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

NOGA, Stanislaw; FUDALI, Pawel

doi:10.2478/ama-2025-0090

Figures & Tables

Model with the nK segment with cyclic symmetry features

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

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

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

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

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

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

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

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

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

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

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

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

		n
		0	1	2	3	4	5	6
m	1	5169	4363	4981	9836	16467	23702	31072
m	2	22007	22655

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

ε_mn		ε₁₁	ε₂₁	ε₂₂	ε₂₃	ε₂₄	ε₃₁	ε₃₂
GR1_1_m	solid185	-6.19	-0.83	-0.19	0.35	1.25	-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

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

		n
		0	1	2	3	4	5	6
m	1	3220	780	1471	3876.9	6905.9	10366	13986
m	2	-	4901.1	8386.7	12587

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

ω_mn		ω₁₁	ω₂₁	ω₂₂	ω₂₃	ω₂₄	ω₃₁	ω₃₂
length of element side [mm]	0.5	639.30	7343	8648.5	11353	15035	14739	17233
	0.7	639.40	7344.6	8650.4	11356	15039	14742	17237
	1	640.06	7350	8658.5	11365	15050	14756	17252
	1.5	640.79	7356	8664.8	11373	15059	14768	17264
	2	642.00	7367.8	8681	11395	15088	14799	17296
	4	648.00	7431	8783	11538	15279	15000	17493
	8	660.68	7553.5	8974	11818	15714	15342	17858

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

ω_mn	ω₁₁	ω₂₁	ω₂₂	ω₂₃	ω₂₄	ω₃₁	ω₃₂
GR1_1	684.38	7460	8760	11440	15020	15390	17600

Description of the FE models of the discussed gears

Model	Type of element	No. of nodes	No. of elements
GR3_1_m	solid185	589498	517008
	solid186	2282912	517008
	solid187	1982857	1311702
GR3_2_m	solid187	88522	57266
GR3_3_m	solid187	13981	11516
	solid186	53317	11516
	solid187	19617	10780
GR3_4_m	solid185	40871	35796
	solid186	158222	35796
	solid187	54010	30838
GR3_5_m	solid185	64379	57024
	solid186	249935	57024
	solid187	83817	48093
GR4_1_m	solid185	531889	459614
	solid186	2050232	459210
	solid187	2344962	1551003
GR4_3_m	solid185	91201	78860
	solid186	352131	78860
	solid187	152873	93255
GR4_4_m	solid185	92403	80593
	solid186	359510	80593
	solid187	156865	96142
GR5_1_m	solid185	211681	194979
	solid186	827639	194979
	solid187	513791	301014
GR5_2_m	solid185	7002	4925
	solid186	26853	4925
	solid187	33275	20285
GR5_3_m	solid185	7296	5215
	solid186	27022	5215
	solid187	15102	8055
GR5_4_m	solid185	19472	15615
	solid186	72705	15615
	solid187	37489	21520
GR5_5_m	solid185	30250	24816
	solid186	114078	24816
	solid187	58919	34468

Calculation statistics data for FE models of the GR_3_1 wheel

Model	WO			MU			RU
Model	solid185	solid186	solid187	solid185	solid186	solid187	solid185	solid186	solid187
GR3_1_m	1	1	1	27.56	36.93	32.39	6.22	23.46	22.08
GR3_2_m	-	-	0.41	-	-	24.17	-	-	11.69
GR3_3_m	0.17	0.49	0.058	4.61	31.74	6.11	2.33	12.31	3.29
GR3_4_m	0.17	1.43	0.055	14.09	14.53	15.16	3.29	14.46	4.31
GR3_5_m	0.39	6.67	0.18	29.77	29.83	32.53	4.54	9.99	8.97

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

Model	WO			MU			RU
Model	solid185	solid186	solid187	solid185	solid186	solid187	solid185	solid186	solid187
GR4_1_m	1	1	1	24.62	48.40	41.78	6.90	31.94	36.65
GR4_3_m	1.14	1.29	0.07	26.14	29.25	15.46	8.32	25.77	7.86
GR4_4_m	1.41	1.45	0.07	26.98	29.85	15.74	8.44	23.89	7.94
GR5_1_m	1	1	1	8.61	24.16	13.18	1.77	14.40	10.41
GR5_2_m	0.44	0.05	0.01	1.43	9.21	8.96	0.99	4.17	2.69
GR5_3_m	0.79	0.06	0.01	1.47	9.46	3.39	1.22	4.89	2.53
GR5_4_m	1.10	0.10	0.02	5.03	18.54	9.84	1.57	5.73	3.05
GR5_5_m	1.26	0.15	0.02	8.61	32.63	16.80	2.01	8.95	3.94

Description of the cyclic symmetry models of the discussed gears

Denotation in the article	Description	Denotation in the article	Description
GR3_2	cyclic symmetry segment model of the GR3_1 wheel, covering one tooth, cut along the centers of the inter – tooth slots (α = 12˚)	GR4_3	cyclic symmetry segment model of the GR4_1 wheel, covering the entire eccentric hole of the wheel disc (α = 60˚, see Fig. 3d)
GR3_3	cyclic symmetry segment model of the GR3_1 wheel without teeth (full ring), covering a 1/50th – circle segment (α = 7.2˚)	GR4_4	cyclic 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_4	cyclic 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_2	cyclic symmetry segment model of the GR5_1 wheel, covering one tooth, cut along the centers of the inter – tooth slots (α = 8.78˚)
GR3_5	cyclic symmetry segment model of the GR3_1 wheel without teeth (full ring), covering a 1/10th – circle segment (α = 36˚)	GR5_3	cyclic symmetry segment model of the GR5_1 wheel without teeth (full ring), covering a 1/50th – circle segment (α = 7.2˚)
GR5_4	cyclic 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_5	cyclic symmetry segment model of the GR5_1 wheel without teeth (full ring), covering a 1/10th – circle segment (α = 36˚)

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

ε_mn	GR5_2_m			GR5_3_m			GR5_4_m			GR5_5_m
ε_mn	solid185	solid186	solid187	solid185	solid186	solid187	solid185	solid186	solid187	solid185	solid186	solid187
ε₁₁	0.48	-0.14	-0.13	-0.69	-1.22	-1.16	-0.69	-1.21	-1.12	-0.69	-1.20	-1.12
ε₁₂	0.37	-0.40	-0.38	0.86	0.43	0.48	0.86	0.43	0.51	0.86	0.44	0.51
ε₁₀	0.33	-0.22	-0.21	-0.34	-0.76	-0.72	-0.35	-0.76	-0.70	-0.35	-0.75	-0.70
ε₁₃	0.22	-0.57	-0.56	1.76	1.52	1.54	1.76	1.52	1.55	1.76	1.53	1.55
ε₁₄	0.30	-0.58	-0.56	1.69	1.46	1.47	1.69	1.46	1.48	1.69	1.46	1.48
ε₂₀	0.33	-0.24	-0.23	-1.07	-1.57	-1.51	-1.11	-1.57	-1.45	-1.08	-1.56	-1.45
ε₂₁	0.32	-0.26	-0.24	-0.46	-0.93	-0.86	-0.48	-0.93	-0.81	-0.45	-0.92	-0.80
ε₁₅	0.43	-0.57	-0.55	1.54	1.27	1.29	1.54	1.27	1.29	1.54	1.28	1.30
ε₂₂	0.29	-0.36	-0.33	0.50	0.01	0.09	0.49	0.01	0.16	0.51	0.02	0.17
ε₁₆	0.5	-0.57	-0.55	1.44	1.12	1.14	1.45	1.12	1.15	-	-	-

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

ε_mn		ε₁₁	ε₂₁	ε₂₂	ε₂₃	ε₂₄	ε₃₁	ε₃₂
length of element side [mm]	0.5	-6.59	-1.57	-1.27	-0.76	0.10	-4.23	-2.09
	0.7	-6.57	-1.55	-1.25	-0.73	0.13	-4.21	-2.06
	1	-6.48	-1.48	-1.16	-0.66	0.20	-4.12	-1.98
	1.5	-6.37	-1.39	-1.09	-0.59	0.26	-4.04	-1.91
	2	-6.19	-1.24	-0.90	-0.39	0.45	-3.84	-1.73
	4	-5.32	-0.39	0.26	0.86	1.72	-2.53	-0.61
	8	-3.46	1.25	2.44	3.30	4.62	-0.31	1.47

Technical data related to the discussed models

No. of wheel	Young modulus E [Pa]	Density ρ [kg/m3]	Poisson’s ratio v	Equivalent Young’s modulus E_z [Pa]	Equivalent density ρ_z [kg/m3]
1	2.0310¹¹	7.8610³	0.3	2.910⁹	4.0583 10³
2					4.3568 10³
3					3.8928 10³
4					4.0387 10³
5					3.9492 10³

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

ε_mn		ε₁₁	ε₂₁	ε₂₂	ε₂₃	ε₂₄	ε₃₁	ε₃₂
length of element side [mm]	0.5	-6.67	-1.64	-1.35	-0.83	0.04	-4.30	-2.17
	0.7	-6.64	-1.63	-1.35	-0.82	0.05	-4.30	-2.16
	1	-6.62	-1.63	-1.33	-0.81	0.05	-4.28	-2.15
	1.5	-6.49	-1.55	-1.27	-0.76	0.09	-4.22	-2.10
	2	-6.62	-1.61	-1.27	-0.73	0.19	-4.24	-2.03
	4	-6.16	-1.44	-1.22	-0.69	0.20	-4.17	-1.91
	8	-6.09	-0.65	-0.06	0.60	1.88	-3.57	-0.96

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

		n
		0	1	2	3	4	5	6
m	1	613	430	955	2639	4944	7796	10859
	2	4700	4944	5678	6250	9370	10267
	3	10510

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

ε_mn		ε₁₁	ε₂₁	ε₂₂	ε₂₃	ε₂₄	ε₃₁	ε₃₂
length of element side [mm]	0.5	-6.48	-1.35	-0.10	-0.49	0.37	15.5	-1.84
	0.7	-6.30	-1.08	-0.69	-0.19	0.67	-3.76	-1.53
	1	-6.02	-0.64	-0.17	0.32	1.19	-3.35	-1.09
	1.5	-4.71	0.58	1.20	1.66	2.55	-1.86	0.28
	2	-4.60	1.45	2.57	2.95	3.87	-1.03	1.34
	4	0.82	9.37	10.8	11.5	13.2	4.22	7.49
	8	-2.92	15.4	18.4	25.9	36.9	9.90	19.6

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

ε_mn	GR1_1_m		GR2_1_m		GR3_1_m		GR4_1_m		GR5_1_m
ε_mn	solid185	solid186	solid185	solid186	solid185	solid186	solid185	solid186	solid185	solid186
ε₁₁	-0.79	0,16	-0.61	0.02	-1.03	-0.26	-1.16	-093	0.62	-0.14
ε₁₀	-0.56	0,12	-0.70	0.02	-2.48	-0.31	-0.51	-0.02	0.57	-0.13
ε₁₂	-1,19	0,24	-1.33	0.01	-1.43	-0.14	-0.94	0.00	0.74	-0.21
ε₁₃	-1,11	0,22	-1.74	0.01	-1.14	-0.01	-1.04	0.02	0.73	-0.23
ε₂₀	-0,85	0,18	-1.60	0.01	-	-	-0.50	0.01	0.59	-0.20
ε₂₁	-1,18	0,21	-0.59	0.02	-0.49	-0.03	-0.56	0.02	0.60	-0.17
ε₁₄	-1,37	0,20	-1.97	0.01	-1.22	-0.06	-1.03	0.03	0.76	-0.25
ε₂₂	-1,50	0,24	-0.86	0.02	-0.53	-0.01	-0.67	0.01	0.62	-0.21
ε₂₃	-1,53	0,22	-1.01	0.02	-0.99	-0.08	-0.67	0.02	-	-
ε₃₀	-0,63	0,12	-	-	-	-	-0.53	0.01	-	-
ε₁₅	-1,62	0,21	-1.18	0.02	-1.40	-0.11	-1.08	0.02	0.82	-0.27
ε₃₁	-2,92	-1,24	-	-	-	-	-	-	-	-
ε₂₄	-1,59	0,19	-	-	-	-	-	-	-	-
ε₂₅	-	-	-	-	-	-	-0.81	0.01	-	-
ε₄₀	-1,16	0,20	-	-	-	-	-	-	-	-
ε₃₂	-1,43	0,23	-	-	-	-	-	-	-	-
ε₁₆	-1,91	0,21	-1.40	0.02	-084	-0.08	-1.09	-0.58	0.89	-0.29

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

ε_mn	GR1_2_m			GR1_3_m			GR1_4_m			GR1_5_m
ε_mn	solid185	solid186	solid187	solid185	solid186	solid187	solid185	solid186	solid187	solid185	solid186	solid187
ε11	-4.08	0.16	0.00	-0.31	0.47	0.47	-0.16	0.31	0.31	-0.31	0.47	0.31
ε10	-1.74	0.25	0.06	-0.56	-0.25	-0.25	-0.56	-0.25	-0.25	-0.62	-0.25	-0.31
ε12	-1.37	0.96	0.72	-1.79	-1.43	-1.49	-1.73	-1.43	-1.43	-1.85	-1.49	-1.55
ε13	-1.00	0.94	0.74	-1.66	-1.37	-1.40	-1.59	-1.33	-1.33	-1.68	-1.40	-1.42
ε20	-1.99	0.56	0.26	-0.02	0.44	0.44	0.06	0.47	0.46	-0.05	0.44	0.37
ε14	-1.19	0.96	0.75	-1.56	-1.24	-1.24	-1.49	-1.19	-1.19	-1.59	-1.24	-1.25
ε22	-3.53	0.17	0.07	-0.98	-0.16	-0.19	-0.86	-0.15	-0.17	-1.02	-0.17	-0.33
ε23	-3.00	0.34	0.25	-0.73	0.04	0.02	-0.63	0.05	0.03	-0.77	0.03	-0.12
ε30	0.39	1.43	1.27	1.30	1.66	1.71	1.27	1.66	1.71	1.28	1.71	1.56
ε15	-1.54	1.00	0.77	-1.48	-1.07	-1.08	-1.41	-1.02	-1.03	-1.53	-1.07	-1.10
ε24	-2.49	0.30	0.21	-0.66	0.05	0.03	-0.59	0.06	0.04	-0.70	0.05	-0.11
ε40	-1.66	1.29	1.05	1.44	1.75	1.76	1.49	1.76	3.56	1.43	1.77	1.69
ε16	-1.98	1.02	0.80	-1.41	-0.90	-0.90	-1.32	-0.85	-0.85	-1.47	-0.90	-0.92

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

ε_mn	GR3_2_m	GR3_3_m			GR3_4_m			GR3_5_m
ε_mn	solid187	solid185	solid186	solid187	solid185	solid186	solid187	solid185	solid186	solid187
ε₁₁	0.90	-3.08	-2.95	-3.08	-2.56	-3.08	-3.21	-2.56	-3.08	-3.08
ε₁₂	-0.36	4.46	4.49	4.46	4.83	4.62	4.49	4.76	4.28	3.54
ε₁₀	-0.37	4.66	4.69	4.66	4.10	4.01	3.85	4.10	4.04	4.01
ε₁₃	-0.36	4.35	4.35	4.35	4.00	3.82	3.74	3.97	3.85	3.85
ε₂₁	-0.37	0.35	0.36	0.35	0.83	0.73	0.63	0.79	0.77	0.75
ε₁₄	-0.36	3.90	3.91	3.90	4.03	4.03	4.01	4.03	4.03	4.03
ε₂₂	-0.32	-1.51	-1.47	-1.51	-1.15	-1.28	-1.37	-1.18	-1.27	-1.25
ε₁₅	-0.68	3.17	3.19	3.17	3.61	3.34	3.30	3.63	3.36	3.35
ε₂₃	-0.39	-1.57	-1.55	-1.57	-1.01	-1.34	-1.39	-1.00	-1.33	-1.37
ε₁₆	-0.34	3.31	3.32	3.31	3.75	3.62	3.60	3.76	3.63	3.61

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

εmn	GR4_3_m			GR4_4_m
εmn	solid185	solid186	solid187	solid185	solid186	solid187
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

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

ε_mn	GR2_2_m	GR2_3_m			GR2_4_m			GR2_5_m
ε_mn	solid187	solid185	solid186	solid187	solid185	solid186	solid187	solid185	solid186	solid187
ε₁₁	0.13	0.53	0.46	0.48	0.54	0.45	0.49	0.54	0.46	0.49
ε₁₂	-0.24	-2.33	-2.88	-2.85	-2.32	-2.87	-2.59	-2.33	-2.87	-2.59
ε₁₀	-0.20	0.31	0.25	0.26	0.32	0.23	0.28	0.31	0.23	0.28
ε₁₃	0.44	-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
ε₂₀	0.32	2.43	2.43	2.41	2.43	2.42	2.42	2.43	2.42	2.41
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
ε₁₅	0.25	-3.80	-4.04	-4.06	-3.79	-4.03	4.01	-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

Calculation statistics data for FE models of the GR_2_1 wheel

Model	WO			MU			RU
Model	solid185	solid186	solid187	solid185	solid186	solid187	solid185	solid186	solid187
GR2_1_m	1	1	1	14.39	31.4	40.61	3.49	21.1	27.8
GR2_2_m	-	-	0.03	-	-	6.50	-	-	5.36
GR2_3_m	0.94	0.15	0.05	2.52	14.18	5.22	3.11	11.94	6.15
GR2_4_m	1.13	0.21	0.05	8.60	29.66	15.57	3.95	14.98	7.17
GR2_5_m	1.06	0.19	0.04	13.49	25.99	14.53	4.77	11.80	7.47

Calculation statistics data for FE models of the GR_1_1 wheel

Model	WO			MU			RU
Model	solid185	solid186	solid187	solid185	solid186	solid187	solid185	solid186	solid187
GR1_1_m	1	1	1	2.57	15.51	10.98	0.5	2.2	2.26
GR1_2_m	0.16	0.58	0.32	0.06	0.86	0.47	0.03	0.38	0.21
GR1_3_m	0.65	0.88	1.72	0.25	1.39	2.23	0.1	0.9	1.12
GR1_4_m	0.68	1.39	1.88	0.88	5.32	6.49	0.42	1.82	2.23
GR1_5_m	0.61	1.05	0.41	1.14	6.92	2.66	0.61	3.21	1.98

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

		n
		0	1	2	3	4	5	6
m	1	6615.3	4794	5630.9	11975	19232	26465	32030
m	2	20610	21263	25065

The toothed gears in question

No. of wheel	Tooth count z	Gear module m₀	Denotation in the article	Description
1	55	2.12	GR1_1	aviation toothed wheel (for aggregates, see Fig. 2a)
2	31	3	GR2_1	aviation toothed wheel (for power transmission, see Fig. 2b)
3	30	6	GR3_1	spiral bevel gear (for power transmission, see Fig. 2c)
4	83	2.5	GR4_1	aviation toothed wheel (for aggregates, see Fig. 2d)
5	41	2.5	GR5_1	aviation toothed wheel (for power transmission, see Fig. 2e)

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

Figures & Tables

Fig. 1.

Fig. 2.

Fig. 3.

Fig. 4.

Fig. 5.

Fig. 6.

Fig. 7.

Fig. 8.

Fig. 9.

Fig. 10.

Fig. 11.

Fig. 12.

Fig. 13.

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

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

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

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

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

Description of the FE models of the discussed gears

Calculation statistics data for FE models of the GR_3_1 wheel

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

Description of the cyclic symmetry models of the discussed gears

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

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

Technical data related to the discussed models

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

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

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

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

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

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

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

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

Calculation statistics data for FE models of the GR_2_1 wheel

Calculation statistics data for FE models of the GR_1_1 wheel

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

The toothed gears in question