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On the Boundary between Nonlinear Jump Phenomenon and Linear Response of Hypoid Gear Dynamics

On the Boundary between Nonlinear Jump Phenomenon and Linear Response of Hypoid Gear Dynamics Hindawi Publishing Corporation Advances in Acoustics and Vibration Volume 2011, Article ID 583678, 13 pages doi:10.1155/2011/583678 Research Article On the Boundary between Nonlinear Jump Phenomenon and Linear Response of Hypoid Gear Dynamics 1 2 Jun Wang and Teik C. Lim Advanced Components Systems Division, Caterpillar Inc., Peoria, IL 61630, USA Mechanical Engineering Program, School of Dynamic Systems, University of Cincinnati, 598 Rhodes Hall, P.O. Box 210072, Cincinnati, OH 45221, USA Correspondence should be addressed to Teik C. Lim, teik.lim@uc.edu Received 27 October 2010; Revised 10 March 2011; Accepted 17 March 2011 Academic Editor: Mohammad Tawfik Copyright © 2011 J. Wang and T. C. Lim. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. A nonlinear time-varying (NLTV) dynamic model of a hypoid gear pair system with time-dependent mesh point, line-of-action vector, mesh stiffness, mesh damping, and backlash nonlinearity is formulated to analyze the transitional phase between nonlinear jump phenomenon and linear response. It is found that the classical jump discontinuity will occur if the dynamic mesh force exceeds the mean value of tooth mesh force. On the other hand, the propensity for the gear response to jump disappears when the dynamic mesh force is lower than the mean mesh force. Furthermore, the dynamic analysis is able to distinguish the specific tooth impact types from analyzing the behaviors of the dynamic mesh force. The proposed theory is general and also applicable to high-speed spur, helical and spiral bevel gears even though those types of gears are not the primary focus of this paper. 1. Introduction past or fully understood. Hence, the goal of this study is to seek out the conditions dictating the transitional boundary Gear dynamics have been studied intensively as evident between nonlinear and linear responses. In other word, we from the discussions in [1–5]. In one of the earlier studies, would like to know precisely when jump phenomenon stops in advance as drive load goes from light to heavy. Comparin and Singh investigated the nonlinear frequency response of an impact pair and located the transition condi- To assist in the above-mentioned analysis, a nonlinear tions for no-impact, single-sided impact, and double-sided time-varying dynamic model with time-dependent mesh impact cases [1]. Later, Kahraman, and Singh examined the position, line-of-action mesh vectors, mesh stiffness, mesh nonlinear dynamics of a spur gear pair [2]aswellasageared damping, and backlash nonlinearity is proposed similar to the one employed by the authors in earlier studies [6, 7]. rotor-bearing system [3] and also studied the interaction between mesh stiffness and clearance nonlinearities [4]. Based on the model, different levels of torques and damping More recently, Cheng and Lim studied the vibratory response ratios are applied so that the transition condition between linear and nonlinear responses can be located. The analysis of a hypoid geared rotor system with nonlinear time-varying mesh characteristics [5]. From those studies, it is now well clearly shows that when the dynamic mesh force is greater known that nonlinear phenomena like jump discontinuity than mean mesh force, the jump phenomena will occur. frequently occurs for lightly loaded gear pairs. As the On the other hand, if the dynamic mesh force is smaller in drive load is progressively increased, the response becomes value than the mean mesh force, the jump behavior tends to disappear making the response appearing to be quite linear. more linear until the jump response completely disappears. Understanding the conditions ahead of time when the geared system will exhibit nonlinear behaviors is highly desirable 2. Mathematical Model since the nonlinear dynamic analysis is typically very time consuming. However, to the best of our knowledge this The two degrees-of-freedom (DOF) lumped parameter transitional phase has never been studied extensively in the torsional vibration model of a hypoid gear pair as shown 2 Advances in Acoustics and Vibration Z Z p g Gear axis Y c k m m Gear axis Pinion axis Pinion axis I p (a) (b) Figure 1: (a) Dynamic model of a hypoid gear pair, and (b) pinion and gear coordinate systems. in Figure 1(a), which was also applied in earlier studies by n r j expressed as λ = · ( × )(l = p, g for pinion and gear l l l the authors [6, 7], is again assumed for the current study. resp.). Also, is the unit vector of the line of action along the This simplified representation is chosen over more complex mesh force direction in the coordinate system S as shown in system to allow for the study to focus on the behavior of Figure 1(b), r is the position vector of mesh point, and j is the boundary between nonlinear jump phenomenon and linear dynamic response. The pinion and gear bodies each are the unit vector along the rotating axes of pinion and gear. By assuming p = δ − e,(1a)and (1b) can be combined allowed to rotate only. The mesh coupling is represented by a pair of mesh damping and stiffness elements. Using either the and simplified into the following form: Newton’s or Lagrangian’s methods, the equations of motion λ T λ T p p g g can be easily derived as ¨ ˙ ¨ m p + c g p + k f p = m + − e , (2a) e m m e I I p g ¨ ˙ I θ + λ c g δ − e ˙ + λ k f (δ − e) = T,(1a) p p p m p m p p − b, p ≥ b, ¨ ˙ ⎨ I θ − λ c g δ − e − λ k f (δ − e) =−T,(1b) g g g m g m g f p = 0, −b< p< b, (2b) (δ − e − b), δ − e ≥ b, ⎪ p + b, p ≤−b, f (δ − e) = 0, −b< δ − e< b, (1c) ⎪ 0, −b< p < b, ⎩ ˙ g p = (2c) (δ − e + b), δ − e ≤−b, p, else, ˙ 2 2 ⎪ δ − e ˙ , δ − e ≥ b, where m = 1/(λ /I + λ /I ). Since mesh damping is gen- ⎪ e p p g g ⎨ erally time varying, proportional mesh damping is assumed g δ − e ˙ = 0, −b< δ − e< b, (1d) here, which is more realistic than simply applying constant ⎪   damping. Thus, the damping model can be expressed as δ − e ˙ , δ − e ≤−b, qk qω m m n ζ = = = , (3) where I and I are the mass moments of inertias of pinion 2m ω 2 p g 2 k m m em em n and gear, T and T are torque loads a on the pinion and gear, p g k and c aremeshstiffness and mesh damping coefficients, m m 2ζ c = qk = k , (4) m m m e is unloaded kinematic transmission error, and 2b is the gear backlash. The dynamic transmission error can be written as 2 2 δ = λ θ − λ θ , while the directional rotation radius can be where ω = k /m , m = 1/(λ /I + λ /I ). p p g g n mm em em p g pm gm Advances in Acoustics and Vibration 3 8 8 7 7 6 6 5 5 4 4 3 3 2 2 1 1 0 0 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 Dimensionless frequency Dimensionless frequency (a) (b) 8 8 7 7 6 6 5 5 4 4 3 3 2 2 1 1 0 0 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 Dimensionless frequency Dimensionless frequency (c) (d) 1 1 0 0 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 Dimensionless frequency Dimensionless frequency (e) (f ) 8 8 7 7 6 6 5 5 4 4 3 3 2 2 1 1 0 0 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 Dimensionless frequency Dimensionless frequency (g) (h) Figure 2: Continued. Dynamic mesh force (mean) Dynamic mesh force (STD) Dynamic mesh force (STD) Dynamic mesh force (STD) Dynamic mesh force (mean) Dynamic mesh force (mean) Dynamic mesh force (STD) Dynamic mesh force (STD) 4 Advances in Acoustics and Vibration 8 8 7 7 6 6 5 5 4 4 3 3 2 2 1 1 0 0 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 Dimensionless frequency Dimensionless frequency (i) (j) Figure 2: Dynamic mesh force response for different input torque loads. Dynamic mesh force (STD) is shown in (a) T = 0.99, (b) T = 1.37, p p (c) T = 2.01, (d) T = 2.81, and (e) T = 3.27, while mean value is given by (f ) T = 0.99, (g) T = 1.37, (h) T = 2.01, (i) T = 2.81, and p p p p p p p (j) T = 3.27. ∗:  no impact; •:  single-sided impact; +: double-sided impact; —–: dynamic mesh force (increasing frequency); ...: dynamic mesh force (decreasing frequency); - - -: theoretical static mesh force. 2 2 2 The time-varying mesh parameters in (2a), (2b), and (2c) where η = λ I /λ I , T = λ T /bω I ,and T = ηT . p g p pm p p g p gm pm n can be described as a superposition of a mean value and a The dimensionless mesh parameters are described as sinusoidal component as shown below: λ = 1+ λ cos ω t + φ , p pa p λ = λ + λ cos ωt + φ , p pm pa p λ = 1+ λ cos ω t + φ , g ga g λ = λ + λ cos ωt + φ , g gm ga g (5) (9) k = k + k cos ωt + φ , k = 1+ k cos ω t + φ , m mm ma k a k e = e cos ωt + φ . a e e = e cos ω t + φ . a e The dynamic mesh force can be computed from Subsequently the dimensionless dynamic mesh force can be F = c g p ˙ + k f p = 2ζ g p ˙ + k f p . (6) derived as d m m m From the torque and directional rotation radii, the mean F = 2ζkg p + kf p , (10) value of the mesh force can be calculated from and the dimensionless mean value mesh force can be T T p g F = = . (7) calculated from λ λ pm gm F = 1+ η T . (11) s p By further assuming dimensionless parameters p = p/b, t = ω t, ω = ω/ω , λ = λ /λ , λ = λ /λ , k = k /k , n n p p pm g g gm m mm The numerical integration method applying an explicit and e = e/b, the dimensionless form of (2a) can be obtained Runge-Kutta (4)and (5) formulation is employed to com- as pute the response from (8a) since there is no analytical 2 2 2 2 λ + ηλ λ + ηλ p g p g method available. The computed time domain response p p +2ζ kg p + kf p 1+ η 1+ η is then applied to calculate the dynamic mesh force using (8a) (10). From the predicted time history response function, the = T λ + T λ − e , p p g g frequency spectrum can be obtained by taking the standard deviation (STD) and mean values of time series data at each p − 1, p ≥ 1, ⎪ operating frequency. From here onwards, the STD value is considered as the amplitude of dynamic mesh force. f p = 0, −1 < p< 1, (8b) p +1, p ≤−1, 3. Dynamic Analysis Using the proposed dynamic model, a typical rear axle 0, −1 < p< 1, hypoid gear set is analyzed. The basic system parameters g p = (8c) p , else, of this rear axle unit are given in Table 1,which were Dynamic mesh force (mean) Dynamic mesh force (mean) Advances in Acoustics and Vibration 5 1.8 2.45 1.75 2.4 1.7 2.35 1.65 2.3 1.6 2.25 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 Dimensionless frequency Dimensionless frequency (a) (b) 3.6 5 3.55 4.95 3.5 4.9 3.45 4.85 3.4 4.8 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 Dimensionless frequency Dimensionless frequency (c) (d) 5.8 5.75 5.7 5.65 5.6 0 0.5 1 1.5 2 2.5 Dimensionless frequency (e) Figure 3: Mean value of mesh force for different input torque loads. (a) T = 0.99, (b) T = 1.37, (c) T = 2.01, (d) T = 2.81, and (e) p p p p T = 3.27. ∗:  no impact; •:  single-sided impact; +: double-sided impact; —–: mean value for increasing frequency; ...: mean value for decreasing frequency; - - -: theoretical static mesh force. also applied in an earlier numerical study by the authors 3.1. Frequency Domain Analysis. The dynamic mesh force [6]. Tooth contact analysis is first conducted to obtain the response calculated using the above procedure for different necessary mesh parameters for five different loads as shown mean loads is shown in Figure 2. Here mesh damping ratio in Table 2. The procedure for computing mesh parameters of ζ = 0.04 is used. As expected, when the torque load obtained from output results of a tooth contact analysis [8] increases, the mean value of mesh force increases too. In is described in an earlier publication by the authors [9]. fact, the mean value of mesh force response is very close The extracted mesh parameters are readily employed in the to the theoretical static mesh force derived from the input dynamic model. For brevity, details of the process will not be torque load and is almost frequency invariant compared repeated here. to the dynamic mesh force that varies significantly with Dynamic mesh force (mean) Dynamic mesh force (mean) Dynamic mesh force (mean) Dynamic mesh force (mean) Dynamic mesh force (mean) 6 Advances in Acoustics and Vibration 12 12 10 10 8 8 6 6 4 4 2 2 0 0 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 Dimensionless frequency Dimensionless frequency (a) (b) 12 12 10 10 8 8 6 6 4 4 2 2 0 0 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 Dimensionless frequency Dimensionless frequency (c) (d) 12 12 10 10 8 8 6 6 4 4 2 2 0 0 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 Dimensionless frequency Dimensionless frequency (e) (f ) 12 12 10 10 8 8 6 6 4 4 2 2 0 0 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 Dimensionless frequency Dimensionless frequency (g) (h) Figure 4: Dynamic mesh force response for different mesh damping ratios. Dynamic mesh force (STD) is shown in (a) ζ = 0.01, (b) ζ = 0.02, (c) ζ = 0.04, and (d) ζ = 0.08, while the mean value is given by (e) ζ = 0.01, (f ) ζ = 0.02, (g) ζ = 0.04, and (h) ζ = 0.08. ∗: no impact; •:  single-sided impact; +: double-sided impact; —–: dynamic mesh force (increasing frequency); ...: dynamic mesh force (decreasing frequency); - - -: theoretical static mesh force. Dynamic mesh force (mean) Dynamic mesh force (mean) Dynamic mesh force (STD) Dynamic mesh force (STD) Dynamic mesh force (mean) Dynamic mesh force (mean) Dynamic mesh force (STD) Dynamic mesh force (STD) Advances in Acoustics and Vibration 7 3.5 3.5 3.45 3.45 3.4 3.4 3.35 3.35 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 Dimensionless frequency Dimensionless frequency (a) (b) 3.5 3.5 3.45 3.45 3.4 3.4 3.35 3.35 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 Dimensionless frequency Dimensionless frequency (c) (d) Figure 5: Mean value of mesh force response for different mesh damping ratios. (a) ζ = 0.01, (b) ζ = 0.02, (c) ζ = 0.04, and (d) ζ = 0.08. ∗: no impact; •:  single-sided impact; +: double-sided impact; —–: mean value for increasing frequency; ...: mean value for decreasing frequency; - - -: theoretical static mesh force. f (p ) N N N 3 2 1 S S 2 1 −11 p Figure 6: Illustration of different tooth impact types. No impact (N , N ,and N ), single-sided impact (S , S ), and double-sided impact 1 2 3 1 2 (D). Dynamic mesh force (mean) Dynamic mesh force (mean) Dynamic mesh force (mean) Dynamic mesh force (mean) 8 Advances in Acoustics and Vibration Table 2: Dimensionless dynamic parameters of a typical automo- tive rear axle hypoid gear pair. Parameter symbols Numerical values T 0.99 1.37 2.01 2.81 3.27 0.9 0.95 η 0.740 0.740 0.740 0.740 0.740 4 λ 0.015 0.014 0.010 0.006 0.004 pa1 1.05 φ 0.25π 0.24π 0.22π 0.15π 0.05π p1 1.1 0.9 λ 0.013 0.013 0.009 0.005 0.004 ga1 0.85 φ 0.25π 0.24π 0.22π 0.15π 0.06π g 1 1 k 0.231 0.137 0.053 0.0462 0.0458 φ −0.33π −0.37π −0.57π −0.93π −0.98π 0 0.5 1 1.5 2 2.5 e 0.49 0.49 0.49 0.49 0.49 a1 Dimensionless frequency φ 0.74π 0.74π 0.74π 0.74π 0.74π e1 Figure 7: Dynamic mesh force (STD) for T = 2.01. ∗,  no impact; •,  single-sided impact; +: double-sided impact; —–: dynamic mesh force (RMS) for increasing frequency; ...:dynamic mesh force (RMS) for decreasing frequency; - - -: theoretical static theoretical static mesh force, virtually no jump discontinuity mesh force. is seen. This is because the mean value of mesh force is able to keep the meshing teeth engaged all the time when its magnitude is higher than the dynamic counterpart that tends to cause the meshing teeth to lose contact. The results Table 1: System parameters for a typical automotive rear axle of mean value of mesh force response for different levels of hypoid gear pair. input torque load are shown as a closed-up view in Figure 3. Number of pinion teeth 10 Again, as seen in these results, as the drive torque load is Number of gear teeth 43 increased, the mean mesh force level increases. However, its Pinion offset (m) 0.0318 variation decreases, yielding a more linear characteristic. Gear pitch radius (m) 0.168 The effect of damping is analyzed next. Figure 4 shows the dynamic mesh force response for different mesh damping Pinion pitch radius (m) 0.048 ratios. From these plots, it can be seen that as ζ is increased, Mass moment of inertia of pinion (kg-m ) 0.002 the dynamic mesh force tends to decrease, while mean value Mass moment of inertia of gear (kg-m)0.05 does not change at all. This is consistent with the results depicted in (11). At the same time, occurrence of jump discontinuity and tooth impacts lessen. Also, the mean value frequency. Also, the dynamic mesh force appears to decrease of mesh force is very close to the theoretical static mesh force with increasing input torque. A more in-depth analysis of the and changes very little. Similarly, the results again show that dynamic mesh force results are discussed next. as dynamic mesh force becomes larger than the mean value At T = 0.99, several jump discontinuities and very p of mesh force, jump response appears, while no evidence of rich single and double-sided tooth impacts can be observed jump response is seen when the dynamic mesh force is lower than the mean value. at the primary resonance and higher frequencies. As T is increased to 1.37, only two jumps are seen in which one The effect of mesh damping on the mean value of mesh force is shown in Figure 5. Note that these plots represent a is a softening type (lower frequency) and the other is a closed-up view of the ones shown in Figure 4.Here, it can hardening one (upper frequency). At the same time, less tooth impacts occur around the primary resonance. For the be seen that as mesh damping ratio ζ is increased, the mean value changes very little and its variation tends to decrease. case of T = 2.01 only a softening type jump, and very few This former result is unlike the previously seen effect of drive single and double-sided tooth impact points appear near the torque load, but the latter observation is the same as the trend primary resonance. At T = 2.81 there is no jump response of drive torque load effect. observed and only a single-sided tooth impact is found. As T is increased further to 3.27, the dynamic response over the entire frequency range shown is completely void of jump 3.2. Time Domain Analysis. Due to the design of backlash in phenomenon and tooth impacts. the hypoid gear set, there are three classes of tooth meshing From the results, an interesting finding is that when the cases that are no impact, single-sided impact, and double- dynamic mesh force is greater than the mean value of mesh sided impact as illustrated in Figure 6. The zone between −1 force or the theoretical static mesh force, jump discontinuity and 1 is backlash zone. So there are three types of no impact can be seen. On the other hand, when the dynamic mesh (N , N ,and N ), two types of single-sided impact (S and 1 2 3 1 force is lower than the mean value of mesh force or the S ), and one type of double-sided impact (D). These tooth Dynamic mesh force (STD) Advances in Acoustics and Vibration 9 8 12 −1 −2 −1 −2 −4 300 350 400 450 500 550 550 600 650 700 750 Dimensionless time Dimensionless time (a) (b) 10 10 1 1 0 0 −1 −1 −2 550 600 650 700 750 250 300 350 400 450 Dimensionless time Dimensionless time (c) (d) −1 −2 0 500 550 600 650 300 350 400 450 500 550 Dimensionless time Dimensionless time (e) (f ) 10 10 −2 550 600 650 700 750 550 600 650 700 750 Dimensionless time Dimensionless time (g) (h) Figure 8: Continued. Dynamic response Dynamic mesh force Dynamic response Dynamic response Dynamic mesh force Dynamic response Dynamic response Dynamic mesh force 10 Advances in Acoustics and Vibration 8 7 −2 0 250 300 350 400 450 500 550 600 650 Dimensionless time Dimensionless time (i) (j) Figure 8: Time history plots of gear response for increasing mesh frequencies. Dynamic response: (a) ω = 0.9, (b) ω = 0.95, (c) ω = 1.0, (d) ω = 1.05, and (e) ω = 1.1. Dynamic mesh force: (f ) ω = 0.9, (g) ω = 0.95, (h) ω = 1.0, (i) ω = 1.05, and (j) ω = 1.1. impact behaviors were first reported by Comparin and Singh the trough valueisabove zero.As ω is decreased to 1.05, the in 1989 [1] for parallel axis gears. For nonparallel axis gears, gear response becomes S type single-sided tooth impact as there are no known prior studies. The current study attempts depicted by the time history response in Figures 9(b) and to fill this gap using the hypoid gear design as the example 9(h) in which the trough value is equal to zero. Similar S case. In the ensuing discussion, let us consider the predicted type single-sided tooth impact response occurs at ω = 1.0as shown in Figures 9(c) and 9(i).At ω = 0.95 and 0.9, double- dynamic mesh force spectrum for T = 2.01 as shown in sided tooth impacts occur and the trough value of dynamic Figure 7. mesh force reaches below zero as shown in Figures 9(d), 9(j), For the case of T = 2.01 as shown in Figure 7, the 9(e),and 9(k), respectively. Note that the negative mesh force time history plots of dynamic response for increasing mesh implies a reversal of its vector along the line of action due to frequencies from 0.9 to 1.1 are shown in a set of plots in the tooth in question back-colliding with the preceding tooth Figure 8. In the plot given by Figure 8(a) for the case of of the mating gear. As ω is further decreased to 0.85 given ω = 0.9, the time history response shown is of type N by Figures 9(f ) and 9(l), N type no tooth impact response where no tooth impact response occurs. The corresponding appears and the trough value of dynamic mesh force is well time history plot of the dynamic mesh force at this frequency above zero. is shown in Figure 8(f ). From this figure, it is seen that From the above analysis, it is shown that for N type no the trough value of dynamic mesh force is above zero. To tooth impact case, the trough value of dynamic mesh force explain this observation, consider (10)and (8b) that show is greater than zero; for S type single-sided tooth impact the dynamic mesh force being dominated by kf (p ). Since case, the trough value of dynamic mesh force is equal to f (p) is greater than zero for N type no impact response, zero; and finally for D type double-sided tooth impact case, the dynamic mesh force must therefore be greater than zero. the trough value of dynamic mesh force reaches below zero. Figures 8(b) and 8(g) for ω = 0.95 shows the time history Furthermore, it can be concluded that for N type no tooth response of type D double-sided tooth impact behavior. It impact case (where no mesh contact at all), the dynamic shows that the trough value of dynamic mesh force reaches mesh force will always be equal to zero; for N type no tooth below zero because f (p )islessthanzerofor p< −1. impact case (where the tooth-tooth engagement happens Figure 8(c) shows the time history of S type single-sided between the current and preceding tooth of the mating gear), toothimpactresponseat ω = 1.0, with the corresponding the peak value of dynamic mesh force will be less than zero dynamic mesh force shown in Figure 8(h). Here, the trough implying a mesh vector in the opposite direction from the valueofdynamic mesh forceisequal to zero because f (p ) = N case; for S type single-sided tooth impact case, the peak 1 2 0for −1 < p< 1. This is found to be similar for the time value of dynamic mesh force will be equal to zero since the history plots of S type single-sided tooth impact response current tooth in question never makes contact with the tooth and dynamic mesh force at ω = 1.05, as shown in Figures on the mating gear it is designed to engage with. 8(d) and 8(i),respectively. At ω = 1.1 as shown in Figures 8(e) and 8(j), the response goes back to N type no tooth impact behavior. 4. Concluding Remarks Time history plots of dynamic response for decreasing mesh frequencies from 1.1 to 0.85 are shown in Figure 9.In A nonlinear time-varying torsional vibration model of a this set of plots, Figure 9(a) shows time history of N type no hypoid gear pair system with time-dependent mesh point, toothimpactresponseat ω = 1.1and Figure 9(g) shows the line-of-action vector, mesh stiffness, mesh damping, and corresponding time history of dynamic mesh force in which backlash nonlinearity is formulated to study the condition Dynamic mesh force Dynamic mesh force Advances in Acoustics and Vibration 11 −1 −1 −2 500 550 600 650 500 550 600 650 700 Dimensionless time Dimensionless time (a) (b) 10 12 1 0 −1 −2 −1 −4 300 350 400 450 500 300 350 400 450 500 Dimensionless time Dimensionless time (c) (d) 12 8 −1 −2 −1 −4 −2 600 650 700 750 800 600 650 700 750 800 850 Dimensionless time Dimensionless time (e) (f ) 0 −2 500 550 600 650 500 550 600 650 700 Dimensionless time Dimensionless time (g) (h) Figure 9: Continued. Dynamic response Dynamic mesh force Dynamic response Dynamic response Dynamic mesh force Dynamic response Dynamic response Dynamic response 12 Advances in Acoustics and Vibration 10 10 −2 300 350 400 450 500 300 350 400 450 500 Dimensionless time Dimensionless time (i) (j) 10 5 −2 −4 0 600 650 700 750 800 600 650 700 750 800 850 Dimensionless time Dimensionless time (k) (l) Figure 9: Time history plots of gear response for decreasing mesh frequencies. Dynamic response: (a) ω = 1.1, (b) ω = 1.05, (c) ω = 1.0, (d) ω = 0.95, (e) ω = 0.9, and (f ) ω = 0.85. Dynamic mesh force: (g) ω = 1.1, (h) ω = 1.05, (i) ω = 1.0, (j) ω = 0.95, (k) ω = 0.9, and (l) ω = 0.85. dictating the boundary between nonlinear jump phenomena Nomenclature and linear responses, that is, when jump discontinuity begins b:Gearbacklash to disappear as mean load is increased from light to heavy c : Mesh damping coefficient levels. An interesting finding is that when the dynamic m e: Unloaded kinematic transmission error mesh force exceeds the mean value of mesh force, jump discontinuity will appear, but when the dynamic mesh force f : Nonlinear displacement function is lower than the mean value of mesh force, jump response is I , I : Mass moments of inertias of pinion and gear p g less likely to occur. : Unit vector along pinion or gear rotating axis Based on the time domain analysis results, it is shown k, k :Meshstiffness that for N type no tooth impact case, the trough value of m : Equivalent mass dynamic mesh force is greater than zero; for N type no tooth impact case, the dynamic mesh force is always equal n : Unit normal vector of mesh point to zero; for N type no toothimpactcase, the peak value p:Difference between dynamic and kinematic of dynamic mesh force is less than zero; for S type single- transmission errors sided tooth impact case, the trough value of dynamic mesh r : Position vector of mesh point force is equal to zero; for S type single-sided tooth impact S : Coordinate system for dynamic formulation case, the peak value of dynamic mesh force is equal to zero; t:Time for D type double-sided tooth impact case, the dynamic T , T : Mean value of pinion and gear torque loads p g mesh force reaches both above and below zero. From these δ: Dynamic transmission error observations, it is clear that different tooth impact types λ : Directional rotation radius can be distinguished by the dynamic mesh force response l behavior. ω: Excitation frequency Dynamic mesh force Dynamic mesh force Dynamic mesh force Dynamic mesh force Advances in Acoustics and Vibration 13 ζ : Mesh damping ratio η: Dimensionless torque ratio φ: Phase angle. Subscripts l: Label for pinion (l = p)and gear (l = g ) a: Fundamental harmonic m:Mean n: Natural. Superscripts ∼: Dimensionless quantities → : Vector quantities : Derivative with respect to time. References [1] R. J. Comparin and R. Singh, “Non-linear frequency response characteristicsofanimpactpair,” Journal of Sound and Vibration, vol. 134, no. 2, pp. 259–290, 1989. [2] A. Kahraman and R. Singh, “Non-linear dynamics of a spur gear pair,” Journal of Sound and Vibration, vol. 142, no. 1, pp. 49–75, 1990. [3] A. Kahraman and R. Singh, “Non-linear dynamics of a geared rotor-bearing system with multiple clearances,” Journal of Sound and Vibration, vol. 144, no. 3, pp. 469–506, 1991. [4] A. Kahraman and R. Singh, “Interactions between time-varying mesh stiffness and clearance non-linearities in a geared system,” Journal of Sound and Vibration, vol. 146, no. 1, pp. 135–156, [5] Y. Cheng and T. C. Lim, “Dynamics of hypoid gear transmission with nonlinear time-varying mesh characteristics,” Journal of Mechanical Design, Transactions of the ASME, vol. 125, no. 2, pp. 373–382, 2003. [6] J. Wang, T. C. Lim, and M. Li, “Dynamics of a hypoid gear pair considering the effects of time-varying mesh parameters and backlash nonlinearity,” Journal of Sound and Vibration, vol. 308, no. 1-2, pp. 302–329, 2007. [7] J. Wang and T. C. Lim, “Effect of tooth mesh stiffness asymmetric nonlinearity for drive and coast sides on hypoid gear dynamics,” Journal of Sound and Vibration, vol. 319, no. 3-5, pp. 885–903, 2009. [8] S. Vijayakar, Contact Analysis Program Package: Calyx, Advanced Numerical Solutions, Hilliard, Ohio, 2003. [9] T. C. Lim and J. Wang, “Effects of assembly errors on hypoid gear mesh and dynamic response,” in Proceedings of the ASME Power Transmission and Gearing Conference (DETC ’05),Long Beach, Calif, USA, 2005. 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On the Boundary between Nonlinear Jump Phenomenon and Linear Response of Hypoid Gear Dynamics

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Hindawi Publishing Corporation
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Copyright © 2011 Jun Wang and Teik C. Lim. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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1687-6261
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10.1155/2011/583678
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Hindawi Publishing Corporation Advances in Acoustics and Vibration Volume 2011, Article ID 583678, 13 pages doi:10.1155/2011/583678 Research Article On the Boundary between Nonlinear Jump Phenomenon and Linear Response of Hypoid Gear Dynamics 1 2 Jun Wang and Teik C. Lim Advanced Components Systems Division, Caterpillar Inc., Peoria, IL 61630, USA Mechanical Engineering Program, School of Dynamic Systems, University of Cincinnati, 598 Rhodes Hall, P.O. Box 210072, Cincinnati, OH 45221, USA Correspondence should be addressed to Teik C. Lim, teik.lim@uc.edu Received 27 October 2010; Revised 10 March 2011; Accepted 17 March 2011 Academic Editor: Mohammad Tawfik Copyright © 2011 J. Wang and T. C. Lim. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. A nonlinear time-varying (NLTV) dynamic model of a hypoid gear pair system with time-dependent mesh point, line-of-action vector, mesh stiffness, mesh damping, and backlash nonlinearity is formulated to analyze the transitional phase between nonlinear jump phenomenon and linear response. It is found that the classical jump discontinuity will occur if the dynamic mesh force exceeds the mean value of tooth mesh force. On the other hand, the propensity for the gear response to jump disappears when the dynamic mesh force is lower than the mean mesh force. Furthermore, the dynamic analysis is able to distinguish the specific tooth impact types from analyzing the behaviors of the dynamic mesh force. The proposed theory is general and also applicable to high-speed spur, helical and spiral bevel gears even though those types of gears are not the primary focus of this paper. 1. Introduction past or fully understood. Hence, the goal of this study is to seek out the conditions dictating the transitional boundary Gear dynamics have been studied intensively as evident between nonlinear and linear responses. In other word, we from the discussions in [1–5]. In one of the earlier studies, would like to know precisely when jump phenomenon stops in advance as drive load goes from light to heavy. Comparin and Singh investigated the nonlinear frequency response of an impact pair and located the transition condi- To assist in the above-mentioned analysis, a nonlinear tions for no-impact, single-sided impact, and double-sided time-varying dynamic model with time-dependent mesh impact cases [1]. Later, Kahraman, and Singh examined the position, line-of-action mesh vectors, mesh stiffness, mesh nonlinear dynamics of a spur gear pair [2]aswellasageared damping, and backlash nonlinearity is proposed similar to the one employed by the authors in earlier studies [6, 7]. rotor-bearing system [3] and also studied the interaction between mesh stiffness and clearance nonlinearities [4]. Based on the model, different levels of torques and damping More recently, Cheng and Lim studied the vibratory response ratios are applied so that the transition condition between linear and nonlinear responses can be located. The analysis of a hypoid geared rotor system with nonlinear time-varying mesh characteristics [5]. From those studies, it is now well clearly shows that when the dynamic mesh force is greater known that nonlinear phenomena like jump discontinuity than mean mesh force, the jump phenomena will occur. frequently occurs for lightly loaded gear pairs. As the On the other hand, if the dynamic mesh force is smaller in drive load is progressively increased, the response becomes value than the mean mesh force, the jump behavior tends to disappear making the response appearing to be quite linear. more linear until the jump response completely disappears. Understanding the conditions ahead of time when the geared system will exhibit nonlinear behaviors is highly desirable 2. Mathematical Model since the nonlinear dynamic analysis is typically very time consuming. However, to the best of our knowledge this The two degrees-of-freedom (DOF) lumped parameter transitional phase has never been studied extensively in the torsional vibration model of a hypoid gear pair as shown 2 Advances in Acoustics and Vibration Z Z p g Gear axis Y c k m m Gear axis Pinion axis Pinion axis I p (a) (b) Figure 1: (a) Dynamic model of a hypoid gear pair, and (b) pinion and gear coordinate systems. in Figure 1(a), which was also applied in earlier studies by n r j expressed as λ = · ( × )(l = p, g for pinion and gear l l l the authors [6, 7], is again assumed for the current study. resp.). Also, is the unit vector of the line of action along the This simplified representation is chosen over more complex mesh force direction in the coordinate system S as shown in system to allow for the study to focus on the behavior of Figure 1(b), r is the position vector of mesh point, and j is the boundary between nonlinear jump phenomenon and linear dynamic response. The pinion and gear bodies each are the unit vector along the rotating axes of pinion and gear. By assuming p = δ − e,(1a)and (1b) can be combined allowed to rotate only. The mesh coupling is represented by a pair of mesh damping and stiffness elements. Using either the and simplified into the following form: Newton’s or Lagrangian’s methods, the equations of motion λ T λ T p p g g can be easily derived as ¨ ˙ ¨ m p + c g p + k f p = m + − e , (2a) e m m e I I p g ¨ ˙ I θ + λ c g δ − e ˙ + λ k f (δ − e) = T,(1a) p p p m p m p p − b, p ≥ b, ¨ ˙ ⎨ I θ − λ c g δ − e − λ k f (δ − e) =−T,(1b) g g g m g m g f p = 0, −b< p< b, (2b) (δ − e − b), δ − e ≥ b, ⎪ p + b, p ≤−b, f (δ − e) = 0, −b< δ − e< b, (1c) ⎪ 0, −b< p < b, ⎩ ˙ g p = (2c) (δ − e + b), δ − e ≤−b, p, else, ˙ 2 2 ⎪ δ − e ˙ , δ − e ≥ b, where m = 1/(λ /I + λ /I ). Since mesh damping is gen- ⎪ e p p g g ⎨ erally time varying, proportional mesh damping is assumed g δ − e ˙ = 0, −b< δ − e< b, (1d) here, which is more realistic than simply applying constant ⎪   damping. Thus, the damping model can be expressed as δ − e ˙ , δ − e ≤−b, qk qω m m n ζ = = = , (3) where I and I are the mass moments of inertias of pinion 2m ω 2 p g 2 k m m em em n and gear, T and T are torque loads a on the pinion and gear, p g k and c aremeshstiffness and mesh damping coefficients, m m 2ζ c = qk = k , (4) m m m e is unloaded kinematic transmission error, and 2b is the gear backlash. The dynamic transmission error can be written as 2 2 δ = λ θ − λ θ , while the directional rotation radius can be where ω = k /m , m = 1/(λ /I + λ /I ). p p g g n mm em em p g pm gm Advances in Acoustics and Vibration 3 8 8 7 7 6 6 5 5 4 4 3 3 2 2 1 1 0 0 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 Dimensionless frequency Dimensionless frequency (a) (b) 8 8 7 7 6 6 5 5 4 4 3 3 2 2 1 1 0 0 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 Dimensionless frequency Dimensionless frequency (c) (d) 1 1 0 0 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 Dimensionless frequency Dimensionless frequency (e) (f ) 8 8 7 7 6 6 5 5 4 4 3 3 2 2 1 1 0 0 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 Dimensionless frequency Dimensionless frequency (g) (h) Figure 2: Continued. Dynamic mesh force (mean) Dynamic mesh force (STD) Dynamic mesh force (STD) Dynamic mesh force (STD) Dynamic mesh force (mean) Dynamic mesh force (mean) Dynamic mesh force (STD) Dynamic mesh force (STD) 4 Advances in Acoustics and Vibration 8 8 7 7 6 6 5 5 4 4 3 3 2 2 1 1 0 0 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 Dimensionless frequency Dimensionless frequency (i) (j) Figure 2: Dynamic mesh force response for different input torque loads. Dynamic mesh force (STD) is shown in (a) T = 0.99, (b) T = 1.37, p p (c) T = 2.01, (d) T = 2.81, and (e) T = 3.27, while mean value is given by (f ) T = 0.99, (g) T = 1.37, (h) T = 2.01, (i) T = 2.81, and p p p p p p p (j) T = 3.27. ∗:  no impact; •:  single-sided impact; +: double-sided impact; —–: dynamic mesh force (increasing frequency); ...: dynamic mesh force (decreasing frequency); - - -: theoretical static mesh force. 2 2 2 The time-varying mesh parameters in (2a), (2b), and (2c) where η = λ I /λ I , T = λ T /bω I ,and T = ηT . p g p pm p p g p gm pm n can be described as a superposition of a mean value and a The dimensionless mesh parameters are described as sinusoidal component as shown below: λ = 1+ λ cos ω t + φ , p pa p λ = λ + λ cos ωt + φ , p pm pa p λ = 1+ λ cos ω t + φ , g ga g λ = λ + λ cos ωt + φ , g gm ga g (5) (9) k = k + k cos ωt + φ , k = 1+ k cos ω t + φ , m mm ma k a k e = e cos ωt + φ . a e e = e cos ω t + φ . a e The dynamic mesh force can be computed from Subsequently the dimensionless dynamic mesh force can be F = c g p ˙ + k f p = 2ζ g p ˙ + k f p . (6) derived as d m m m From the torque and directional rotation radii, the mean F = 2ζkg p + kf p , (10) value of the mesh force can be calculated from and the dimensionless mean value mesh force can be T T p g F = = . (7) calculated from λ λ pm gm F = 1+ η T . (11) s p By further assuming dimensionless parameters p = p/b, t = ω t, ω = ω/ω , λ = λ /λ , λ = λ /λ , k = k /k , n n p p pm g g gm m mm The numerical integration method applying an explicit and e = e/b, the dimensionless form of (2a) can be obtained Runge-Kutta (4)and (5) formulation is employed to com- as pute the response from (8a) since there is no analytical 2 2 2 2 λ + ηλ λ + ηλ p g p g method available. The computed time domain response p p +2ζ kg p + kf p 1+ η 1+ η is then applied to calculate the dynamic mesh force using (8a) (10). From the predicted time history response function, the = T λ + T λ − e , p p g g frequency spectrum can be obtained by taking the standard deviation (STD) and mean values of time series data at each p − 1, p ≥ 1, ⎪ operating frequency. From here onwards, the STD value is considered as the amplitude of dynamic mesh force. f p = 0, −1 < p< 1, (8b) p +1, p ≤−1, 3. Dynamic Analysis Using the proposed dynamic model, a typical rear axle 0, −1 < p< 1, hypoid gear set is analyzed. The basic system parameters g p = (8c) p , else, of this rear axle unit are given in Table 1,which were Dynamic mesh force (mean) Dynamic mesh force (mean) Advances in Acoustics and Vibration 5 1.8 2.45 1.75 2.4 1.7 2.35 1.65 2.3 1.6 2.25 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 Dimensionless frequency Dimensionless frequency (a) (b) 3.6 5 3.55 4.95 3.5 4.9 3.45 4.85 3.4 4.8 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 Dimensionless frequency Dimensionless frequency (c) (d) 5.8 5.75 5.7 5.65 5.6 0 0.5 1 1.5 2 2.5 Dimensionless frequency (e) Figure 3: Mean value of mesh force for different input torque loads. (a) T = 0.99, (b) T = 1.37, (c) T = 2.01, (d) T = 2.81, and (e) p p p p T = 3.27. ∗:  no impact; •:  single-sided impact; +: double-sided impact; —–: mean value for increasing frequency; ...: mean value for decreasing frequency; - - -: theoretical static mesh force. also applied in an earlier numerical study by the authors 3.1. Frequency Domain Analysis. The dynamic mesh force [6]. Tooth contact analysis is first conducted to obtain the response calculated using the above procedure for different necessary mesh parameters for five different loads as shown mean loads is shown in Figure 2. Here mesh damping ratio in Table 2. The procedure for computing mesh parameters of ζ = 0.04 is used. As expected, when the torque load obtained from output results of a tooth contact analysis [8] increases, the mean value of mesh force increases too. In is described in an earlier publication by the authors [9]. fact, the mean value of mesh force response is very close The extracted mesh parameters are readily employed in the to the theoretical static mesh force derived from the input dynamic model. For brevity, details of the process will not be torque load and is almost frequency invariant compared repeated here. to the dynamic mesh force that varies significantly with Dynamic mesh force (mean) Dynamic mesh force (mean) Dynamic mesh force (mean) Dynamic mesh force (mean) Dynamic mesh force (mean) 6 Advances in Acoustics and Vibration 12 12 10 10 8 8 6 6 4 4 2 2 0 0 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 Dimensionless frequency Dimensionless frequency (a) (b) 12 12 10 10 8 8 6 6 4 4 2 2 0 0 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 Dimensionless frequency Dimensionless frequency (c) (d) 12 12 10 10 8 8 6 6 4 4 2 2 0 0 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 Dimensionless frequency Dimensionless frequency (e) (f ) 12 12 10 10 8 8 6 6 4 4 2 2 0 0 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 Dimensionless frequency Dimensionless frequency (g) (h) Figure 4: Dynamic mesh force response for different mesh damping ratios. Dynamic mesh force (STD) is shown in (a) ζ = 0.01, (b) ζ = 0.02, (c) ζ = 0.04, and (d) ζ = 0.08, while the mean value is given by (e) ζ = 0.01, (f ) ζ = 0.02, (g) ζ = 0.04, and (h) ζ = 0.08. ∗: no impact; •:  single-sided impact; +: double-sided impact; —–: dynamic mesh force (increasing frequency); ...: dynamic mesh force (decreasing frequency); - - -: theoretical static mesh force. Dynamic mesh force (mean) Dynamic mesh force (mean) Dynamic mesh force (STD) Dynamic mesh force (STD) Dynamic mesh force (mean) Dynamic mesh force (mean) Dynamic mesh force (STD) Dynamic mesh force (STD) Advances in Acoustics and Vibration 7 3.5 3.5 3.45 3.45 3.4 3.4 3.35 3.35 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 Dimensionless frequency Dimensionless frequency (a) (b) 3.5 3.5 3.45 3.45 3.4 3.4 3.35 3.35 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 Dimensionless frequency Dimensionless frequency (c) (d) Figure 5: Mean value of mesh force response for different mesh damping ratios. (a) ζ = 0.01, (b) ζ = 0.02, (c) ζ = 0.04, and (d) ζ = 0.08. ∗: no impact; •:  single-sided impact; +: double-sided impact; —–: mean value for increasing frequency; ...: mean value for decreasing frequency; - - -: theoretical static mesh force. f (p ) N N N 3 2 1 S S 2 1 −11 p Figure 6: Illustration of different tooth impact types. No impact (N , N ,and N ), single-sided impact (S , S ), and double-sided impact 1 2 3 1 2 (D). Dynamic mesh force (mean) Dynamic mesh force (mean) Dynamic mesh force (mean) Dynamic mesh force (mean) 8 Advances in Acoustics and Vibration Table 2: Dimensionless dynamic parameters of a typical automo- tive rear axle hypoid gear pair. Parameter symbols Numerical values T 0.99 1.37 2.01 2.81 3.27 0.9 0.95 η 0.740 0.740 0.740 0.740 0.740 4 λ 0.015 0.014 0.010 0.006 0.004 pa1 1.05 φ 0.25π 0.24π 0.22π 0.15π 0.05π p1 1.1 0.9 λ 0.013 0.013 0.009 0.005 0.004 ga1 0.85 φ 0.25π 0.24π 0.22π 0.15π 0.06π g 1 1 k 0.231 0.137 0.053 0.0462 0.0458 φ −0.33π −0.37π −0.57π −0.93π −0.98π 0 0.5 1 1.5 2 2.5 e 0.49 0.49 0.49 0.49 0.49 a1 Dimensionless frequency φ 0.74π 0.74π 0.74π 0.74π 0.74π e1 Figure 7: Dynamic mesh force (STD) for T = 2.01. ∗,  no impact; •,  single-sided impact; +: double-sided impact; —–: dynamic mesh force (RMS) for increasing frequency; ...:dynamic mesh force (RMS) for decreasing frequency; - - -: theoretical static theoretical static mesh force, virtually no jump discontinuity mesh force. is seen. This is because the mean value of mesh force is able to keep the meshing teeth engaged all the time when its magnitude is higher than the dynamic counterpart that tends to cause the meshing teeth to lose contact. The results Table 1: System parameters for a typical automotive rear axle of mean value of mesh force response for different levels of hypoid gear pair. input torque load are shown as a closed-up view in Figure 3. Number of pinion teeth 10 Again, as seen in these results, as the drive torque load is Number of gear teeth 43 increased, the mean mesh force level increases. However, its Pinion offset (m) 0.0318 variation decreases, yielding a more linear characteristic. Gear pitch radius (m) 0.168 The effect of damping is analyzed next. Figure 4 shows the dynamic mesh force response for different mesh damping Pinion pitch radius (m) 0.048 ratios. From these plots, it can be seen that as ζ is increased, Mass moment of inertia of pinion (kg-m ) 0.002 the dynamic mesh force tends to decrease, while mean value Mass moment of inertia of gear (kg-m)0.05 does not change at all. This is consistent with the results depicted in (11). At the same time, occurrence of jump discontinuity and tooth impacts lessen. Also, the mean value frequency. Also, the dynamic mesh force appears to decrease of mesh force is very close to the theoretical static mesh force with increasing input torque. A more in-depth analysis of the and changes very little. Similarly, the results again show that dynamic mesh force results are discussed next. as dynamic mesh force becomes larger than the mean value At T = 0.99, several jump discontinuities and very p of mesh force, jump response appears, while no evidence of rich single and double-sided tooth impacts can be observed jump response is seen when the dynamic mesh force is lower than the mean value. at the primary resonance and higher frequencies. As T is increased to 1.37, only two jumps are seen in which one The effect of mesh damping on the mean value of mesh force is shown in Figure 5. Note that these plots represent a is a softening type (lower frequency) and the other is a closed-up view of the ones shown in Figure 4.Here, it can hardening one (upper frequency). At the same time, less tooth impacts occur around the primary resonance. For the be seen that as mesh damping ratio ζ is increased, the mean value changes very little and its variation tends to decrease. case of T = 2.01 only a softening type jump, and very few This former result is unlike the previously seen effect of drive single and double-sided tooth impact points appear near the torque load, but the latter observation is the same as the trend primary resonance. At T = 2.81 there is no jump response of drive torque load effect. observed and only a single-sided tooth impact is found. As T is increased further to 3.27, the dynamic response over the entire frequency range shown is completely void of jump 3.2. Time Domain Analysis. Due to the design of backlash in phenomenon and tooth impacts. the hypoid gear set, there are three classes of tooth meshing From the results, an interesting finding is that when the cases that are no impact, single-sided impact, and double- dynamic mesh force is greater than the mean value of mesh sided impact as illustrated in Figure 6. The zone between −1 force or the theoretical static mesh force, jump discontinuity and 1 is backlash zone. So there are three types of no impact can be seen. On the other hand, when the dynamic mesh (N , N ,and N ), two types of single-sided impact (S and 1 2 3 1 force is lower than the mean value of mesh force or the S ), and one type of double-sided impact (D). These tooth Dynamic mesh force (STD) Advances in Acoustics and Vibration 9 8 12 −1 −2 −1 −2 −4 300 350 400 450 500 550 550 600 650 700 750 Dimensionless time Dimensionless time (a) (b) 10 10 1 1 0 0 −1 −1 −2 550 600 650 700 750 250 300 350 400 450 Dimensionless time Dimensionless time (c) (d) −1 −2 0 500 550 600 650 300 350 400 450 500 550 Dimensionless time Dimensionless time (e) (f ) 10 10 −2 550 600 650 700 750 550 600 650 700 750 Dimensionless time Dimensionless time (g) (h) Figure 8: Continued. Dynamic response Dynamic mesh force Dynamic response Dynamic response Dynamic mesh force Dynamic response Dynamic response Dynamic mesh force 10 Advances in Acoustics and Vibration 8 7 −2 0 250 300 350 400 450 500 550 600 650 Dimensionless time Dimensionless time (i) (j) Figure 8: Time history plots of gear response for increasing mesh frequencies. Dynamic response: (a) ω = 0.9, (b) ω = 0.95, (c) ω = 1.0, (d) ω = 1.05, and (e) ω = 1.1. Dynamic mesh force: (f ) ω = 0.9, (g) ω = 0.95, (h) ω = 1.0, (i) ω = 1.05, and (j) ω = 1.1. impact behaviors were first reported by Comparin and Singh the trough valueisabove zero.As ω is decreased to 1.05, the in 1989 [1] for parallel axis gears. For nonparallel axis gears, gear response becomes S type single-sided tooth impact as there are no known prior studies. The current study attempts depicted by the time history response in Figures 9(b) and to fill this gap using the hypoid gear design as the example 9(h) in which the trough value is equal to zero. Similar S case. In the ensuing discussion, let us consider the predicted type single-sided tooth impact response occurs at ω = 1.0as shown in Figures 9(c) and 9(i).At ω = 0.95 and 0.9, double- dynamic mesh force spectrum for T = 2.01 as shown in sided tooth impacts occur and the trough value of dynamic Figure 7. mesh force reaches below zero as shown in Figures 9(d), 9(j), For the case of T = 2.01 as shown in Figure 7, the 9(e),and 9(k), respectively. Note that the negative mesh force time history plots of dynamic response for increasing mesh implies a reversal of its vector along the line of action due to frequencies from 0.9 to 1.1 are shown in a set of plots in the tooth in question back-colliding with the preceding tooth Figure 8. In the plot given by Figure 8(a) for the case of of the mating gear. As ω is further decreased to 0.85 given ω = 0.9, the time history response shown is of type N by Figures 9(f ) and 9(l), N type no tooth impact response where no tooth impact response occurs. The corresponding appears and the trough value of dynamic mesh force is well time history plot of the dynamic mesh force at this frequency above zero. is shown in Figure 8(f ). From this figure, it is seen that From the above analysis, it is shown that for N type no the trough value of dynamic mesh force is above zero. To tooth impact case, the trough value of dynamic mesh force explain this observation, consider (10)and (8b) that show is greater than zero; for S type single-sided tooth impact the dynamic mesh force being dominated by kf (p ). Since case, the trough value of dynamic mesh force is equal to f (p) is greater than zero for N type no impact response, zero; and finally for D type double-sided tooth impact case, the dynamic mesh force must therefore be greater than zero. the trough value of dynamic mesh force reaches below zero. Figures 8(b) and 8(g) for ω = 0.95 shows the time history Furthermore, it can be concluded that for N type no tooth response of type D double-sided tooth impact behavior. It impact case (where no mesh contact at all), the dynamic shows that the trough value of dynamic mesh force reaches mesh force will always be equal to zero; for N type no tooth below zero because f (p )islessthanzerofor p< −1. impact case (where the tooth-tooth engagement happens Figure 8(c) shows the time history of S type single-sided between the current and preceding tooth of the mating gear), toothimpactresponseat ω = 1.0, with the corresponding the peak value of dynamic mesh force will be less than zero dynamic mesh force shown in Figure 8(h). Here, the trough implying a mesh vector in the opposite direction from the valueofdynamic mesh forceisequal to zero because f (p ) = N case; for S type single-sided tooth impact case, the peak 1 2 0for −1 < p< 1. This is found to be similar for the time value of dynamic mesh force will be equal to zero since the history plots of S type single-sided tooth impact response current tooth in question never makes contact with the tooth and dynamic mesh force at ω = 1.05, as shown in Figures on the mating gear it is designed to engage with. 8(d) and 8(i),respectively. At ω = 1.1 as shown in Figures 8(e) and 8(j), the response goes back to N type no tooth impact behavior. 4. Concluding Remarks Time history plots of dynamic response for decreasing mesh frequencies from 1.1 to 0.85 are shown in Figure 9.In A nonlinear time-varying torsional vibration model of a this set of plots, Figure 9(a) shows time history of N type no hypoid gear pair system with time-dependent mesh point, toothimpactresponseat ω = 1.1and Figure 9(g) shows the line-of-action vector, mesh stiffness, mesh damping, and corresponding time history of dynamic mesh force in which backlash nonlinearity is formulated to study the condition Dynamic mesh force Dynamic mesh force Advances in Acoustics and Vibration 11 −1 −1 −2 500 550 600 650 500 550 600 650 700 Dimensionless time Dimensionless time (a) (b) 10 12 1 0 −1 −2 −1 −4 300 350 400 450 500 300 350 400 450 500 Dimensionless time Dimensionless time (c) (d) 12 8 −1 −2 −1 −4 −2 600 650 700 750 800 600 650 700 750 800 850 Dimensionless time Dimensionless time (e) (f ) 0 −2 500 550 600 650 500 550 600 650 700 Dimensionless time Dimensionless time (g) (h) Figure 9: Continued. Dynamic response Dynamic mesh force Dynamic response Dynamic response Dynamic mesh force Dynamic response Dynamic response Dynamic response 12 Advances in Acoustics and Vibration 10 10 −2 300 350 400 450 500 300 350 400 450 500 Dimensionless time Dimensionless time (i) (j) 10 5 −2 −4 0 600 650 700 750 800 600 650 700 750 800 850 Dimensionless time Dimensionless time (k) (l) Figure 9: Time history plots of gear response for decreasing mesh frequencies. Dynamic response: (a) ω = 1.1, (b) ω = 1.05, (c) ω = 1.0, (d) ω = 0.95, (e) ω = 0.9, and (f ) ω = 0.85. Dynamic mesh force: (g) ω = 1.1, (h) ω = 1.05, (i) ω = 1.0, (j) ω = 0.95, (k) ω = 0.9, and (l) ω = 0.85. dictating the boundary between nonlinear jump phenomena Nomenclature and linear responses, that is, when jump discontinuity begins b:Gearbacklash to disappear as mean load is increased from light to heavy c : Mesh damping coefficient levels. An interesting finding is that when the dynamic m e: Unloaded kinematic transmission error mesh force exceeds the mean value of mesh force, jump discontinuity will appear, but when the dynamic mesh force f : Nonlinear displacement function is lower than the mean value of mesh force, jump response is I , I : Mass moments of inertias of pinion and gear p g less likely to occur. : Unit vector along pinion or gear rotating axis Based on the time domain analysis results, it is shown k, k :Meshstiffness that for N type no tooth impact case, the trough value of m : Equivalent mass dynamic mesh force is greater than zero; for N type no tooth impact case, the dynamic mesh force is always equal n : Unit normal vector of mesh point to zero; for N type no toothimpactcase, the peak value p:Difference between dynamic and kinematic of dynamic mesh force is less than zero; for S type single- transmission errors sided tooth impact case, the trough value of dynamic mesh r : Position vector of mesh point force is equal to zero; for S type single-sided tooth impact S : Coordinate system for dynamic formulation case, the peak value of dynamic mesh force is equal to zero; t:Time for D type double-sided tooth impact case, the dynamic T , T : Mean value of pinion and gear torque loads p g mesh force reaches both above and below zero. From these δ: Dynamic transmission error observations, it is clear that different tooth impact types λ : Directional rotation radius can be distinguished by the dynamic mesh force response l behavior. ω: Excitation frequency Dynamic mesh force Dynamic mesh force Dynamic mesh force Dynamic mesh force Advances in Acoustics and Vibration 13 ζ : Mesh damping ratio η: Dimensionless torque ratio φ: Phase angle. Subscripts l: Label for pinion (l = p)and gear (l = g ) a: Fundamental harmonic m:Mean n: Natural. Superscripts ∼: Dimensionless quantities → : Vector quantities : Derivative with respect to time. References [1] R. J. Comparin and R. Singh, “Non-linear frequency response characteristicsofanimpactpair,” Journal of Sound and Vibration, vol. 134, no. 2, pp. 259–290, 1989. [2] A. Kahraman and R. Singh, “Non-linear dynamics of a spur gear pair,” Journal of Sound and Vibration, vol. 142, no. 1, pp. 49–75, 1990. [3] A. Kahraman and R. Singh, “Non-linear dynamics of a geared rotor-bearing system with multiple clearances,” Journal of Sound and Vibration, vol. 144, no. 3, pp. 469–506, 1991. [4] A. Kahraman and R. Singh, “Interactions between time-varying mesh stiffness and clearance non-linearities in a geared system,” Journal of Sound and Vibration, vol. 146, no. 1, pp. 135–156, [5] Y. Cheng and T. C. Lim, “Dynamics of hypoid gear transmission with nonlinear time-varying mesh characteristics,” Journal of Mechanical Design, Transactions of the ASME, vol. 125, no. 2, pp. 373–382, 2003. [6] J. Wang, T. C. Lim, and M. Li, “Dynamics of a hypoid gear pair considering the effects of time-varying mesh parameters and backlash nonlinearity,” Journal of Sound and Vibration, vol. 308, no. 1-2, pp. 302–329, 2007. [7] J. Wang and T. C. Lim, “Effect of tooth mesh stiffness asymmetric nonlinearity for drive and coast sides on hypoid gear dynamics,” Journal of Sound and Vibration, vol. 319, no. 3-5, pp. 885–903, 2009. [8] S. Vijayakar, Contact Analysis Program Package: Calyx, Advanced Numerical Solutions, Hilliard, Ohio, 2003. [9] T. C. Lim and J. Wang, “Effects of assembly errors on hypoid gear mesh and dynamic response,” in Proceedings of the ASME Power Transmission and Gearing Conference (DETC ’05),Long Beach, Calif, USA, 2005. 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