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Frontiers of Mechanical Engineering

ISSN 2095-0233

ISSN 2095-0241(Online)

CN 11-5984/TH

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Front. Mech. Eng.    2020, Vol. 15 Issue (1) : 133-150    https://doi.org/10.1007/s11465-019-0557-7
RESEARCH ARTICLE
Computation and investigation of mode characteristics in nonlinear system with tuned/mistuned contact interface
Houxin SHE1, Chaofeng LI1,2(), Qiansheng TANG1, Hui MA1,2, Bangchun WEN1,2
1. School of Mechanical Engineering and Automation, Northeastern University, Shenyang 110819, China
2. Key Laboratory of Vibration and Control of Aero-Propulsion System (Ministry of Education), Northeastern University, Shenyang 110819, China
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Abstract

This study derived a novel computation algorithm for a mechanical system with multiple friction contact interfaces that is well-suited to the investigation of nonlinear mode characteristic of a coupling system. The procedure uses the incremental harmonic balance method to obtain the nonlinear parameters of the contact interface. Thereafter, the computed nonlinear parameters are applied to rebuild the matrices of the coupling system, which can be easily solved to calculate the nonlinear mode characteristics by standard iterative solvers. Lastly, the derived method is applied to a cycle symmetry system, which represents a shaft–disk–blade system subjected to dry friction. Moreover, this study analyzed the effects of the tuned and mistuned contact features on the nonlinear mode characteristics. Numerical results prove that the proposed method is particularly suitable for the study of nonlinear characteristics in such nonlinear systems.
Keywords coupling vibration      nonlinear mode      original algorithm      contact interface     
Corresponding Author(s): Chaofeng LI   
Just Accepted Date: 19 November 2019   Online First Date: 19 December 2019    Issue Date: 21 February 2020
 Cite this article:   
Houxin SHE,Chaofeng LI,Qiansheng TANG, et al. Computation and investigation of mode characteristics in nonlinear system with tuned/mistuned contact interface[J]. Front. Mech. Eng., 2020, 15(1): 133-150.
 URL:  
https://academic.hep.com.cn/fme/EN/10.1007/s11465-019-0557-7
https://academic.hep.com.cn/fme/EN/Y2020/V15/I1/133
Fig.1  Diagram of the lumped SDB model.
Fig.2  Structure diagram of the dry friction.
Fig.3  Algorithm of the nonlinear analysis.
Fig.4  (a) Amplitude–frequency response of the blade root, (b) nonlinear damping, and (c) nonlinear stiffness of an SDB system.
Fig.5  Amplitude–frequency response curves for different excitation levels when W= 200 rad/s: (a) Blade body; (b) blade root.
Fig.6  Nonlinear parameters of the contact interface in Sector 1 for W= 200 rad/s, F =500 N: (a) Nonlinear stiffness; (b) nonlinear damping.
Fig.7  Variations of the natural frequencies for W= 200 rad/s and F = 500 N: (a) Blade body; (b) blade root.
Sector Natural frequency (ω = 200 Hz)/Hz Natural frequency (ω = 250 Hz)/Hz
Blade Body Blade Root Shaft Disk Blade Body Blade Root Shaft Disk
1 253.2769 1836.3942 153.5851 2313.1122 229.2292 977.9689 152.6156 2277.6773
2 253.2769 1836.3942 – 2313.1122 229.2292 977.9689 – 2277.6773
3 253.3133 1855.3368 – 2875.0239 229.2587 978.5021 – 2858.6677
4 253.3133 1855.3368 – 2875.0239 229.2587 978.5021 – 2858.6677
5 253.3342 1860.8601 – 3510.4298 229.2757 978.7749 – 3499.9044
6 253.3342 1860.8601 – 3510.4298 229.2757 978.7749 – 3499.9044
7 253.3446 1862.9076 – 4048.8739 229.2841 978.9029 – 4040.6643
8 253.3446 1862.9076 – 4048.8739 229.2841 978.9029 – 4040.6643
9 253.3494 1863.7412 – 4125.2422 229.2880 978.9605 – 4123.3330
10 253.3494 1863.7412 – 4401.7604 229.2880 978.9605 – 4394.5509
11 253.3508 1863.9731 – 4401.7604 229.2892 978.9772 – 4394.5509
12 262.5346 1878.8707 – 4524.0797 239.6277 983.6187 – 4517.1575
Tab.1  Natural frequency of the tuned coupling system for ω = 200 Hz and ω = 250 Hz
Fig.8  Mode shapes of the tuned coupling system for ω = 200 Hz. (a) Sector 1; (b) Sector 2; (c) Sector 3; (d) Sector 4; (e) Sector 5; (f) Sector 6; (g) Sector 7; (h) Sector 8; (i) Sector 9; (j) Sector 10; (k) Sector 11; (l) Sector 12.
Fig.9  Mode shapes of the tuned coupling system for ω= 250 Hz. (a) Sector 1; (b) Sector 2; (c) Sector 3; (d) Sector 4; (e) Sector 5; (f) Sector 6; (g) Sector 7; (h) Sector 8; (i) Sector 9; (j) Sector 10; (k) Sector 11; (l) Sector 12.
Sector Random mistuning friction coefficient
1 0.311
2 0.326
3 0.310
4 0.311
5 0.297
6 0.289
7 0.271
8 0.306
9 0.280
10 0.261
11 0.318
12 0.299
Tab.2  Values of the random mistuning friction coefficient μ in the different contact interfaces
Fig.10  Amplitude–frequency response curves for F = 500 N and W= 200 rad/s under mistuned friction coefficient case: (a) Blade body; (b) blade root.
Fig.11  Nonlinear parameters of the contact interfaces in the different sectors for W= 200 rad/s and F = 500 N under mistuned friction coefficient case: (a) Nonlinear stiffness; (b) nonlinear damping.
Fig.12  Variations of the natural frequencies for W= 200 rad/s and F = 500 N under mistuned friction coefficient case: (a) Blade body, (b) blade root.
Sector Natural frequency (ω = 200 Hz)/Hz Natural frequency (ω = 250 Hz)/Hz
Blade body Blade root Shaft Disk Blade body Blade root Shaft Disk
1 253.2769 1836.3942 153.5851 2313.1122 224.9295 925.3268 152.5889 2277.5969
2 253.2769 1836.3942 – 2313.1122 225.6163 930.8274 – 2277.6629
3 253.3133 1855.3368 – 2875.0239 227.1697 950.9701 – 2858.5518
4 253.3133 1855.3368 – 2875.0239 227.7164 955.5698 – 2858.7146
5 253.3342 1860.8601 – 3510.4298 228.5753 966.5974 – 3499.8755
6 253.3342 1860.8601 – 3510.4298 229.5426 982.2695 – 349.9880
7 253.3446 1862.9076 – 4048.8739 229.7237 983.1077 – 4040.5895
8 253.3446 1862.9076 – 4048.8739 229.9858 987.5018 – 4040.6938
9 253.3494 1863.7412 – 4125.2422 230.4309 989.6337 – 4123.3277
10 253.3494 1863.7412 – 4401.7604 231.1232 997.7133 – 4394.5169
11 253.3508 1863.9731 – 4401.7604 231.6311 1008.8343 – 4394.5436
12 262.5346 1878.8707 – 4524.0797 239.5725 1014.4355 – 4517.1376
Tab.3  Natural frequency of the coupling system for ω = 200 Hz and ω = 250 Hz under mistuned friction coefficient case
Fig.13  Mode shapes of the SDB system with mistuned friction coefficient for ω = 200 Hz. (a) Sector 1; (b) Sector 2; (c) Sector 3; (d) Sector 4; (e) Sector 5; (f) Sector 6; (g) Sector 7; (h) Sector 8; (i) Sector 9; (j) Sector 10; (k) Sector 11; (l) Sector 12.
Fig.14  Mode shapes of the SDB system with mistuned friction coefficient for ω = 250 Hz. (a) Sector 1; (b) Sector 2; (c) Sector 3; (d) Sector 4; (e) Sector 5; (f) Sector 6; (g) Sector 7; (h) Sector 8; (i) Sector 9; (j) Sector 10; (k) Sector 11; (l) Sector 12.
Sector Random mistuning contact stiffness coefficient/(106 m?s–1)
1 8.206
2 7.713
3 7.986
4 8.045
5 7.956
6 7.183
7 8.240
8 8.155
9 8.194
10 7.802
11 8.307
12 7.849
Tab.4  Values of the random mistuning contact stiffness coefficients kt in the different contact interfaces
Fig.15  Amplitude–frequency response curves for F = 500 N and W= 200 rad/s under mistuned contact stiffness case: (a) Blade body; (b) blade root.
Fig.16  Nonlinear parameters of the contact interfaces in different sectors for W= 200 rad/s and F = 500 N under mistuned contact stiffness case: (a) Nonlinear stiffness; (b) nonlinear damping.
Fig.17  Variations of the natural frequencies for W= 200 rad/s and F = 500 N under mistuned contact stiffness case: (a) Blade body; (b) blade root.
Sector ω = 200 Hz ω = 250 Hz
Blade Body Blade Root Shaft Disk Blade Body Blade Root Shaft Disk
1 252.5143 1767’6760 153.5837 231.2356 228.9848 974.8533 152.6094 2277.6532
2 253.0586 1823.3982 — 2313.5381 229.0257 975.3624 — 2277.6628
3 253.1508 1833.8829 — 2874.6105 229.0434 975.6548 — 2858.6492
4 253.2028 1839.6541 — 2875.2533 229.0502 975.7721 — 2858.6591
5 253.3003 1852.8047 — 3509.9887 229.0792 976.1677 — 3499.8861
6 253.3580 1855.4957 — 3510.7562 229.0841 976.2503 — 3499.9020
7 253.3931 1865.3426 — 4048.6993 229.0875 976.2760 — 4040.6524
8 253.4973 1872.5862 — 4048.9612 229.2182 977.6082 — 4040.6588
9 253.5149 1880.6315 — 4125.2322 229.2674 978.6491 — 4123.3310
10 253.5453 1882.9722 — 4401.6487 229.2935 978.9403 — 4394.5410
11 253.5975 1887.2574 — 4401.7967 229.4777 980.5990 — 4394.5449
12 262.5051 1896.7463 — 4524.0438 239.5138 983.3419 — 4517.1499
Tab.5  Natural frequency (Hz) of the coupling system for ω = 200 Hz and ω = 250 Hz under mistuned contact stiffness case
Fig.18  Mode shapes of the SDB system with mistuned contact stiffness for ω = 200 Hz. (a) Sector 1; (b) Sector 2; (c) Sector 3; (d) Sector 4; (e) Sector 5; (f) Sector 6; (g) Sector 7; (h) Sector 8; (i) Sector 9; (j) Sector 10; (k) Sector 11; (l) Sector 12.
Fig.19  Mode shapes of the SDB system with mistuned contact stiffness for ω = 250 Hz. (a) Sector 1; (b) Sector 2; (c) Sector 3; (d) Sector 4; (e) Sector 5; (f) Sector 6; (g) Sector 7; (h) Sector 8; (i) Sector 9; (j) Sector 10; (k) Sector 11; (l) Sector 12.
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