this issue, or go to the journal homepage for more
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WCCM/APCOM 2010 IOP Conf. Series: Materials Science and Engineering 10 (2010) 012150
IOP Publishing doi:10.1088/1757-899X/10/1/012150
An efficient finite element solution for gear dynamics CG Cooley1 , RG Parker2,4 , and SM Vijayakar3 The Ohio State University, 201 W. 19th St., Columbus, OH 43210, USA State Key Lab for Mechanical Systems and Vibration, University of Michigan-Shanghai Jiao Tong University Joint Institute, Shanghai Jiao Tong University 3 Advanced Numerical Solutions, LLC, 3956 Brown Park Dr. Suite B, Hilliard, OH 43026, USA 1
2
E-mail: 4
[email protected] Abstract. A finite element formulation for the dynamic response of gear pairs is proposed. Following an established approach in lumped parameter gear dynamic models, the static solution is used as the excitation in a frequency domain solution of the finite element vibration model. The nonlinear finite element/contact mechanics formulation provides accurate calculation of the static solution and average mesh stiffness that are used in the dynamic simulation. The frequency domain finite element calculation of dynamic response compares well with numerically integrated (time domain) finite element dynamic results and previously published experimental results. Simulation time with the proposed formulation is two orders of magnitude lower than numerically integrated dynamic results. This formulation admits system level dynamic gearbox response, which may include multiple gear meshes, flexible shafts, rolling element bearings, housing structures, and other deformable components.
1. Introduction Gear vibration is transfered to the surrounding structure by shafts and bearings, creating unwanted noise. Dynamic tooth loads at or near resonant gear speeds generate high stresses in the teeth and bearings and can cause premature failure. For these reasons the dynamic behavior of gears is essential to the proper design of geared systems. Although gear dynamics has been studied for decades, few studies present a formulation intended for the dynamic response of full gear-train systems that contain multiple gear meshes, flexible shafts, bearings, and housing structures. There are few reliable computational tools for the dynamic analysis of general gear configurations. Some models exist, but they are limited by simplified modeling of gear tooth mesh interfaces, two-dimensional models that neglect out-of-plane behavior, and models specific to a single gear configuration. General threedimensional finite element models for dynamic response are rare because they require significant computational effort. This is compounded by the many time steps required for the transient response to diminish so that steady state data can be obtained. This study attempts to fill this gap with a general finite element formulation that can be used for full gearbox dynamic analyses. An approach that uses the static solution to approximate the dynamic excitation is proposed. The use of static transmission error as excitation is common in gear dynamics modeling [1, 2, 3, 4, 5]. The present work offers superior static solution and average mesh stiffness calculation from a nonlinear, three-dimensional finite element/contact mechanics formulation of the complete system. The tooth contact forces and local tooth deformations are calculated c 2010 Published under licence by IOP Publishing Ltd
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WCCM/APCOM 2010 IOP Conf. Series: Materials Science and Engineering 10 (2010) 012150
IOP Publishing doi:10.1088/1757-899X/10/1/012150
from a specialized contact model suited to elastic body contact. The tooth surface geometry, including surface modifications, is represented with high precision. Using this formulation the variations of the static solution as the gears rotate are determined at multiple locations over one mesh cycle. This varying static solution is then used as harmonic excitation to a full system finite element model. Spur and helical gears are handled naturally by this formulation. This formulation is examined for single gear pairs where benchmark experiments are available. Both spur and helical gears are examined. Comparisons are made to numerically integrated, time domain finite element solutions that do not use any approximations of the excitation based on static solutions and to experimental data found in the literature. Typical analyses model gear dynamics using lumped parameter models. These models commonly assume the gear blanks to be rigid bodies with lumped spring elements to model the gear mesh interface. The complexity of the gear mesh interface varies for each model. Accurate gear mesh modeling is essential to capturing the dynamics of gears, and lumped parameter models suffer due to the various approximations made in modeling the gear mesh interface. A review of gear dynamic models prior to 1988 can be found in [6]. More recent gear dynamics studies are found in the review article by Wang et al. [7] and in the thorough bibliography of [8]. Lumped parameter mesh models for helical gears are more complex than the single lumped stiffness element used for spur gears. Some models use many lumped stiffness elements that are placed along discretized contact lines of a helical gear. Examples of lumped parameter models for helical gear dynamics can be found in [2, 8, 9] Kahraman and Blankenship experimentally obtained the nonlinear response of spur gear pairs with varying involute contact ratios [10]. Recently, a finite element/contact mechanics approach has been used to study the dynamic properties of gears [11, 12, 13]. All dynamic results using this formulation have been restricted to two-dimensional analyses that assume plane strain behavior of the gears in mesh. The major advantage of the finite element/contact mechanics approach is that no restrictions or assumptions about the gear mesh forces are made and the contact forces are calculated at each instant as the gears move kinematically. 2. Finite element analysis 2.1. Time domain finite element/contact mechanics formulation Parker et al. [12] formulated a two-dimensional approach for the finite element/contact mechanics dynamic modeling for gears. Their model is limited to two-dimensional spur gear modeling and cannot capture three-dimensional affects from helical gears. This work develops a finite element/contact mechanics model for three-dimensional gear dynamics. The equations of motion for an arbitrary single gear body i can be written as
Mf f i Mf ri MTfri Mrri
Df f i Df ri ¨fi x x˙ f i + ¨ ri x DTfri Drri x˙ ri Kf f i Kf ri xf i f + = fi KTfri Krri xri fri
(1)
where Mf f i and Kf f i are the finite element mass and stiffness matrices, respectively. The vector xf i contains the finite element degrees of freedom, and the vector xri contains the six rigid body degrees of freedom of the ith gear reference frame. The damping matrix is found using a Rayleigh damping model, Df f i = αMf f i + βKf f i . Any added lumped mass or inertia is placed in Mrri . Stiffness and damping terms from any lumped element bearing matrices are added to Krri and Drri . The off-diagonal matrices couple the elastic and rigid body degrees of freedom. Assembling the equations of motion for each gear into a single matrix equation gives 2
WCCM/APCOM 2010 IOP Conf. Series: Materials Science and Engineering 10 (2010) 012150
IOP Publishing doi:10.1088/1757-899X/10/1/012150
M¨ x + Dx˙ + Kx = f
(2)
Equation (2) is solved using Newmark’s time discretization method. After discretization in time, Equation (2) can be written in the compact form ˆ x = ˆf Kˆ where
(3)
ˆ = M + γ∆tD + β∆t2 K, K ˆf = − [−2M + (1 − 2γ)∆tD + 1 − 2β + γ ∆t2 K]xn 2 1 − [M − (1 − γ)∆tD + + β − γ ∆t2 K]xn−1 2 1 1 − 2β + γ fn + + β − γ fn−1 ], + ∆t2 [βfn+1 + 2 2
(4)
xn = x(to + n∆t), fn = f(to + n∆t), and β = 12 and γ = 14 are discretization parameters chosen to give numerical stability ˆ and ˆf in Equations (3) and (4) as the effective without adding damping. We can think of K stiffness matrix and forcing vector, although both contain dynamic effects through the temporal ˆ = xn+1 . discretization. The unknown dynamic solution vector at time to +(n+1)∆t is given by x For a static simulation the time derivatives of x in (2) are zero, leaving Kx = f
(5)
which is in the same form as (3). Both the static and dynamic equations are solved similarly. Only the static case will be detailed and distinctions from the dynamic will be pointed out. The response vector, x, is decomposed by a linear transformation that separates the elastic (qφ ) and rigid body (qθ ) modal coordinates according to q qφ . φ (6) = Tφ .. Tθ x=T qθ qθ where the matrix T contains the eigenvectors of the symmetric matrix K. Tθ contains eigenvectors with zero eigenvalue. Substitution of Equation (6) into (5) and premultiplying by the transpose of T breaks the discretized matrix equation of motion into two separate equations [12] Kφφ qφ = TTφ f 0 = TTθ f
(7)
where Kφφ = TTφ KTφ and TTθ KTθ = 0 because the eigenvalues associated with each eigenvector in Tθ are zero. Equation (7b) is the solvability condition of (5) that ensures each gear is in static ˆ contains no zero eigenvalues and Tθ = 0. Therefore, equilibrium. For dynamic simulations, K Equation (7b) vanishes for dynamic simulations. The contact modeling is the highlight of the finite element/contact mechanics formulation. A combined finite element and semi-analytical formulation (a surface integration of the Bousinesq solution for a point load on an elastic half-space) is used to model the contact between the gear 3
WCCM/APCOM 2010 IOP Conf. Series: Materials Science and Engineering 10 (2010) 012150
IOP Publishing doi:10.1088/1757-899X/10/1/012150
teeth [14, 15, 16]. This combined formulation eliminates the usual need for a highly refined finite element mesh near the tooth surfaces. Instead, the tooth surface is modeled using special finite elements that use Hermite polynomials as interpolation functions, allowing the use of a relatively coarse finite element gear tooth mesh while maintaining the precise tooth surface geometry needed for gear dynamic modeling [11]. Because contact is modeled by an analytical formulation, the tooth contact surface geometry can be modeled with extremely high resolution; there is no restriction to using nodal coordinates and shape functions to define the geometry. The contact zone is broken into two parts; a near field where relative displacements are calculated using the analytical formulation, and a far field where conventional finite element methods calculate absolute displacements [11]. The two solutions are matched at an interface that is far enough from the contact zone for the analytical solution to remain accurate. The tooth surface is subdivided into a number of candidate contact points along the profile and lead directions over the face of the gear teeth. The separation at each candidate contact point along its normal is written as [12] d = Ap + Cqθ +
(8)
where A is a compliance matrix that gives the inner (semi-analytical) and outer (finite element) contributions to separation from contact, p is a vector of contact loads, C is a kinematic matrix that adds the rigid body contribution to separation, qθ is a vector of rigid body degrees of freedom, and is the separation of the bodies in their unloaded state (typically from tooth surface modifications). The forcing vector from the right-hand side of Equation (5) is separated into external noncontact loads and contact loads as f = Ep + fo
(9)
where E is a matrix that transforms the loading calculated from the semi-analytical formulation due to contact to the finite element nodes on the tooth. Applied loads and bearing reaction loads are contained in the force vector fo . Substitution of Equation (9) into Equation (7b) and rearranging gives the force balance on the finite element mesh as CT p = λ
(10)
where λ = −TTθ fo is the transformed external force vector and CT = TTθ E. Equations (8) and (10) are solved by linear programming methods [15] for qθ , d, and p subject to the constraints that each of the k components of d and p are non-negative and one (but not both) of dk and pk are zero for every k. The calculated p is substituted into Equation (9) to get f . That result is substituted into Equation (7a) to find qφ . Substitution of qφ and the calculated qθ into Equation (6) gives the dynamic finite element displacement vector x. The main limitation of a full three-dimensional finite element dynamic simulation is the computational cost of running such a model. This is a multi-body dynamic formulation using full gear models that rotate according to prescribed nominal kinematic motions. The contact vectors d and p are large for three-dimensional simulations because the contact algorithm now must calculate displacements and pressures along the profile direction and along the facewidth of the gear teeth. The time per integration step of the gear pairs in this study varied between 10 and 30 seconds. Additionally, a number of computation steps are required to remove transient response at constant speed and for a period of acceleration between two nearby constant speeds. For the gear pairs tested in this study 400 steps was sufficient to accelerate between two nearby constant speeds while 200 to 400 steps were needed to diminish the transient response. A total 4
WCCM/APCOM 2010 IOP Conf. Series: Materials Science and Engineering 10 (2010) 012150
IOP Publishing doi:10.1088/1757-899X/10/1/012150
of over four hours of computation time is required to get to steady state response at each mesh frequency. These two areas, which are crucial to the dynamic finite element simulation, consume significant computation time. 2.2. Frequency domain finite element/contact mechanics formulation To reduce the computation time required to calculate the dynamic response of gear systems a frequency domain formulation is developed. This formulation seeks to accurately and efficiently represent the gear tooth mesh interface by using the nonlinear static results from the combined finite element/contact mechanics formulation to approximate the dynamic excitation. The frequency domain method begins by solving Equation (5) for the gear tooth mesh stiffness and static deflections at many points over a mesh cycle. Equations (8) and (10) give the components of d and p, each of which contains positive or zero elements with nonzero entries in p corresponding to zero entries in d. If we partition p such that pb contains all nonzero entries of p, then the corresponding elements of db will be zero. The partitioned equation (8) is then written as db = 0 = Abb pb + Cb qθ + b
(11)
−1 pb = −A−1 bb Cb qθ − Abb b
(12)
Solution of (11) for pb gives
In general, Abb is a large matrix and calculation of its inverse is computationally intensive. In this case, however, A−1 bb is calculated during the linear programming solution of d and p from Equation (8) and (10). Therefore, A−1 bb is available with no additional computations. Substitution of Equation (12) into the nonzero partition of Equation (10) (where the partition . is p = [ pT .. 0T ]T ) and rearranging gives b
λb + λ Kqq qθ = −λ
(13)
T −1 where Kqq = CTb A−1 bb Cb and λ = −Cb Abb b . The matrix Kqq in Equation (13) represents the contact stiffness of the gear mesh interface at a specific instance in the mesh cycle. The second forcing term in (13), λ , is the excitation that occurs due to tooth surface modifications; it vanishes for gear teeth that have no modifications. Because Equation (13) involves a stiffness matrix that depends on the changing contact conditions as the gears rotate kinematically through a mesh cycle, multiple steps are analyzed over a mesh period. The mesh cycle is divided into N steps and a corresponding Kqq is found at each step to give a set {Klqq }N l=1 . This mesh stiffness depends only on the position in the mesh cycle and is transformed into a time dependent stiffness Kqq (t) that varies periodically over a mesh period depending on the gear speed. The static displacement vector xs is found at each of the N steps and is transformed using the gear speed to give xs (t). The total degrees of freedom are reduced using static condensation. Equation (5) is partitioned as Krr Kro xr f = r (14) Kor Koo xo 0
where xr is the set of retained degrees of freedom and xo contains the removed (or “slave”) degrees of freedom. The retained degrees of freedom are chosen by the mesh generator based on the connectivity of the finite element mesh. Degrees of freedom subjected to loadings are always
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WCCM/APCOM 2010 IOP Conf. Series: Materials Science and Engineering 10 (2010) 012150
IOP Publishing doi:10.1088/1757-899X/10/1/012150
retained in xr . Solution of the lower partition of Equation (14) relates the retained degrees of freedom to the removed ones by xo = −K−1 (15) oo Kor xr Substitution of (15) into the static displacement vector x = [xTr xTo ]T gives the non-square, frequency independent transformation matrix as I x = Wxr = xr (16) −K−1 oo Kor where the transformation matrix W relates the total degrees of freedom to the retained degrees of freedom. Reduction in coordinates by static condensation is achieved by substitution of (16) into Equation (2) with use of (6) to give ˆ qr + D ˆ q˙ r + Lq ˆ r = ˆf M¨ where
ˆ = (TW)T MTW M ˆ = (TW)T KTW L
(17)
ˆ = (TW)T DTW D ˆf = (TW)T f
(18)
The gear pair models in this study were reduced from approximately 100,000 degrees of freedom in x to 1,000 degrees of freedom in qr . ˆ does not contain mesh stiffness from gear tooth contact and has The stiffness matrix L components ˆ φφ 0 K ˆ L= (19) 0 0 Augmenting Equation (19) with the stiffness due to contact, that is, Kqq (t) from Equation (13), leaves the mass and damping matrices unchanged, but the stiffness matrix becomes ˆ φφ K 0 ˆ= L (20) 0 Kqq (t) The mesh stiffness associated with the contact can be broken into mean and time varying e qq (t). The components of K ˆ φφ do not change with time. The stiffness parts, Kqq (t) = Kqq + K matrix is similarly decomposed into mean and time varying parts as ˆ =L ˆm + L ˆ v (t) L ˆm = L
ˆ φφ K 0 0 Kqq
ˆ v (t) = L
0 0 e qq (t) 0 K
Substitution of the stiffness matrices in Equation (21) into Equation (17) gives ˆ qr + D ˆ q˙ r + L ˆm + L ˆ v (t) qr = ˆf M¨
(21)
(22)
Equation (22) is a linear time-varying equation of motion for the geared system with fluctuating mesh stiffness. Solution of (22) would involve numerical integration and consequently long solution times for removing transients. To this point, the only approximation made to get Equation (22) is the static condensation transformation in Equation (16), and solution of Equation (22) would give accuracy similar to numerical integration of (3). For computational efficiency, numerical integration of Equation (22) is avoided by an approximation of the excitation. It has been shown in the literature that use of the static 6
WCCM/APCOM 2010 IOP Conf. Series: Materials Science and Engineering 10 (2010) 012150
IOP Publishing doi:10.1088/1757-899X/10/1/012150
transmission error as excitation can accurately predict the dynamic response of geared systems [1, 2, 3, 4, 5]. Following a similar approach, the solution of Equation (22) is found from the approximation of the forcing as follows to get a linear time-invariant system of equations. The static solution vector is found by removal of dynamic terms in (22) to give ˆm + L ˆ v (t) qr ≈ ˆf L (23) The solution of Equation (23) at each instant in the mesh cycle is the static solution qrs . Rearranging, the approximation for the right-hand side of Equation (22) becomes ˆf = L ˆ v (t)qrs + L ˆ m qrs ≈ L ˆ v (t)qr + L ˆ m qrs
(24)
Substitution of the approximation in Equation (24) into Equation (22) gives the final approximate equation for the dynamic response of the system ˆ qr + D ˆ q˙ r + L ˆ m qr = L ˆ m qrs (t) M¨
(25)
Equation (25) is a forced, linear, time-invariant equation. The eigenvalue problem associated with Equation (25) comes from the undamped homogeneous solution with the assumed harmonic displacement qr = aejωt as ˆ +L ˆ ma = 0 −ω 2 Ma
(26)
Equation (26) is solved by conventional methods for the gear system natural frequencies (ωn ) and mode shapes (an ). ˆ m qrs (t) in Equation (25), is repeated as each tooth passes through the The forcing term, L gear mesh interface; with this periodicity, it contains frequency components at integer multiples of the tooth mesh frequency fm . Typically, only a few harmonics of mesh frequency are used in the analysis because the Fourier coefficients of the higher harmonics of the static solution are negligible compared to the first few. Generally three to five terms are used in the dynamic calculation. More terms can be added as necessary. The M -term truncated Fourier expression for the right-hand side of (25) is M X
ˆ m qrs = L ˆm L
Qkrs ejkωt
(27)
k=−M
Qkrs
kth
where is the Fourier coefficient and the mean value is removed. The resulting steady state dynamic solution is qr =
M X R X
k=−M n=1
an
ˆ m Qk aTn L rs ejkωt (ωn2 − k2 ω 2 ) + j2ζn ωn kω
(28)
where R is the total number of modes from the eigenvalue problem (26). The frequency domain method is computationally efficient. Long time integration simulations are avoided by using a frequency domain method. Static condensation reduces the total number of degrees of freedom. The reduction matrix (16) is independent of the forcing frequency. Therefore, the reduced mass, damping, and stiffness matrices are calculated once and used repeatedly to calculate dynamic response over a range of mesh frequencies.
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WCCM/APCOM 2010 IOP Conf. Series: Materials Science and Engineering 10 (2010) 012150
IOP Publishing doi:10.1088/1757-899X/10/1/012150
3. Results and discussion 3.1. Example 1: spur gear pair dynamics A finite element model of the involute contact ratio (ICR) 1.37 spur gear pair from the experiments by Kahraman and Blankenship [10] is used as a benchmark for the time and frequency domain methods. The time domain finite element calculation of natural frequency from impulse response for the ICR 1.37 gear pair at 170 N-m torque gives 2662 Hz, within 3% of the experimentally reported natural frequency in [10]. The frequency domain calcuation of natural freuqency is 2664 Hz, within 2.8% of the experimental natural frequency. The dynamic response of the unity ratio spur gear pair with ICR of 1.37 is shown in Figure 1 at 170 N-m torque for the time and frequency domain methods. The dynamic response is shown for a wide range of mesh frequencies. The experiments show nonlinear response at the primary resonance (fm ≈ fn ) and excitation at higher harmonics (fm ≈ fn /l for integer l > 1). The time and frequency domain finite element results accurately capture the dynamics of the gear pair as compared to the experimental data in [10]. The time domain calculation of dynamic response is able to capture the the well-known softening type nonlinearity at the primary resonance due to tooth separation [10, 12]. The time domain finite element prediction of jump frequencies match the experimental data. The resonant peaks at the natural frequency and higher harmonics are predicted accurately by the frequency domain method. The off-resonant amplitudes of dynamic response correlate well with the experimental data. Amplitudes at resonance, however, do not correlate well with experiments due to the strong nonlinear behavior for this lightly damped gear set. Nonlinear phenomena cannot be captured by the proposed frequency domain solution because the linear formulation assumes the static tooth forces approximate the dynamic tooth forces. This is a poor assumption for large amplitude vibration. In such cases, especially with tooth contact loss, deviations from the frequency domain method and nonlinear gear response are expected. The overall correlation in Figure 1 is strong for the frequency domain method. The natural frequency and off-resonant regimes are predicted accurately. Additionally, resonances due to excitations with higher harmonics are captured. While the time domain solution’s ability to capture the strongly nonlinear three-dimensional spur gear dynamics with contact loss is encouraging, the analysis takes approximately five hours to analyze one mesh frequency on a conventional dual-core computer. It took a total of 200 hours of computation time to obtain the dynamic solution in Figure 1. With this kind of time investment, the time domain finite element formulation is not practical in most cases. Additionally, this is only for a spur gear pair, meaning this method of solution is not feasible for a system level analysis where multiple gears are coupled together with shafts, bearings, and a housing. 3.2. Example 2: helical gear pair dynamics The geometric properties of the helical gear pair used in this study are the same as those for the ICR 1.37 spur gear pair except for the addition of a 30◦ helix angle. Nonlinear tooth contact loss is not common in moderately to heavily loaded helical gear pairs. Consequently, the frequency domain method should perform best for these systems. Figure 2 shows a comparison of the time domain and frequency domain finite element calculation of dynamic response for the helical gear pair at 170 N-m torque. The frequency domain solution accurately predicts the dynamic response compared to the benchmark time domain solution, including at the higher harmonic excitation zones near 791 Hz and 527 Hz. The natural frequency calculation for the time and frequency domain method is 1593 Hz and 1582 Hz, a negligible difference. The time domain prediction of dynamic response does not show the nonlinear softening behavior due to contact loss for this gear pair. Figure 2b shows 8
WCCM/APCOM 2010 IOP Conf. Series: Materials Science and Engineering 10 (2010) 012150
IOP Publishing doi:10.1088/1757-899X/10/1/012150
RMS of oscillating DTE component (µm)
30
25
20
15
10
5
0 500
1000
1500 2000 2500 Mesh frequency (Hz)
3000
3500
Figure 1. Dynamic response of a spur gear pair with ICR 1.37 at 170 N-m torque using the time domain method (4) and frequency domain method (solid) compared to experimental results (◦) [10] .
9 8
150 b)
a) Total solution time (hours)
RMS of oscillating DTE component (µm)
a comparison of the total solution time for both methods. The time domain simulation in this case took five days to analyze, whereas the frequency domain method took less than one hour.
7 6 5 4 3 2
132.2 hours
100
50
1 0
400
600
0.28 hours 0 Frequency domain method Time domain method
800 1000 1200 1400 1600 1800 Mesh frequency (Hz)
Figure 2. a) Finite element calculation of RMS of dynamic transmission error for a threedimensional profile ICR 1.37 helical gear pair with 30◦ helix angle at 170 N-m torque using the time domain solution (◦) and frequency domain solution (solid). b) Comparison of time domain and frequency domain finite element total solution time for the helical gear pair dynamic response.
4. Conclusions A computationally efficient frequency domain formulation for gear dynamics has been developed. This finite element/contact mechanics formulation accurately predicts the dynamic response of three-dimensional spur and helical gear pairs. The main conclusions are:
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WCCM/APCOM 2010 IOP Conf. Series: Materials Science and Engineering 10 (2010) 012150
IOP Publishing doi:10.1088/1757-899X/10/1/012150
(i) The time domain finite element/contact mechanics formulation accurately models the strongly nonlinear dynamic behavior of lightly damped spur gear pairs in three-dimensions. The analyses require significant computation time and are not suited for system level models. (ii) The frequency domain method accurately predicts spur gear dynamic response for low to moderate amplitude vibration. Agreement in off-resonant amplitudes was shown for multiple contact ratios and torques as compared to experimental data and time domain finite element data. (iii) The frequency domain method is well suited for the dynamic analysis of helical gears, where the nonlinearity due to contact loss is weaker than for spur gears. Dynamic response at resonant and off-resonant speeds are accurately captured using the frequency domain method with much less computational effort than the time domain solution. (iv) The time savings of the frequency domain method could be used to dynamically model complete gearbox systems. These models could include multiple gear meshes, flexible shafts, rolling element bearings, and housing structures. References [1] Gregory, R. W., Harris, S. L., and Munro, R. G., 1963, “Dynamic behavior of spur gears,” Proceedings of the institution of mechanical engineers, 178, pp. 261–266. [2] Kahraman, A., 1993, “Effect of axial vibrations on the dynamics of a helical gear pair,” Journal of Vibration and Acoustics, 115(1), pp. 33–39. [3] Lin, H. H., Oswald, F., and Townsend, D., 1994, “Dynamic loading of spur gears with linear or parabolic tooth profile modifications,” Mechanism and Machine Theory, 29(8), pp. 1115–1129. [4] Ozguven, H. N. and Houser, D. R., 1988, “Dynamic analysis of high speed gears by using loaded static transmission error,” Journal of Sound and Vibration, 125(1), pp. 71–83. [5] Velex, P. and Ajmi, M., 2006, “On the modelling of excitations in geared systems by transmission errors,” Journal of Sound and Vibration, 290(3-5), pp. 882–909. [6] Ozguven, H. N. and Houser, D. R., 1988, “Mathematical models used in gear dynamics – a review,” Journal of Sound and Vibration, 121, pp. 383–411. [7] Wang, J., Li, R., and Peng, X., 2003, “Survey of nonlinear vibration of gear transmission systems,” Applied Mechanics Reviews, 56(3), pp. 309–329. [8] Velex, P. and Maatar, M., 1996, “A mathematical model for analyzing the influence of shape deviations and mounting errors on gear dynamic behavior,” Journal of Sound and Vibration, 191(5), pp. 629–660. [9] Eritenel, T. and Parker, R., 2009, “Three-dimensional dynamic behavior of gear pairs using a nonlinear tooth mesh model,” Journal of Sound and Vibration, submitted. [10] Kahraman, A. and Blankenship, G., 1996, “Gear dynamics experiments: Part ii: Effect of involute contact ratio,” ASME Power Transmission and Gearing Conference, San Diego. [11] Parker, R. G., Agashe, V., and Vijayakar, S. M., 2000, “Dynamic response of a planetary gear system using a finite element/contact mechanics model,” Journal of Mechanical Design, 122(3), pp. 304–310. [12] Parker, R. G., Vijayakar, S. M., and Imajo, T., 2000, “Non-linear dynamic response of a spur gear pair: Modelling and experimental comparisons,” Journal of Sound and Vibration, 237(3), pp. 435–455. [13] Taminana, V. K., Kahraman, A., and Vijayakar, S. M., 2007, “A study of the relationship between the dynamic factors and the dynamic transmission error of spur gear pairs,” Journal of Mechanical Design, 129(1), pp. 75–84. [14] Vijayakar, S. M., 1991, “A combined surface integral and finite-element solution for a three-dimensional contact problem,” International Journal for Numerical Methods in Engineering, 31(3), pp. 525–545. [15] Vijayakar, S. M., Busby, H. R., and Houser, D. R., 1988, “Linearization of multibody frictional contact problems,” Computers and Structures, 29(4), pp. 569–576. [16] Vijayakar, S. M., Busby, H. R., and Wilcox, L., 1989, “Finite element analysis of three-dimensional conformal contact with friction,” Computers and Structures, 33(1), pp. 49–61.
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