Nonlinear frequencies and forced responses of pipes conveying fluid via a coupled Timoshenko model

Abstract

In this paper, a nonlinear Timoshenko model of the coupled vibration of a pipe conveying fluid is established to distinguish it from the Euler-Bernoulli coupled model and the Timoshenko model of the transverse vibration in terms of application scope and accuracy. The generalized Hamilton’s principle is utilized to derive the coupled Timoshenko model. The coupled Timoshenko model can be degenerated into three other models, namely the integro-partial differential Timoshenko transverse model, the partial differential Timoshenko transverse model, and the Euler-Bernoulli coupled model. The finite difference method (FDM) is developed to calculate the time history of the free vibration of the pipe. Based on the time history, the nonlinear frequency is obtained by using the discrete Fourier transform method (DFT). Furthermore, the amplitude-frequency curves of the forced vibration of the viscoelastic pipe are studied. The necessity for the Timoshenko coupled pipe model is shown by comparing with the other three models. Through extremely time-consuming calculations, comparisons show that these nonlinear models agree well at a low flow velocity and slight vibration. However, when the flow velocity or the initial displacement amplitude is large, the relative error of the nonlinear frequencies of the free vibration may exceed 30%. Moreover, the results indicate that the nonlinear coefficient has great influence on the nonlinear frequency when the flow velocity or the initial amplitude is large. Interestingly, the shear coefficient has a significant impact on the nonlinear frequency when the initial amplitude is small. In general, this paper shows that a large flow velocity, a large vibration amplitude or a shorter pipe length makes the coupled Timoshenko theory more necessary for modeling pipes conveying fluid.