Parametric and Internal Resonances of an Axially Moving Beam with Time-Dependent Velocity

Table 1

Variation of natural frequencies for the first two modes with mean travelling velocity 𝑣₀ showing 3 : 1 internal resonance for 𝑣_𝑓=0.2.

𝑣0	𝜔1	𝜔2	3𝜔1	𝜀𝜎1=𝜔2−3𝜔1	(𝜀𝜎1/𝜔1)%
0.4500	3.3149	9.7032	9.9447	−0.2415	−7.2853
0.4600	3.2969	9.6857	9.8907	−0.2050	−6.2180
0.4700	3.2786	9.6677	9.8358	−0.1681	−5.1272
0.4800	3.2599	9.6493	9.7797	−0.1304	−4.0001
0.4900	3.2407	9.6305	9.7221	−0.0916	−2.8265
0.5000	3.2211	9.6113	9.6633	−0.0520	−1.6144
0.5100	3.2011	9.5917	9.6033	−0.0116	−0.3624
0.5110	3.1991	9.5897	9.5973	−0.0076	−0.2376
0.5120	3.1970	9.5878	9.5910	−0.0032	−0.1001
0.5130	3.1950	9.5858	9.5850	0.0008	0.0250
0.5140	3.1930	9.5838	9.5790	0.0048	0.1503
0.5600	3.0948	9.4876	9.2844	0.2032	6.5659
0.5800	3.0493	9.4431	9.1479	0.2952	9.6809
0.6000	3.0020	9.3968	9.0060	0.3908	13.0180