Construction of Semianalytical Solutions to Spur Gear Dynamics Given Periodic Mesh Stiffness and Sliding Friction Functions

Transcription

1 Song He Acoustics and Dynamics Laboratory, Department of Mechanical Engineering, The Ohio State University, Columbus, OH Todd Rook Goodrich Aerospace, 0 Waco Street, Troy, OH todd.rook@goodrich.com Rajendra Singh Fellow ASME, Acoustics and Dynamics Laboratory, Department of Mechanical Engineering, The Ohio State University, Columbus, OH singh.3@osu.edu Construction of Semianalytical Solutions to Spur Gear Dynamics Given Periodic Mesh Stiffness and Sliding Friction Functions Gear dynamic models with time-varying mesh stiffness, viscous mesh damping, and sliding friction forces and moments lead to complex periodic differential equations. For example, the multiplicative effect generates higher mesh harmonics. In prior studies, time-domain integration and fast Fourier transform analysis have been utilized, but these methods are computationally sensitive. Therefore, semianalytical single- and multiterm harmonic balance methods are developed for an efficient construction of the frequency responses. First, an analytical single-degree-of-freedom, linear time-varying system model is developed for a spur gear pair in terms of the dynamic transmission error. Harmonic solutions are then derived and validated by comparing with numerical integration results. ext, harmonic solutions are extended to a six-degree-of-freedom system model for the prediction of (normal) mesh loads, friction forces, and pinion/gear displacements (in both line-of-action and off-line-of-action directions). Semianalytical predictions compare well with numerical simulations under nonresonant conditions and provide insights into the interaction between sliding friction and mesh stiffness. DOI: 0.5/ Introduction Sliding friction acts as an excitation to spur and helical gear dynamics, as described by Houser et al., Velex and Cahouet, Velex and Sainsot 3, and Lundvall et al. 4. Earlier, Vaishya and Singh 5 7 developed a single-degree-of-freedom SDOF spur gear model with rectangular mesh stiffness k t and sliding friction profiles; they solved the dynamic transmission error DTE t response by using the Floquet theory 5 and multiterm harmonic balance method MHBM 6. Their work was recently refined and extended to helical gears in our papers 8,9 where closed-form solutions of t for a SDOF system are derived under friction excitation. The equal load sharing assumption 5 9 yields simplified expressions and analytically tractable solutions, but these do not describe realistic conditions. This particular deficiency was partially overcome in our articles 0, where we proposed a multidegree-of-freedom MDOF model with realistic time-varying k t and sliding friction functions. However, we utilized numerical integration and fast Fourier transform FFT analysis methods in prior studies 8 that are often computationally sensitive. Hence, a semianalytical algorithm based on MHBM is highly desirable for an efficient construction of frequency responses without any loss of generality. Recently, Velex and Ajmi implemented a similar harmonic analysis to approximate the dynamic factors in helical gears based on tooth loads and quasistatic transmission errors. Their work, however, does not describe the multidimensional system dynamics or include the frictional effect, which may lead to multiplicative terms as described earlier. The prime objective of this article is thus to extend prior publications 5,0. In particular, we intend to develop semianalytical harmonic balance solutions to the 6DOF spur gear model 0 with realistic mesh stiffness k t and sliding friction functions. Contributed by the Power Transmission and Gearing Committee of ASME for publication in the JOURAL OF MECHAICAL DESIG. Manuscript received October 6, 007; final manuscript received June 0, 008; published online October 7, 008. Review conducted by Philippe Velex. Problem Formulation Gear dynamic models are usually described by periodic differential equations 3 5 due to significant variations in mesh stiffness k t and mesh damping c t within the fundamental period t c one mesh cycle. Additionally, dynamic friction force F f t and torque M f t also undergo periodic variations, with the same period t c, due to changes in normal mesh loads and coefficient of friction, as well as a reversal in the direction of F f t at the pitch point 5 0, especially in spur and helical gears. For the sake of illustration, a unity ratio ASA spur gear with tip relief and contact ratio around.6 is used as an sample case for this study; refer to Table of Ref. 0 for its parameters. Major periodic excitations, namely, the mesh stiffness k t of the two meshing tooth pairs and the frictional functions f t, are shown for the sample case in Figs. a and b, respectively. Here, points A and D are the starting and ending points of one complete mesh event; points B and C are the lowest and highest points of single tooth contact; P is the pitch point, where friction force changes direction. Refer to Fig. a for geometric locations of the points along the line-of-action LOA direction, while the off-line-ofaction OLOA direction is the frictional direction. Define tooth pairs and 0 as the pairs rolling along lines AC and CD, respectively. The effective stiffness function k i t of each meshing tooth pair can be computed by k i t = s,i t / r b s,i t, i=0, 0, where r b is the base radius and s,i t and s,i t are the static normal contact force and angular deflection computed by a static finite element/contact mechanics FE/CM analysis 6. Modifications to the tooth profiles, such as the linear tip relief, are characterized by changes in the effective s,i t and s,i t profiles. Hence the resulting k i t includes the effect of the tip relief as well. Refer to Ref. 0 for the calculation procedure of k i t. Derivations of f t will be explained later in Eqs. 4a 4e along with key definitions of the system model. The fundamental nature of the linear time-varying LTV system is illustrated in Fig. b, and the system model is described in our paper 0. Key assumptions include the following: i The Journal of Mechanical Design Copyright 008 by ASME DECEMBER 008, Vol. 30 / 60-

2 Fig. a Periodic mesh stiffness functions of the spur gear pair example with tip relief given nominal pinion torque T p =550 lb in. Key: blue dashed line, k 0 t ; red solid line, k t. b Periodic frictional functions. Key: blue dashed line, f 0 t ; green dashed-dotted line, f t ; red solid line, f t. pinion and gear are rigid disks. ii Vibratory motions are small in comparison to the nominal motion; this would lead to a LTV model. Only nonresonant conditions are considered since significant dynamic motions at resonance may not only change the contact pattern such as the effective center distance but also cause teeth separation, which would introduce nonlinear interactions and further complicate the formulation. iii The Coulomb friction is assumed with a constant although the sign is reversed at the pitch point. The governing SDOF equation in terms of DTE t =r bp p t r bg g t is given below, where subscripts p and g correspond to the pinion and gear, respectively; is the vibratory component of the rotation; T and are the nominal torque and rotation speed; is the base pitch; and J is the moment of inertia. ote that J e is defined differently from that in Ref. 0 for a compact formulation of Eq. a, S S J e t + J b c i t t + k i t t + i=0 i=0 sgn mod pr bp t, + S j L AP c i t t + k i t t X pi t J g r bp + X gi t J p r bg = T e + T t a J e = J p J g J b = J g r bp + J p r bg T e = T pj g r bp + T gj p r bg b c d Fig. a Snap shot of the contact pattern at t=0 for the sample spur gear pair. b ormal mesh and friction forces of the analytical spur gear system model. X pi t = L XA + S i +mod pr bp t,, X gi t = L YC + i mod gr bg t,, S S T t = J b c i t t + k i t t + i=0 i=0 i =0,...,S = floor e i =0,...,S = floor f sgn mod pr bp t, + S j L AP c i t t + k i t t X pi t J g r bp + X gi t J p r bg g Here, the jth tooth pair passes though the pitch point P during the meshing event, and the reversal at P is characterized by the sign function sgn with a constant coefficient of Coulomb friction 5. The modulus function, defined as mod t,t c =t t c floor t/t c, is used to formulate the periodic friction force F f t and the moment arm X t at the fundamental period of t c. The floor function rounds off the contact ratio to the nearest integers toward a lower value. Also, T t of Eq. g represents the time-varying component of the forcing function due to the unloaded static manufacturing transmission error t. Finally, geometric lengths L XA, L YC, and L AP of Fig. a could be found as follows, where u is the gear ratio, l is the center distance, wt is the transverse operating pressure angle, and r a is the outside radius, 60- / Vol. 30, DECEMBER 008 Transactions of the ASME

3 L XA = l sin wt rag r bg a L YC = rag r bg + b L AP = rag r bg ul sin wt c u + Observe that Eqs. a g significantly differs from the classical Hill s equation 3 in several ways. First, the periodic k i t is not confined to a rectangular wave assumed by Vaishya and Singh 5 7 or a simple sinusoid as in Mathieu s equation. Instead, Eqs. a g should describe realistic, yet continuous, profiles of Fig. a resulting from a detailed FE/CM analysis 6. Hence, multiple harmonics of k i t should be considered. Second, the periodic viscous c i t term should dissipate vibratory energy due to the sliding friction besides its kinematic effect. Third, the S i= i t c i t and S i= i t k i t terms of Eqs. a g incorporate combined but phase correlated contributions from all yet changing tooth pairs in contact. Consequently, the relative phase between neighboring tooth pairs should play an important role in the resulting response t. Fourth, multiplicative effects between k i t, c i t, X i t, and t should result in higher mesh harmonics, which poses difficulty in constructing closed-form solutions. Lastly, the coupling between T t and t in Eq. g indicates that the frictional forces and moments reside on both sides of Eq. a as either periodically varying parameters or external excitations, thus posing further mathematical complications. Each periodic function, k i, c i, f i, or, now has a period of T=. Figure b shows typical f 0, f, and f functions, which describe the periodic moment arm and sliding friction excitations for the sample case Direct Application of Multiterm Harmonic Balance. Define the Fourier series expansions of the periodic k i and c i in Eqs. 4a 4e up to mesh harmonics as follows, where the angular frequency n = n in rad/s and n is the mesh order: k i = A ki0 + A kin cos n + B kin sin n A ki0 = 0 k i d A kin k in cos n d = 0 B kin k in sin n d = 0 c i = ki I e = A ci0 + A cin cos n + B cin sin n 5a 5b 5c 5d 3 Semianalytical Solutions to the SDOF Spur Gear Dynamic Formulation Consider the sample case with only the mean load T e, i.e., T t =0 or t =0. Equations a g can be rewritten over one complete mesh cycle 0 t t c, as follows for the sample case with two mesh tooth pairs, where m e and F e are the effective mass and force; E, E, and E 3 are gear constants, A di0 = 0 c i d A cin c in cos n d = 0 6a 6b 6c m e t + c 0 t t + k 0 t t +E + E 3 t + c t t + k t t + E + E 3 t sgn t L AP pr bp = F e E = L XA + J g r bp + L YC J p r bg /J b E = L XA J g r bp + L YC + J p r bg /J b E 3 = pr bp J g r bp J p r bg /J b 3a 3b 3c 3d B cin c in sin n d 6d = 0 The f i functions could be expanded explicitly as shown below, f 0 = n sin n 7a m e = J e /J b 3e F e = T e /J b 3f ext, reformulate Eqs. 3a 3f in terms of the dimensionless time =t/t c, such that =d /d =t c t and =d /d =t c t, m e + t c c 0 + t c k 0 +E + t c E 3 f 0 + t c c + t c k +E f + t c E 3 f = t c F e 4a f 0 =mod, f = sgn mod, P = sgn f 0 P f =mod, sgn mod, P = f 0 f 4b 4c 4d f = P 4 sin n f = P P cos n n n 4 cos n P n sin n 7b n P sin n P cos n P cos n + n P cos n P sin n P 0.5 n sin n 7c P = L AP / 4e Finally, assume that the periodic dynamic response is of Journal of Mechanical Design DECEMBER 008, Vol. 30 / 60-3

4 the following form: = A 0 + A n cos n + B n sin n 8 Substitute Fourier series expansions of Eqs. 5a 5d, 6a 6d, 7a 7c, and 8 into Eqs. 4a 4e and balance the mean and harmonic coefficients of sin n and cos n. This converts the linear periodic differential equation into easily solvable linear algebraic equations as expressed below where K= h is a square matrix of dimension + consisting of known coefficients of k i, c i, and f i. By calculating the inverse of K= h, the + Fourier coefficients of could be computed at any gear mesh harmonic n, A B K= h A A 0 B = Fe Semianalytical Solutions Based on One-Term HBM. ext, we construct one-term HBM 7 solutions to conceptually illustrate the method. Set the harmonic order = only the fundamental mesh, in addition to the mean term in Eqs. 5a 5d, 6a 6d, 7a 7c, 8, and 9 and balance the harmonic terms in Eqs. 4a 4e. This leads to a K= h matrix of dimension 3. Three of its typical coefficients are given as follows and the rest could be found in a similar manner: K h = t ca k A f E 3 + B k B f E t c B k0 E 3 / +A k00 +A k0 + A k A f E +A k00 E + t c A k00 E 3 +A k0 A f0 E Fig. 3 Semianalytical versus numerical solutions for the SDOF model, expressed by Eq. 3, given T p=550 lb in., Ω p =500 rpm, and =0.04. a Time-domain responses; b mesh harmonics in frequency domain. Key: blue solid line and, numerical simulations; black dashed line and, semianalytical solutions using one-term HBM; red dashed-dotted line and, semianalytical solutions using five-term HBM. +t c A k0 A f0 E 3 + t c B k B f E 3 0a K h = A k0 + A k + A 0 E + A k0 A f E + t c A k0 A f E 3 + A k A f0 E + t c A k A f0 E 3 + t c A k0 E 3 / 0b K h3 = B k A f0 E + B k + t c B k A f0 E 3 + A k0 B f E + t c A k0 B f E 3 + B k0 E + B k0 t c A k00e 3 + t cb k0 E 3 0c The Fourier series coefficients of are then obtained by inverting K= h. Figure 3 shows that the one-term HBM solution predicts the overall tendency mean and first harmonic fairly well when compared with numerical simulations at T p=550 lb in. and p=500 rpm. This confirms that the one-term HBM and likewise the MHBM approach coverts the periodic differential Eqs. 4a 4e with multiple interacting coefficients into simpler algebraic calculations that are computationally more efficient than numerical integrations and subsequent FFT analyses. Thus, the semianalytical solution provides an effective design tool. Also, most coefficients of Eqs. 0a 0c show sideband effects that are introduced by k t or c t and f i t functions. 3.3 Iterative MHBM Algorithm. When 5, we can utilize a symbolic software 8 to balance multiple harmonic terms and calculate K= h. However, the computational cost involved with each element of K= h increases by 3 due to the triple multiplication of periodic coefficients in Eqs. 4a 4e. Consequently, for higher say, 5, a direct computation of K= h becomes inefficient and thus inadvisable. Instead, we apply a matrix-based iterative MHBM algorithm 9,0. First, define mean and vibratory speed variables and, where is the subharmonic index, = / t c = t 0, =mod /, Also, define the differential operator ˆ as ˆ = d d = a b c a = ˆ b Equations 3a 3f are then converted into the following form, where E 4 = E 3 : m e ˆ + c0 ˆ + k 0 +E + E 4 + c ˆ + k + E + E 4 sgn /L AP = F e or expressed more compactly as m e ˆ + C ˆ + K = Fe 3 4a C = c 0 +E + E 4 + c + E + E 4 sgn /L AP 4b 60-4 / Vol. 30, DECEMBER 008 Transactions of the ASME

5 K = k 0 +E + E 4 + k + E + E 4 sgn /L AP 4c For the MHBM, the discrete Fourier transform DFT is applied to transform from the time domain to the frequency domain, and a discretized Fourier expansion matrix F= transforms the frequency domain back to the time domain, sin cos... sin cos sin cos... sin cos F= = ] ] ] ] ] sin M cos M... sin M cos M 5a ] M = F= ˆ ˆ ˆ ] ˆ M = F= D= 5b 5c Fig. 4 Semianalytical versus numerical solutions for the SDOF model as a function of pinion speed with =0.04. a Mesh order, b n=, c n=3, and d n=4. Key: red, numerical simulations; blue solid line, semianalytical solutions using five-term HBM. ˆ ˆ ˆ ] ˆ M = F= D= 5d Here, i =i /M and M +, is a vector of + Fourier coefficients, and the Fourier differentiation matrix D= is given as D= = 0 5e ] ] ] D= = f ] ] ] Applying the DFT to the equation of motion yields the following MHBM equations where F= + is the Moore Penrose or pseudoinverse of the DFT matrix, m e D= + F= + C= F= D= + F= + K= F= = F= + F e 6a K= diag K K K M 6b C= diag C C C M 6c F e = F e T 6d Figure 3 shows that the five-term HBM solutions compare well with numerical simulations. Likewise, an increase in captures higher frequency components around the tenth mesh harmonic, as observed in the numerical simulations. The semianalytical solutions are efficiently used in Fig. 4 for parametric studies of at the gear mesh harmonics over a range of p, and these calculations are indeed achieved with reduced computational cost. 4 Semianalytical Solutions to 6DOF Spur Gear Dynamic Formulation A careful examination of the 6DOF model reported in Ref. 0 reveals that the dynamic bearing displacement y in the OLOA direction depends on the LOA dynamics since the friction force k acts in the OLOA direction. Conversely, the dynamic bearing displacement x in the LOA direction is not influenced by i.e., is decoupled from the OLOA motion y. This is because of the following reasons: First, no off-diagonal term exists in the effective shaft-bearing stiffness matrix that couples the LOA and OLOA dynamics 8. Second, the friction force described by the Coulomb model is independent of y. Third, recall our assumption that the dynamic responses such as y have no influence on the kinematics such as the center distance. Such one-way coupling between LOA and OLOA dynamics implies that we can reduce the 6DOF model into a simpler 3DOF model in terms of x p, x g, and. Dynamic responses of the 3DOF model could further be used to determine the friction forces that excite the OLOA dynamics. For the 3DOF system, when the composite DTE =r bp p r bg g +x p x g can be approximated by the semianalytical solution =r bp p r bg g of the SDOF model, a similar one-way coupling is created where is decoupled from x p and x g, while x p and x g are excited by in terms of the normal loads. This suggests that when the excitation mesh frequencies do not excite any coupled transverse-torsional modes in the LOA direction, the harmonic solutions of the SDOF system could be extended to predict translational responses in both LOA and OLOA directions by utilizing the one-way coupling effects. ote that the semianalytical method analyzes the 6DOF system 0 as a 5DOF model as it calculates the t rather than absolute angular displacements p t or p t. Journal of Mechanical Design DECEMBER 008, Vol. 30 / 60-5

6 In order to quantify the nonresonant condition under which the one-way coupling exists in the LOA direction, we use a 6DOF linear time-invariant spur gear model of Fig. 5 a and focus on its subset of a unity gear pair 3DOF model to study the natural frequency distribution. We define the following parameters for the sample case: mass of pinion gear m p =m g =M; moment of inertia J p =J g =J; basic radius r bp =r bg =R; the equivalent mass m e =J/ R ; time-averaged mesh stiffness k m T p =550 lb in. ; shaftbearing stiffness K Bp =K Bg =K B. The natural frequencies of the 3DOF system are found as follows :,3 = k mm + k m + K B m e km M + k m + K B m e 4Mm e k m K B Mm e 7a = K B 7b M Mode and Mode 3 3 correspond to the first and second coupled transverse-torsional modes, respectively, and Mode is the purely transverse mode. Also, the natural frequency of the corresponding SDOF torsional system can be estimated as S = km /m e. Figure 5 b compares the natural frequencies of the 3DOF and SDOF models as a function of the stiffness ratio K B /k m. For an easy comparison with the excitation mesh frequency n, which is determined by the nominal pinion rpm, all natural frequencies are converted in Fig. 5 b from rad/s into rpm units. Observe that of the SDOF model asymptotically approaches only one of the natural frequencies of the 3DOF model. Consequently, when the excitation frequency coincides with either of the two resonances not predicted by the SDOF, larger errors are expected when extending the solutions of the SDOF model into other dimensions. Such nonresonant restriction of the proposed algorithm is, however, consistent with our assumption that vibratory motions are small in comparison to the mean motion. When K B /k m or K B /k m 0, S asymptotically approaches 3 or. Moreover, n does not excite any resonance in Zone I n and Zone III n 3. Additionally, nonresonant Zone II could be found for both soft K B /k m, n S and stiff K B /k m 0, S n shaft-bearing cases. In these nonresonant zones, the semianalytical solution of the SDOF model could be extended to the 6DOF system. Figure 6 compares the five-term HBM prediction of for the SDOF system with numerical simulation for a 6DOF model 0 for two limiting values of K B /k m. When K B /k m =00, prediction matches well with numerical simulation. However, when K B /k m =0.37 i.e., nominal case of Ref. 0, a good correlation is observed only away from system resonances in Zones I III. The dynamic normal mesh loads i of the pinion and gear are equal in magnitude but opposite in direction 0. To account Fig. 5 a 6DOF spur gear pair model and its subset of unity gear pair 3DOF model used to study the natural frequency distribution. b atural frequencies Ω asafunctionofthe stiffness ratio K B /k m. Key: black solid line, Ω S of SDOF system torsional only, in terms of DTE ; blue dashed line, Ω of 3DOF system; green dotted line, Ω of 3DOF system; red dashed-dotted line, Ω 3 of 3DOF system. Fig. 6 Semianalytical versus numerical solutions for the 6DOF model as a function of Ω p with =0.04. a Mesh order, b n=, c n=3, and d n=4. Key: blue solid line, predictions using five-term HBM; green, numerical simulations 4 with nominal K B K B /k m =0.37 ; red, numerical simulations with stiff K B K B /k m = / Vol. 30, DECEMBER 008 Transactions of the ASME

7 for interactions between k i and as well as between c i and, Fourier series is expanded to find the i terms up to mesh harmonics, i = k i + c i = A i0 + A in cos n + B in sin n 8 Similarly, the dynamic friction forces F fi are expanded in Eqs. 9a and 9b. Since tooth pair is associated with a periodic change of friction force at the pitch point, F f is expanded up to 3 mesh harmonics due to a multiplication of k,, and f, F f0 = 0 = A F00 + A F0n cos n + B F0n sin n 9a F f = f = A F0 + A Fn cos n 3 + B Fn sin n 3 9b Fig. 7 Semianalytical versus numerical solutions of the LOA displacement x p for the 6DOF model as a function of Ω p with K B /k m =00, =0.04. a Mesh order, b n=, c n=3, and d n=4. Key: red, numerical simulations; blue solid line, predictions using five-term HBM. In the LOA or x direction, the transfer function at frequency n with p as the input and x p as the output is found by using the corresponding linear time-invariant model as follows 0, where K pbx and pbx are the shaft-bearing stiffness and damping terms: X p t c = p m p + K pbx t c + j t c pbx KpBx m p 0a Magnitude M pxn and phase pxn at the nth mesh order, n = n rad/s, are M px n = t c n m p + K pbx t c +4 n t c pbx K pbx m p 0b px n = tan nt c pbx KpBx m p 0c n m p K pbx t c Thus the pinion bearing displacement x p in the LOA or x direction is expanded as x p = M px 0 A 00 + A 0 + M px n A 0n + A n cos n + px n + M px n B 0n + B n sin n + px n In the OLOA or y direction, the magnitude and phase of the transfer function from friction force F f to pinion displacement y p could be found at n as M py n = t c n m p + K pby t c +4 n t c pby K pby m p a py n = tan nt c pby KpBy m p b n m p K pby t c Thus the pinion displacement y p in the OLOA or y direction is expanded as y p = M py 0 A F00 + M py 0 A F0 + M py n A F0n cos n + py n + M py n B F0n sin n + py n 3 + M py n A Fn cos n + py n 3 + M py n B Fn sin n + py n 3 Equations and 3 confirm that multiplications between periodic coefficients k i or c i, or ċ i, and f i lead to higher mesh harmonic components, which are commonly observed in spur gears. Figure 7 compares the semianalytical x p with numerical prediction 0 as a function of p with K B /k m =00. Good correlations are observed up to 6,000 rpm in Zone with high K B /k m including the LOA shaft-bearing resonances at n=3 and n=4. Likewise, the semianalytical y p in the OLOA direction compares well with numerical simulations in Fig. 8, where the shaft-bearing resonances are also observed at n=3 and n=4. Although good correlations are usually expected for stiff shaft bearings, it is worthwhile to examine the case where K B is comparable or less than k m. By using the nominal parameters of Ref. 0, where K B /k m =0.37, semianalytical predictions are compared with numerical simulations in Figs. 9 and 0 for LOA and OLOA responses, respectively. Observe that semianalytical x p matches numerical simulations only from system resonances, as explained earlier in Fig. 5. onetheless, good correlation is observed in the OLOA direction over the operating speed range in Fig. 0 since the bearing resonance dictates the y p dynamics. Finally, semianalytical predictions are used to examine the interactions between the sliding friction and tip relief embedded in the realistic mesh stiffness function k t. Predicted displacement ratios of y p to x p for the first four harmonics are displayed in Fig. for the sample case with long tip relief 0 as compared to Journal of Mechanical Design DECEMBER 008, Vol. 30 / 60-7

8 Fig. 8 Semianalytical versus numerical solutions of the OLOA displacement y p for the 6DOF model as function of Ω p with K B /k m =00, =0.04. a Mesh order, b n=, c n=3, and d n=4. Key: red, numerical simulations; blue solid line, predictions using five-term HBM. Fig. 0 Semianalytical versus numerical solutions of the OLOA displacement y p for the 6DOF model as a function of Ω p with K B /k m =0.37, =0.04. a Mesh order, b n=, c n =3, and d n=4. Key: green, numerical simulations; blue solid line, predictions using five-term HBM. the perfect involute profile case without any tip relief as a function of the mean torque T p. Observe that for the perfect involute profile, the y p /x p ratios remain almost constant over the entire torque range, while applying tip relief significantly alters the y p /x p ratios. A peak ranging from 0.8 up to 5 is observed for the long tip relief case around the optimal load of 550 lb in., implying that an amplification due to the sliding friction takes place. This can be explained by the fact that the application of tip relief not only minimizes x p at the design load but also amplifies the harmonic of y p due to an out-of-phase relationship between the dynamic normal load and dynamic friction forces F f 0. 5 Conclusion An application of the harmonic balance method to the gear dynamic models with multiple periodic coefficients leads to an efficient algorithm to evaluate the effect of sliding friction on spur gear dynamics as compared with the numerical integration method 0. Multiterm harmonic balance solutions based on the SDOF linear time-varying model are successfully extended to a 6DOF system model for the predictions of normal mesh loads, friction forces, and bearing displacements in the LOA and OLOA directions. A simplified 3DOF linear time-invariant model is used Fig. 9 Semianalytical versus numerical solutions of the LOA displacement x p for the 6DOF model as a function of Ω p with K B /k m =0.37, =0.04. a Mesh order, b n=, c n=3, and d n=4. Key: green, numerical simulations; blue solid line, predictions using five-term HBM. Fig. Semianalytical predictions for the ratio of OLOA displacement x p to the LOA displacement x p for the 6DOF model asafunctionoft p with K B /k m =0.37, =0.04. a Mesh order, b n=, c n=3, and d n=4. Key: red dashed line with, perfect involute gear; blue solid line with, gear with tip relief / Vol. 30, DECEMBER 008 Transactions of the ASME

9 to explain the nonresonant operating conditions under which the proposed algorithm is applicable. Semianalytical solutions also provide new insights into the multiplicative interactions between time-varying mesh stiffness say, with contributions from tip relief and sliding friction including friction force excitation and varying torque. Such dynamic interactions explain the higher harmonic components and the phase relationship between sliding friction and mesh stiffness function. These are the main contributions of our paper. The methods of this article could be extended to multimesh spur gear dynamics. Acknowledgment This article is based on a three-year study that was supported by the U.S. Army Research Office. We acknowledge the Graduate School of The Ohio State University for awarding the Presidential Fellowship to the first author. References Houser, D. R., Vaishya, M., and Sorenson, J. D., 00, Vibro-Acoustic Effects of Friction in Gears: An Experimental Investigation, SAE Paper o Velex, P., and Cahouet, V., 000, Experimental and umerical Investigations on the Influence of Tooth Friction in Spur and Helical Gear Dynamics, ASME J. Mech. Des., 4, pp Velex, P., and Sainsot, P., 00, An Analytical Study of Tooth Friction Excitations in Spur and Helical Gears, Mech. Mach. Theory, 37, pp Lundvall, O., Strömberg,., and Klarbring, A., 004, A Flexible Multi-Body Approach for Frictional Contact in Spur Gears, J. Sound Vib., 78 3, pp Vaishya, M., and Singh, R., 00, Analysis of Periodically Varying Gear Mesh Systems With Coulomb Friction Using Floquet Theory, J. Sound Vib., 43 3, pp Vaishya, M., and Singh, R., 00, Sliding Friction-Induced on-linearity and Parametric Effects in Gear Dynamics, J. Sound Vib., 48 4, pp Vaishya, M., and Singh, R., 003, Strategies for Modeling Friction in Gear Dynamics, ASME J. Mech. Des., 5, pp He, S., Gunda, R., and Singh, R., 007, Inclusion of Sliding Friction in Contact Dynamics Model for Helical Gears, ASME J. Mech. Des., 9, pp He, S., and Singh, R., 008, Dynamic Transmission Error Prediction of Helical Gear Pair Under Sliding Friction Using Floquet Theory, ASME J. Mech. Des., 30 5, He, S., Gunda, R., and Singh, R., 007, Effect of Sliding Friction on the Dynamics of Spur Gear Pair With Realistic Time-Varying Stiffness, J. Sound Vib., 30, pp He, S., and Singh, R., 007, Dynamic Interactions Between Sliding Friction and Tip Relief in Spur Gears, DETC , Tenth International Power Transmission and Gearing Conference, Las Vegas, Sept Velex, P., and Ajmi, M., 007, Dynamic Tooth Loads and Quasi-Static Transmission Errors in Helical Gears Approximate Dynamic Factor Formulae, Mech. Mach. Theory, 4, pp Richards, J. A., 983, Analysis of Periodically Time-Varying Systems, Springer, ew York. 4 Jordan, D. W., and Smith, P., 004, onlinear Ordinary Differential Equations, 3rd ed., Oxford University Press, ew York. 5 Thomsen, J. J., 003, Vibrations and Stability, nd ed., Springer, ew York. 6 EXTERALD, CALYX software, 003, Helical3D User s Manual, ASOL Inc., Hilliard, OH, 7 Kim, T. C., Rook, T. E., and Singh, R., 005, Super- and Sub-Harmonic Response Calculations for a Torsional System With Clearance on-linearity Using Harmonic Balance Method, J. Sound Vib., 8 3 5, pp MAPLE 0, symbolic software, 005, Waterloo Maple Inc., Waterloo, O. 9 Padmanabhan, C., Barlow, R. C., Rook, T. E., and Singh, R., 995, Computational Issues Associated With Gear Rattle Analysis, ASME J. Mech. Des., 7, pp Duan, C., and Singh, R., 005, Super-Harmonics in a Torsional System With Dry Friction Path Subject to Harmonic Excitation Under a Mean Torque, J. Sound Vib., , pp Kahraman, A., and Singh, R., 99, Error Associated With A Reduced Order Linear Model of Spur Gear Pair, J. Sound Vib., 49 3, pp Singh, R., 005, Dynamic Analysis of Sliding Friction in Rotorcraft Geared Systems, Army Research Office, Grant o. DAAD Journal of Mechanical Design DECEMBER 008, Vol. 30 / 60-9