Pressure-based and potential-based mixed Ritz-differential quadrature formulations for free and forced vibration of Timoshenko beams in contact with fluid

Abstract

Two general mixed formulations, i.e., the pressure-based mixed Ritz-differential quadrature method and the potential-based mixed Ritz-differential quadrature method, are proposed to study the free and forced vibration of Timoshenko beams in contact with fluid. The proposed mixed methods use the Ritz method and the differential quadrature method to discretize the governing partial differential equation of motion of the beam and that of the fluid, respectively. As a result, application of proposed mixed methods gives two sets of ordinary differential equations; one belongs to the structure (beam) and the other to the fluid. These two sets of ordinary differential equations are coupled with each other and can be expressed as a system of ordinary differential equations which can be further discretized in time using various time integration schemes. The proposed mixed formulations, in general, enjoy from analytical characteristics of the Ritz method and simplicity of the differential quadrature method. Their accuracy, reliability and efficiency are shown through numerical simulations.

Appendix: Explicit analytical solution for natural frequencies of simply supported Timoshenko beams in contact with a bounded incompressible fluid with Dirichlet boundary conditions

Consider the beam–fluid system shown in Fig. 1. When pressure-based formulation is adopted, the governing partial differential equations for forced vibration of the system are given in Eqs. (1)–(3). In the case of free vibrations, the dynamic lateral load is equal to zero. The solutions of harmonic vibration of the system can then be expressed as

$$ w(x,t) = \tilde{w}(x)e^{i\omega t} ,\quad \Uppsi (x,t) = \tilde{\Uppsi }(x)e^{i\omega t} ,\quad p(x,z,t) = \tilde{p}(x,z)e^{i\omega t} $$

(102)

Ignoring the compressibility of the fluid and substituting Eq. (102) into Eqs. (1)–(3), we obtain the following dimensionless governing partial differential equations for free vibration of the beam–fluid system

$$ \eta_{1} \frac{\partial }{\partial X}\left( {\frac{{\partial \tilde{w}}}{\partial X} - \tilde{\Uppsi }} \right) + \eta_{2} \omega^{2} \tilde{w} = \eta_{3} \tilde{p}(X,Z = 0) $$

(103)

$$ \eta_{4} \frac{{\partial^{2} \tilde{\Uppsi }}}{{\partial X^{2} }} + \eta_{5} \left( {\frac{{\partial \tilde{w}}}{\partial X} - \tilde{\Uppsi }} \right) + \eta_{2} \omega^{2} \tilde{\Uppsi } = 0 $$

(104)

$$ \frac{{\partial^{2} \tilde{p}}}{{\partial X^{2} }} + \eta_{6}^{2} \frac{{\partial^{2} \tilde{p}}}{{\partial Z^{2} }} = 0 $$

(105)

where

$$ X=\frac{x}{L},\quad Z = \frac{z}{H}, \quad \eta_{1} = \frac{k}{2(1 + \mu )}\left( \frac{L}{r} \right)^{2} , \quad \eta_{2} = \frac{{\rho_{s} AL^{4} }}{EI}, \quad \eta_{3} = \frac{{bL^{4}}}{EI}, \quad \eta_{4} = \left( \frac{L}{r} \right)^{2}, \quad \eta_{5} = \frac{k}{2(1 + \mu )}\left( \frac{L}{r} \right)^{4} = \eta_{1} \eta_{4} ,\quad \eta_{6} = \frac{L}{H},\quad r = {\sqrt{ \frac{I}{A}}} = \frac{h}{{\sqrt{12}}} $$

(106)

where r being the radius of gyration of the beam cross-section.

According to boundary conditions of the problem, the normal mode functions of the beam–fluid system are chosen as [31, 44, 45]

$$ \tilde{w}_{n} (X) = \hat{w}_{n} \sin n\pi X,\quad \tilde{\Uppsi }_{n} (X) = \hat{\Uppsi }_{n} \cos n\pi X $$

(107)

$$ \tilde{p}_{n} (x,z) = \hat{p}_{n} \sin n\pi X\sinh n\pi \vartheta (1 - Z),\quad \vartheta = \frac{1}{{\eta_{6} }} = \frac{H}{L} $$

(108)

where $\tilde{w}_{n}$, $\tilde{\Uppsi }_{n}$ and $\tilde{p}_{n}$ are constants. Imposing the beam–fluid interface boundary condition [see Eq. (7)], we obtain

$$ \hat{p}_{n} = - \frac{{\rho_{f} H\hat{w}_{n} }}{n\pi \vartheta \cosh n\pi \vartheta }\omega_{n}^{2} $$

(109)

Upon introducing Eq. (109) to Eq. (108), the hydrodynamic pressure on the beam–fluid interface can be expressed as

$$ \tilde{p}_{n} (x,0) = - \hat{w}_{n} \bar{p}_{n} \omega_{n}^{2} \sin n\pi X $$

(110)

where

$$ \bar{p}_{n} = \frac{{\rho_{f} H}}{n\pi \vartheta }\tanh n\pi \vartheta $$

(111)

Substituting Eqs. (107) and (110) into Eqs. (103) and (104) gives

$$ \left[ {\begin{array}{*{20}c} {n^{2} \pi^{2} \eta_{1} - \eta_{7} \omega_{n}^{2} } & { - n\pi \eta_{1} } \\ {n\pi \eta_{5} } & {\eta_{2} \omega_{n}^{2} - n^{2} \pi^{2} \eta_{4} - \eta_{5} } \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {\hat{w}_{n} } \\ {\hat{\Uppsi }_{n} } \\ \end{array} } \right\} = \left\{ {\begin{array}{*{20}c} 0 \\ 0 \\ \end{array} } \right\},\quad \eta_{7} = \eta_{2} + \eta_{3} \bar{p}_{n} $$

(112)

Setting the determinant of the matrix in Eq. (112) to zero gives the characteristic equation of the fluid-loaded Timoshenko beam as

$$ \eta_{2} \eta_{7} \omega_{n}^{4} - (\eta_{5} \eta_{7} + n^{2} \pi^{2} \eta_{4} \eta_{7} + n^{2} \pi^{2} \eta_{1} \eta_{2} )\omega_{n}^{2} + n^{4} \pi^{4} \eta_{1} \eta_{4} = 0 $$

(113)

Equation (113) can also be rewritten as

$$ \eta_{8} \tilde{\omega }_{n}^{2} - ((\eta_{5} + n^{2} \pi^{2} \eta_{4} )\eta_{8} + n^{2} \pi^{2} \eta_{1} )\tilde{\omega }_{n} + n^{4} \pi^{4} \eta_{1} \eta_{4} = 0 $$

(114)

where

$$ \eta_{8} = 1 + \frac{\lambda }{n\pi }\tanh n\pi \vartheta ,\quad \lambda = \frac{{\rho_{f} L}}{{\rho_{s} h}},\quad \tilde{\omega }_{n} = \eta_{2} \omega_{n}^{2} = \omega_{n}^{2} \frac{{\rho_{s} AL^{4} }}{EI} $$

(115)

Equation (114) is a quadratic equation in $\tilde{\omega }_{n}$ and gives two values of $\tilde{\omega }_{n}$ for any value of n. The roots of this equation are given by

$$ \tilde{\omega }_{{n_{1} ,n_{2} }} = \frac{1}{2\alpha }( - \beta \pm \sqrt {\beta^{2} - 4\alpha \gamma } ) $$

(116)

where

$$ \alpha = \eta_{8} ,\quad \beta = - ((\eta_{5} + n^{2} \pi^{2} \eta_{4} )\eta_{8} + n^{2} \pi^{2} \eta_{1} ),\quad \gamma = n^{4} \pi^{4} \eta_{1} \eta_{4} $$

(117)

It is noted that the smaller value of $\tilde{\omega }_{n}$ corresponds to the bending deformation mode, while the larger value corresponds to the shear deformation mode.