Comparative Analysis of Free Optical Vibration of Lamination Composite Optical Beams Using the Boubaker Polynomials Expansion Scheme and the Differential Quadrature Method

Abstract

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 [1–12]. 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 [7–12].

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:𝜎_𝑥_𝑗=𝐸_𝑥_𝑗⋅𝜀_𝑥_𝑗.(1) According to Bernoulli-Euler hypotheses, the deformation at a certain distance from neutral plane is𝜀_𝑥=𝑧𝜌,(2) where 𝜌 is the curvature of the beam. The relationship between normal stress and bending moment is given by𝑀=2^0ℎ/2𝜎_𝑥⋅𝑧⋅𝑏⋅𝑑𝑧,(3) or𝑀=2𝑏3𝜌^𝑁/2_𝑗=1𝐸_𝑥_𝑗𝑧^3𝑗−𝑧^3𝑗−1,(4) where ℎ and 𝑏 are the height and the width of the beam, 𝑁 is the number of layer and 𝑧_𝑗 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:𝑀=𝐸^ef⋅𝐼_𝑦𝑦𝜌=𝐸^ef𝐼_𝑦𝑦𝑑²𝑤𝑑𝑥²𝐸_ef,=8ℎ^3𝑁/2_𝑗=1𝐸_𝑥_𝑗𝑧^3𝑗−𝑧^3𝑗−1,(5) where 𝐸_ef is the effective elasticity modulus and 𝐼_𝑦𝑦 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:𝐸_ef⋅𝐼_𝑦𝑦⋅𝜕⁴𝑤𝜕𝑥⁴+𝜌⋅𝐴⋅𝜕²𝑤𝜕𝑡²=0.(6) As a solution of (6), it can be used a separation of variables solution for harmonic free vibration:𝑤(𝑥,𝑡)=𝑒^{𝑖𝜔𝑛𝑡}𝑊(𝑥),(7) where 𝜔_𝑛 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)𝑑𝑥⁴−𝜁²𝑊(𝑥)=0,(8) where 𝜆 is the dimensionless frequency of the beam vibrations given by𝜁=𝜔^2𝑛𝜌_𝑚𝐴𝐸_ef𝐼_𝑦𝑦.(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.

Figure 1 

Cantilever composite laminated optical beam.

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<⋯<𝑥_𝑁=𝐿, where 𝑁 is the number of grid points. Application of the DQM to discrete the derivative in (8) leads to^𝑁_𝑗=1𝐴^(4)𝑖𝑗𝑊_𝑗−𝜁²𝑊_𝑖=0,𝑖=3,4,…,(𝑁−2),(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,^𝑁_𝑗=1𝐴^(1)2𝑗𝑊_𝑗=0.(14) Similarly, the boundary conditions at 𝑥=𝐿 can be discretized as^𝑁_𝑗=1𝐴^{(2)(𝑁−1)𝑗}𝑊_𝑗=0,(15)_𝑁_𝑗=1𝐴^(3)𝑁𝑗𝑊_𝑗=0.(16) The assembly of (13) through (15) yields the following set [14] of linear equations:⎡⎢⎢⎣𝑆𝑆_{𝑏𝑏𝑑𝑏}𝑆_𝑏𝑑𝑆_𝑑𝑑⎤⎥⎥⎦⎧⎪⎨⎪⎩𝑊^𝑏𝑊_𝑑⎫⎪⎬⎪⎭=⎧⎪⎨⎪⎩𝜁{𝑊0}_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:𝑆𝑆_𝑏𝑏𝑊_𝑏+𝑆_𝑏𝑑𝑊_𝑑={0},_𝑑𝑏𝑊_𝑏+𝑆_𝑑𝑑𝑊_𝑑=𝜁²𝑊_𝑑.(18) From the first part of (18), one obtains𝑊_𝑏=−𝑆_𝑏𝑑⁻¹𝑆_𝑏𝑑𝑊_𝑑.(19) Back-substituting (19) into the second part of (18), one gets[𝑆]𝑊_𝑑−𝜆²[𝐼]𝑊_𝑑={0},(20) where [𝑆] is of order (𝑁−4)×(𝑁−4) and given by[𝑆]=−𝑆_𝑑𝑏𝑆_𝑏𝑏⁻¹𝑆_𝑏𝑑+𝑆_𝑑𝑑.(21) Both the eigenvalues being the frequency squared values and the eigenvectors {𝑊_𝑑} describing the mode shapes of the freely vibrating beam may be obtained simultaneously from the [𝑆] matrix.

Figure 2 

A one-dimensional quadrature grid with adjacent 𝛿-points.

4. BPES Solution

The BPES [16–23] is applied to (8) through setting the expression𝑊(𝑥)=12𝑁_0𝑁⁰_𝑘=1𝜆_𝑘×𝐵_4𝑘𝑥×𝑟^𝑘𝐿,(22) where 𝐵_4𝑘 are the 4𝑘-order Boubaker polynomials, 𝑥∈[0,𝐿] is the normalized time, 𝑟_𝑘 are 𝐵_4𝑘 minimal positive roots, 𝑁₀ is a prefixed integer, and 𝜆_𝑘|_{𝑘=1,…,𝑁0} are unknown pondering real coefficients.

Consequently, it comes for (8) that12𝑁₀𝑟𝐿^𝑘^4𝑁0_𝑘=1𝜆_𝑘×𝑑𝐵^4𝑘−𝜁₂𝑑𝑥⁴𝑥×𝑟𝐿^𝑘12𝑁₀𝑟𝐿^𝑘^4𝑁0_𝑘=1𝜆_𝑘×𝐵_4𝑘𝑥×𝑟^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 are^𝑁_𝑞=1𝐵_4𝑞(𝑥)|||||_𝑥=0=−2𝑁≠0,_𝑁_𝑞=1𝐵_4𝑞(𝑥)|||||_{𝑥=𝑟𝑞}=0,_𝑁_𝑞=1𝑑𝐵_4𝑞(𝑥)𝑑𝑥|||||_𝑥=0=0,_𝑁_𝑞=1𝑑𝐵_4𝑞(𝑥)𝑑𝑥|||||_{𝑥=𝑟𝑞}=^𝑁_𝑞=1𝐻_𝑞with𝐻_𝑛=𝐵^′_4𝑛𝑟_𝑛=4𝑟^𝑛2−𝑟^2𝑛×∑^𝑛𝑞=1𝐵^24𝑞𝑟_𝑛𝐵_4(𝑛+1)𝑟_𝑛+4𝑟^3𝑛.(24) Boundary conditions are inherently verified:𝑑𝑊(𝑥)𝑑𝑥|||_𝑥=0=12𝑁_0𝑁⁰_𝑘=1𝜆_𝑘×𝑑𝐵^4𝑘(𝑥)𝑑𝑥||||_𝑥=0𝑑𝑊(𝑥)=0,𝑑𝑥|||_𝑥=𝐿=12𝑁_0𝑁⁰_𝑘=1𝜆_𝑘×𝑑𝐵^4𝑘(𝑥)𝑑𝑥||||_{𝑥=𝑟𝑘}=0,_𝑁0_𝑘=1𝜆_𝑘×𝐻_𝑛=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 𝜆̃_𝑘|_{𝑘=1,…,𝑁0} that minimizes the absolute difference 𝐷_𝑁0: 𝐷_𝑁0=|||||⎛⎜⎜⎝12𝑁_0𝑁⁰_𝑘=1𝜆̃_𝑘×Λ_𝑘⎞⎟⎟⎠−𝜁⎛⎜⎜⎝12𝑁_0𝑁⁰_𝑘=1𝜆̃_𝑘×Λ^𝑘Λwith_𝑘⎞⎟⎟⎠|||||=𝑟^𝑘𝐿⁴^𝐿0𝑑𝐵_4𝑘Λ𝑑𝑥⁴𝑥×𝑟𝐿^𝑘𝑑𝑥,_′𝑘=^𝐿0𝐵_4𝑘𝑥×𝑟^𝑘𝐿𝑑𝑥.(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.

Table 1 

Geometry and material properties of the composite materials.

Figure 3 

Comparison of BPES and DQM frequencies for cross-ply laminated beam.

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.