Research On The Periodic Motions Of Micro-scale Cantilever Pipe

Pipes conveying fluid are key components of many structures in engineering, the dynamics of which have been studied extensively and deeply so far. With the development in science and technology, the behavior existing in the micro structures has attracted enormous attentions. Meanwhile, the small-scale fluid-conveying pipes are commonly used in Micro-Electro-Mechanical Systems (MEMS). Therefore, it is crucial to know the dynamical behaviors of micro-tubes conveying fluid. In this dissertation, the planar and spatial vibrations of cantilever conveying-fluid pipes at the micro-scale will be investigated. Attentions are paid to its bifurcations, types of periodic motions and the effects of length scale parameter. The present paper is organized as follow:In Chapter 1, first of all, the research methods , new results and hot points in the nonlinear dynamics are introduced . Secondly, A literature review on the history and present researches on the planar linear and nonlinear vibrations, spatial vibrations of the pipes conveying fluid is given. Finally, many recent investigations on the micro-scale structures of the type of beam, are explored. Based on the statements above, the main problems are presented, and then the outline of our work is given.Chapter 2 deals with the dynamics of planar nonlinear oscillation of micro-scale cantilever tubes conveying fluid, the governing equation of motion is derived by using the Von Karman geometrical relation in conjunction with the modified couple stress theory. An eigenvalue analysis for linear equation is performed to examine the effect of internal material length scale parameter on the graph of critical flow velocity versus mass ratio. It is found that these curves of critical flow velocity versus mass ratio for different material length scale parameters may intersect each other . At each degenerate point, the first Lyapunov's coefficient and the derivation of the critical eigenvalue with respect to flow velocity are calculated by employing the projection methods based on the center manifold theory and normal form method, which demonstrates that the bifurcation is supercritical. The dynamics on fold and intersection point of the curve of critical flow velocity versus mass ratio are also investigated and then bifurcation curves towards to different direction are detected.In Chapter 3, based on the geometrical analysis of three dimensional motion of the pipe conveying fluid and the modified couple stress theory, the governing equations, which is a system of coupled intergro-differential equations, are derived.Utilizing the center manifold theory, normal form method and O(2) symmetry, the original governing equations can be theoretically reduced to a two-degree-of-freedom vibration system. Two types of periodic motions together with their stabilities are analyzed by the averaging methods. Since the planar periodic motion remains planar periodic motion under the action of symmetry group O(2), we deduce that the averaged equations have an eigenvalue zero, which agrees exactly with the result presented by direct calculation. Numerical studies are carried out to investigate the effect of size dependence on the periodic motions. The results confirm that: the larger the dimensional material length scale parameter is, the wider the region of stable planar periodic motion is; and that for mass ratio corresponding to fold of the curve of critical flow velocity, the bifurcating periodic motions may have different stabilities.In Chapter 4, with a progressively increasing number of mode in the Galerkin discretization, the diagrams of critical flow velocity versus mass ratio can be depicted.Comparsion between the results of the Galerkin approximation and that obtained by solving the two-point boundary problem shows that only the 8-mode discretization has a good agreement with the exact solution. The methods of projection are employed to calculate the coefficients which determine the stabilities and the bifurcation's features of our system. Comparing these finite-dimensional analysis results in this chapter with that obtained by infinite-dimensional analysis in Chapter 2 and Chapter 3, we conclude that: the 2-mode discretization is not enough to exhibit the dynamical behaviors of pipe conveying fluid, especially for the three dimensional vibration of pipe, which cannot predict stable spatial periodic motion; the dynamics predicted by the 4-mode discretization are valid for small mass ratio; for planar motion of pipe, the 6-mode approximate solutions have an excellent quantitative agreement with that obtained by infinite-dimensional analysis; but for spatial problem with large mass ratio, the 6-mode approximation cannot capture stable planar periodic motion; the 8-mode discretization can give the critical flow velocities which coincide with the exact solution, and can predict all of the qualitative dynamical characteristics of motion of pipe conveying fluid, except for very few degenerate points.Finally, the main research results of this dissertation are summarized, and a brief plan for future studies is given.