Experimental and Theoretical Study of Gear Dynamical Transmission Characteristic Considering Measured Manufacturing Errors

In the research of gear transmission, the vibration and noise problem has received many concerns all the times. Scholars use tooth modification technique to improve the meshing state of gearings in order to reduce the vibration and noise. However, few of researchers consider the influence of measured manufacturing errors when they do the study of tooth modification. In order to investigate the efficiency of the tooth modification in the actual project, this paper proposes a dynamic model of a helical gear pair including tooth modification and measured manufacturing errors to do a deterministic analysis on the dynamical transmission performance. In this analysis, based on the measured tooth deviation, a real tooth surface (including modification and measured tooth profile error) is fitted by a bicubic B-spline.With the tooth contact analysis (TCA) and loaded tooth contact analysis (LTCA) on the real tooth surface, the loaded transmission error, tooth surface elastic deformation, and load distribution can be determined. Based on the results, the time-varying mesh stiffness and gear mesh impact are computed. Taking the loaded transmission error, measured cumulative pitch error, eccentricity error, time-varying mesh stiffness, and gear mesh impact as the internal excitations, this paper establishes a 12-degree-of-freedom (DOF) dynamic model of a helical gear pair and uses the Fourier series method to solve it. In two situations of low speed and high speed, the gear system dynamic response is analyzed in the time and frequency domains. In addition, an experiment is performed to validate the simulation results. &e study shows that the proposed technique is useful and reliable for predicting the dynamic response of a gear system.


Introduction
In the gear machining process, the design deviation, manufacturing error, and fixing error of the machine cutting tool can cause a change in conjugate conditions and produce a tooth profile error, pitch error, eccentric error, etc. ese errors affect the contact characteristics of gears, worsen the stability and bearing capability of gear transmission, and cause enormous noise and vibration.us, many scholars pay close attention to the prediction and control of gear vibration and noise in their transmission designs [1][2][3][4][5][6][7][8].After an extensive study, some researchers have used the tooth profile modification to compensate the elastic deformation of gear teeth to reduce the gear vibration [9][10][11][12][13][14].However, there are errors remained during the theoretical profile modification in the machining process.erefore, the accuracy and efficiency of the tooth profile modification must be checked to avoid bad working conditions of the gear trains.
In the related studies, there are some cases which address the gear errors in the study of gear dynamics analysis.Case one: the effect of errors is ignored.Case two: although the errors are considered, the range of every error is estimated in theory, and errors supposedly change according to a particular curve.For example, Hu et al. [15] researched the effect of tooth modification of a high-speed gear system and set the variation of the tooth profile error and pitch error as a sine function.Kubo and Kiyono [16] analyzed the effect of tooth form errors on the helical gear dynamics, and they assume a composite tooth form error with four types of shape.Wei et al. [17] established a coupled nonlinear dynamics model which is used for planetary gear transmission systems in wind turbines and incorporates the effects of the time-varying mesh stiffness, dynamic transmission error, gear mesh impact, and input varying load; thus, the transmission error changes as a sine wave with time.Case three: the errors are shown as random processes.Chen et al. [18] used a stochastic volatility model to simulate the random wind velocity and analyzed the effect on the dynamic characteristics of wind turbine gear transmission system.Wang and Zhang [19] developed a dynamic and stochastic simulation model to analyze the vibration of the gear transmission system; they considered the gear transmission errors as the displacement excitation which was decomposed into harmonic and random components.ey used a secondorder Markov process with time-variant parameters considering the influence of the rotational speed to simulate a random component.Chen et al. [20] proposed a rotational nonlinear dynamic model of a planetary gear transmission system, which included the time-varying mesh stiffness and synthetic mesh error with random fluctuation; the synthetic mesh error was resolved into a sine signal and white noise.Nevertheless, the most related studies did not consider the low-frequency error (such as eccentricity error, cumulative pitch error, etc.) and hardly showed the real manufacturing errors of gear, so the simulation results might produce an obviously different dynamic response from the experiment.
In this paper, the main focus is to investigate the impact of the measured tooth profile error and cumulative pitch error on the dynamic response of gear system with tooth modification.
e main objective is to do dynamics analysis of a helical gear pair based on the real tooth surface (including tooth modification and tooth profile error) and the measured cumulative pitch error, so that the transmission performance of the machined gear system can be predicted.e internal excitations, including time-varying mesh stiffness, gear mesh impact, loaded transmission error, measured cumulative pitch error, and eccentric errors, will be considered in the analysis.At the end, the proposed method is compared with the traditional method and verified by a noise and vibration experiment.

Structure of Real Tooth Surface
A pair of helical gears is used as an example, which parameters are shown in Table 1.e tooth surface is finish machined by grinding, and the processing precision is level 6 (adopting the standard of ISO 1328-1:1997).e pinion (drive wheel) has profile and lead modification, and the gear (driven wheel) is a standard involute gear.e theoretical modification curve is shown in Figure 1. Figure 2 is the actual modification curve measured by a Klingelnberg P100 gear measuring device.e gear's measured surface deviation from the theoretical involute is displayed in Figure 3. e tooth surface deviation of the gear is notably small; thus, it is considered as the theoretical involute in the following calculation, and only the real tooth surface of the pinion must be structured.

2.1.
Fitting of Surface Deviation.Some scholars have applied the bicubic B-spline method to many studies.Gardner and Gardner [21] used it to model a magnetohydrodynamic duct flow and obtained excellent consistency between the numerical result and the analytic solution even for large values of the applied magnetic field.Na [22] used the method to fit a complex curved surface, and the fitting surface was C 2 conforming. is paper also uses a bicubic B-spline to form the deviation surface.
Before fitting the surface, some discrete points must be set on the measured modification curve in Figure 2. To improve the fitting accuracy, we select more discrete points at both ends of the measured modification curve than in the middle.Finally, 42 points are selected in the meshing line direction and 47 points are selected in the lead direction.
en, the deviations of 42 points in the meshing line direction must be converted into the deviations along the tooth profile direction.According to the meshing relationship shown in Figure 4, the relationship between the tooth depth and the length of the meshing line is as follows: where Y profile is the tooth depth, Y meshing is the length of the meshing line, and r bp is the base radius of the pinion.As shown in Equation ( 2), the 42 × 47 discrete points are arranged in a matrix.
Based on the 42 × 47 discrete points, we fit the cubic B-spline curves in the directions of lead and tooth depth.e control vertices of the cubic B-spline curve are computed by using the method of reverse vertices, which is presented in Appendix [22].
en, the cubic B-spline curve equations generated by these vertices are as follows: e lead direction,

Shock and Vibration
e direction of tooth depth, 1 4 1 0 where V i, j and W(s) i, j are the vertex of the cubic B-spline curve in the direction of lead and tooth depth.
If the expressions of s and v are plugged into (4), the surface deviation δ i,j (x,y) can be obtained, which is on the rotary projection of the tooth surface.When i changes from −1 to m−2 and j changes from −1 to n−2, we can obtain m × n patches constituting the deviation surface, shown in Figure 5.

Superposition of Deviation Surface and eoretical Tooth
Surface. Figure 6 shows the coordinates of the skew parabolic rack-cutter applied to the machined pinion.Here, M is a point on the surface of the rack-cutter, which coordinate is (u, −a c u 2 , 0) in the coordinate system S b .Parameters u (Figure 6(a)) and l |O c O a | (Figure 6(b)) are the surface parameters of the skew rack-cutter [23].d p is the location parameter of the parabolic pole.β is the helix angle.e tooth surface of the pinion is enveloped by the rack-cutter surface during processing.According to the meshing principle and enveloping principle [24], the position vector and normal vector of the theoretical tooth surface can be derived and denoted as R t (u, l) and n t (u, l). e real tooth surface is constituted by superposing the deviation surface and theoretical tooth surface.en, the position vector and normal vector of the real tooth surface are [25] where δʹ (u, l) is the deviation of the tooth surface, which is converted from the deviation of the rotary projection of the tooth surface δ i,j (x, y). e conversion can be realized by the relation between (u, l) and (x, y) shown in the following equation: where R tx (u, l), R ty (u, l), and R tz (u, l) are the coordinate components of R t (u, l).Next, the partial derivatives zδ ′ (u, l)/zu and zδ ′ (u, l)/zl in (6) can be computed using the following equation:

Internal Excitations
3.1.Time-Varying Mesh Sti ness.Using the TCA [24] and LTCA [26] on the real tooth surface of the pinion and the involute tooth surface of the gear, the time-varying mesh sti ness was computed.Using the TCA, the transmission error and backlash between teeth were obtained at zero load.us, the LTCA uses the nite element method and nonlinear programming method to obtain the comprehensive contact deformations Δ n in the normal direction at n contact points of the tooth surface in a meshing period (the comprehensive contact deformation Δ n simultaneously considers the e ect of other meshed teeth).en, the comprehensive mesh sti ness k m is calculated by the following equation: where F t is the tangent force, T g is the load torque, r bg is the base radius of the gear, F n is the normal force, and β b is the base helix angle.Finally, the comprehensive mesh sti ness k m consists of n discrete values in a meshing period, so it must be interpolated by the piecewise cubic-Hermite interpolation function and expressed as a periodic function by 4 Shock and Vibration the Fourier series.Based on this idea, the comprehensive time-varying mesh sti ness was computed at 830 N•m of load torque as shown in Figure 7.

Gear Meshing Impact.
e corner mesh impact mainly includes the meshing-in and meshing-out impacts.Seireg and Houser [27] have experimentally veri ed that the meshing-in impact had an obviously greater e ect of than the meshing-out impact on the gear transmission behavior.us, we only consider the meshing-in impact.Wu et al. [28] and Wang et al. [29] solved the meshing-in impact of unmodi ed and modi ed gear pairs, respectively.
is paper refers to the theory and method of these papers to calculate the meshing-in impact of the practical modi ed gear pair.e speci c solution steps are shown below.
After the modi cation, in the initial phase of the meshing process, the meshing points obtained by TCA do not bear load in the LTCA.us, the actually original position of the meshing-in contact is nearly the rst point which the LTCAcalculated load sharing factor between teeth is not zero as shown in Figure 8. Shock and Vibration meshing line, ignoring the manufacturing errors and the deformation of tooth; M is the actual original position of the meshing-in contact; M 1 M 2 is the instantaneous meshing line through point M; v p and v g are the tangential speeds of the pinion and gear; v np and v ng are the normal speeds of the pinion and gear; r bp and r bg are the theoretical base radii of the pinion and gear; and r′ bg is the actual base radius of the gear.e shock velocity with v s can be computed using the following equation: where R M , n M are the position vector and normal vector of M, respectively, which can be obtained by the TCA; ρ M is the vector from O g to M, with ρ M R M − E ; E is the vector from O f to O g ; w p , w g are the angular speeds of pinion and gear.
Because of the gear mass and inertia, if the shock speed v s emerges between the pinion and the gear, the impact must be generated at the meshing point M. As shown in Figure 9(b), m ep and m eg are the equivalent masses of particles for the pinion and gear: According to the theory of impact dynamics, the impact kinetic energy and elastic potential energy are: where m e is the total equivalent mass of the gear pair; F s is the maximal meshing-in impact; W s is the deformation caused by F s ; and K s is the sti ness at the original meshing point M, which value is the single tooth sti ness at M. Based on the law of conservation of energy, the relation can be obtained as follows: By solving Equations ( 12)-( 14), the maximal impact force can be computed as follows: Because the di erence between r′ bg and r bg is small, this paper uses an approximation solution for the impact force by substituting r bg for r′ bg [28].
We assume that the impact force is a half sine wave; then, it can be expressed as where ω s is the angular frequency, with ω s π/t s ; t s is the working time, which is de ned as the process of changing the shock speed from versus to 0. According to the momentum theorem, the following relation can be obtained: 6 Shock and Vibration Using ( 18), the working time t s [30] can be solved: us, the impact force is calculated at 830 N•m of load torque as shown in Figure 10.

Error Excitation.
We consider the cumulative pitch error, eccentric error, and loaded transmission error as the error excitation.Figures 11(a) and 11(b) show the measured cumulative pitch errors of the pinion and gear, so the relative cumulative pitch error of the gear pair is equal to the difference between the gear and pinion.Because the installed eccentricity of the gear system is difficult to be measured, we define the eccentric errors of the pinion and gear as 15 μm and 20 μm based on the radial run-out tolerance F r (adopting the standard of ISO 1328-2:1997).Figure 12 shows the relationship of the eccentric error and the line of action on the end face for the pinion and gear.E i (i � p, g) is the eccentric error of the pinion or gear, and Φ i is the angle between E i and the line of action on the end face.As shown in Figure 12, Φ i can be obtained as follows: where α t is the operating pressure angle; φ is the position angle of the center line; ω i is the angular velocity of the pinion or gear; and c i is the initial phase of the eccentric error.us, the equivalent error of E i projected onto the meshing line direction is where β b is the base helix angle.Finally, the relative cumulative pitch error projected onto the meshing line direction, eccentric error, and loaded transmission error obtained by the LTCA are superimposed, and the result is the displacement excitation of the gear system, which value in one period is shown in Figure 13.

Dynamic Model and Solution Method
4.1.Dynamic Model. Figure 14 depicts a 12-DOF dynamic model of a helical gear pair.In this picture, the pinion is lefthanded and rotates about the z-axis counterclockwise.Both gears are treated as rigid discs, which are connected to each other by a time-varying mesh spring k m in the plane of action along the normal direction.e displacement excitation e is shown in Figure 13 on the line of action.Both gears have radial motions in the x and y directions and axial motion in the z direction.Moreover, they have the swinging motion around the x-axis and y-axis and torsional motion around the z-axis.us, the equations of motion of the gear pair are [31]  where m i is the mass of gear i (i p, g); I xi , I yi , and I zi are the moments of inertia around the x-, y-and z-axes for gear i; r bi is the base radius of gear i; β b is the base helix angle; f s is the impact force; T p is the driving torque; and T g is the load torque.k m and c m are the time-varying mesh sti ness and mesh damping of the engagement pair: where ξ is the mesh damping ratio, which value is 0.07 in this paper; k m is the mean mesh sti ness of the engagement pair.k xi , k yi , k zi , k uxi , k uyi , k uzi and c xi , c yi , c zi , c uxi , c uyi , c uzi are the support sti ness and support damping in the directions of the 6-DOF, respectively, for gear i (i p, g).In Equation ( 21), δ is the relative displacement between the gear pair in the direction of the meshing line, and it is expressed as follows: where ψ is the angle between the plane of action and the positive y-axis, which is de ned as ψ α t − φ, counterclockwise rotation for pinion, α t + φ, clockwise rotation for pinion, where α t is the operating pressure angle, and φ is the position angle of the center line in Figure 14.Equation ( 21) is written in the matrix form as

Shock and Vibration
where M is the mass matrix; K b and C b are the supporting stiffness matrix and supporting damping matrix; K m and C m are the meshing stiffness matrix and meshing damping matrix; and V is the projection vector of displacement in the directions of the 12-DOF, which convert to the meshing line direction.

Solution Method.
ere are two main types of methods to solve the gear system dynamic equation: numerical integration methods and parse methods.Numerical integration methods include the Gill method, Runge-Kutta method, Ritz method, etc. Parse methods include the harmonic balance method, Fourier series method, mode superposition method, etc. Considering of the low-frequency error, the computation period of the dynamic equation has n meshing periods, and n is equal to the smallest common multiple of teeth number of the pinion and gear.However, distinctive features of the parse method are the quick calculation speed and the ability to directly obtain the steadystate solution.us, this paper uses the method of Fourier series to solve the dynamic equations.
e general idea is that the response and exciting force are expanded as Fourier series; the coefficient of each Fourier component of the response is ordered to be equal to that of the exciting force; finally, the stable solution is obtained.
First, let q, k m , and e be expressed as follows: where q 0 , k m0 , and e 0 are the mean value of the dynamic transmission error, time-varying mesh stiffness, and error excitation; Δq, Δk m , and Δe are the fluctuating components.
By substituting Equation ( 26) into (25), we obtain a new equation: where Equation ( 27) is expressed in terms of a time-varying part and a time-invariant part as follows: For the term ΔK•(q 0 +Δq) on the right side of Equation ( 28), q 0 can be computed using Equation (28); Δq is unknown, so it is approximated by the fluctuating component of the loaded transmission error obtained by the LTCA, which is denoted as Δq s .Let F be equal to the right side of Equation (28), that is, where F is the exciting force of the gear system.So, the dynamic equation (25) has been converted to the constant coefficient differential equation (29). 10

Shock and Vibration
Second, the dynamic response Δq and exciting force F are expanded into n order Fourier series: where ω is the fundamental frequency, with ω � 2πn p /60z g ; n p is the speed of the pinion; and z g is the number of gear teeth.e deciding rule of n order is that the exciting force F that is expanded as a Fourier series should be consistent with the one before expansion; here, n � 5000.
Finally, by plugging the ith-order (i � 1, 2, . .., n) expressions of the response and the exciting force into Equation ( 29), the linear algebraic equations can be obtained.

−ω
where ω i is the frequency of the ith-order harmonic, and ω i � iω.When A i and B i are calculated by Equation ( 31), the responses Δq and q can be obtained using Equations ( 30) and ( 26).

Comparison of Solution Methods.
To verify the accuracy and calculation speed of the Fourier series method, a simple example is solved using this method and the Runge-Kutta method.e basic parameters and dynamic equation of the example are shown in Table 2 and the following equation: where e is the sum of the eccentric errors of the pinion and gear. is paper calculates the relative rotational displacement between the pinion and gear in two cases: case one: the amplitude of the eccentric error of the pinion and gear is 0 μm; case two: both amplitudes are 10 μm. e results of two cases are shown in Figure 15.
Figure 15 shows that only the amplitude of the mesh frequency wave slightly varies between the results of Fourier series method and Runge-Kutta method, which illustrate that the gear system dynamic equation can be solved using the Fourier series method.In terms of computing time, there is no difference between these two methods in case one; in case two, the computing time of the Fourier series method is approximately 5 s, whereas, the computing time of the Runge-Kutta method is approximately 60 s.If the example in Table 1 is solved by the Runge-Kutta method, the computational cycle will be approximately 20 times longer than that of the example in this section, so the computing time will also significantly increase.To improve the calculation efficiency, this paper adopts the Fourier series method to solve the system response in the following analysis.

Example Analysis.
is paper provides two calculation examples using the proposed method.e example at low speed (50.8 rpm) shows the computation of the quasistatic transmission error, and the example at a high speed (2104 rpm) shows the analysis of the vibration response of the gear system.e basic parameters of the examples are shown in Table 1; other system parameters are listed in Table 3.
Based on the above examples, the equations of motion in Section 4.1 are solved, and the relative vibratory displacement along the line of action on the end face can be obtained.e specific results in the time and frequency domains are shown in Figures 18-25.

Example Calculation by Using the Traditional Method.
In order to validate the effectiveness and feasibility of the proposed method, we compare it with the traditional method.e novelty of the proposed method lies in considering the measured manufacturing error in the study of tooth modification.So, in this section, we give two examples by using the traditional method: one is without regard to the manufacturing errors and the other is considering the given value and variation of the manufacturing errors according to the manufacturing tolerance.
Case 1. Ignoring the effect of the manufacturing errors.
In this case, the gears only have the tooth modification and eccentric errors.e value of modification is the theoretical modification curve in Figure 1. e eccentric errors are consistent with the above examples in Section 5.1.We do TCA and LTCA on the theoretical modification tooth surface fitted by a bicubic B-spline and compute the timevarying mesh stiffness and meshing-in impact based on the results of TCA and LTCA.Finally, taking the time-varying mesh stiffness, meshing-in impact, loaded transmission error and eccentric errors as the internal excitations, we do dynamic analysis for the gear system.

Shock and Vibration
Case 2. Considering the given value and variation of the manufacturing errors.
On the basis of Case one, we increase tooth profile error and cumulative pitch error to the gear system.Refer to the literature [15], we set the variation of the tooth profile error and cumulative pitch error as a harmonic function, shown in the following equation: where E fα is the amplitude of the tooth profile error of pinion; E fp1 and E fp2 are the amplitudes of the cumulative pitch errors of pinion and gear.Referring to the standard of ISO 1328-1:1997, we find out E fα � 13 μm, E fp1 � 28 μm, and E fp2 � 36 μm; ω m is the mesh frequency; ω r1 and ω r2 are the rotation frequencies of pinion and gear; and φ i (i � fα, fp1, fp2) are the phase angles, which are set as φ fα � 0, φ fp1 � 3π/2, and φ fp2 � π/2.
In the process of dynamic analysis, we add the manufacturing errors to the equations of motion directly like (22) in reference [15], whereas the proposed method brings the tooth profile error to the loaded transmission error by TCA and LTCA.
All the two cases are calculated at 50.8 rpm and 2104 rpm.e basic parameters of gears, shafts, and bearings are consistent with the above examples in Section 5.1.e results are compared and analyzed in Section 5.4.

Experimental Validation.
A test system in Figure 16 was assembled to measure the system dynamic response.e test system can produce a circulating power flow through a close loop between the test and reaction gears.e test and reaction gears were connected by flexible shafts and elastic couplings, respectively, which were used to transmit a torque to the loop.Circular gratings were installed in the axle end of each test gear to measure the circumferential turning angle, which precision was ±5″.e grating signal was collected using a data acquisition card and an M + P acquisition system as shown in Figure 17.
e turning angles of the test gears were converted from the grating signal, and the gear transmission error along the line of action on the end face was calculated using the following equation: where ϕ p and ϕ g were the actual turning angles of the pinion and gear; r bp and r bg were the base radii.e specific results in the time and frequency domains are shown in Figures 18-25  e amplitudes corresponding to the fundamental frequencies in Figure 19 are summarized in Table 4, where f zge/zgs is the rotation frequency of gear; f zpe/zps is the rotation frequency of pinion; and f me/ms is the mesh frequency.
Comparing the results in Figures 18 and 19, we can see that the transmission error curves in both experimental and simulation results include the fluctuations of the rotation frequency and mesh frequency, and the values of these fundamental frequencies in the simulation results are largely consistent with ones in the experimental results.
However, combining Table 4, some other conclusions can be drawn: (1) For the case of the given manufacturing errors, the vibration amplitudes of the fundamental frequencies are clearly larger than the experimental result, which indicates that the amplitudes of the given cumulative pitch error and tooth profile error are so large as to be not practical.is case's results are certainly not desirable.(2) For the case of no manufacturing errors, the amplitude corresponding to mesh frequency is far less than the experimental result.So, it is also not advisable to predict the transmission performance of the machined gear system on the basis of the result of this case.(3) Only in the case of the measured manufacturing errors, the amplitudes of the base frequency components are close to the ones in the experimental results.Having some differences between the two results is because the internal and external excitations in the experiment are more complex than the simulation.By and large, the proposed method can serve as an estimate when the gear transmission experiment cannot be carried out.
rough the above analysis, we learn that the proposed method is viable and practical.So, using a Chebyshev highpass filter, we handle the results of the experiment and the simulation calculated by using the proposed method.e new curves and their FFT spectra after filtering are shown in Figures 20 and 21. e displacement curves for both the experimental and simulation results in Figure 20 have the modulation envelope.

Shock and Vibration 13
f me + 2f zge , f me + f zpe , 2f me + 2f zge , 2f me + f zpe , 2f me + 2f zpe , f ms + f zgs , f ms + f zps , 2f ms + f zgs , and 2f ms + f zps are the upper sidebands.According to the results, the mesh frequency and its harmonic components are all modulated by the rotation frequency signal of the pinion and gear.However, because of the more complex low-frequency excitations in experiment, the distribution of the sidebands is symmetrical for the simulation, but not for the experiment.6 summarizes the vibration amplitudes of the main frequencies for four kinds of results, where the meanings of f zge/zgs , f zpe/zps , and f me/ms are same as Table 4.
Comparing with the results calculated at 50.8 rpm, we can see that the vibration amplitudes of mesh frequency and rotation frequencies opinion and gear increase a lot for the experiment.However, to the simulations of three cases, only the vibration amplitude of mesh frequency grows a little.Next, we analyze these phenomena: (1) For the simulations of three cases, the cumulative pitch error at 2104 rpm is same as one at 50.8 rpm, so the vibration of rotation frequencies is almost unchanged.With the increase of speed, the meshingin impact force grows, which leads to the vibration of mesh frequency intensi ed.(2) For the experiment, because of speed rise, the amplitude corresponding to rotation frequency of gear nearly doubles the one at 50.8 rpm, which may be caused by low-frequency excitations other than the cumulative pitch error.About the vibration of mesh frequency, the extent of increase is much larger than one in the simulations, which may result from meshing-in impact and other new high-frequency excitations.
Based on the above analysis, we can draw some conclusions as follows: (1) For the case of the given measured manufacturing errors, although the amplitudes corresponding to  16 Shock and Vibration mesh frequency and rotation frequency of gear are close to the ones in the experimental results, the simulation result is still undesirable.is is because the vibration intensi ed in the experiment is not caused by the cumulative pitch error and tooth pro le error.e simulation of this case is distorted.(2) For the case of no manufacturing errors, the vibration of high frequency is still far less than one in the experimental result.So, it is also totally impracticable.(3) About the case of the measured manufacturing errors, the waveform and trend of the dynamic transmission error are roughly identical to ones in the experimental results.In a future study, if we attempt to consider other factors such as the bearing dynamic force, dynamic load, and assemble error of gear system, the simulation results would approximate the experiment results more and more.So the proposed method is feasible and worth studying.
We analyze the high-frequency components for the experiment and the simulation computed by using the proposed method further on.Figures 24 and 25 show the relative vibration displacement and its FFT spectra after ltering out the low-frequency components.e sidebands around the meshing frequency in the experimental results are more complex than those in the simulation results.It suggests that some new low-frequency excitations come into being in the experiment.Table 7 shows the corresponding sidebands in the experiment and simulation.e simulation result 1 is corresponding to the case of the measured manufacturing errors, 2 is corresponding to the case of no manufacturing errors, and 3 is corresponding to the case of the given manufacturing errors.e simulation result 1 is corresponding to the case of the measured manufacturing errors, 2 is corresponding to the case of no manufacturing errors, and 3 is corresponding to the case of the given manufacturing errors.

Figure 4 :
Figure 4: Relationship between tooth depth and length of meshing line.

Figure 9 :
Figure 9: Sketch map of meshing-in impact (on end face): (a) shock velocity principle, (b) dynamic model of meshing-in impact.
yp c zp c uxp c uyp c uzp c xg c yg c zg c uxg c uyg c uzg  , C b � diag c xp c yp c zp c uxp c uyp c uzp c xg c yg c zg c uxg c uyg c uzg  , P � 0 0 0 0 0 .

5. 4 .
Comparison and Analysis 5.4.1.Result Analysis at 50.8 rpm.Figures 18 and 19 show the quasistatic transmission error of the helical gear pair and its spectrum for the above three examples and the experiment.

Figure 15 :
Figure 15: Relative rotational displacement in (a) case one and (b) case two.
inertia of pinion and gear include the mass and inertia of the corresponding shaft.Direct current motor Reaction gears Coupling Torque speed sensor Test gears Loader Circular gratings

Figure 16 :Figure 17 :
Figure 16: Gear transmission test system to implement proposed measurement scheme.
The quasi-static transmission error (µm)ExperimentNo manufacturing errorsThe measured manufacturing errors The given manufacturing errors

Figure 19 :
Figure 19: Spectrum and partial enlarged view of quasistatic transmission error: (a) experiment; (b) the measured manufacturing errors; (c) no manufacturing errors; (d) the given manufacturing errors.

Figure 24 :
Figure 24: Mesh frequency component of dynamic transmission error.

Figure 25 :
Figure 25: Spectrum and partial enlarged view of gear frequency component of dynamic transmission error: (a) experiment; (b) simulation by using the proposed method.

Table 1 :
Parameters of pair of helical gears.
Table 5 summarizes the sidebands from the experiment and simulation according to Figure 21.In this table, f me − f zpe + f zge , 2f me − f zpe − f zge , f ms − f zps , f ms − f zgs , 2f ms − f zps , and 2f ms − f zgs are the lower sidebands;

Table 2 :
Parameters of simple example.

Table 3 :
Parameters of example system.

Table 5 :
Sidebands in experimental and simulation results at 50.8 rpm.

Table 4 :
e amplitudes corresponding to the fundamental frequencies in experimental and simulation results at 50.8 rpm.

Table 6 :
e amplitudes corresponding to the fundamental frequencies in experimental and simulation results at 2104 rpm.

Table 7 :
Sidebands in experimental and simulation results at 2104 rpm.