Internal resonance of a supercritically axially moving beam subjected to the pulsating speed

The present work explores nonlinear dynamics of a supercritically moving beam under the 3:1 internal resonance condition. Responses are very different with those without internal resonance. Based on the direct multiple scale method, resonances for the first-two natural modes are identified to exist in three cases. The first one is that the pulsating speed frequency closes to two times of the first natural frequency. Under this condition, responses for natural modes are distinctly coupled as response curves are twisted. The internal resonance plays an essential role in energy transmission between related modes. It not only arouses a double-jumping phenomenon, but also reduces the typical parametric responses to zero astoundingly. Besides, the internal resonance changes the critical pulsating speed and produces some saddle-node bifurcations. In the case of the pulsating speed frequency closing to two times of the second natural frequency, only the second natural mode could be excited. The response occurs in the form of a typical parametric resonance. The third case is the pulsating speed frequency closing to the sum of the first-two natural frequencies. Different with the first two cases, quasi-periodic responses are found in the form of beat vibrations. Amplitude and the frequency of beats are affected by the pulsating speed, the internal resonance condition and also the pulsating frequency. Contributions of them are quite different. This paper is instructive to the study of vibration of other gyroscopic continuous systems, such as pipes conveying fluid and rotation continua.