Skip to main content
Have a personal or library account? Click to login
ANALYSIS OF TRANSVERSE NATURAL VIBRATION OF AVIATION TRANSMISSIONS TOOTHED WHEELS WITH USING CYCLIC SYMMETRY PROPERTIES Cover

ANALYSIS OF TRANSVERSE NATURAL VIBRATION OF AVIATION TRANSMISSIONS TOOTHED WHEELS WITH USING CYCLIC SYMMETRY PROPERTIES

Acta Mechanica et Automatica
Volume 19 (2025): Issue 4 (December 2025)
By:  and    
Open Access
|Dec 2025
Figures & tables

Figures & Tables

Fig. 1.

Model with the nK segment with cyclic symmetry features

Fig. 2.

Geometric models of the gears under investigation: a) GR1_1, b) GR2_1, c) GR3_1, d) GR4_1, e) GR5_1

Tab. 1.

The toothed gears in question

No. of wheelTooth count zGear module m0Denotation in the articleDescription
1552.12GR1_1aviation toothed wheel (for aggregates, see Fig. 2a)
2313GR2_1aviation toothed wheel (for power transmission, see Fig. 2b)
3306GR3_1spiral bevel gear (for power transmission, see Fig. 2c)
4832.5GR4_1aviation toothed wheel (for aggregates, see Fig. 2d)
5412.5GR5_1aviation toothed wheel (for power transmission, see Fig. 2e)
Tab. 2.

Description of the cyclic symmetry models of the discussed gears

Denotation in the articleDescriptionDenotation in the articleDescription
GR1_2cyclic symmetry segment model of the GR1_1 wheel, covering one tooth, cut along the centers of the inter – tooth slots (α = 6.55˚)GR2_2cyclic symmetry segment model of the GR2_1 wheel, covering one tooth, cut along the centers of the inter – tooth slots (α = 11.61˚))
GR1_3cyclic symmetry segment model of the GR1_1 wheel without teeth (full ring), covering a 1/50th – circle segment (α = 7.2˚)GR2_3cyclic symmetry segment model of the GR2_1 wheel without teeth (full ring), covering a 1/50th – circle segment (α = 7.2˚)
GR1_4cyclic symmetry segment model of the GR1_1 wheel without teeth (full ring), covering a 1/16th – circle segment (α = 22.5˚), see Fig. 3a)GR2_4cyclic symmetry segment model of the GR2_1 wheel without teeth (full ring), covering a 1/16th – circle segment (α = 22.5˚), see Fig. 3a)
GR1_5cyclic symmetry segment model of the GR1_1 wheel without teeth (full ring), covering a 1/10th – circle segment (α = 36˚)GR2_5cyclic symmetry segment model of the GR2_1 wheel without teeth (full ring), covering a 1/10th – circle segment (α = 36˚)
Fig. 3.

Examples of simplified geometric models of the discussed gears with full ring: a) GR1_4, b) GR2_4, c) GR3_4, d) GR4_3, e) GR4_4, f) GR5_4

Tab. 3.

Description of the cyclic symmetry models of the discussed gears

Denotation in the articleDescriptionDenotation in the articleDescription
GR3_2cyclic symmetry segment model of the GR3_1 wheel, covering one tooth, cut along the centers of the inter – tooth slots (α = 12˚)GR4_3cyclic symmetry segment model of the GR4_1 wheel, covering the entire eccentric hole of the wheel disc (α = 60˚, see Fig. 3d)
GR3_3cyclic symmetry segment model of the GR3_1 wheel without teeth (full ring), covering a 1/50th – circle segment (α = 7.2˚)GR4_4cyclic symmetry segment model of the GR4_1 wheel where the boundary areas of the segment are led through the centers of symmetry of the eccentric holes of the wheel (α = 60˚, see Fig. 3e)
GR3_4cyclic symmetry segment model of the GR3_1 wheel without teeth (full ring), covering a 1/16th – circle segment (α = 22.5˚, see Fig. 3a)GR5_2cyclic symmetry segment model of the GR5_1 wheel, covering one tooth, cut along the centers of the inter – tooth slots (α = 8.78˚)
GR3_5cyclic symmetry segment model of the GR3_1 wheel without teeth (full ring), covering a 1/10th – circle segment (α = 36˚)GR5_3cyclic symmetry segment model of the GR5_1 wheel without teeth (full ring), covering a 1/50th – circle segment (α = 7.2˚)
GR5_4cyclic symmetry segment model of the GR5_1 wheel without teeth (full ring), covering a 1/16th – circle segment (α = 22.5˚, see Fig. 3a)GR5_5cyclic symmetry segment model of the GR5_1 wheel without teeth (full ring), covering a 1/10th – circle segment (α = 36˚)
Fig. 4.

FEM models (reference FEM models) of the gears under investigation: a) GR1_1_m, b) GR2_1_m, c) GR3_1_m, d) GR4_1_m, e) GR5_1_m

Fig. 5.

Examples of simplified FE models of the discussed gears with full ring: a) GR1_4_m, b) GR2_4_m, c) GR3_4_m, d) GR4_3_m, e) GR4_4_m, f) GR5_4_m

Tab. 4.

Description of the FE models of the discussed gears

ModelType of elementNo. of nodesNo. of elements
GR1_1_msolid1856724848690
solid18625201848690
solid187302701175996
GR1_2_msolid185986474
solid186106841740
solid18779373912
GR1_3_msolid18544152756
solid186155922756
solid1872634815784
GR1_4_msolid185125449295
solid186460179295
solid1876171539127
GR1_5_msolid1851986711476
solid1865772611476
solid1873524619867
GR2_1_msolid185301199274231
solid1861159443269157
solid18718647651255089
GR2_2_msolid1872481413141
GR2_3_msolid185100406988
solid186368236984
solid1871976710445
GR2_4_msolid1852795822633
solid18610579722529
solid1875049029057
GR2_5_msolid1854204234980
solid18615908834500
solid1877591344480
Tab. 5.

Description of the FE models of the discussed gears

ModelType of elementNo. of nodesNo. of elements
GR3_1_msolid185589498517008
solid1862282912517008
solid18719828571311702
GR3_2_msolid1878852257266
GR3_3_msolid1871398111516
solid1865331711516
solid1871961710780
GR3_4_msolid1854087135796
solid18615822235796
solid1875401030838
GR3_5_msolid1856437957024
solid18624993557024
solid1878381748093
GR4_1_msolid185531889459614
solid1862050232459210
solid18723449621551003
GR4_3_msolid1859120178860
solid18635213178860
solid18715287393255
GR4_4_msolid1859240380593
solid18635951080593
solid18715686596142
GR5_1_msolid185211681194979
solid186827639194979
solid187513791301014
GR5_2_msolid18570024925
solid186268534925
solid1873327520285
GR5_3_msolid18572965215
solid186270225215
solid187151028055
GR5_4_msolid1851947215615
solid1867270515615
solid1873748921520
GR5_5_msolid1853025024816
solid18611407824816
solid1875891934468
Tab. 6.

Technical data related to the discussed models

No. of wheelYoung modulus E [Pa]Density ρ [kg/m3]Poisson’s ratio vEquivalent Young’s modulus Ez [Pa]Equivalent density ρz [kg/m3]
12.0310117.861030.32.91094.0583 103
24.3568 103
33.8928 103
44.0387 103
53.9492 103
Tab. 7.

Calculation statistics data for FE models of the GR_1_1 wheel

ModelWOMURU
solid185solid186solid187solid185solid186solid187solid185solid186solid187
GR1_1_m1112.5715.5110.980.52.22.26
GR1_2_m0.160.580.320.060.860.470.030.380.21
GR1_3_m0.650.881.720.251.392.230.10.91.12
GR1_4_m0.681.391.880.885.326.490.421.822.23
GR1_5_m0.611.050.411.146.922.660.613.211.98
Tab. 8.

Calculation statistics data for FE models of the GR_2_1 wheel

ModelWOMURU
solid185solid186solid187solid185solid186solid187solid185solid186solid187
GR2_1_m11114.3931.440.613.4921.127.8
GR2_2_m--0.03--6.50--5.36
GR2_3_m0.940.150.052.5214.185.223.1111.946.15
GR2_4_m1.130.210.058.6029.6615.573.9514.987.17
GR2_5_m1.060.190.0413.4925.9914.534.7711.807.47
Tab. 9.

Calculation statistics data for FE models of the GR_3_1 wheel

ModelWOMURU
solid185solid186solid187solid185solid186solid187solid185solid186solid187
GR3_1_m11127.5636.9332.396.2223.4622.08
GR3_2_m--0.41--24.17--11.69
GR3_3_m0.170.490.0584.6131.746.112.3312.313.29
GR3_4_m0.171.430.05514.0914.5315.163.2914.464.31
GR3_5_m0.396.670.1829.7729.8332.534.549.998.97
Tab. 10.

Calculation statistics data for FE models of the GR_4_1 and GR5_1 wheels

ModelWOMURU
solid185solid186solid187solid185solid186solid187solid185solid186solid187
GR4_1_m11124.6248.4041.786.9031.9436.65
GR4_3_m1.141.290.0726.1429.2515.468.3225.777.86
GR4_4_m1.411.450.0726.9829.8515.748.4423.897.94
GR5_1_m1118.6124.1613.181.7714.4010.41
GR5_2_m0.440.050.011.439.218.960.994.172.69
GR5_3_m0.790.060.011.479.463.391.224.892.53
GR5_4_m1.100.100.025.0318.549.841.575.733.05
GR5_5_m1.260.150.028.6132.6316.802.018.953.94
Tab. 11.

Values of natural frequencies Wmn[Hz] (reference FE model of the GR1_1 wheel)

n
0123456
m116136371676458485491331318668
26570p>731286151130714970
3124331446817164
415004
Tab. 12.

Values of natural frequencies Wmn[Hz] (reference FE model of the GR2_1 wheel)

n
0123456
m16615.347945630.911975192322646532030
2206102126325065
Tab. 13.

Values of natural frequencies Wmn[Hz] (reference FE model of the GR3_1 wheel)

n
0123456
m1322078014713876.96905.91036613986
2-4901.18386.712587
Tab. 14.

Values of natural frequencies Wmn[Hz] (reference FE model of the GR4_1 wheel)

n
0123456
m161343095526394944779610859
24700494456786250937010267
310510
Tab. 15.

Values of natural frequencies ?mn[Hz] (reference FE model of the GR5_1 wheel)

n
0123456
m15169436349819836164672370231072
22200722655
Tab. 16.

Values of natural frequencies Wmn[Hz] (reference FE model of the GR1_1 wheel)

εmnGR1_1_mGR2_1_mGR3_1_mGR4_1_mGR5_1_m
solid185solid186solid185solid186solid185solid186solid185solid186solid185solid186
ε11-0.790,16-0.610.02-1.03-0.26-1.16-0930.62-0.14
ε10-0.560,12-0.700.02-2.48-0.31-0.51-0.020.57-0.13
ε12-1,190,24-1.330.01-1.43-0.14-0.940.000.74-0.21
ε13-1,110,22-1.740.01-1.14-0.01-1.040.020.73-0.23
ε20-0,850,18-1.600.01---0.500.010.59-0.20
ε21-1,180,21-0.590.02-0.49-0.03-0.560.020.60-0.17
ε14-1,370,20-1.970.01-1.22-0.06-1.030.030.76-0.25
ε22-1,500,24-0.860.02-0.53-0.01-0.670.010.62-0.21
ε23-1,530,22-1.010.02-0.99-0.08-0.670.02--
ε30-0,630,12-----0.530.01--
ε15-1,620,21-1.180.02-1.40-0.11-1.080.020.82-0.27
ε31-2,92-1,24--------
ε24-1,590,19--------
ε25-------0.810.01--
ε40-1,160,20--------
ε32-1,430,23--------
ε16-1,910,21-1.400.02-084-0.08-1.09-0.580.89-0.29
Tab. 17.

Values of the frequency error εmn [%] (comparison results comes from simplified FE models of GR1_1 wheel)

εmnGR1_2_mGR1_3_mGR1_4_mGR1_5_m
solid185solid186solid187solid185solid186solid187solid185solid186solid187solid185solid186solid187
ε11-4.080.160.00-0.310.470.47-0.160.310.31-0.310.470.31
ε10-1.740.250.06-0.56-0.25-0.25-0.56-0.25-0.25-0.62-0.25-0.31
ε12-1.370.960.72-1.79-1.43-1.49-1.73-1.43-1.43-1.85-1.49-1.55
ε13-1.000.940.74-1.66-1.37-1.40-1.59-1.33-1.33-1.68-1.40-1.42
ε20-1.990.560.26-0.020.440.440.060.470.46-0.050.440.37
ε14-1.190.960.75-1.56-1.24-1.24-1.49-1.19-1.19-1.59-1.24-1.25
ε22-3.530.170.07-0.98-0.16-0.19-0.86-0.15-0.17-1.02-0.17-0.33
ε23-3.000.340.25-0.730.040.02-0.630.050.03-0.770.03-0.12
ε300.391.431.271.301.661.711.271.661.711.281.711.56
ε15-1.541.000.77-1.48-1.07-1.08-1.41-1.02-1.03-1.53-1.07-1.10
ε24-2.490.300.21-0.660.050.03-0.590.060.04-0.700.05-0.11
ε40-1.661.291.051.441.751.761.491.763.561.431.771.69
ε16-1.981.020.80-1.41-0.90-0.90-1.32-0.85-0.85-1.47-0.90-0.92
Tab. 18.

Values of the frequency error εmn [%] (comparison results comes from simplified FE models of GR2_1 wheel)

εmnGR2_2_mGR2_3_mGR2_4_mGR2_5_m
solid187solid185solid186solid187solid185solid186solid187solid185solid186solid187
ε110.130.530.460.480.540.450.490.540.460.49
ε12-0.24-2.33-2.88-2.85-2.32-2.87-2.59-2.33-2.87-2.59
ε10-0.200.310.250.260.320.230.280.310.230.28
ε130.44-3.13-3.48-3.47-3.12-3.49-3.29-3.12-3.47-3.29
ε14-0.63-3.77-4.07-4.06-3.76-4.07-3.99-3.76-4.07-3.99
ε200.322.432.432.412.432.422.422.432.422.41
ε21-0.19-0.04-0.04-0.06-0.04-0.04-0.05-0.03-0.04-0.05
ε22-0.45-1.25-1.62-1.61-1.25-1.61-1.46-1.25-1.62-1.47
ε150.25-3.80-4.04-4.06-3.79-4.034.01-3.79-4.03-4.01
ε23-0.54-2.64-2.83-2.82-2.63-2.82-2.68-2.64-2.83-2.68
ε16-0.22-4.11-4.34-4.31-4.11-4.33-4.23-4.11-4.34-4.24
Tab. 19.

Values of the frequency error εmn [%] (comparison results comes from simplified FE models of GR3_1 wheel

εmnGR3_2_mGR3_3_mGR3_4_mGR3_5_m
solid187solid185solid186solid187solid185solid186solid187solid185solid186solid187
ε110.90-3.08-2.95-3.08-2.56-3.08-3.21-2.56-3.08-3.08
ε12-0.364.464.494.464.834.624.494.764.283.54
ε10-0.374.664.694.664.104.013.854.104.044.01
ε13-0.364.354.354.354.003.823.743.973.853.85
ε21-0.370.350.360.350.830.730.630.790.770.75
ε14-0.363.903.913.904.034.034.014.034.034.03
ε22-0.32-1.51-1.47-1.51-1.15-1.28-1.37-1.18-1.27-1.25
ε15-0.683.173.193.173.613.343.303.633.363.35
ε23-0.39-1.57-1.55-1.57-1.01-1.34-1.39-1.00-1.33-1.37
ε16-0.343.313.323.313.753.623.603.763.633.61
Tab. 20.

Values of the frequency error εmn [%] (comparison results comes from simplified FE models of GR4_1 wheel

εmnGR4_3_mGR4_4_m
solid185solid186solid187solid185solid186solid187
ε11-3.26-3.26-3.26-3.26-3.26-3.24
ε10-4.42-4.26-4.26-4.47-4.24-4.26
ε12-4.21-4.31-4.10-4.48-4.33-4.21
ε13-3.85-3.82-3.79-3.89-3.83-3.81
ε20-1.35-1.16-1.18-1.38-1.17-1.20
ε21-2.36-2.14-2.18-2.38-2.14-2.17
ε22-3.44-3.35-3.32-3.48-3.37-3.32
ε23-0.93-0.78-0.77-0.90-0.77-0.77
ε15-3.86-3.77-3.68-3.82-3.66-3.78
ε25-2.84-2.59-2.58-2.82-2.56-2.56
ε30-1.89-1.56-1.63-1.94-1.52-1.63
ε16-3.28-3.49-3.51-3.21-3.45-3.48
Tab. 21.

Values of the frequency error εmn [%] (comparison results comes from simplified FE models of GR5_1 wheel

εmnGR5_2_mGR5_3_mGR5_4_mGR5_5_m
solid185solid186solid187solid185solid186solid187solid185solid186solid187solid185solid186solid187
ε110.48-0.14-0.13-0.69-1.22-1.16-0.69-1.21-1.12-0.69-1.20-1.12
ε120.37-0.40-0.380.860.430.480.860.430.510.860.440.51
ε100.33-0.22-0.21-0.34-0.76-0.72-0.35-0.76-0.70-0.35-0.75-0.70
ε130.22-0.57-0.561.761.521.541.761.521.551.761.531.55
ε140.30-0.58-0.561.691.461.471.691.461.481.691.461.48
ε200.33-0.24-0.23-1.07-1.57-1.51-1.11-1.57-1.45-1.08-1.56-1.45
ε210.32-0.26-0.24-0.46-0.93-0.86-0.48-0.93-0.81-0.45-0.92-0.80
ε150.43-0.57-0.551.541.271.291.541.271.291.541.281.30
ε220.29-0.36-0.330.500.010.090.490.010.160.510.020.17
ε160.5-0.57-0.551.441.121.141.451.121.15---
Fig. 6.

The laboratory test stand

Tab. 22.

Experimental values of natural frequencies ωmn[Hz] (the GR1_1 wheel)

ωmnω11ω21ω22ω23ω24ω31ω32
GR1_1684.387460876011440150201539017600
Tab. 23.

Values of the frequency error εmn [%] (the comparison results come from the experimental data of the GR1_1 wheel)

εmnε11ε21ε22ε23ε24ε31ε32
GR1_1_msolid185-6.19-0.83-0.190.351.25-3.24-1.08
solid186-7.07-2.19-1.90-1.38-0.52-4.82-2.70
solid187-6.92-1.98-1.66-1.16-0.33-5.99-2.48
Tab. 24.

Values of the natural frequencies ωmn[Hz] (for the different values of the length of element side of the GR1_1 wheel, solid187)

ωmnω11ω21ω22ω23ω24ω31ω32
length of element side [mm]0.5639.3073438648.511353150351473917233
0.7639.407344.68650.411356150391474217237
1640.0673508658.511365150501475617252
1.5640.7973568664.811373150591476817264
2642.007367.8868111395150881479917296
4648.007431878311538152791500017493
8660.687553.5897411818157141534217858
Tab. 25.

Values of the frequency error εmn [%] (for the different values of the length of element side of the GR1_1 wheel, solid187)

εmnε11ε21ε22ε23ε24ε31ε32
length of element side [mm]0.5-6.59-1.57-1.27-0.760.10-4.23-2.09
0.7-6.57-1.55-1.25-0.730.13-4.21-2.06
1-6.48-1.48-1.16-0.660.20-4.12-1.98
1.5-6.37-1.39-1.09-0.590.26-4.04-1.91
2-6.19-1.24-0.90-0.390.45-3.84-1.73
4-5.32-0.390.260.861.72-2.53-0.61
8-3.461.252.443.304.62-0.311.47
Tab. 26.

Values of the frequency error εmn [%] (for the different values of the length of element side of the GR1_1 wheel, solid186)

εmnε11ε21ε22ε23ε24ε31ε32
length of element side [mm]0.5-6.67-1.64-1.35-0.830.04-4.30-2.17
0.7-6.64-1.63-1.35-0.820.05-4.30-2.16
1-6.62-1.63-1.33-0.810.05-4.28-2.15
1.5-6.49-1.55-1.27-0.760.09-4.22-2.10
2-6.62-1.61-1.27-0.730.19-4.24-2.03
4-6.16-1.44-1.22-0.690.20-4.17-1.91
8-6.09-0.65-0.060.601.88-3.57-0.96
Tab. 27.

Values of the frequency error εmn [%] (for the different values of the length of element side of the GR1_1 wheel, solid185)

εmnε11ε21ε22ε23ε24ε31ε32
length of element side [mm]0.5-6.48-1.35-0.10-0.490.3715.5-1.84
0.7-6.30-1.08-0.69-0.190.67-3.76-1.53
1-6.02-0.64-0.170.321.19-3.35-1.09
1.5-4.710.581.201.662.55-1.860.28
2-4.601.452.572.953.87-1.031.34
40.829.3710.811.513.24.227.49
8-2.9215.418.425.936.99.9019.6
Fig. 7.

Mode shapes related to frequency ω11: a) experimental results, b) full FE model

Fig. 8.

Mode shapes related to frequency ω21: a) experimental results, b) full FE model

Fig. 9.

Mode shapes related to frequency ω22: a) experimental results, b) full FE model

Fig. 10.

Mode shapes related to frequency ω23: a) experimental results, b) full FE model

Fig. 11.

Mode shapes related to frequency ω24: a) experimental results, b) full FE model

Fig. 12.

Mode shapes related to frequency ω31: a) experimental results, b) full FE model

Fig. 13.

Mode shapes related to frequency ω32: a) experimental results, b) full FE model

DOI: https://doi.org/10.2478/ama-2025-0090 | Journal eISSN: 2300-5319 | Journal ISSN: 1898-4088
Journal RSS Feed
Language: English
Page range: 812 - 826
Submitted on: Oct 29, 2025
Accepted on: Dec 21, 2025
Published on: Dec 31, 2025
Published by: Bialystok University of Technology
In partnership with: Paradigm Publishing Services
Keywords:
cyclic symmetry modeling,
toothed gears transverse vibration,
circular plate vibration,
FEM solution,
simplified toothed ring model
Related subjects:
Engineering,
Mechanical engineering,
Mechanics,
Bioengineering and biomedical engineering,
Biomechanics,
Civil engineering,
Environmental engineering

© 2025 Stanislaw NOGA, Pawel FUDALI, published by Bialystok University of Technology
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Volume 19 (2025): Issue 4 (December 2025)
Previous article