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BY 4.0 license Open Access Published by De Gruyter Open Access January 3, 2022

Modal characteristics of harmonic gear transmission flexspline based on orthogonal design method

  • Leyu Wei EMAIL logo , Jie Yang , Changlu Wang , Xingxing Wang and Xinmeng Liu
From the journal Open Physics

Abstract

The flexspline features several modal characteristics that are related to its structural parameters, which could impose an impact on the stability and reliability of the system of harmonic gear transmission. In this article, we have integrated the method of orthogonal design with the approach of finite element, so as to study the modal characteristics of flexspline under varying structural parameters. Furthermore, we have analyzed the ways of how varying structural parameters could impose an impact on the first eight-order natural frequencies of flexspline. In addition, by adopting the method of range and variance analysis for the obtained data, we have examined the influence imposed by varying structural parameters on the modal characteristics of flexspline, which could lay a foundation for the optimal design of the structural parameters of flexspline.

1 Introduction

The transmission of harmonic gear is based on the theory of coordination of elastic deformation. Unlike the conventional approach to gear transmission, the method of harmonic gear transmission has its unique advantages due to its compact structure, high speed ratio, optimal accuracy and efficiency of transmission, smooth operation, less noise, and optimal coaxiality [1,2,3]. As a result, the method has been widely applied in spatial technology, military science, robotics, and optical instruments [4,5,6].

During the harmonic gear transmission, as a thin-wall member, the flexspline is expected to generate a vibration response under the action of periodic excitation, which is prone to undermine the kinetic accuracy of the mechanism and compromise the transmission performance. In particular, when the excitation frequency is close to the natural frequency, local or overall resonance will occur, resulting in more serious consequences. Therefore, it is necessary to study the modal characteristics of flexspline to improve the dynamic characteristics of the system. During their study, Pan and Yang examined the dynamic characteristics of a certain type of harmonic reducer, and obtained the first four-order modal shapes of flexspline, laying a foundation for the optimal design of flexspline [7]. Oh adopted a sort of composite of steel and reinforced fiber to fabricate flexspline, facilitating improvement in the dynamic properties of flexspline [8]. Based on the sensitivity analysis, Deng identified the mechanism of how varying structural parameters of the flexspline may impose an impact on the fatigue strength coefficient as well as their degree of sensitivity [9].

Judging from our study, the structural parameters of flexspline serve as critical factors contributing to the modal characteristics of flexspline, whereas varying structural parameters could impose different impacts on the modal characteristics accordingly. However, among existing studies, few scholars have carried out research on the mechanism of how structural parameters could influence the modal characteristics of flexspline. By integrating the method of orthogonal design and the approach of finite element, we have analyzed the inherent law of how different structural parameters could affect the modal characteristics of flexspline, so as to lay a foundation for selection of optimal structural parameters [10,11].

2 Modal theory

The method of modal analysis is an analytical approach used in structural dynamics. At present, the modal analysis method has evolved into a vital approach to address issues of engineering vibration. Major methods related to the modal analysis include the subspace iteration method, the iterative Lanczos method, and the iterative Ritz vector method. In the software of finite element analysis, the iterative Lanczos method has found extensive applications. Specifically, the basic step of this approach is to select the initial vector and form m Lanczos vectors through multiple times of inverse iterations, orthogonization, and normalization. The orthogonal and normalized coefficients could form a tridiagonal matrix, whose eigensolution features certain relations to the first higher order eigensolutions of the original generalized eigenproblem. By leveraging such relations, we are able to obtain the first higher order eigensolutions of the original generalized eigenproblem [12,13].

With respect to the generalized eigenvalue of the n order:

(1) K ϕ = λ M ϕ .

In terms of the structure, suppose that the stiffness K is positive definite, then S = K −1 M. First, we have selected a proper initial vector q 1 to satisfy the equation of (q 1) T M q 1 = 1. Subsequently, suppose that k refers to the number of iterations, then we have conducted the following iteration of k = 1, 2, …, m:

  1. Inverse iteration of vector:

    (2) q ¯ k + 1 = S q k .

  2. Orthogonization of vector. We have performed the orthogonization of q ˜ k + 1 and two Lanczos vectors, with orthogonal coefficients of α k and β k , as specified in the following formulas:

    (3) α k = q ¯ k + 1 α k q k β k q k 1 ,

    (4) β k = ( q ¯ k + 1 ) T M q k 1 ,

    (5) β 1 = 0 .

  3. Normalization vector. We have performed normalization of q ˜ k + 1 to form the (k + 1)th Lanczos vector, as specified in the following formula:

(6) q k + 1 = q ˜ k + 1 / γ k + 1 ,

where,

(7) γ k + 1 = ( q ˜ k + 1 ) T M q ˜ k + 1 .

According to the calculation of generalized inner product, after we have completed the aforementioned procedures of orthogonalization and normalization, we have formed m Lanczos vectors q i (i = 1, 2, …, m), accordingly. The tridiagonal matrix is specified in the following formula:

(8) T m = α 1 β 2 γ 2 α 2 β 3 γ 3 O O O O α m 1 β m λ m α m .

Solving the tridiagonal eigenvalue,

(9) T m z i = ρ z i ,

we have obtained the approximate solutions of the first m-order eigenvalues of the original generalized eigenproblem [12,13].

(10) λ ¯ i = 1 ρ i , i = 1 , 2 , , m .

Let

(11) Q m = [ q 1 q 2 q m ] .

Then, the approximate solution of the first m-order eigenvector of the original problem is,

(12) ϕ ¯ i = Q m z i , i = 1 , 2 , , m .

For dynamic analysis, only the first m order eigenvalues are often required. Therefore, the solution of the m-order tridiagonal matrix T m eigenvalue problem is much simpler than the original generalized eigenvalue problem.

3 Orthogonal design

The structural parameters of flexspline include: the length L, tooth width B, tooth root transition fillet R, tooth ring thickness δ, and wall thickness δ1, and dz is the diameter of the wave generator, (as illustrated in Figure 1), which could jointly impose an impact on the inherent properties of flexspline. Given that there is a large number of structural parameters, and that each parameter amounts to varying levels, one separate analysis for each level of combination of parameters will increase the time of analysis [14,15]. Without affecting the conclusion, the orthogonal design method can reduce the analysis time to a maximum extent and improve the efficiency. Therefore, we have adopted the method of orthogonal design to perform the finite element modal analysis [16].

Figure 1 
               Structural parameters of flexspline.
Figure 1

Structural parameters of flexspline.

We have opted for five discrete values for each structural parameter of flexspline in a certain harmonic reducer. On this basis, we have selected the table of L 25(56) orthogonal design, while meeting the condition of statistical accuracy [17]. Moreover, these five structural parameters featured five discrete values, whereas the remaining one factor was adopted for error analysis for 25 times. The table of orthogonal design generated through the software of SPSS is specified in Table 1.

Table 1

Orthogonal design table

Model No. Length [L] (mm) Tooth ring thickness [δ] (mm) Wall thickness [δ 1] (mm) Tooth width [B] (mm) Tooth root transition fillet [R] (mm) Error column
1 19 0.75 0.5 12 4 2
2 44 0.75 0.65 9 8 4
3 19 0.65 0.45 9 0 1
4 25 0.85 0.65 13 4 1
5 19 0.8 0.65 11 6 5
6 38 0.75 0.55 11 2 1
7 44 0.8 0.55 13 0 2
8 38 0.65 0.5 13 8 5
9 25 0.75 0.6 10 0 5
10 31.5 0.85 0.5 11 0 4
11 31.5 0.75 0.45 13 6 3
12 19 0.85 0.55 10 8 3
13 38 0.85 0.6 9 6 2
14 44 0.85 0.45 12 2 5
15 44 0.7 0.5 10 6 1
16 38 0.7 0.65 12 0 3
17 25 0.7 0.45 11 8 2
18 44 0.65 0.6 11 4 3
19 31.5 0.8 0.6 12 8 1
20 19 0.7 0.6 13 2 4
21 38 0.8 0.45 10 4 4
22 31.5 0.7 0.55 9 4 5
23 31.5 0.65 0.65 10 2 2
24 25 0.65 0.55 12 6 4
25 25 0.8 0.5 9 2 3

4 Analysis of modal characteristics

4.1 Analytical model

There are rather intricate conditions in the boundary of flexspline. In case the established contact model features tooth characteristics, then the grid, memory occupancy, and calculation time will be subject to significant increases. Therefore, we need to ensure that the treatments of the tooth of flexspline are equivalent. By adopting the equivalent method, we intend to reduce the tooth ring of flexspline to a smooth ring with a certain equivalent thickness, whereas the equivalent tooth ring thickness δ amounted to 1.67 3 [18,19,20] of the smooth ring wall thickness at the tooth root. In this study, we have applied the ABAQUS software into our analysis of the modal of flexspline, and the grid division of no. 10 flexspline is illustrated in Figure 2.

Figure 2 
                  Grid division of no. 10 flexspline.
Figure 2

Grid division of no. 10 flexspline.

4.2 Natural frequency analysis

We have opted for the 42CrMo as the material of flexspline. In terms of the ABAQUS material properties, we suppose that elastic modulus E = 2.06 × 10 5 MPa , Poisson’s ratio μ = 0.3 , and density ρ = 7,850 kg/m 3 . After dividing the hexahedral grid, we have set the type of programs as the frequency analysis, and we have imposed constraints on the bottom end surface of the flexspline cup. We are able to carry out the natural frequency analysis after submitting the task. Based on our analysis of each model, we have obtained the first eight-order natural frequencies of each model (as specified in Table 2) and the first eight-order modal diagrams of Model 1 (as illustrated in Figure 1).

Table 2

The first eight order natural frequencies of each model

Model No. Modal 1 (Hz) Modal 2 (Hz) Modal 3 (Hz) Modal 4 (Hz) Modal 5 (Hz) Modal 6 (Hz) Modal 7 (Hz) Modal 8 (Hz)
1 777 785 874 874 1,165 1,165 1,419 1,419
2 333 335 508 508 785 785 899 899
3 356 356 391 433 454 945 945 1,596
4 644 645 798 798 1,247 1,247 1,351 1,351
5 918 919 992 992 1,549 1,549 1,556 1,556
6 156 156 253 331 340 809 809 1,433
7 294 297 535 535 673 673 964 964
8 322 325 494 494 734 734 865 865
9 627 629 767 767 1,259 1,259 1,274 1,274
10 401 403 683 683 815 815 1,198 1,222
11 384 389 651 651 754 754 1,100 1,147
12 772 773 1,002 1,002 1,352 1,352 1,651 1,651
13 381 383 622 622 873 873 1,104 1,104
14 274 277 536 536 574 574 964 1,044
15 242 246 284 284 450 450 636 636
16 394 396 524 524 905 905 916 916
17 482 485 690 690 927 927 1,175 1,175
18 312 315 459 459 781 781 797 797
19 196 196 290 293 297 702 702 1,236
20 977 982 1,156 1,156 1,709 1,709 1,747 1,747
21 312 316 592 592 640 640 1,044 1,065
22 444 447 611 611 993 993 1,039 1,039
23 511 513 578 578 948 948 1,118 1,118
24 237 237 308 393 409 898 898 1,482
25 528 530 770 770 1,013 1,013 1,331 1,331

Judging from Table 2, it could be noted that the natural frequency of different models varied to a significant extent, indicating that the structural parameters could impose a huge impact on the modal characteristics of flexspline. Judging from Figure 3, it could be noted that the natural frequency of each order of Model 1 was the equivalent, whereas the difference lies in the amplitude and direction of vibration. Specifically, the vibration deformation of the first six order took place at the tooth ring, and the vibration deformation of the last second order took place at the bottom of the cup, all of which were obviously symmetric. The modals of the remaining models showed similar characteristics, which we would not elaborate further in this article.

Figure 3 
                  The first eight-order modals of Model 1. (a) Modal 1, (b) Modal 2, (c) Modal 3, (d) Modal 4, (e) Modal 5, (f) Modal 6, (g) Modal 7, and (h) Modal 8.
Figure 3

The first eight-order modals of Model 1. (a) Modal 1, (b) Modal 2, (c) Modal 3, (d) Modal 4, (e) Modal 5, (f) Modal 6, (g) Modal 7, and (h) Modal 8.

4.3 Impact imposed by varying parameters on the modal characteristics of flexspline

4.3.1 Influence imposed by the length on the modal characteristics

The discrete value of the length L amounted to 19, 25, 31.5, 38, and 44 mm (length-to-diameter ratio reaching 0.3, 0.4, 0.5, 0.6, and 0.7, respectively). After taking the mean value of the varying L values of the same level, we have performed the polynomial fitting of the varying levels and mean values of L to identify the rules of how the values of L could affect the natural frequency of each order, as illustrated in Figure 4. Judging from Figure 4, the first eight-order natural frequencies of flexspline experienced declines with the increase in L. In addition, they decreased significantly in the range of 19–31.5 mm, and decreased slightly in the range of 31.5–44 mm, showing a similar trend of variation. Moreover, the first-order natural frequency and the second-order natural frequency were close to each other, whereas the third-order natural frequency and the fourth-order natural frequency were different.

Figure 4 
                     Influence imposed by the length on the natural frequency of each order.
Figure 4

Influence imposed by the length on the natural frequency of each order.

4.3.2 Influence imposed by the tooth ring thickness on the modal characteristics

The discrete value of the tooth ring thickness δ amounted to 0.65, 0.70, 0.75, 0.80, and 0.85 mm. The relationship between the δ and the natural frequency of each order is illustrated in Figure 5. Judging from Figure 5, the δ and the natural frequency curve of each order were not monotonic. In addition, the first 7-order natural frequencies experienced gradual increase in the range of 0.65–0.70 mm, decline in the range of 0.70–0.80 mm, and increase again in the range of 0.80–0.85 m. The trends of variation in the first seven-order natural frequencies were similar, which were opposite to those of the first eight-order natural frequencies.

Figure 5 
                     Influence imposed by the tooth ring thickness on the natural frequency of each order.
Figure 5

Influence imposed by the tooth ring thickness on the natural frequency of each order.

4.3.3 Influence imposed by the wall thickness on the modal characteristics

The discrete value of the wall thickness δ 1 amounted to 0.45, 0.50, 0.55, 0.60, and 0.65 mm. The relationship between the δ 1 and the natural frequency of each order is illustrated in Figure 6. Judging from Figure 6, the trends of variation in the first four-order natural frequencies were similar, whereas the trends of variation in the last four-order natural frequencies were different. In addition, the trend of variation in the fifth-order natural frequency was contrary to that of the eighth-order natural frequency, whereas the sixth-order natural frequency revealed an increasing trend. Moreover, the seventh-order natural frequency experienced slight changes.

Figure 6 
                     Influence imposed by the wall thickness on the natural frequency of each order.
Figure 6

Influence imposed by the wall thickness on the natural frequency of each order.

4.3.4 Influence imposed by the tooth width on the modal characteristics

The discrete value of the tooth width B amounted to 9, 10, 11, 12, and 13 mm. The relationship between B and the natural frequency of each order is illustrated in Figure 7. Judging from Figure 7, the trends of variation in the first seven-order natural frequencies were similar, all of which experienced increase first, then decrease, and then increase again. Furthermore, the maximum value of the first seven-order natural frequencies appeared when B was approaching 10 mm, and the minimum value of the first seven-order natural frequencies appeared when B was approaching 12 mm. The trend of variation in the eighth-order natural frequency was contrary to that of the seventh-order natural frequency, whereas the sixth-order natural frequency and the seventh-order natural frequency experienced slight changes.

Figure 7 
                     Influence imposed by the tooth width on the natural frequency of each order.
Figure 7

Influence imposed by the tooth width on the natural frequency of each order.

4.3.5 Influence imposed by the tooth root transition fillet on the modal characteristics

The discrete value of the tooth root transition fillet R amounted to 0, 2, 4, 6, and 8 mm. The relationship between the values of R and the natural frequency of each order is illustrated in Figure 8. Judging from Figure 8, the trends of variation in the natural frequencies of each order were similar, all of which experienced slow increase first, then slight decrease, and then slight increase again. In addition, the natural frequencies of each order experienced slight fluctuations, reaching the peak value when the value of R was approaching 2 mm.

Figure 8 
                     Influence imposed by the tooth root transition fillet on the natural frequency of each order.
Figure 8

Influence imposed by the tooth root transition fillet on the natural frequency of each order.

Based on the trends of how the aforementioned five parameters imposed an impact on the natural frequency of each order of flexspline, we may conclude that the mechanisms of how the values of L and R could affect the natural frequency of each order were similar, but the range of variation in the natural frequency of each order was subject to significant impact of the value of L, and slightly influenced by the value of R. In addition, the influencing mechanisms of the values of δ and B on the natural frequency of each order were also similar. For instance, the trends of variation in the first seven-order natural frequencies were similar, whereas the trends of variation in the eighth-order natural frequency were contrary to those of the first seven-order natural frequency. In addition, the influence imposed by the value of δ 1 on the natural frequency of each order showed greater variations.

5 Correlation analysis

The value of each structural parameter could impose an impact on the natural frequency of flexspline. To study the influence imposed by each structural parameter on the natural frequency of flexspline, the authors have conducted the correlation analysis, which is oftentimes related to the methods of range analysis and variance analysis. To facilitate the calculation, the effects imposed by the interaction between structural parameters have been excluded. The authors have calculated the mean value of the natural frequency of each order at varying levels of structural parameters by using the data of Table 2. The ranges of the first eight-order natural frequencies at varying levels of structural parameters are specified in Table 3.

Table 3

Ranges of the first eight-order natural frequencies at varying levels of structural parameters

Parameter First order range Second order range Third order range Fourth order range Fifth order range Sixth order range Seventh order range Eighth order range
L 469 469 419 427 593 691 612 726
δ 160 162 282 257 332 136 329 172
δ 1 198 197 138 106 417 319 122 219
B 148 149 220 203 353 175 226 88
R 84 85 95 86 158 111 135 200

The curve of ranges illustrated based on Table 3 is shown in Figure 9. Judging from Figure 9, at the natural frequencies of all orders, the range of the value of L was the maximum, whereas the range of the value of R was the minimum, and the range of the values of δ1, δ, and B intertwined with each other. The range in the order from large to small was L, δ1, δ, B, and R in the scope of the first order and the second order; L, δ, B, δ1, and R in the scope of the third order and the fourth order; and L, δ1, B, δ, and R in the scope of the fifth order and sixth order.

Figure 9 
               Curve of ranges.
Figure 9

Curve of ranges.

The variance analysis was carried out by using the data listed in Table 2, and the obtained value of F was specified as shown in Table 4. The curve of F illustrated according to Table 4 is shown in Figure 10. The value of F in the order from large to small was L, δ1, δ, B, and R in the scope of the first order, the second order, and the fifth order; L, δ, δ1, B, and R in the scope of the third order and the fourth order; the value of F was the maximum in the scope of the sixth order, the seventh order, and the eighth order, and showed no evident trend in the scope of the remaining parameters. For the natural frequencies of all orders, in case α = 0.10 was set, we could obtain F α (4, 4) = 4.11 by looking up the table, and it could be noted that the F value of L obtained through variance analysis exceeded 4.11. Therefore, the value of L could impose a significant impact on the natural frequency of flexspline. The F value of the other parameters was less than 4.11, which could impose no significant impact on the natural frequency of flexspline. The F value of each parameter could reveal the influencing degree on the natural frequency of flexspline.

Table 4

F value obtained through variance analysis

F value First order Second order Third order Fourth order Fifth order Sixth order Seventh order Eighth order
L 6.12 6.13 4.74 4.86 9.99 12.2 9.70 12.4
δ 0.66 0.67 1.80 1.46 2.90 0.47 2.28 0.75
δ 1 1.13 1.12 0.56 0.36 4.78 3.23 0.38 1.09
B 0.61 0.62 1.10 0.93 2.89 0.71 1.20 0.19
R 0.26 0.27 0.33 0.32 0.84 0.37 0.62 0.99
Figure 10 
               Curve of F value through variance analysis.
Figure 10

Curve of F value through variance analysis.

6 Conclusion

By integrating the method of orthogonal design and the method of finite element, the mechanism of how varying structural parameters could impose an impact on the natural frequencies of each order of flexspline, was analyzed. The authors performed the correlation analysis and came to the following conclusions:

  1. The modal characteristics of the flexspline show regular vibration deformation in the radial direction and are evenly distributed in the circumferential direction, mostly in the sixth natural frequency. In the axial direction, the vibration deformation shows the inclination of the cup bottom plane, which mostly occurs above the sixth natural frequency. The damage of the flexspline mostly occurs at the maximum deformation in the radial direction of the ring gear and the axial direction of the cup body.

  2. The value of L and the natural frequency of all orders experienced monotonical declines, whereas the value of L could impose a significant impact on the natural frequency of all orders.

  3. The values of δ1, δ, B, and R could impose no significant impact on the natural frequency of all orders, whereas the trends of variation revealed a certain degree of randomness.

  4. The first-order natural frequency of flexspline was rather low. During the design of varying parameters, we need to calculate the first-order natural frequency so as to prevent the natural frequency from approaching the working frequency as much as possible and to avoid resonance.

  1. Funding information: This research project was funded by the Research on Anti-fatigue Design and Manufacturing of Harmonic Reducer Based on Flexible Deformation of the High-level Talent Scientific Research Start-up Project of North China University of Water Resources and Electric Power (Project No.: 201909008).

  2. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

  3. Conflict of interest: The authors state no conflict of interest.

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Received: 2021-06-17
Revised: 2021-10-20
Accepted: 2021-11-20
Published Online: 2022-01-03

© 2021 Leyu Wei et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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