Three-Dimensional Free Vibration Analysis of Gears with Variable Thickness Using the Chebyshev–Ritz Method

To reflect vibration more comprehensively and to satisfy the machining demand for high-order frequencies, we presented a threedimensional free vibration analysis of gears with variable thickness using the Chebyshev–Ritz method based on three-dimensional elasticity theory. We derived the eigenvalue equations. We divided the gear model into three annular parts along the locations of the step variations, and the admissible function was a Ritz series that consisted of a Chebyshev polynomial multiplying boundary function. The convergence study demonstrated the high accuracy of the present method. We used a hammering method for a modal experiment to test two annular plates and one gear’s eigenfrequencies in a completely free condition. We also applied the finite element method to solve the eigenfrequencies. Through a comparative analysis of the frequencies obtained by these three methods, we found that the results achieved by the Chebyshev–Ritz method were close to those obtained from the experiment and finite elementmethod.The relative errors of four sets of data were greater than 4%, and the errors of the other 48 sets were less than 4%. Thus, it was feasible to use the Chebyshev–Ritz method to solve the eigenfrequencies of gears with variable thickness.


Introduction
Gears have a wide range of applications in industrial production and are essential for transmitting power [1,2].Ultrasonic machining has many advantages compared with traditional machining, such as reducing cutting force and surface roughness and improving machining accuracy [3][4][5].Therefore, it is essential to introduce ultrasonic machining into gear machining to improve the gear's machining quality.The quality of ultrasonic gear machining mainly depends on the vibration characteristic of the ultrasonic machining system.During ultrasonic gear machining, the gear is not only a workpiece but also a part of the ultrasonic machining system [6,7].In past research, the gear was simplified as a uniform plate or an annular stepped plate, while regarding the gear's pitch diameter as the plate's diameter, to solve the gear's natural frequency [8,9].
There are several plate theories according to different thickness-diameter ratios [10,11].Kirchhoff established the foundation of a thin-plate theory with research on the vibration of a thin plate [12].The theoretical results of thin-plate theory are close to accurate natural frequencies when the thickness-diameter ratio is less than 0.2, but the relative error increases rapidly when the thickness-diameter ratio is greater than 0.2 [13,14].Mindlin considered shear deformation to lay the foundation for a moderately thick-plate theory [15].Some investigations of the vibration characteristics of a stepped plate based on Mindlin theory have been presented [16,17].Only the frequencies of transverse bending vibration mode have been given, however, as Mindlin theory assumes that the plate displacement varies linearly in thickness direction of plate with the plate thickness remaining unchanged and this theory also ignores the stress in the thickness direction of plate.Leissa summarized previous work to form an analytical system of different kinds of plates [18].Thereafter, many researchers had deeper study of plate theory and established a vibration model, that is, the derivation of an analytical formula and the solving of a numerical solution under different conditions for engineering applications [19][20][21].The finite element method and calculus method also have been used to analyze the plate's vibration characteristic [22][23][24][25].
These theories and methods have restrictions, however; for example, different kinds of plates require different theories for analysis, high-order frequencies cannot be obtained, and the finite element method cannot obtain a solution in theory.
Thus, some research has been conducted based on threedimensional elasticity theory using the Ritz method for more accurate solutions.The Ritz method was established by Ritz in 1909 [26] to efficiently obtain accurate frequencies.Since then, many scholars have used this method to analyze different objects' vibration characteristics, including uniform and nonuniform beams, rectangular plates, and thin plates, according to one-dimensional or two-dimensional theory [27][28][29][30].Then, Leissa and Kang used the Ritz method with an admissible function composed of algebraic polynomial multiplying boundary functions based on three-dimensional elasticity theory to analyze the vibration characteristic of thick, linearly tapered, annular plates [31,32].This method can be used to analyze the vibration characteristics of plates with arbitrary thickness.The numerical stability of an algebraic polynomial is bad, however, and high-order frequencies cannot be obtained.Because the Chebyshev polynomial possesses the simplicity of algebraic polynomial and avoids the numerical instability of high-order algebraic polynomial, Zhou applied the Chebyshev-Ritz method, which uses the Chebyshev polynomial to replace the algebraic polynomial in admissible functions to obtain more accurate eigenfrequencies of plates with constant thickness [33][34][35].Although Zhou obtained more accurate eigenfrequencies, he did not apply this method to plates with variable thickness.In engineering, gears usually have variable thickness, so further research by the Chebyshev-Ritz method is needed.
To reflect the vibration more comprehensively and satisfy the machining demand for high-order frequencies, we presented a three-dimensional free vibration analysis of gears with variable thickness using the Chebyshev-Ritz method based on three-dimensional elasticity theory.We derived the eigenvalue equations.To achieve the eigenfrequencies, we divided the gear model into three parts along the locations of the step variations and verified the convergence of this method.We used a hammering method for the modal experiment to test two annular plates and a gear's frequencies in a completely free condition.We also applied the finite element method to solve the eigenfrequencies.Through a comparative analysis of the frequencies obtained by the three methods discussed earlier, the results achieved according to the Chebyshev-Ritz method were close to those achieved by the experiment and finite element method.This finding indicated that it is feasible to use the Chebyshev-Ritz method to solve the eigenfrequencies of gears with variable thickness.Because three-dimensional elasticity theory can be used for any kinds of plates, the analysis method has a broad application in engineering.

Method of Analysis
To omit keyway, cast fillet, machining chamfer, and hole to reduce the weight of a straight-toothed spur gear and helicalspur gear, a gear with variable thickness can be simplified as an annular plate, as shown in Figure 1.A cylindrical coordinate (, , ) is defined.The coordinate origin is coincident with the center of the plate's neutral surface.The radial () and axial () coordinates are measured from the central axis, and  is the circumferential angle.The corresponding displacement components at a generic point are , V, and  in the , , and  direction, respectively.Thus, the ranges of the coordinate are given for the annular plate by the following: 0 ≤  ≤  4 , 0 ≤  ≤ 2, and 0 ≤  ≤  1 /2.
To analyze vibration characteristics, the annular plate shown in Figure 1 can be regarded as a combination of three parts that are the annular plates shown in Figure 2. Therefore, the annular plate According to the three-dimensional linear elasticity theory, the strain energy  of the annular plate Ω in Figure 1 is given in integral form as follows: where  is the shear modulus and ] is Poisson's ratio.The strain components   (,  = , , ) are defined as follows: The kinetic energy  of the plate is given as where  is the mass density per unit volume and  denotes time.
For simplicity and convenience in the mathematical formation, three dimensionless parameters are given as follows: For free vibration, the displacement function can be expressed as where  is the eigenfrequency of the annular plate in Figure 1 and  = √ −1.
In consideration of the annular plate in Figure 1, which can be regarded as three parts, and substituting (4) and ( 5) into ( 1) and ( 3), the maximum strain energy  max and kinetic energy  max can be given as follows: where where where ; ; Each of the displacement amplitude functions is composed of a double series of Chebyshev polynomials multiplied by boundary functions as follows: where , , , , , and  are the truncation orders of the Chebyshev polynomial series and   ,   , and   are coefficients yet to be determined.  () ( = 1, 2, 3, . . .;  = , ) is the one-dimensional th Chebyshev polynomial, which can be written as follows: The boundary functions   ,  V , and   should enable the displacement components , V, and  to satisfy the geometric boundary conditions of the rod.For example, (1) completely free: inner edge fixed, remaining boundaries free: outer edge fixed, remaining boundaries free: both ends fixed: The Chebyshev polynomial series   () ( = 1, 2, 3, . ..) is a set of complete and orthogonal series in the interval [−1, 1].Thus, the double series   ()  () (,  = 1, 2, 3, . ..) is also a complete and orthogonal set.The Chebyshev polynomial has excellent properties in the approximation of functions.Therefore, its numerical stability is better than the algebraic polynomial series.
The energy functional of the plate is defined as follows: Minimizing the previous function with respect to the coefficient, leads to the following eigenvalue equation: in which Λ = ( 4 − 1 )√/ and  and  are the stiffness and mass matrices resulting from the maximum strain energy ( max ) and the maximum kinetic energy ( max ).The vector  is given as

Convergence Study
To verify the accuracy of frequencies obtained by the previous method, we conducted a convergence study.A convergence study is based on the fact that all of the frequencies obtained by the Ritz method should converge to their exact value in an upper bound manner when ( 13) and ( 14) are used.If the results did not converge properly, it would be likely that the assumed admissible function might not be the right choice.Table 1 gives the results of the convergence study of the completely free annular plate in Figure 1, with  1 = 15,  2 = 25,  3 = 70,  4 = 100,  1 = 25, and  2 = 15 when s = 1. 1 ,  2 ,  3 ,  4 ,  1 , and  2 have same unit.Table 1 lists the first five eigenfrequencies in  4 √/ with Poisson's ratio ] = 0.3 and retains four significant digits.To make the study of convergence less complicated, equal numbers of polynomial terms were taken in the  ( =  = ) and  ( =  = ) directions.
Table 1 shows the convergence of the first five eigenfrequencies.The values of  and  begin with six and eight, respectively, increasing  from 6 to 16 and  from 8 to 14.This result showed that the first and second eigenvalues were the easiest to converge to the accurate results. = 14 and  = 14 were the smallest terms to obtain accurate values in the  and  directions.The Chebyshev polynomial series needed additional terms for higher order eigenfrequencies.When  ×  = 16×12, the first five eigenfrequencies all converged to the accurate values.This result proved that the Chebyshev-Ritz method can be used to analyze the vibration characteristic of the annular plate in Figure 1 and to achieve good convergence.
In Table 2, we listed three sets of convergence comparison with the Chebyshev-Ritz method and the Ritz method using an algebraic polynomial as the admissible function from  ×  = 10 × 10 to  ×  = 12 × 12.The eigenfrequencies worked by the two methods in  ×  = 10 × 10 were quite close.The eigenfrequencies worked by the Chebyshev-Ritz method became increasingly accurate with an increase of  and  from  ×  = 10 × 10 to  ×  = 12 × 12.When  ×  = 11 × 11, however, the eigenfrequencies worked by the Ritz method using an algebraic polynomial were ill-conditioned and the eigenfrequencies began to diverge.When  ×  = 12 × 12, the eigenfrequencies all became complex numbers.This result indicated that the eigenfrequencies using an algebraic polynomial as the admissible function could not converge to the accurate values for gears with variable thickness.It is well known that ill-conditioned eigenfrequencies will occur with an increase in the number of terms.Sooner or later, the occurrence of ill-conditioned eigenfrequencies will depend on the admissible function adopted in the analysis.Because the Chebyshev polynomial possesses better numerical stability, the occurrence of ill-conditioned eigenfrequencies was delayed so that accurate eigenfrequencies could be obtained.

The Comparative Analysis of the Eigenfrequencies Using the Chebyshev-Ritz Method, Finite Element Method, and a Modal Experiment
To validate the accuracy of the eigenfrequencies worked out by the Chebyshev-Ritz method, we designed two annular plates and a gear for the modal experiment.The two annular plates are shown in Figure 3, and the gear is shown in Figure 4(a).The dimension parameters of the plates are listed in Table 3.
To test the eigenfrequencies, we adopted the hammering method and established the vibration measurement system to receive the hammering pulse of the annular plates.To simulate the completely free condition, we hung the annular plate with a rope.We used 13 measuring channels of the data acquisition instrument to test the frequencies.One channel was an incentive channel, and the others were parallel corresponding channel.The sensor on the hammer was connected to the data acquisition instrument and the data acquisition instrument was connected to the laptop with a program called Coinv DASP V 10.0.We placed three three-dimensional acceleration sensors on the plate in equal intervals in the  and  directions.The three-dimensional acceleration sensors were connected to the data acquisition instrument and the frequencies collected by data acquisition instrument were limited between 1 kHz and 24 kHz.During the experiment, we established an analysis model by inputting the coordinate point of the annular plate to DASP V 10.0.We used the hammer beating the annular plate's surface and read the data from DASP V 10.0.The experimental platform was established as shown in Figure 4.
The eigenfrequencies of the two annular plates are listed in Tables 4 and 5 according to the Chebyshev-Ritz method and the experiment.CR represents the eigenfrequencies solved by Chebyshev-Ritz method, EX represents eigenfrequencies by experiment, and FE represents eigenfrequencies by finite element method.The finite element analysis type was a modal analysis with the Block Lanczos method (using Ansys 15.0).The unit type was SOLID 95 with 20 nodes, and   Mathematical Problems in Engineering 7 We also designed the gear shown in Figure 4(a) to validate this method.Parameter R 4 is equal to the pitch radius.The gear was made of No. 45 steel with Poisson's ratio ] = 0.27, Young's modulus  = 210 GPa, and density  = 7800 kg/m 3 .The dimension parameters are listed in Table 6.
Figure 5 shows that the CREs in total are greater than 4% for each plate and the gear is very low.This low percentage indicated that it was feasible to use the Chebyshev-Ritz method based on three-dimensional elasticity theory to analyze the eigenfrequencies of gears with variable thickness.The low percentage of FEX greater than 4% indicated that the experimental scheme was reasonable and that the experimental results were reliable.The FEXs were mainly concentrated in the range of 1%-4% and rarely appeared in   Mathematical Problems in Engineering 9 the range that was not greater than 1%.This phenomenon showed that it was difficult to achieve high precision using the experimental method because of the influence of the experimental platform's systematic error.In Figure 5, beyond 65% of the CREs were not greater than 1% for each plate and gear, and less than 13% of the CREs were greater than 4%.These values proved that the accuracy of results derived according to the Chebyshev-Ritz method and the finite element method was basically at the same level.We regarded the distribution of the percentage of CRE, in total, to be a linear distribution.This phenomenon provided additional guidance and a theoretical basis for the establishment of a system error compensation model for the experimental platform.

Conclusion
We should have repeated the finite element method's preprocessing steps to analyze the vibration characteristics, including modeling for different gears and different mesh division for accurate frequencies.The Chebyshev-Ritz method could be used to omit these preprocessing steps.By using the Chebyshev-Ritz method, we could substitute only the material property parameters and the dimension parameters into the computing program to analyze the vibration characteristics of different gears with variable thickness.It was easy to achieve parametric programming and analysis.We were able to conduct three-dimensional free vibration analysis of gears with variable thickness using the Chebyshev-Ritz method to analyze the eigenfrequencies in different models with different s.This analysis method can be used for plates with any thickness-diameter ratios to solve their eigenfrequencies.The CREs, in total, that were greater than 4%, were low, which indicated that it was feasible to use the Chebyshev-Ritz method based on three-dimensional elasticity theory to analyze the eigenfrequencies of gears with variable thickness.The accuracy of results according to the Chebyshev-Ritz method and finite element method were basically at the same level because the CREs were mainly concentrated in a range not greater than 1%.The experimental method was reasonable, and the experimental results were reliable.This method can be used to analyze different types of stepped objects and also to omit the preprocessing steps for different analysis objects using the finite element method.

Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.

Figure 4 :
Figure 4: (a) Location of three-dimensional acceleration sensors and (b) data acquisition instrument.

Table 3 :
Dimension parameters of two annular plates.

Table 4 :
Frequencies between 1 kHz and 24 kHz of the brass annular plate obtained by the Chebyshev-Ritz method and the experiment compared with those obtained by finite element method. represents axisymmetric mode and 0  represents torsional mode.we adopted an intellectual finite element mesh division with four-stage precision for the mesh division.CRE represented relative error as (|  −   |/  )%, and FEX was (|  −   |/  )%.The brass annular plate's Poisson's ratio was ] = 0.36, Young's modulus was  = 88 GPa, and density  = 8557 kg/m 3 .The grey cast iron annular plate's Poisson's ratio was ] = 0.3, Young's modulus was  = 130 GPa, and density was  = 7410 kg/m 3 . 0

Table 5 :
Frequencies between 1 kHz and 24 kHz of the grey cast iron annular plate obtained by the Chebyshev-Ritz method and the experiment compared with those obtained by finite element method.
0  represents axisymmetric mode and 0  represents torsional mode.

Table 7 :
Frequencies between 1 kHz and 24 kHz of the gear obtained by the Chebyshev-Ritz method and the experiment compared with those obtained by finite element method.