, , and

#### 1. Introduction

The laminated composite plates, as important structural components, are often supported by elastic foundations. Therefore, it is necessary to understand the vibration of these laminated plates. Shen [1] carried out the postbuckling analysis of simply supported composite laminated plates on Pasternak-type elastic foundation. Xiang et al. [2] analyzed the vibration of moderately thick simply supported rectangular laminates on Pasternak foundation. Huang et al. [3] presented a finite strip method for a three-span simply supported plate resting on elastic foundations. A lot of research has been done for the plates resting on foundations, but the study of the dependence of the vibration for the moderately thick laminated composite plate on the boundary conditions is relatively few.

This paper presents an improved Fourier series method for the free vibration analysis of the moderately thick laminated plate with different boundary conditions on Pasternak foundations.

##### Figure 1

Rectangular plate resting on elastic foundations.

#### 2. Methods

As shown in Fig. 1, five types of springs are used to describe the boundary conditions. K w and K s are linear Winkler foundation and linear Pasternak foundation, respectively. The Lagrangian function is:

L=TUfVplateVspring

(1)

According to Refs. [4, 5], we can easily get the total kinetic energy (T) and strain energy (V p l a t e ) and V s p r i n g u n i f o r m . For the case considered, the total energy V s p r i n g p o i n t s , due to multi-points supports, and the strain energy U f , due to the Pasternak foundations, are given by:

Vspringspoints=12r=0NRQ(0,yr)TKx0Q(0,yr)+Q(a,yr)TKxaQ(a,yr)+12s=0NSQ(xs,0)TKy0Q(xs,0)+Q(xs,b)TKybQ(xs,b)

(2)

Uf=12AKww2+Kswx2+wy2dA

(3) where K i j and Q are springs matrix and displacements matrix, according to Refs. [5, 6]. Finally, we arrive at the following matrix equation, from which the natural frequencies can be obtained by solving a standard matrix Eigen problem:

(Kω2M)G=0

(4)

#### 3. Results

The first four frequency parameters Ω=( ωa2 )[ ρ /E2h2 ]1/2 for the isotropic laminated plates with different boundary constraints presented in the Table 1 agree well with the Finite Element Method (FEM) numerical results. In Figs. 2 and 3, the variations of the frequency parameters Ω against the number of layers n and points q are depicted. When n=10, the frequency parameters may reach their crest and then remain unchanged. When the total numbers of the clamped points reaches 28, the frequency parameters become stable, which can effectively approximate the uniformly clamped boundary condition.

##### Figure 2

Dependence of frequency parameter Ω on layers number n for plate.

##### Figure 3

Dependence of frequency parameter Ω on points number q for plate.

##### Table 1

Frequency parameter Ω for plate with different BC.

 BC Methods Model sequence 1 2 3 4 Free Present 3.6627 5.4012 6.5182 8.4358 FEM 3.5986 5.3627 6.4682 8.3880 Uniform elastic Present 3.6438 5.3827 6.5191 8.4369 FEM 3.6012 5.3162 6.4705 8.3909 8-points elastic Present 3.6499 5.3898 6.5248 8.4439 FEM 3.6012 5.3262 6.4705 8.3909

#### 4. Conclusion

In this paper, we can easily find that multi-points supports can effectively approximate the uniformly clamped boundary conditions, which is the basic idea of grid division in FEM.

#### Acknowledgements

This paper is funded by the International Exchange Program of Harbin Engineering University for Innovation-oriented Talents Cultivation. This paper is also supported by National Natural Science Foundation China (No 51505096), National Natural Science Foundation of China (No U1430236), and Natural Science Foundation of Heilongjiang Province of China (E2016024).

#### References

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2

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3

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