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Experimental and Analytical Approach to Study the Effect of Large Vibration Amplitude of Rectangular Plates

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Abstract

The effect of geometrical non-linearities of thin clamped and simply supported rectangular plates is considered in this paper. The formulation is made basing on Von Kármán theory, Lagrange equations, and the harmonic balance method. In addition, the calculations are made and allowed the application of the semi-analytical model which involved three tensors (mass (\(m_{ij}\)), linear (\(k_{ij}\)) and non-linear rigidity (\(b_{ijkl}\))) which are reduced from the potential and kinetic energy. The solution of non-linear algebraic equations is derived through an approximation solution which is obtained explicitly. The stability of forced vibration is studied for different degrees of freedom (dof) which is reached at four dof using the multimode approach. To valid our numerical results we made the experimental measurements. First, the geometric imperfections are investigated experimentally intending to measure the deviation of our plate. Second, our structure is investigated on a test rig with heavy boundaries conditions which are made from steel alloy, to avoid the external auto-excitation. The harmonic excitation force is generated by an electrodynamic exciter which is applied at a point of the examined plate. The geometric imperfection parameter does not take into account in the formulation of the problem. This assumption is based on the reasons that are cited in this paper.

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Notes

  1. Fully free (FFFF), simply supported plates with immovable and movable edges (SSSS), and fully clamped (CCCC) rectangular plates.

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Authors and Affiliations

  1. Centre d’Etude Doctorale, École Nationale Supérieure de l’Electricité et de Mécanique, Hassan II University of Casablanca, Casablanca, 8118, Morocco

    A. Majid & E. Abdeddine

  2. Laboratoire Mécanique, Productique et Génie Industriel, École Supérieure Technologie, Hassan II University of Casablanca, Casablanca, 8012, Morocco

    A. Majid, E. Abdeddine, Kh. Zarbane & Z. Beidouri

Authors
  1. A. Majid
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  2. E. Abdeddine
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Correspondence to A. Majid.

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Appendix A Numerical Details of CSSS Rectangular Plates (Boundaries Conditions)

Appendix A Numerical Details of CSSS Rectangular Plates (Boundaries Conditions)

A.1 C-SS Beam Functions in x Direction

The function of C-SS beam are given below:

$$\begin{aligned} {w}_{j}(x)=\frac{\mathrm{ch}(\beta _{j}x)-\mathrm{cos}(\beta _{j}x)}{\mathrm{ch}(\beta _{j})-\mathrm{cos}(\beta _{j})}-\frac{\mathrm{sh}(\beta _{j}x)-\mathrm{sin}(\beta _{j}x)}{\mathrm{sh}(\beta _{j})-\mathrm{sin}(\beta _{j})} \end{aligned}$$

where the eigenvalue parameters (\(\beta _{j}\)) are obtained from solution of the transcendental equation tg\((\beta )- \mathrm{th}(\beta )\) and are given in Table 3, Figs. 10, 11.

Table 3 Symmetric (a) and antisymmetric (b) eigenvalue parameters for a C-SS beam
Full size table
Fig. 10
figure 10

C-SS beams functions for i = 1, 2, 3, 4, 5, and 6

Full size image

where \({w}_{i}\) for \(i=1, 2,\ldots \) are the mode shapes for a \(C-SS\) beams.

A.2 SS–SS Beam Functions in y Direction

$$\begin{aligned} {w}_{j}(y)=\mathrm{sin}\left( \frac{j\pi y}{b}\right) \end{aligned}$$

where \({w}_{j}\) for \(j=1,2,\ldots \) are the mode shapes for a SS–SS beams.

Fig. 11
figure 11

SS–SS beams functions for j=1, 2, 3, 4, 5, and 6

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Majid, A., Abdeddine, E., Zarbane, K. et al. Experimental and Analytical Approach to Study the Effect of Large Vibration Amplitude of Rectangular Plates. J. Vib. Eng. Technol. 10, 3011–3023 (2022). https://doi.org/10.1007/s42417-022-00533-9

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