Internal resonance of an axially moving unidirectional plate partially immersed in fluid under foundation displacement excitation
Introduction
The dynamic interaction between thin plates and fluid medium is of great concern in various scientific and engineering applications such as aerospace and aeronautical industries, aircraft construction, shipping and marine engineering. Many researches on fluid‐structure interaction problem have been carried out by using theoretical and numerical techniques which obtain considerable fruits. It is well known that the natural frequencies of plates in contact with fluid are different from those in air. Because of inertia increase due to the fluid motion caused by the structural vibration, the natural frequencies of a plate in fluid are significantly lower than those in air. Furthermore, the fluid also influences significantly on the vibration response of plate, therefore research on nonlinear dynamics of interaction between an elastic plate and fluid is necessary.
Research on the natural vibration characteristics of a structure immersed in fluid was initiated by Lamb [1], who calculated the first bending mode shape of a circular plate in contact with fluid. The developed method was based on a calculation of the kinetic energy of the fluid. The natural frequency is obtained using Rayleigh’s method. Powell and Roberts [2] experimentally verified Lamb’s results. Fu and Price [3] studied the dynamic behavior of a vertical or horizontal cantilever plate totally or partially immersed in fluid. They obtained a fluid-added mass matrix by solving the integral equation numerically. Amabili and Kwak [4], [5] studied vibration characteristics of circular or annular plates, and the added mass effect was represented in analytical form. Ergin and Uğurlu [6] calculated the fluid-added mass matrix of the cantilever plate partially submerged in a liquid using the boundary integral equation method together with the method of image. In their analysis, the fluid is assumed to be ideal, and fluid forces are associated with inertial effects of the surrounding fluid. Kwak et al. [7] investigated the free flexural vibration of a cantilever plate partially submerged in a fluid. The virtual mass matrix is then combined with the dynamic model of a thin rectangular plate obtained by using the Rayleigh‐Ritz method. The natural properties of a unidirectional vibrating steel strip partially submerged in fluid and under tension was studied by Li et al. [8], who used the velocity potential and Bernoulli’s equation to describe the fluid pressure acting on the steel strip. The effect of fluid on vibration of the strip was equivalent to added mass of the strip.
Main concern of above-mentioned documents is the free vibration of structures immersed in the fluid, however, studies related to nonlinear dynamics of structures in contact with fluid are scarce. Paak et al. [9], who studied the nonlinear dynamics and stability of cantilevered circular cylindrical shells subjected to internal fluid flow by developing a nonlinear theoretical model. The resulting coupled nonlinear ODEs were integrated numerically, and bifurcation analyses were performed using the AUTO software. They found that the stationary solution of the system lost stability through a supercritical Hopf bifurcation. Tubaldi et al. [10] investigated the geometrically non‐linear vibration of thin infinitely long rectangular plates subjected to axial flow and concentrated harmonic excitation for different flow velocities. The fluid is modeled by potential flow and the flow perturbation potential is derived by applying the Galerkin technique. They obtained the bifurcation diagrams of the static solutions, frequency‐response curves, time histories, and phase-plane diagrams. They discussed the effect of system parameters, such as flow velocity and geometric imperfections, on the stability of plate and its geometrically non‐linear vibration response to harmonic excitation.
The nonlinear dynamics of plates has been studied for a long time due to its importance in many engineering applications. Amabili and co‐worker [11], [12], [13], [14], [15] profoundly investigated nonlinear vibrations of plates. Theory and experiments for nonlinear vibrations of rectangular plates with different boundary conditions was presented in [11]. Amabili [12] used theory and experiments to investigate large-amplitude (geometrically nonlinear) vibrations of rectangular plates subjected to harmonic excitation. Alijani et al. [13], [14] investigated nonlinear vibrations of completely free laminated and sandwich rectangular plates. Static deflection as well as free and forced nonlinear vibration of thin square plates made of hyperelastic materials were investigated in [15]. Ribeiro and Akhavan [16] analyzed nonlinear vibrations of variable stiffness composite laminated plates.
The literature concerning the dynamics study of axially moving plates is also very extensive. Hatami et al. [17] used classical plate theory to study the free vibration of axially moving symmetrically laminated plates subjected to in-plan forces. Banichuk et al. [18] studied analytically the instability of an axially moving plate, which showed that the onset of instability takes place in a divergence form for some critical value of the transport velocity when the frequency of the plate vibrations is equal to zero. Marynowski and Crabski [19] analyzed dynamic analysis of axially moving thermally loaded two-dimensional system by using the Hamilton׳s principle. The nonlinear dynamics of an axially moving plate was investigated in [20], [21], who obtained frequency‐response curves, time histories, and phase-plane diagrams, and highlighted the effect of system parameters, such as the axial speed and the pretension, on the resonant responses. Moreover, they examined the geometrically non‐linear dynamics of an axially moving plate by constructing the bifurcation diagrams of Poincaré maps.
In this paper, the analysis is based on the assumption that the fluid is inviscid and incompressible, and its motion is irrotational. The velocity potential and Bernoulli’s equation are employed to express the fluid pressure acting on the plate. The effect of fluid on vibration of the plate is equivalent to added mass of the plate. Therefore the plate partially immersed in fluid is simplified into a stepped plate of constant rigidity while with different density and damping. Singular function theory is introduced to deal with the discontinuous characteristics of the stepped plate. The dynamic characteristics and bifurcation phenomenon of the axially moving plate partially immersed in fluid under foundation displacement excitation are researched using analytic method and numerical method.
Section snippets
Theoretical formulation
Consider the physical model of a rectangular partially immersed plate with simple support on two opposite edges and free on the other two edges stated as in Fig. 1, where and represent the width and length of the plate respectively, and is the thickness. denotes the length of the rectangular plate out of the liquid. indicates the axially moving velocity of the plate. Axial tension is applied to the plate with two simple supports along loaded sides. The plate is assumed to be made
Analysis process
The averaging method is introduced to analyze the nonlinear dynamic characteristics of axially moving plate partially immersed in fluid. New variables are introduced and defined as
So Eq. (5) is converted to state equation and can be expressed compactly in matrix form as followswhere
Analytic results
Fig. 4 shows relationships between excitation frequency and excitation amplitude. It can be observed that excitation amplitude is not a constant value, but increases together with excitation frequency.
By Eqs. (16) and (20), frequency‐response curves on averaging plane which are respectively near the first mode and the second mode are obtained, and they are shown in Fig. 5, Fig. 6. It can be observed that, due to the fact that excitation amplitudes increase together with excitation frequency,
Bifurcations
Through discussion mentioned above, it is found that changes of parameter will lead to the change of topological structure in the system. This section studies global dynamic bifurcations of averaging equations for the system, and will show specifically global dynamic characteristics of averaging equations in different parameter sets among the whole parameter space. Variables are introduced with following changes
Case1.: We can let
Through Eq. (21) we can get
Conclusions
This paper studies characteristics of nonlinear dynamics for an axially moving unidirectional plate partially immersed in fluid under 1:3 internal resonance. Because of the plate mainly occurred the flexural vibration under foundation displacement excitation, the influences of shear deformation and rotary inertia are neglected. It is found that, due to the influence of foundation displacement, frequency‐response curve shows special structural form. In the first resonance area, the first mode
Acknowledgments
This work was supported by the Started Foundation of the Ministry of Education of China (Grant No. N130305004) and the National Natural Science Foundation of China (Grant No. 11172063).
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