A Normalized Transfer Matrix Method for the Free Vibration of Stepped Beams: Comparison with Experimental and FE(3D) Methods

Table 7

Percentage error between the computational and experimental results for single-step test samples (Group S2). Lowest four nonzero free-free modes.

			Method	Modal frequencies in Hz
				1	2	3	4
20/40	40	0.2	(A) EXP	1848.940	5028.140	9854	16303
			(B) NTM	1873.981	5194.229	10211.79	16776.400
			(C) FE(3D)	1859.490	5056.749	9893.345	16380.170
			(D) % error (A, B)	1.354	3.303	3.630	2.903
			(E) % error (A, C)	0.570	0.568	0.399	0.473
	100	0.5	(A) EXP	2153.900	7713.530	13701.800	19845.900
			(B) NTM	2244.601	7923.127	13661.990	20230.700
			(C) FE(3D)	2142.012	7700.169	13663.580	19793.360
			(D) % error (A, B)	4.211	2.717	−0.290	1.938
			(E) % error (A, C)	−0.551	−0.173	−0.278	−0.264
	140	0.7	(A) EXP	3596.820	8103.960	15773.100	20733.800
			(B) NTM	3813.426	8160.003	15739.160	21174.340
			(C) FE(3D)	3581.521	8090.723	15705.070	20634.360
			(D) % error (A, B)	6.022	0.691	−0.215	2.124
			(E) % error (A, C)	−0.425	−0.163	−0.431	−0.479
25/40	40	0.2	(A) EXP (Figure 11)	2430.240	6510.630	12186.700	19359.300
			(B) NTM	2444.111	6638.069	12492.520	19588.580
			(C) FE(3D)	2430.512	6518.094	12223.080	19314.540
			(D) % error (A, B)	0.570	1.957	2.509	1.184
			(E) % error (A, C)	0.011	0.114	0.298	−0.231
	100	0.5	(A) EXP	2796.050	8693.500	14542.200	22134
			(B) NTM	2912.321	8781.311	14633.780	22182.430
			(C) FE(3D)	2806.782	8698.963	14551.070	22095.920
			(D) % error (A, B)	4.158	1.010	0.629	0.218
			(E) % error (A, C)	0.383	0.062	0.060	−0.172
	140	0.7	(A) EXP	4015.620	8665.980	16238.9	22684
			(B) NTM	4167.385	8797.248	16074.330	22814.460
			(C) FE(3D)	4021.237	8655.476	16159.230	22610.690
			(D) % error (A, B)	3.779	1.514	−1.013	0.575
			(E) % error (A, C)	0.139	−0.121	−0.490	−0.323
30/40	40	0.2	(A) EXP	3023.690	7906	14381.400	21856.600
			(B) NTM	3023.321	7946.457	14429.850	21806.140
			(C) FE(3D)	3014.442	7878.443	14307.960	21781.750
			(D) % error (A, B)	−0.012	0.511	0.336	−0.230
			(E) % error (A, C)	−0.305	−0.348	−0.510	−0.342
	100	0.5	(A) EXP	3401.940	9329.910	15681.400	23607.700
			(B) NTM	3478.703	9293.555	15691.770	23326.460
			(C) FE(3D)	3405.324	9313.349	15645.690	23513.650
			(D) % error (A, B)	2.256	−0.389	0.066	−1.191
			(E) % error (A, C)	0.099	−0.177	−0.227	−0.398
	140	0.7	(A) EXP	4201.680	9322.630	16596.400	23967.700
			(B) NTM	4268.272	9431.643	16437.830	23729.240
			(C) FE(3D)	4211.100	9322.286	16530.200	23881.400
			(D) % error (A, B)	1.584	1.169	−0.955	−0.994
			(E) % error (A, C)	0.224	−0.003	−0.398	−0.360
35/40	40	0.2	(A) EXP	3608.280	9126.360	16128.600	23817.3
			(B) NTM	3607.895	9104.530	16000.810	23500.960
			(C) FE(3D)	3607.465	9106.544	16054.460	23720.110
			(D) % error (A, B)	−0.010	−0.239	−0.792	−1.328
			(E) % error (A, C)	−0.022	−0.217	−0.459	−0.408
	100	0.5	(A) EXP	3885.030	9778.800	16719.900	24538.800
			(B) NTM	3906.518	9705.047	16577.640	24125.780
			(C) FE(3D)	3882.419	9759.382	16658.180	24417.41
			(D) % error (A, B)	0.553	−0.754	−0.850	−1.683
			(E) % error (A, C)	−0.067	−0.198	−0.369	−0.495
	140	0.7	(A) EXP	4248.240	9888.540	17028	24738.400
			(B) NTM	4253.660	9878.822	16849.850	24311.230
			(C) FE(3D)	4247.271	9870.286	16963.900	24616.760
			(D) % error (A, B)	0.127	−0.098	−1.046	−1.726
			(E) % error (A, C)	−0.022	−0.184	−0.376	−0.491