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X-Ritz Solution for Nonlinear Free Vibrations of Plates with Embedded Cracks

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Abstract

The analysis of large amplitude vibrations of cracked plates is considered in this study. The problem is addressed via a Ritz approach based on the first-order shear deformation theory and von Kármán’s geometric nonlinearity assumptions. The trial functions are built as series of regular orthogonal polynomial products supplemented with special functions able to represent the crack behaviour (which motivates why the method is dubbed as eXtended Ritz); boundary functions are used to guarantee the fulfillment of the kinematic boundary conditions along the plate edges. Convergence and accuracy are assessed to validate the approach and show its efficiency and potential. Original results are then presented, which illustrate the influence of cracks on the stiffening effect of large amplitude vibrations. These results can also serve as benchmark for future solutions of the problem.

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Authors and Affiliations

  1. Department of Engineering, University of Palermo, Viale delle Scienze, 90128, Palermo, Italy

    Ivano Benedetti, Vincenzo Gulizzi & Alberto Milazzo

Authors
  1. Ivano Benedetti
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  2. Vincenzo Gulizzi
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  3. Alberto Milazzo
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Correspondence to Alberto Milazzo.

A Plate inertia properties and mass matrix

A Plate inertia properties and mass matrix

The inertia matrices are defined as

$$\begin{aligned} \varvec{J}_0 = \left[ \begin{array}{lll} J_0 &{} 0 &{} 0 \\ 0 &{} J_0 &{} 0\\ 0 &{} 0 &{} J_0 \end{array}\right] , \quad \varvec{J}_1 = \left[ \begin{array}{ll} J_1 &{} 0 \\ 0 &{} J_1 \\ 0 &{} 0 \end{array}\right] , \quad \varvec{J}_2 = \left[ \begin{array}{lll} J_2 &{} 0 &{} 0 \\ 0 &{} J_2 &{} 0\\ 0 &{} 0 &{} J_2 \end{array}\right] , \end{aligned}$$
(A1)

where denoted by \(\rho\) the material density, \(J_0\), \(J_1\) and \(J_2\) are the plate mass moments of inertia given by

$$\begin{aligned} J_k =\int _h \rho \, x_3^k \, dx_3 \end{aligned}$$
(A2)

The mass matrix is deduced applying the variational calculus to the second integral of the functional \(\varPi\) after its discretization via Eqs. (2) To this aim, the plate generalized displacements approximation is formally rearranged in compact matricial form as

figure a

where the Ritz coefficients \({C_{\chi }}_{mn}^{\langle k\rangle }\) (\(\chi \in \{u_1,u_2,u_3,\theta _1,\theta _2\}\) and \(k=0,1,2,3,4\)) are arranged in the column vectors \(\varvec{X}_{\chi }\) with a corresponding arrangement of the approximation functions in the row vectors \(\varvec{\varPhi }_{\chi }\). Substituting this expression in the second integral of \(\varPi\), one obtains

$$\begin{aligned}&\frac{1}{2}\, \omega ^2\int _{\varOmega } \left[ \, \varvec{u}^T \varvec{J}_0 \varvec{u} + \varvec{u}^T \varvec{J}_1 \varvec{\vartheta } + \varvec{\vartheta }^T \varvec{J}_1^T \varvec{u} + \varvec{\vartheta }^T \varvec{J}_2 \varvec{\vartheta } \,\right] d\varOmega \nonumber \\&\quad =\frac{1}{2}\, \omega ^2 \varvec{X}^T\int _{\varOmega } \left[ \, \varvec{\varPsi }_u^T \varvec{J}_0 \varvec{\varPsi }_u + \varvec{\varPsi }_u^T \varvec{J}_1 \varvec{\varPsi }_\vartheta \right. \nonumber \\&\left. \qquad + \varvec{\varPsi }_\vartheta ^T \varvec{J}_1^T \varvec{\varPsi }_\vartheta + \varvec{\varPsi }_\vartheta ^T \varvec{J}_2 \varvec{\varPsi }_\vartheta \,\right] d\varOmega \varvec{X}= \frac{1}{2}\, \omega ^2 \varvec{X}^T\varvec{M}\varvec{X} \end{aligned}$$
(A4)

which define the mass matrix as

$$\begin{aligned} \varvec{M}= \int _{\varOmega } \left[ \, \varvec{\varPsi }_u^T \varvec{J}_0 \varvec{\varPsi }_u + \varvec{\varPsi }_u^T \varvec{J}_1 \varvec{\varPsi }_\vartheta + \varvec{\varPsi }_\vartheta ^T \varvec{J}_1^T \varvec{\varPsi }_\vartheta + \varvec{\varPsi }_\vartheta ^T \varvec{J}_2 \varvec{\varPsi }_\vartheta \,\right] d\varOmega \end{aligned}$$
(A5)

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Benedetti, I., Gulizzi, V. & Milazzo, A. X-Ritz Solution for Nonlinear Free Vibrations of Plates with Embedded Cracks. Aerotec. Missili Spaz. 98, 75–83 (2019). https://doi.org/10.1007/s42496-019-00006-5

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