The free vibrational characteristics of die springs are examined by Riccati transfer matrix method in this study. The warping deformation of spring’s cross section, as a new design factor, is incorporated into the differential equation of motion. Numerical simulations show that the warping deformation is a significant role of the behavior of natural frequencies of die springs and should be considered carefully. Approximately 40% of the errors may occur if warping is neglected. The change laws of warping effect with the parameter variations of springs are also explored, including the height-to-width ratio of the cross section, the cylinder diameter, the helix pitch angle, and the number of coils. The warping effect exhibits the most remarkable changes with the variation in the height-to-width ratio of the cross section. However, this effect is not fairly sensitive to the changes in other parameters, and it is particularly significant when the cross section is relatively narrow regardless of the changes in other parameters. This study evidently answers the key scientific question: “under what working condition should the warping effect be considered or ignored?” The analysis results can be used to guide spring designers in engineering.
Die spring usually refers to a cylindrical helical metal spring with rectangular cross section (see Figure
The studies on the free vibration of cylindrical helical springs have been carried by many scholars [
Based on the naturally curved and twisted beam theory [
The analysis results in this paper show that if the cross section of a spring wire is relatively long and narrow, it will induce 43.06% error to the fundamental frequency with the warping being ignored. On the contrary, the error can be reduced to 0.37% if the warping effect was considered. Numerical results show that the warping effect exhibits the most remarkable changes with the variation of the height-to-width ratio of the cross section. Therefore, the suggestions can be given as follows: when the aspect ratio of the rectangular cross section is
Consider a naturally curved and twisted beam in space in Figure
Geometry of a naturally curved and twisted beam.
Simultaneously solving (
Assuming that the cross section has infinite rigidity in its own plane but is free to warp out of plane and the deformation of the beam is caused by the extension, bending and torsion can be expressed as [
Let
For the case of isotropic beam under consideration, the stress components can be expressed as follows:
The resultant forces
The external force and moments per unit length of the beam axis can be expressed as
The geometrical properties of the die springs are given in Figure
Geometry of a typical cylindrical helical spring.
Cylindrical helical spring with rectangular cross section.
The differential equations of die spring can be simplified as follows by simultaneously solving (
Equations (
The equations become a “stiff-conditioning” system of equations because of the increase by two degrees of freedom containing the generalized warping coordinate and the generalized warping moment. Therefore, using the method of [
For any matrix
After getting the element transfer matrix by Scaling-Squaring method and Padé approximation formula, the state vector of the motion differential equations (
Then the unit transfer relationship can be rewritten as [
Substituting Riccati transformation
Several numerical results are given in this section.
Firstly, as shown in Table
Viability and effectiveness of the proposed model.
Mode number (Hz) | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|
FEM [ |
314.1 | 337.2 | 368.2 | 376.0 | 609.0 |
Reference [ |
313.2 | 336.8 | 366.7 | 375.5 | 607.9 |
Present study | 314.0 | 338.3 | 368.9 | 376.8 | 609.2 |
Springs with rectangular cross section and clamped-clamped ends are shown in Examples
Utilizing ANSYS program in this analysis, the die spring is partitioned into 720 solid elements (the type is solid 45 in ANSYS); the total number of node is 1452. Let
First five mode shapes of a die spring: (1) first mode (523.25 Hz); (2) second mode (897.71 Hz); (3) third mode (914.36 Hz); (4) fourth mode (973.88 Hz); (5) fifth mode (1118.41 Hz).
First five normal mode shapes of the generalized warping coordinate of a die spring: (1) first mode (523.25 Hz); (2) second mode (897.71 Hz); (3) third mode (914.36 Hz); (4) fourth mode (973.88 Hz); (5) fifth mode (1118.41 Hz).
Table
Comparison of the first five frequencies of the spring.
Mode number (Hz) | FEM | Present study (PS) | |||
---|---|---|---|---|---|
Warping ignored | Errors (%) | Warping included | Errors (%) | ||
1 | 521.34 | 745.84 | 43.06 | 523.25 | 0.37 |
2 | 899.54 | 1009.30 | 12.20 | 897.71 | 0.20 |
3 | 915.97 | 1044.17 | 13.99 | 914.36 | 0.17 |
4 | 970.21 | 1122.22 | 15.67 | 973.88 | 0.38 |
5 | 1129.5 | 1338.65 | 18.52 | 1118.41 | 0.98 |
Utilizing ANSYS code in the analysis, the die springs of this example are partitioned into 720 solid 45 elements; the total number of node is 1452. Let
From Table The natural frequencies of the spring increase with increase of the cross-sectional areas. If spring’s cross-sectional areas keep fixed, the different arrangements of the cross section will have significant effect on the natural frequencies. When the cross section is narrower, the differences between the natural frequencies are greater. It is suggested to consider the warping deformation in die spring’s dynamic analysis when the aspect ratio of the rectangular cross section is
Change law of the warping effect on natural frequencies with respect to the aspect ratio (
Mode number (Hz) | 1 | 2 | 3 | 4 | 5 | |
---|---|---|---|---|---|---|
|
FEM | 521.34 | 899.54 | 915.97 | 970.21 | 1129.5 |
Warping ignored | 745.84 | 1009.3 | 1044.1 | 1122.2 | 1338.6 | |
Error (%) | 43.06 | 12.20 | 13.99 | 15.67 | 18.52 | |
Warping included | 523.25 | 897.71 | 914.36 | 973.88 | 1118.4 | |
Error (%) | 0.37 | 0.20 | 0.17 | 0.38 | 0.98 | |
|
||||||
|
FEM | 718.70 | 1136.7 | 1170.0 | 1209.3 | 1339.7 |
Warping ignored | 817.31 | 1128.6 | 1207.1 | 1251.5 | 1511.9 | |
Error (%) | 13.72 | 0.71 | 3.16 | 3.49 | 12.85 | |
Warping included | 725.9 | 1126.5 | 1165.3 | 1203.8 | 1351.2 | |
Error (%) | 1.01 | 0.89 | 0.41 | 0.45 | 0.86 | |
|
||||||
|
FEM | 888.27 | 1135.3 | 1318.4 | 1369.8 | 1654.8 |
Warping ignored | 901.25 | 1133.4 | 1316.8 | 1366.6 | 1677.5 | |
Error (%) | 1.46 | 0.16 | 0.12 | 0.23 | 1.40 | |
Warping included | 881.50 | 1132.6 | 1309.2 | 1358.4 | 1641.9 | |
Error (%) | 0.76 | 0.23 | 0.70 | 0.82 | 0.77 | |
|
||||||
|
FEM | 1006.7 | 1142.0 | 1396.9 | 1454.8 | 1863.0 |
Warping ignored | 996.82 | 1136.7 | 1382.5 | 1437.9 | 1845.2 | |
Error (%) | 0.98 | 0.46 | 1.03 | 1.16 | 0.95 | |
Warping included | 996.66 | 1136.4 | 1382.2 | 1437.8 | 1845.0 | |
Error (%) | 0.99 | 0.49 | 1.05 | 1.17 | 0.96 | |
|
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|
FEM | 886.69 | 913.84 | 1152.0 | 1200.5 | 1618.1 |
Warping ignored | 902.91 | 913.55 | 1149.1 | 1196.3 | 1633.7 | |
Error (%) | 1.83 | 0.03 | 0.25 | 0.35 | 0.96 | |
Warping included | 882.74 | 912.38 | 1146.2 | 1192.5 | 1610.8 | |
Error (%) | 0.45 | 0.16 | 0.47 | 0.67 | 0.45 | |
|
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|
FEM | 688.94 | 708.39 | 885.52 | 924.77 | 1279.0 |
Warping ignored | 685.18 | 819.72 | 891.44 | 934.60 | 1291.5 | |
Error (%) | 0.55 | 15.72 | 0.67 | 1.06 | 0.98 | |
Warping included | 684.82 | 729.04 | 881.56 | 919.73 | 1279.3 | |
Error (%) | 0.59 | 2.92 | 0.45 | 0.55 | 0.02 | |
|
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|
FEM | 458.34 | 528.46 | 599.48 | 627.81 | 864.93 |
Warping ignored | 485.90 | 609.77 | 627.59 | 772.46 | 869.75 | |
Error (%) | 6.02 | 15.86 | 4.69 | 23.04 | 0.56 | |
Warping included | 457.66 | 527.83 | 598.45 | 625.73 | 862.19 | |
Error (%) | 0.15 | 0.11 | 0.17 | 0.33 | 0.32 |
Consider
Consider
Consider
Tables
Change law of the warping effect on natural frequencies with respect to the diameter of the cylinder.
Mode number (Hz) | 1 | 2 | 3 | 4 | 5 | |
---|---|---|---|---|---|---|
|
FEM | 836.61 | 1419.6 | 1446.5 | 1553.7 | 1760.4 |
Warping ignored | 1162.25 | 1574.61 | 1629.74 | 1752.98 | 2089.33 | |
Error (%) | 38.92 | 10.92 | 12.67 | 12.83 | 18.68 | |
Warping included | 818.03 | 1402.37 | 1428.08 | 1522.45 | 1746.51 | |
Error (%) | 2.22 | 1.21 | 1.27 | 2.01 | 0.79 | |
|
||||||
|
FEM | 366.39 | 625.73 | 638.26 | 680.81 | 778.55 |
Warping ignored | 517.72 | 701.37 | 726.15 | 780.18 | 930.54 | |
Error (%) | 41.30 | 12.09 | 13.77 | 14.60 | 19.51 | |
Warping included | 363.61 | 623.34 | 635.98 | 676.02 | 777.59 | |
Error (%) | 0.76 | 0.38 | 0.36 | 0.70 | 0.12 | |
|
||||||
|
FEM | 205.48 | 351.70 | 358.68 | 382.02 | 438.26 |
Warping ignored | 291.77 | 394.48 | 408.34 | 439.17 | 523.85 | |
Error (%) | 41.99 | 12.16 | 13.85 | 14.96 | 21.58 | |
Warping included | 204.59 | 350.92 | 357.63 | 380.31 | 437.40 | |
Error (%) | 0.43 | 0.22 | 0.29 | 0.45 | 0.20 | |
|
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|
FEM | 131.12 | 224.78 | 229.20 | 243.94 | 280.38 |
Warping ignored | 186.48 | 252.72 | 261.06 | 281.29 | 335.25 | |
Error (%) | 42.22 | 12.43 | 13.90 | 15.31 | 19.57 | |
Warping included | 130.43 | 224.46 | 228.59 | 243.07 | 280.32 | |
Error (%) | 0.53 | 0.14 | 0.27 | 0.36 | 0.02 |
Change law of the warping effect on natural frequencies with respect to the number of coils.
Mode number (Hz) | 1 | 2 | 3 | 4 | 5 | |
---|---|---|---|---|---|---|
|
FEM | 984.59 | 1400.9 | 1738.5 | 2168.0 | 2259.2 |
Warping ignored | 1332.77 | 1713.92 | 1848.66 | 2286.35 | 2336.40 | |
Error (%) | 35.36 | 22.34 | 6.34 | 5.46 | 3.42 | |
Warping included | 973.17 | 1406.73 | 1746.22 | 2263.04 | 2271.58 | |
Error (%) | 1.16 | 0.42 | 0.44 | 4.38 | 0.55 | |
|
||||||
|
FEM | 521.34 | 899.54 | 915.97 | 970.21 | 1129.5 |
Warping ignored | 745.84 | 1009.30 | 1044.17 | 1122.22 | 1338.65 | |
Error (%) | 43.06 | 12.20 | 13.99 | 15.67 | 18.52 | |
Warping included | 523.25 | 897.71 | 914.36 | 973.88 | 1118.41 | |
Error (%) | 0.37 | 0.20 | 0.17 | 0.38 | 0.98 | |
|
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|
FEM | 359.89 | 521.12 | 529.55 | 697.79 | 771.62 |
Warping ignored | 504.25 | 574.71 | 588.36 | 765.78 | 963.49 | |
Error (%) | 40.11 | 10.28 | 11.10 | 9.76 | 24.87 | |
Warping included | 353.34 | 515.50 | 523.97 | 685.21 | 764.24 | |
Error (%) | 1.82 | 1.08 | 1.05 | 1.80 | 0.96 | |
|
||||||
|
FEM | 270.56 | 330.04 | 335.05 | 533.20 | 582.48 |
Warping ignored | 359.73 | 364.84 | 393.39 | 577.06 | 731.52 | |
Error (%) | 32.96 | 10.54 | 17.41 | 8.23 | 25.59 | |
Warping included | 265.39 | 326.26 | 331.55 | 523.17 | 764.24 | |
Error (%) | 1.91 | 1.14 | 1.04 | 1.88 | 0.86 |
Change law of the warping effect on natural frequencies with respect to the helix pitch angle.
Mode number (Hz) | 1 | 2 | 3 | 4 | 5 | |
---|---|---|---|---|---|---|
|
FEM | 533.77 | 966.74 | 980.62 | 993.31 | 1120.5 |
Warping ignored | 748.56 | 1082.97 | 1099.23 | 1137.02 | 1345.44 | |
Error (%) | 40.24 | 12.02 | 12.09 | 14.47 | 20.07 | |
Warping included | 525.88 | 957.13 | 971.50 | 978.46 | 1112.27 | |
Error (%) | 1.50 | 0.99 | 0.93 | 1.49 | 0. 73 | |
|
||||||
|
FEM | 528.43 | 840.48 | 862.36 | 981.13 | 1125.3 |
Warping ignored | 740.62 | 936.78 | 973.11 | 1122.27 | 1330.55 | |
Error (%) | 40.15 | 11.46 | 12.84 | 14.39 | 18.24 | |
Warping included | 521.71 | 834.46 | 855.53 | 970.49 | 1119.82 | |
Error (%) | 1.27 | 0.72 | 0.79 | 1.08 | 0.49 | |
|
||||||
|
FEM | 520.35 | 722.19 | 751.51 | 966.25 | 1118.0 |
Warping ignored | 722.42 | 809.66 | 856.58 | 1119.07 | 1307.31 | |
Error (%) | 38.83 | 12.11 | 13.95 | 15.82 | 16.93 | |
Warping included | 515.27 | 718.51 | 747.84 | 957.19 | 1115.40 | |
Error (%) | 0.97 | 0.51 | 0.51 | 0.94 | 0.23 | |
|
||||||
|
FEM | 509.13 | 625.76 | 663.87 | 949.78 | 1106.1 |
Warping ignored | 676.70 | 703.33 | 787.78 | 1113.24 | 1273.52 | |
Error (%) | 32.91 | 12.40 | 18.66 | 17.21 | 15.14 | |
Warping included | 504.25 | 622.41 | 659.50 | 941.67 | 1102.38 | |
Error (%) | 0.96 | 0.54 | 0.66 | 0.85 | 0.34 |
This study presents the change laws of warping effect on natural frequencies with respect to spring parameters. The warping effect exhibits the most remarkable changes with the variation of the height-to-width ratio of the cross section. However, this effect is not fairly sensitive to the changes in other parameters, and it is particularly significant when the cross section is relatively narrow regardless of changes in other parameters. In this case, approximately 40% of errors may occur if warping is neglected.
The authors declare that they have no competing interests.
This work is partially supported by National Natural Science Foundation of China (11402090 and 51209094) and the Science and Technology Planning Project of Zhengzhou City of China (no. 153PKJGG111).