Issue 
Int. J. Simul. Multidisci. Des. Optim.
Volume 12, 2021



Article Number  15  
Number of page(s)  7  
DOI  https://doi.org/10.1051/smdo/2021016  
Published online  24 August 2021 
Research Article
Effect of lamination schemes on natural frequency and modal damping of fiber reinforced laminated beam using Ritz method
^{1} National Institute of Technology Karnataka Surathkal, Mangalore 575 025, India
^{2} Vignan's Institute of Information Technology, Visakhapatnam, Andhra Pradesh 530051, India
^{3} Vellore Institute of Technology Chennai, Tamilnadu 600 127, India
^{4} Aditya Engineering College, Surampalem, Andhra Pradesh, India
^{*} email: bnaidus@gmail.com
Received:
8
January
2021
Accepted:
6
August
2021
The current study focussed on analysing natural frequency and damping of laminated composite beams (LCBs) by varying fiber angle, aspect ratio, material property and boundary conditions. Ritz method with displacement field based on the shear and normal deformable theory is used and the modal damping is calculated using modal strain energy method. Effects of symmetric angleply and crossply, anti symmetric crossply, balanced and quasiisotropic lay up schemes on modal damping are presented for the first time. Results revealed that influence of layup scheme on natural frequencies is significant for the thin beams while the modal damping of the thin beams are not sensitive to layup scheme. However, the layup scheme influences the damping significantly for the thick beams. Similarly, high strength fiber reinforced LCBs have higher natural frequency while low strength fiber reinforced LCBs have higher damping due to the better fibermatrix interaction.
Key words: Ritz method / free vibration / damping / aspect ratio / LCB
© S.N. Balireddy et al., Published by EDP Sciences, 2021
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
Fibre reinforced laminated structures are always in demand because of high strength and very less weight. The laminated composite beam is a very common structural element used in various engineering applications such as mechanical, automobile, marine and aircraft industries [1]. The structures made of laminated composite materials have higher damping compared to conventional metallic structures due to the fillermatrix interaction [2]. Rajesh and Jeyaraj [3] through experiments demonstrated that, for a fibre reinforced composite beam, modal damping is influenced by nature of reinforcement. Senthilkumar et al. [4] shown that fibre length and its content influences the natural frequency of LCB significantly.
Various theorems presented by several researchers to analyze the free vibration frequencies of LCBs, using numerical and analytical methods, are recently reviewed by Sayyad and Ghugal [5]. Vo et al. [6] presented a shear and normal deformation model to analyse natural frequencies of LCBs using Ritz method. Nguyen et al. [7] formulated a unified model to study the static and dynamic behaviours of LCBs using Ritz method based on different theorems. Jeyaraj et al. [8] analysed sound radiation behaviour of a laminated composite plate using finite element method and found that increase in modal damping significantly reduces the vibration response at the resonances. Eltaher and Mohamed [9] studied stability characteristics of composite sandwich beams using differential quadrature method. Li et al. [10] used a unified higher order theorem based method to analyse natural frequencies of LCBs under the axial compression load. Nguyen et al. [11] presented an analytical model for the analysis of static and dynamic behaviours of LCBs using Ritz method.
Modal damping is capable of controlling vibration and sound levels when the system is excited at the resonant frequencies. Damping plays a vital role in the design of engineering components subjected to vibration and other dynamic loadings. Chandra et al. [12] presented a detailed study on damping of laminated composites and reported that modal strain energy method is used in general to estimate the damping theoretically. In aerospace applications, FRP composites are preferred due to higher inherent damping associated with it. The increase in inherent damping reduces the peak forced vibration responses significantly [13]. Ni and Adams [14] presented a method to estimate damping of a LCB considering bendingtwisting coupling and compared the numerical results with the experiments. Recently, Ozer et al. [15] developed a finite element for the damping analysis of laminated composites. Different elastic constants are defined in the complex form in order to calculate the modal damping of the composite structure numerically. Li et al. [16] presented an energy based theoretical model to analyse the damping of thin FRP plate. Lin et al. [17] predicted modal damping of carbon and glass FRP plates using modal strain energy and finite element method. Bruyneel et al. [18] highlighted the importance of designing laminated composite structures with optimised parameters to withstand against the buckling load. Irhirane et al. [19] presented different failure modes associated with LCBs and concluded that still necessary studies needed to find the suitable failure criteria for the given LCB. Liao et al. [20] studied LCBs made of carbonepoxy to analyse the interleaving effect on the damping behaviour.
Literature study revealed that damping analysis of LCBs is very important for its design considering vibration and other dynamic effects. Some researchers analysed vibration and damping behaviour of LCBs using numerical, analytical and experimental methods. However, there is no comprehensive study reported so far considering different types of lamination schemes, aspect ratio, boundary condition and material of the FRP on the combined vibration and damping behaviour of the LCBs. Hence, the effect of symmetric, unsymmetric, crossply, angleply, balancedply and quasiisotropic lamination schemes on the vibration and damping of LCBs is analysed. The effect of aspect ratio, structural boundary conditions and type of FRP material (carbonepoxy and glassepoxy) on the natural frequency and damping of LCBs also investigated in this work. Numerical method formulated based shear and normal deformation theorem and Ritz method is used to obtain the natural frequencies while modal strain energy based method is used to obtain the modal damping of the LCBs.
2 Methodology
A LCB having length, width and thickness as L,b and h receptively as shown in Figure 1 is considered. The shear and normal deformation theorem is used to define the strain displacement relation [6] and is as follows(1a) (1b)
In the above equation u, w_{b}, w_{s} and φ_{z} are the four unknown displacements of the midplane of the beam. The strains in axial, normal and shear are:(2a) (2b) (2c)
For orthotropic lamina, the stressstrain relation is given by(3)
Strain energy variation ∂U is given by:
variation in the kinetic energy δk of the beam is,
The equation of equilibrium are obtained by following the Hamilton's principle. The present study used Ritz procedure in finding the solution. The displacement functions for variables u(x), w_{b}(x), w_{s}(x) and w_{z}(x) are given as [1],(6a) (6b) (6c) (6d)
The terms A_{j},B_{j},C_{j} and D_{j} are undetermined coefficients, θ_{j}(x), ϕ(x), ζ_{j}(x) and ψ_{j}(x) are trial functions. The coefficients such as p_{u}, q_{u}, p_{wb}, q_{wb}, p_{ws}, q_{ws} and q_{wz} vary according to the end conditions of the LCB analysed. Substitution of the displacement functions in the equilibrium equations leads to the following typical eigenvalue problem which is used to calculate the natural frequencies of the LCB's studied.(7)
In the above equation, K and M are stiffness and mass matrices respectively. ω_{k} is the natural frequency and φ_{k} is the mode shape [2]. It should be noted that the elastic properties of the material used in the present work are of complex in nature. Hence, the stiffness matrix evaluated also will be of complex in nature i.e., it consists of real and imaginary parts represented as K_{I} and K_{R} respectively.
Based on the modal strain energy method, the modal loss factor (η_{k}) of kth mode is obtained. as follows,(8)
In the above equation, φ_{k} is the kth mode shape and K_{I} is the imaginary part of the stiffness matrix. The reader is referred to reference [11] for more details regarding the relations stiffness and mass matrix coefficients.
Fig. 1 Geometry of the analysed laminated beam. 
3 Validation study
3.1 Natural frequency validation
LCB beam analysed by Nguyen et al. [11] is considered for the validation of natural frequency calculation using the present study. Both Nguyen et al. [11] and present methods used same theorem and Ritz method to evaluate the nondimensional fundamental frequency. The results of the both the studies shown good agreement as shown in Table 1.(9)
4 Results
The study intended to predict the effect of boundary conditions, aspect ratio and various types of laminate schemes on natural frequency and damping of glass epoxy and carbon epoxy LCBs. The beams are investigated for SS, CC and CF boundary conditions. Symmetric angleply and crossply, antisymmetric crossply, balanced and quasiisotropic laminates are also analysed additionally. A LCB having a cross section of b × h and of length(L) 0.5 m is considered in the study. It is assumed that the cross section is square and thickness is varied with respect to the given aspect ratio (L/h). In order to analyse thick and thin beam cases, the LCB is investigated for two aspect ratios (L/h = 5 and 20). The material properties of both glass epoxy (GE) and carbon epoxy HMS (CE) used in the study are presented in Table 2 [17]. For both the materials, resin DX210 is used. An increment of 15° for θ is considered for the symmetric (0°/θ°/0°) laminates and corresponding variation in natural frequency and damping values for GELCB are presented in Tables 3 and 4 for the two different L/h ratios. The LCBs are analysed for their first three bending modes. A comprehensive study on widely used laminate schemes for natural frequency and damping of fundamental mode is presented in Tables 5 and 6 for the two L/h ratios respectively. Symmetric angleply and crossply, antisymmetric crossply, balanced and quasiisotropic laminates are analysed for their first fundamental mode. The considered laminated schemes are analysed for both glass epoxy and carbon epoxy materials.
Table 3 is tabulated for natural frequency and damping of symmetric (0°/θ°/0°) LCB with an aspect ratio of 5. The results clearly depicts effect of boundary condition is more influential than effect of fiber angle. The depiction is due to, stiffness of the beam is directly influenced to the change in boundary condition. CC beams are more stiffer than other two beams and hence these are having higher natural frequency as anticipated. The variation in the values of natural frequency and damping observes to be marginal along the fiber angle variation. But, the frequency amplitude of CC beams are starting from high frequency band and that of SS and CF beams are falling in between lower frequency band and medium frequency band. Modal damping of SSLCB modes are much higher than the other LCBs under other two boundary conditions due to the less structural stiffness.
The variation in natural frequency and modal damping of symmetric (0°/θ°/0°) LCB with L/h = 20 is presented in Table 4. Unlike the LCB with L/h = 5, there is marginal variation in natural frequency for all the boundary conditions. However, there is significant change in the damping values as observed for L/h = 5 cases. It is also observed that due to the higher aspect ratio the LCBs with L/h = 20 have lower damping values compared to the corresponding LCBs with L/h = 5, except for few cases.
Effect of lamination scheme, boundary condition and aspect ratio on the GELCB and CELCB is reported in Tables 5 and 6, respectively. From the results of Table 5 it is clear that there is a greater variation in natural frequencies and damping values with regard to GELCB and CELCB. In general, natural frequencies of CELCB are higher than that of the GELCB due to higher elastic modulus associated with CE material. However, the modal damping values of GELCBs are much higher than the CELCBs due to the better fibrematrix interaction, this can be clearly observed for thick beams. The fibermatrix interaction is better in GELCBs as the relatively weaker glass fiber able to interact more with the the matrix compared to the strong carbon fiber. In the case of layup scheme, higher natural frequency is obtained for CELCBs with symmetric crosply (0°/90°/90°/0°) for both the thin and thick beams. Similarly higher damping is observed for GELCB with antisymmetric angleply (45°/–45°/45°/–45°).
Natural frequencies and modal damping of 0°/θ°/0° GELCB beam with L/h = 5.
Natural frequencies and modal damping of 0°/θ°/0° GELCB beam with L/h = 20.
Influence of lamination scheme on frequency and modal damping of the fundamental mode of LCB with L/h = 5.
Influence of lamination scheme on frequency and modal damping of the fundamental mode of LCB with L/h = 20.
5 Conclusion
Free vibration and damping investigation on glass epoxy and carbon epoxy LCBs with various configurations including boundary conditions, aspect ratio, fiber angle and lay up is presented. The solutions in the study are obtained using Ritz method with polynomial displacement field. From the results the following conclusions are drawn:
The type of boundary conditions significantly effecting the natural frequency and damping of LCBs.
Natural frequencies of thick beams are not much sensitive to lamination scheme while, damping of thick beams is highly sensitive to the lamination scheme.
Modal damping of thin beams are not much influenced by the lamination scheme while the natural frequency of thin beams is very much sensitive to lamination scheme.
Natural frequencies and modal damping are also highly sensitive to the nature of the reinforced material. The relatively strong fiber reinforcement enhances natural frequencies due to the higher elastic modulus associated with it. The relatively weak fiber reinforcement enhances modal damping due to the better fibermatrix interaction.
References
 A. Karamanli, M. Aydogdu, Buckling of laminated composite and sandwich beams due to axially varying inplane loads, Compos. Struct. 210, 391–408 (2019) [CrossRef] [Google Scholar]
 M.P. Arunkumar, M. Jagadeesh, J. Pitchaimani, K.V. Gangadharan, M.L. Babu, Sound radiation and transmission loss characteristics of a honeycomb sandwich panel with composite facings: effect of inherent material damping, J. Sound Vibr. 383, 221–232 (2016) [CrossRef] [Google Scholar]
 M. Rajesh, J. Pitchaimani, Experimental investigation on buckling and free vibration behavior of woven natural fiber fabric composite under axial compression, Compos. Struct. 163, 302–311 (2017) [CrossRef] [Google Scholar]
 K.S. Kumar, I. Siva, P. Jeyaraj, J.W. Jappes, S.C. Amico, N. Rajini, Synergy of fiber length and content on free vibration and damping behavior of natural fiber reinforced polyester composite beams, Mater. Des. (1980–2015) 56, 379–386 (2014) [CrossRef] [Google Scholar]
 A.S. Sayyad, Y.M. Ghugal, Bending, buckling and free vibration of laminated composite and sandwich beams: a critical review of literature, Compos. Struct. 171, 486–504 (2017) [CrossRef] [Google Scholar]
 T.P. Vo, H.T. Thai, M. Aydogdu, Free vibration of axially loaded composite beams using a fourunknown shear and normal deformation theory, Compos. Struct. 178, 406–414 (2017) [CrossRef] [Google Scholar]
 T.K. Nguyen, B.D. Nguyen, T.P. Vo, H.T. Thai, A novel unified model for laminated composite beams, Compos. Struct. 238, 111943 (2020) [CrossRef] [Google Scholar]
 P. Jeyaraj, N. Ganesan, C. Padmanabhan, Vibration and acoustic response of a composite plate with inherent material damping in a thermal environment, J. Sound Vibr. 320, 322–338 (2009) [CrossRef] [Google Scholar]
 M.A. Eltaher, S.A. Mohamed, Buckling and stability analysis of sandwich beams subjected to varying axial loads, Steel Compos. Struct. 34, 241–260 (2020) [Google Scholar]
 J. Li, X. Hu, X. Li, Free vibration analyses of axially loaded laminated composite beams using a unified higherorder shear deformation theory and dynamic stiffness method, Compos. Struct. 158, 308–322 (2016) [CrossRef] [Google Scholar]
 N.D. Nguyen, T.K. Nguyen, T.P. Vo, H.T. Thai, Ritzbased analytical solutions for bending, buckling and vibration behavior of laminated composite beams, Int. J. Struct. Stab. Dyn. 18, 1850130 (2018) [CrossRef] [Google Scholar]
 R. Chandra, S.P. Singh, K. Gupta, Damping studies in fiberreinforced composites − a review, Compos. Struct. 46, 41–51 (1999) [CrossRef] [Google Scholar]
 P. Jeyaraj, N. Ganesan, C. Padmanabhan, Vibration and acoustic response of a composite plate with inherent material damping in a thermal environment, J. Sound Vib. 320, 322–338 (2009) [CrossRef] [Google Scholar]
 R.G. Ni, R.D. Adams, The damping and dynamic moduli of symmetric laminated composite beams − theoretical and experimental results, J. Compos. Mater. 18, 104–121 (1984) [CrossRef] [Google Scholar]
 M.S. Ozer, H. Koruk, K.Y. Sanliturk, Development of an equivalent shell finite element for modelling damped multilayered composite structures, Compos. Struct. 254, 112828 (2020) [CrossRef] [Google Scholar]
 H. Li, Y. Niu, Z. Li, Z. Xu, Q. Han, Modeling of amplitudedependent damping characteristics of fiber reinforced composite thin plate, Appl. Math. Model. 80, 394–407 (2020) [CrossRef] [Google Scholar]
 D.X. Lin, R.G. Ni, R.D. Adams, Prediction and measurement of the vibrational damping parameters of carbon and glass fiberreinforced plastics plates, J. Compos. Mater. 18, 132–152 (1984) [CrossRef] [Google Scholar]
 M. Bruyneel, B. Colson, P. Jetteur, C. Raick, A. Remouchamps, S. Grihon, Recent progress in the optimal design of composite structures: industrial solution procedures on case studies, Int. J. Simul. Multidiscipl. Des. Optim. 2, 283–288 (2008) [CrossRef] [Google Scholar]
 E.H. Irhirane, J. Echaabi, M. Hattabi, M. Aboussaleh, A. Saouab, A comparative study of failure criteria applied to composite materials, Int. J. Simul. Multidiscipl. Des. Optim. 2, 141–147 (2008) [CrossRef] [Google Scholar]
 F.S. Liao, A.C. Su, T.C. Hsu, Vibration damping of interleaved carbon fiberepoxy composite beams, J. Compos. Mater. 28, 1840–1854 (1994) [CrossRef] [Google Scholar]
Cite this article as: Somi Naidu Balireddy, Pitchaimani Jeyaraj, Lenin Babu Mailan Chinnapandi, Ch V.S.N. Reddi, Effect of lamination schemes on natural frequency and modal damping of fiber reinforced laminated beam using Ritz method, Int. J. Simul. Multidisci. Des. Optim. 12, 15 (2021)
All Tables
Influence of lamination scheme on frequency and modal damping of the fundamental mode of LCB with L/h = 5.
Influence of lamination scheme on frequency and modal damping of the fundamental mode of LCB with L/h = 20.
All Figures
Fig. 1 Geometry of the analysed laminated beam. 

In the text 
Current usage metrics show cumulative count of Article Views (fulltext article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 4896 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.