Keywords

1 Introduction

Rectangular plates are commonly used as structural components in many engineering fields. The knowledge of their natural frequencies and mode shapes is necessary to determine their response under the working loads and to estimate properly the induced strains and stresses in order to make an optimal design. In spite of the very large number of research work devoted to this topic, due to the variety of edge conditions, there are still numerous situations which are not yet covered by the literature, particularly in the non-linear regime. Free vibration of rectangular plates punctually supported at a corner has been studied by numerous researches using various laborious methods, such as [1, 2], but yet the studies were restricted to linear vibration. The present work investigated the vibration of plates simply supported at two adjacent edges and free at the two other edges with a point support at the free corner, denoted as (SSFFRPSC). The Rayleigh-Ritz method (RRM) has been used to study the plate linear vibrations for various values of the aspect ratio. Benamar’s method (BM) [3, 4] has then been to investigate the nonlinear vibration for large vibration amplitudes, leading to the plotted backbone curves.

2 General Formulation

The studied SSFFRPSC is plotted in Fig. 1. The transverse displacement function \(\mathrm{W}\) of the current point P(x,y) is given by the following expression:

$$\mathrm{W}\left(\mathrm{x},\mathrm{y},\mathrm{t}\right)= \sum\nolimits_{\mathrm{k}=1}^{\mathrm{N}}{\mathrm{a}}_{\mathrm{k}}{\mathrm{w}}_{\mathrm{k}}\left(\mathrm{x},\mathrm{y}\right)\mathrm{sin}({\upomega}{\rm t})$$
(1)
Fig. 1.
figure 1

Plates simply supported at the edges \({{x}}={{y}}=0\) and free at the edges \({{x}}={{a}}\) and \({{y}}={{b}}\) with a point support at the corner \({{x}}={{a}},\,{{y}}={{b}}\) (SSFFRPSC)

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\({\mathrm{a}}_{\mathrm{k}}\) represents the contribution coefficient of test plate function \({\mathrm{w}}_{\mathrm{k}}\). This test functions \({\mathrm{w}}_{\mathrm{k}}\) used, for \(k=\mathrm{1,2}\dots {n}^{2}\), are obtained as products of n linear simply supported-free beam mode \({X}_{i}\) in the x- and y- directions:

$${\mathrm{w}}_{\mathrm{k}}\left(\mathrm{x},\mathrm{y}\right)= {\mathrm{w}}_{\mathrm{ij}}\left(\mathrm{x},\mathrm{y}\right)= {\mathrm{X}}_{\mathrm{i}}\left(\mathrm{x}\right) {\mathrm{X}}_{\mathrm{j}}\left(\mathrm{y}\right),\mathrm{ k}=\mathrm{n}\left(\mathrm{i}-1\right)+\mathrm{j}$$
(2)

With:

$${\mathrm{X}}_{\mathrm{i}}\left(\mathrm{x}\right)= {\mathrm{C}}_{1}\,\mathrm{ sin }\left(\mathrm{\lambda x}\right)+{\mathrm{C}}_{2}\,\mathrm{ cos }\left(\mathrm{\lambda x}\right)+{\mathrm{C}}_{3}\,\mathrm{ sinh }\left(\mathrm{\lambda x}\right)+{\mathrm{C}}_{4}\,\mathrm{ cosh }(\mathrm{\lambda x})$$
(3)

n being the number of trial beam functions used. The simply supported-free beam frequency parameters \(\lambda =\frac{\mathrm{L}}{\uppi }\sqrt[4]{\frac{\rho {\omega }_{b}^{2}}{EI}}\), listed in Table 1, are analytical solution of the differential equation which governs the beam vibration [5].

Table 1. Lowest ten frequency parameters \(\uplambda =\frac{\mathrm{L}}{\uppi }\sqrt[4]{\frac{\uprho {\upomega }_{\mathrm{b}}^{2}}{\mathrm{EI}}}\) of a simply supported-free beam.
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The nonlinear vibrations have been described in many papers by Benamar and al [6,7,8] by a set of nonlinear algebraic equation involving the rigidity and the mass matrices \([K]\) and \([M\)] and a fourth order tensor \(\left[B\left(\left\{A\right\}\right)\right]\) expressing the structure nonlinearity. A similar study is applied in the present work leads to:

$$\left[\mathrm{K}\right]\left\{\mathrm{A}\right\}-{\Omega }^{2}\left[\mathrm{M}\right] \cdot \left\{\mathrm{A}\right\}+\frac{2}{3}\left[\mathrm{B}\left(\left\{\mathrm{A}\right\}\right)\right]\left\{\mathrm{A}\right\}=0$$
(4)

\(\{A\}\) being the vector of the trial function contribution coefficients. The expressions for \([M]\) and \([B]\) are identical to those given in [6] but the parameters are calculated here using the functions defined above which satisfy the present edge conditions. On the other hand, the strain energy due to the point support, modeled by a translational spring of rigidity \({K}_{Sp}\), is given by:

$${V}_{Spr}=\frac{{K}_{Sp}}{2} {W}^{2}\left(a,b\right)=\frac{1}{2} {{a}_{i}{a}_{j}K}_{ij}^{Sp}\mathrm{sin}\left(\omega t\right)=\frac{1}{2}{a}_{i}{a}_{j} {K}_{Sp} {w}_{i}\left(a,b\right) \cdot {w}_{j}\left(a,b\right) $$
(5)

\({K}_{ij}^{Sp}\) is the rigidity term associated to the elastic energy stored in the spring to be included in the tensor \([K]\). For very large values of \({K}_{Sp}\), \(W(a,b)\) tends to zero.

3 Numerical Results

3.1 Numerical Results for Linear Vibration

The free linear vibration of the point supported plate is found by eliminating the nonlinear tensor B({A}). The Eq. (4) becomes:

$$\left[\mathrm{K}\right]\left\{\mathrm{A}\right\}-{\Omega }^{2}\left[\mathrm{M}\right] \cdot \left\{\mathrm{A}\right\}=0$$
(6)

The results obtained from solution of Eq. (6) are summarized giving The lowest seven frequency parameters of SSFFRPSC are listed in Table 2 for several values of \(\alpha =\frac{a}{b}\). These results are compared with those given by Li in [2]. The differences remain less than \(0.6\%\). The lowest four mode shapes are shown in Fig. 2 for \(\alpha =0.6\). The cross-sections at the plate diagonal \(\frac{x}{a}=\frac{y}{b}\) are given in Fig. 3, showing a zero displacement at the simply supported corner.

Table 2. Lowest seven frequency parameters \({\Omega }_{\mathrm{i}}={\upomega }_{\mathrm{i}}\sqrt[]{\frac{\uprho \mathrm{{H}}}{\mathrm{D}}} {\mathrm{a}}^{2}\) of a SSFFRPSC for several values of \(\mathrm{\alpha }=\mathrm{a}/\mathrm{b}\)
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Fig. 2.
figure 2

Lowest four modes of SSFFRPSC for \(\alpha =0.6\)

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Fig. 3.
figure 3

Cross-sections of the four modes at the plate diagonal \((x/a=y/b).\) \(\alpha =0.6.\)

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3.2 Numerical Results for Nonlinear Vibration

In order to investigate the nonlinear vibrations of the studied plate the fourth order tensor \(\left[\mathrm{B}\left(\left\{\mathrm{A}\right\}\right)\right]\) is not neglected in Eq. (4). Benamar’s method is applied to find the amplitude dependent nonlinear fundamental frequency parameters for plates having different values of the plate aspect ratio \(\alpha =\frac{a}{b}\). Equation (4), allows to calculate the frequency parameters: both sides of the equation are pre-multiplied by \({\left\{\mathrm{A}\right\}}^{\mathrm{T}}\) which becomes [6]:

$${\Omega }_{\mathrm{NL}}^{2}=\frac{{\mathrm{a}}_{\mathrm{i}} {{\mathrm{a}}_{\mathrm{j}}\mathrm{ m}}_{\mathrm{ij}} +1.5 {\mathrm{a}}_{\mathrm{i}} {{\mathrm{a}}_{\mathrm{j}} {\mathrm{a}}_{\mathrm{k}} {\mathrm{a}}_{\mathrm{l}}\mathrm{ b}}_{\mathrm{ijkl}}}{{\mathrm{a}}_{\mathrm{i}} {{\mathrm{a}}_{\mathrm{j}}\mathrm{ k}}_{\mathrm{ij}}}.$$
(7)

The so-called second formulation (SF) was used in this work to solve Eq. (4) and find the \({a}_{i}\)’s, i = 2 to N, for various values of the first component \({a}_{11}\), and consequently the amplitude dependent nonlinear mode shapes. The bases of this approximate method, developed by El Kadiri–Benamar, are detailed in Ref [6] for the nonlinear free vibration problem formulated in the MFB (see Appendix B in Ref [6]), leads to \(\left(m\times n\right)-1\) equations, the \({i}^{th}\) equation is:

$$-{\Omega }^{2} \,\, {\overline{a}}_{s} \,\,{\overline{m}}_{si}+{\overline{a}}_{s} \,\, {\overline{k}}_{si}+1.5 \,\, {\overline{a}}_{s} \,\, {\overline{a}}_{u} \,\, {\overline{a}}_{v} \,\, {\overline{b}}_{suvi}=0, i=\mathrm{2, 3}, 4,\,...\,n\times m$$
(8)

Examples of numerical results obtained by the SF are listed in Table 3 corresponding to a SSFFRPC for\(\alpha =0.6\), and in Table 4 for α \(=0.3, 0.6, 0.9\) and for several values of \({a}_{11}\) showing the nonlinear amplitude dependence of the contribution coefficient. Figure 4 plots the backbone curves of SSFFR by the second formulation for several values of the aspect ratio indicating a nonlinear behavior of the hardening type. The aspect ratio influence on the fundamental mode shape is shown in Fig. 5. It appears that the hardening effect is more accentuated with increasing the aspect ratio\(\alpha \). Figure 6 gives the corresponding normalized cross sections of the amplitude dependent nonlinear first mode, for increasing values of the first component\({a}_{11}\), at the plate diagonal \(\frac{y}{b}=\frac{x}{a}\) and at the plate middle line \(y=\frac{b}{2}\), showing a nonlinear increase of curvature at the simply supporter corner with increasing the vibration amplitude.

Table 3. \({\Omega }_{\mathrm{NL}}\)/.\(\Omega \), \({\mathrm{W}}_{\mathrm{max}}\)and \({\mathrm{a}}_{\mathrm{ij}}\) as functions of \({\mathrm{a}}_{11}\) for \(\alpha =0.6\) \((\Omega =5.10929)\)
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Table 4. \({\Omega }_{\mathrm{NL}}\)/\(\Omega \) and \({\mathrm{W}}_{\mathrm{max}}\) as functions of \({\mathrm{a}}_{11}\) for several values of \(\alpha \)
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Fig. 4.
figure 4

Backbone curves of SSFFRPSC by the second formulation with various aspect ratios

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Fig. 5.
figure 5

Fundamental mode shape of SSFFRPSC with various aspect ratios \(\alpha =0.2, 0.6, 1\).

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Fig. 6:
figure 6

Normalized cross sections of the amplitude dependent nonlinear first mode shape of SSFFRPSC for increasing values of \({a}_{11}\); (a) at the diagonal \(\frac{y}{b}=\frac{x}{a}, \)(b) at middle plate \(y=\frac{b}{2}\)

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4 Conclusion

Linear frequencies and mode shapes of point supported rectangular plates at the free corner whose opposite edges are simply supported has been studied using the Rayleigh-Ritz method with appropriate plate functions. The efficiency of the Rayleigh-Ritz method used in such plate is established by the accuracy of found results which are compared with the available bibliography. The results are given for several values of the aspect ratio. The mode shapes not only were given up to the fourth mode, but their normalized cross sections are plotted. The Benamar’s method was applied to study the nonlinear vibrations. The nonlinear frequency parameter \({\Omega }_{NL}\) has been calculated by the second formulation developed by Benamar and al for various values of the maximum displacement. The Backbone has been plotted for many values of the plate aspect ratio α = a/b. The first nonlinear mode shape which depends on the amplitude has been determined and plotted for several values of first component \({a}_{11}\) of the vector of the trial function contribution coefficients \(\{A\}.\)