The effects of stacking sequences of composite laminated optical beams on free vibration frequencies are investigated using two methods: the Boubaker Polynomials Expansion Scheme (pbes) and the Differential Quadrature Method (dqm). In the last decades, these two techniques have been separately performed for obtaining accurate numerical solutions to several initial boundary value problems (Vo et al. 2010, Li et al. 2008, Chen 2003, Hu et al. 2008, Karami et al. 2003, Malekzadeh et al. 2004, Khare et al. 2004, Della and Shu 2005, Ramtekkar et al. 2002, Adam 2003). Conjointly yielded results are compared and discussed.

1. Introduction

Free optical vibration of generally laminated beams has been of increasing interest in the last decades’ literature [112]. Vo et al. [1] investigated free vibration of axially loaded thin-walled composite beams with arbitrary lay-ups. The proposed model was based on equations of motion for flexural-torsional coupled vibration which were derived from the Hamilton’s principle. In the same context, Li et al. [2] studied the free vibration and buckling behaviors of axially loaded laminated composite beams using the dynamic stiffness method. The model took into account influences of axial force, Poisson effect, axial deformation, shear deformation, and rotary inertia. Hu et al. [4] Karami et al. [5], and Malekzadeh et al. [6] proposed a differential quadrature element method (DQEM) by using Hamilton’s principle for free vibration analysis of arbitrary nonuniform Timoshenko beams on elastic supports.

Many other analytical methods of analysis have been used to study the vibration of plates, shells, and beams [712].

In this paper, a model on the vibration analysis of laminated composite beam has been developed and studied using two resolution protocols. For the beam used, it is assumed that Bernoulli-Euler hypothesis is valid. The results obtained by the two methods are compared. It has been concluded that all of the results are very close to each other.

2. Problem Formalization

The normal stress in 𝑗th layer of a composite laminated beam shown in Figure 1 can be written in the following way:𝜎𝑥𝑗=𝐸𝑥𝑗𝜀𝑥𝑗.(1) According to Bernoulli-Euler hypotheses, the deformation at a certain distance from neutral plane is𝜀𝑥=𝑧𝜌,(2) where 𝜌 is the curvature of the beam. The relationship between normal stress and bending moment is given by𝑀=20/2𝜎𝑥𝑧𝑏𝑑𝑧,(3) or𝑀=2𝑏3𝜌𝑁/2𝑗=1𝐸𝑥𝑗𝑧3𝑗𝑧3𝑗1,(4) where and 𝑏 are the height and the width of the beam, 𝑁 is the number of layer and 𝑧𝑗 is the distance between the outer face of 𝑗th layer, and the neutral plane. The relationship between the bending moment and the curvature can be written as follows:𝐸𝑀=ef𝐼𝑦𝑦𝜌=𝐸ef𝐼𝑦𝑦𝑑2𝑤𝑑𝑥2,𝐸ef=83𝑁/2𝑗=1𝐸𝑥𝑗𝑧3𝑗𝑧3𝑗1,(5) where 𝐸ef is the effective elasticity modulus and 𝐼𝑦𝑦 is the cross-sectional inertia moment of the beam. Flexural motion of a linear elastic laminated composite beam without shear or rotary inertia effects is described by Bernoulli-Euler equation:𝐸ef𝐼𝑦𝑦𝜕4𝑤𝜕𝑥4𝜕+𝜌𝐴2𝑤𝜕𝑡2=0.(6) As a solution of (6), it can be used a separation of variables solution for harmonic free vibration:𝑤(𝑥,𝑡)=𝑒𝑖𝜔𝑛𝑡𝑊(𝑥),(7) where 𝜔𝑛 is the frequency and 𝑊(𝑥)  is the mode shape function of the lateral deflection. Substitution of this solution into (6) eliminates the time dependency and yields the following characteristic value problem:𝑑4𝑊(𝑥)𝑑𝑥4𝜁2𝑊(𝑥)=0,(8) where 𝜆 is the dimensionless frequency of the beam vibrations given by𝜁=𝜔2𝑛𝜌𝑚𝐴𝐸ef𝐼𝑦𝑦.(9) For a cantilever composite laminated beam shown in Figure 1, the boundary conditions at the two ends are𝑊=𝑑𝑊(𝑥)𝑑𝑥=0at𝑥=0,(10) due to the deflection and rotation both being zero at the clamped end, and𝑑2𝑊(𝑥)𝑑𝑥2=0at𝑥=𝐿,(11) due to the bending moment and shear force both vanishing at the free end.

The analytical solution of (8) subjected to (10) and (11) yields the frequency equation:cos(𝛽𝐿)cosh(𝛽𝐿)+1=0,𝛽=𝜁4,(12) which may be found in the relevant literature [13].

3. DQM Solution

DQM method is carried out for the approximate solution of the characteristic value problem in (8) with the boundary conditions given by (10) and (11) by first discretizing the interval [0,𝐿] such that 0=𝑥1<𝑥2<<𝑥𝑁=𝐿, where 𝑁 is the number of grid points. Application of the DQM to discrete the derivative in (8) leads to𝑁𝑗=1𝐴(4)𝑖𝑗𝑊𝑗𝜁2𝑊𝑖=0,𝑖=3,4,,(𝑁2),(13) where 𝐴(4)𝑖𝑗 are the weighting coefficients of the fourth-order derivative which can be calculated using the explicit relations given by Shu [14]. Note that we have two boundary conditions specified at both ends. These two conditions at the same point provoke a great challenge for the DQM, because we have only one quadrature equation at one point in the DQM, which prevents implementing the two boundary conditions. We use 𝛿-point technique to eliminate the difficulties in implementing two conditions at a single boundary point (Figure 2). Following the same approach presented in [15], the boundary conditions at 𝑥=0 can be discretized as𝑊1=0,𝑁𝑗=1𝐴(1)2𝑗𝑊𝑗=0.(14) Similarly, the boundary conditions at 𝑥=𝐿 can be discretized as𝑁𝑗=1𝐴(2)(𝑁1)𝑗𝑊𝑗=0,(15)𝑁𝑗=1𝐴(3)𝑁𝑗𝑊𝑗=0.(16) The assembly of (13) through (15) yields the following set [14] of linear equations:𝑆𝑏𝑏𝑆𝑏𝑑𝑆𝑑𝑏𝑆𝑑𝑑𝑊𝑏𝑊𝑑=𝜁𝑊{0}𝑑,(17) where the subscripts 𝑏 and 𝑑 indicate the grid points used for writing the quadrature analog of boundary conditions and the governing differential equation, respectively. By matrix substructuring of (17), one has the following two equations:𝑆𝑏𝑏𝑊𝑏+𝑆𝑏𝑑𝑊𝑑=𝑆{0},𝑑𝑏𝑊𝑏+𝑆𝑑𝑑𝑊𝑑=𝜁2𝑊𝑑.(18) From the first part of (18), one obtains𝑊𝑏𝑆=𝑏𝑑1𝑆𝑏𝑑𝑊𝑑.(19) Back-substituting (19) into the second part of (18), one gets[𝑆]𝑊𝑑𝜆2[𝐼]𝑊𝑑={0},(20) where [𝑆] is of order (𝑁4)×(𝑁4) and given by[𝑆]𝑆=𝑑𝑏𝑆𝑏𝑏1𝑆𝑏𝑑+𝑆𝑑𝑑.(21) Both the eigenvalues being the frequency squared values and the eigenvectors {𝑊𝑑} describing the mode shapes of the freely vibrating beam may be obtained simultaneously from the [𝑆] matrix.

4. BPES Solution

The BPES [1623] is applied to (8) through setting the expression1𝑊(𝑥)=2𝑁0𝑁0𝑘=1𝜆𝑘×𝐵4𝑘𝑟𝑥×𝑘𝐿,(22) where 𝐵4𝑘 are the 4𝑘-order Boubaker polynomials, 𝑥[0,𝐿] is the normalized time, 𝑟𝑘 are 𝐵4𝑘 minimal positive roots, 𝑁0 is a prefixed integer, and 𝜆𝑘|𝑘=1,,𝑁0 are unknown pondering real coefficients.

Consequently, it comes for (8) that12𝑁0𝑟𝑘𝐿4𝑁0𝑘=1𝜆𝑘×𝑑𝐵4𝑘𝑑𝑥4𝑟𝑥×𝑘𝐿𝜁212𝑁0𝑟𝑘𝐿4𝑁0𝑘=1𝜆𝑘×𝐵4𝑘𝑟𝑥×𝑘𝐿=0.(23) The related boundary conditions expressed through (10) and (12). The BPES protocol ensures their validity regardless main equation features. In fact, thanks to Boubaker polynomials first derivatives properties are𝑁𝑞=1𝐵4𝑞|||||(𝑥)𝑥=0=2𝑁0,𝑁𝑞=1𝐵4𝑞|||||(𝑥)𝑥=𝑟𝑞=0,𝑁𝑞=1𝑑𝐵4𝑞(𝑥)|||||𝑑𝑥𝑥=0=0,𝑁𝑞=1𝑑𝐵4𝑞(𝑥)|||||𝑑𝑥𝑥=𝑟𝑞=𝑁𝑞=1𝐻𝑞with𝐻𝑛=𝐵4𝑛𝑟𝑛=4𝑟𝑛2𝑟2𝑛×𝑛𝑞=1𝐵24𝑞𝑟𝑛𝐵4(𝑛+1)𝑟𝑛+4𝑟3𝑛.(24) Boundary conditions are inherently verified:𝑑𝑊(𝑥)|||𝑑𝑥𝑥=0=12𝑁0𝑁0𝑘=1𝜆𝑘×𝑑𝐵4𝑘(𝑥)||||𝑑𝑥𝑥=0=0,𝑑𝑊(𝑥)|||𝑑𝑥𝑥=𝐿=12𝑁0𝑁0𝑘=1𝜆𝑘×𝑑𝐵4𝑘(𝑥)||||𝑑𝑥𝑥=𝑟𝑘=0,𝑁0𝑘=1𝜆𝑘×𝐻𝑛=0.(25) The BPES solution is obtained through five steps:(i)Integrating, for a given value of 𝑁0, the whole expression given by (23) along the interval [0,𝐿].(ii)Determining the set of coefficients where ̃𝜆𝑘|𝑘=1,,𝑁0 that minimizes the absolute difference 𝐷𝑁0: 𝐷𝑁0=|||||12𝑁0𝑁0𝑘=1̃𝜆𝑘×Λ𝑘1𝜁2𝑁0𝑁0𝑘=1̃𝜆𝑘×Λ𝑘|||||Λwith𝑘=𝑟𝑘𝐿4𝐿0𝑑𝐵4𝑘𝑑𝑥4𝑟𝑥×𝑘𝐿Λ𝑑𝑥,𝑘=𝐿0𝐵4𝑘𝑟𝑥×𝑘𝐿𝑑𝑥.(26)(iii)Deducing the corresponding frequency using (9).(iv)Incrementing 𝑁0.(v) Ranging the obtained frequencies.

5. Results and Discussion

Natural frequencies of the symmetric laminated composite cantilever beam have been estimated using the Boubaker Polynomials Expansion Scheme (PBES) and the Differential Quadrature Method (DQM), and for parameters values indicated in Table 1. Figure 3 presents the obtained values. The results have been evaluated as quite close to each other.

The natural frequency alteration as a direct result of the change in the stacking sequence causes resonance if the changed frequency becomes closer to the working frequency. Hence, selection of the stacking sequences in the laminated composite beams has to be outlined.

6. Conclusion

This work deals with two protocols for the calculation of natural frequency of the symmetric laminated composite cantilever beam. Calculations performed by means of Boubaker Polynomials Expansion Scheme PBES and Differential Quadrature Method DQM yielded coherent and similar results.

All considered results have been seen to be in accordance with each other. Changes in the stacking sequence, which likely allow tailoring of the material to achieve desired natural frequencies and respective mode shapes without changing its geometry, are the subject of following studies.