Vibration Analysis of Cracked Composite Laminated Plate and Beam Structures
Main Article Content
Abstract
The composite structures usually experiences vibration during service. Presence of delamination in a structure not only significantly impacts the stiffness of the structure with declining trends but also affect their vibration properties. The purpose of this paper is to review the literature studies on the effect of vibration on composite delamianted structures. The literature studies have been classified on the basis of methodology i.e experimental, analytical and finite element methods. In this review, particular attention has been given to the study of results, comparison with the work conducted by other researchers in the field of the free and forced vibration behavior of the composite laminates subjected to delamination or crack.
Downloads
Download data is not yet available.
Article Details
How to Cite
[1]
2018. Vibration Analysis of Cracked Composite Laminated Plate and Beam Structures. Romanian Journal of Acoustics and Vibration. 15, 1 (Aug. 2018), 3–13.
Section
Articles
Journal author rights
In order for Romanian Journal of Acoustics and Vibration to publish and disseminate research articles, we need publishing rights. This is determined by a publishing agreement between the author and Romanian Journal of Acoustics and Vibration. This agreement deals with the transfer or license of the copyright to Romanian Journal of Acoustics and Vibration and authors retain significant rights to use and share their own published articles. Romanian Journal of Acoustics and Vibration supports the need for authors to share, disseminate and maximize the impact of their research and these rights, in Romanian Journal of Acoustics and Vibration proprietary are defined below:
For subscription articles:
Authors transfer copyright to the publisher as part of a journal publishing agreement, but have the right to: Share their article for personal use (manuscript version); retain patent, trademark and other intellectual property rights (including research data); proper attribution and credit for the published work.
For open access articles:
Authors sign an exclusive license agreement, where authors have copyright but license exclusive rights in their article to the publisher. In this case authors have the right to: share their article in the same ways permitted to third parties under the relevant user license; retain patent, trademark and other intellectual property rights (including research data); proper attribution and credit for the published work.
Rights granted to Romanian Journal of Acoustics and Vibration
For both subscription and open access articles, published in proprietary titles, Romanian Journal of Acoustics and Vibration is granted the following rights:
- The exclusive right to publish and distribute an article, and to grant rights to others, including for commercial purposes;
- For open access articles, Romanian Journal of Acoustics and Vibration will apply the relevant third party user licence (Open access licencses) where Romanian Journal of Acoustics and Vibration publishes the article on its online platforms;
- The right to provide the article in all forms and media so the article can be used on the latest technology even after publication;
- The authority to enforce the rights in the article, on behalf of an author, against third parties, for example in the case of plagiarism or copyright infringement.
How to Cite
[1]
2018. Vibration Analysis of Cracked Composite Laminated Plate and Beam Structures. Romanian Journal of Acoustics and Vibration. 15, 1 (Aug. 2018), 3–13.
References
[1] Friedrich, K. and Almajid, A.A., Manufacturing aspects of advanced polymer composites for automotive applications, Applied Composite Materials, Vol.20, No. 2, 2013, pp. 107-128
[2] Sharma, A., Sharma, A.K., Raghav, A.K., and Kumar, V., Vibrational study of square plate with thermal effect and circular variation in density, Romanian Journal of Acoustics and Vibration, Vol.13, No. 2, 2016, pp. 146-152
[3] Muhammad Imran, R.K., Saeed Badshah, Finite Element Analysis to Investigate the Influence of Delamination Size, Stacking Sequence and Boundary Conditions on the Vibration Behavior of Composite Plate, Iranian Journal of Materials Science and Engineering, 2018,
[4] Corum, J., Battiste, R., Liu, K., and Ruggles, M., Basic Properties of Reference Cross ply Carbon-Fiber Composite, Oak Ridge National Laboratory, ORNL/TM-2000, 2000,
[5] Muhammad Imran, R.K., Saeed Badshah, Vibrtaion Analysis of Cracked Composite Laminated Plate, Pakistan Journal of Scientific and Industrial Research Series A: Physical Sciences Vol.61(A), No. 2, 2018,
[6] Imran, M., Pre-stress and Free Vibration Optimization of Composite Ocean Current Turbine Blade, International Journal of Science, Engineering and Innovative Research, Vol.3, 2015, pp. 1-5
[7] Ghiasi, H., Fayazbakhsh, K., Pasini, D., and Lessard, L., Optimum stacking sequence design of composite materials Part II: Variable stiffness design, Composite Structures, Vol.93, No. 1, 2010, pp. 1-13
[8] Cavallo, T., Zappino, E., and Carrera, E., Free-vibration analysis of space vehicle structures made by composite materials, Composite Structures, 2017,
[9] Sajid Riaz, Z.A., Buckling Analysis of an Ocean Current Turbine Blade, International Journal of Science, Engineering and Innovative Research, Vol.2, 2015, pp. 6-11
[10] Liu, Y., Xiao, J., and Shu, D., Free Vibration of Delaminated Beams with an Edge Crack, Procedia Engineering, Vol.75, 2014, pp. 78-82
[11] Abbas, S., Abbas, Z., and Pasha, R.A., Analytical Estimation and Numerical Simulation of Vibration based Piezoelectric Energy Harvesters, International Journal of Science, Engineering and Innovative Research, Vol.13, No. 13, 2016, pp. 1-9
[12] Luo, H. and Hanagud, S. Delamination detection using dynamic characteristics of composite plates. in 36th Structures, Structural Dynamics and Materials Conference. 1995.
[13] Luo, H. and Hanagud, S., Delamination modes in composite plates, Journal of Aerospace Engineering, Vol.9, No. 4, 1996, pp. 106-113
[14] Ghoneam, S.M., Dynamic analysis of open cracked laminated composite beams, Composite Structures, Vol.32, No. 1, 1995, pp. 3-11
[15] Shu, D. and Della, C.N., Vibrations of multiple delaminated beams, Composite Structures, Vol.64, No. 3–4, 2004, pp. 467-477
[16] Parhi, P., Bhattacharyya, S., and Sinha, P., Finite element dynamic analysis of laminated composite plates with multiple delaminations, Journal of reinforced plastics and composites, Vol.19, No. 11, 2000, pp. 863-882
[17] Della, C.N. and Shu, D., Vibration of beams with double delaminations, Journal of Sound and Vibration, Vol.282, No. 3–5, 2005, pp. 919-935
[18] Della, C.N. and Shu, D., Free vibration analysis of composite beams with overlapping delaminations, European Journal of Mechanics - A/Solids, Vol.24, No. 3, 2005, pp. 491-503
[19] Mujumdar, P.M. and Suryanarayan, S., Flexural vibrations of beams with delaminations, Journal of Sound and Vibration, Vol.125, No. 3, 1988, pp. 441-461
[20] Sharma, A., Kumar, V., and Raghav, A.K., Vibrational frequency of circular tapered square plate, Romanian Journal of Acoustics and Vibration, Vol.14, No. 1, 2017, pp. 21
[21] Lestari, W. and Hanagud, S. Health monitoring of structures-Multiple delamination dynamics in composite beams. in 40th Structures, Structural Dynamics, and Materials Conference and Exhibit. 1999.
[22] Chattopadhyay, A., Gu, H., and Dragomir-Daescu, D., Dynamics of delaminated composite plates with piezoelectric actuators, AIAA journal, Vol.37, No. 2, 1999, pp. 248-254
[23] Shen, M.-H. and Grady, J., Free vibrations of delaminated beams, AIAA journal, Vol.30, No. 5, 1992, pp. 1361-1370
[24] Lee, S., Park, T., and Voyiadjis, G.Z., Free vibration analysis of axially compressed laminated composite beam-columns with multiple delaminations, Composites Part B: Engineering, Vol.33, No. 8, 2002, pp. 605-617
[25] Wang, J.T.-S. and Lin, C.-C., Engineering analysis of buckling of delaminated beam plates, Composite structures, Vol.34, No. 4, 1996, pp. 397-407
[26] Kim, H.-Y. and Hwang, W., Effect of debonding on natural frequencies and frequency response functions of honeycomb sandwich beams, Composite Structures, Vol.55, No. 1, 2002, pp. 51-62
[27] Burlayenko, V.N. and Sadowski, T., Influence of skin/core debonding on free vibration behavior of foam and honeycomb cored sandwich plates, International Journal of Non-Linear Mechanics, Vol.45, No. 10, 2010, pp. 959-968
[28] Shu, D., Vibration of sandwich beams with double delaminations, Composites Science and Technology, Vol.54, No. 1, 1995, pp. 101-109
[29] Dey, S. and Karmakar, A., Free vibration analyses of multiple delaminated angle-ply composite conical shells – A finite element approach, Composite Structures, Vol.94, No. 7, 2012, pp. 2188-2196
[30] Marjanović, M. and Vuksanović, D., Layerwise solution of free vibrations and buckling of laminated composite and sandwich plates with embedded delaminations, Composite Structures, Vol.108, 2014, pp. 9-20
[31] Brandinelli, L. and Massabò, R., Free vibrations of delaminated beam-type structures with crack bridging, Composite Structures, Vol.61, No. 1–2, 2003, pp. 129-142
[32] Luo, S.-N., Yi-Ming, F., and Zhi-Yuan, C., Non-linear vibration of composite beams with an arbitrary delamination, Journal of Sound and Vibration, Vol.271, No. 3–5, 2004, pp. 535-545
[33] Chen, D.-L., Dynamic Analysis of Composite Laminated Circular Plate with Circular Delamination, Journal of Mechanics, Vol.32, No. 6, 2016, pp. 683-692
[34] Lee, J., Free vibration analysis of delaminated composite beams, Computers & Structures, Vol.74, No. 2, 2000, pp. 121-129
[35] Wang, J., Liu, Y., and Gibby, J., Vibrations of split beams, Journal of sound and vibration, Vol.84, No. 4, 1982, pp. 491-502
[36] Hirwani, C.K., Patil, R.K., Panda, S.K., Mahapatra, S.S., Mandal, S.K., Srivastava, L., and Buragohain, M.K., Experimental and numerical analysis of free vibration of delaminated curved panel, Aerospace Science and Technology, Vol.54, 2016, pp. 353-370
[37] Nanda, N. and Sahu, S.K., Free vibration analysis of delaminated composite shells using different shell theories, International Journal of Pressure Vessels and Piping, Vol.98, 2012, pp. 111-118
[38] Yam, L.H., Wei, Z., Cheng, L., and Wong, W.O., Numerical analysis of multi-layer composite plates with internal delamination, Computers & Structures, Vol.82, No. 7–8, 2004, pp. 627-637
[39] Lin, D., Ni, R., and Adams, R., Prediction and measurement of the vibrational damping parameters of carbon and glass fibre-reinforced plastics plates, Journal of composite materials, Vol.18, No. 2, 1984, pp. 132-152
[40] Wei, Z., Yam, L.H., and Cheng, L., Detection of internal delamination in multi-layer composites using wavelet packets combined with modal parameter analysis, Composite Structures, Vol.64, No. 3–4, 2004, pp. 377-387
[41] Alnefaie, K., Finite element modeling of composite plates with internal delamination, Composite Structures, Vol.90, No. 1, 2009, pp. 21-27
[42] Krawczuk, M., Ostachowicz, W., and Zak, A., Analysis of natural frequencies of delaminated composite beams based on finite element method, Structural Engineering and Mechanics, Vol.4, No. 3, 1996, pp. 243-255
[43] Krawczuk, M., Ostachowicz, W., and Zak, A., Natural vibration frequencies of delaminated composite beams, Computer Assisted Mechanics and Engineering Sciences, Vol.3, No. 3, 1996, pp. 233-243
[44] Krawczuk, M., Ostachowicz, W., and Zak, A., Dynamics of cracked composite material structures, Computational Mechanics, Vol.20, No. 1, 1997, pp. 79-83
[45] Kim, H.S., Chattopadhyay, A., and Ghoshal, A., Dynamic analysis of composite laminates with multiple delamination using improved layerwise theory, AIAA journal, Vol.41, No. 9, 2003, pp. 1771-1779
[46] Thornburgh, R. and Chattopadhyay, A. Modeling the dynamic effects of delamination in adaptive composite laminate. in 43rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference. 2002.
[47] Ju, F., Lee, H., and Lee, K., Free-vibration analysis of composite beams with multiple delaminations, Composites Engineering, Vol.4, No. 7, 1994, pp. 715-730
[48] Kitipornchai, S., Ke, L.L., Yang, J., and Xiang, Y., Nonlinear vibration of edge cracked functionally graded Timoshenko beams, Journal of Sound and Vibration, Vol.324, No. 3–5, 2009, pp. 962-982
[49] Qu, G.M., Li, Y.Y., Cheng, L., and Wang, B., Vibration analysis of a piezoelectric composite plate with cracks, Composite Structures, Vol.72, No. 1, 2006, pp. 111-118
[50] Chang, T.-P., Hu, C.-Y., and Jane, K.-C., Vibration Analysis of Delaminated Composite Plates Under Axial Load∗, Journal of Structural Mechanics, Vol.26, No. 2, 1998, pp. 195-218
[51] Tenek, L.H., Henneke, E.G., and Gunzburger, M.D., Vibration of delaminated composite plates and some applications to non-destructive testing, Composite Structures, Vol.23, No. 3, 1993, pp. 253-262
[52] Wang, J. and Lin, C.-C. Vibration of beam-plates having multiple delaminations. in 36th Structures, Structural Dynamics and Materials Conference. 1995.
[53] Jafari-Talookolaei, R.-A., Analytical solution for the free vibration characteristics of the rotating composite beams with a delamination, Aerospace Science and Technology, Vol.45, 2015, pp. 346-358
[54] Kulkarni, S.V. and Frederick, D., Frequency as a parameter in delamination problems—a preliminary investigation, Journal of composite materials, Vol.5, No. 1, 1971, pp. 112-117
[55] Hammami, M., El Mahi, A., Karra, C., and Haddar, M., Experimental analysis of the linear and nonlinear behaviour of composites with delaminations, Applied Acoustics, Vol.108, 2016, pp. 31-39
[56] Jian, X., Tzou, H., Lissenden, C., and Penn, L., Damage detection by piezoelectric patches in a free vibration method, Journal of Composite Materials, Vol.31, No. 4, 1997, pp. 345-359
[57] Paolozzi, A. and Peroni, I., Experimental assessment of debonding damage in a carbon-fibre reinforced plastic sandwich panel by frequency variations, Composite Structures, Vol.35, No. 4, 1996, pp. 435-444
[58] Diaz Valdes, S.H. and Soutis, C., Delamination detection in composite laminates from variations of their modal characteristics, Journal of Sound and Vibration, Vol.228, No. 1, 1999, pp. 1-9
[59] Kessler, S.S., Spearing, S.M., Atalla, M.J., Cesnik, C.E.S., and Soutis, C., Damage detection in composite materials using frequency response methods, Composites Part B: Engineering, Vol.33, No. 1, 2002, pp. 87-95
[60] Penn, L., Jump, J., Greenfield, M., and Blandford, G., Use of the free vibration spectrum to detect delamination in thick composites, Journal of Composite Materials, Vol.33, No. 1, 1999, pp. 54-72
[61] Hou, J. and Jeronimidis, G., Vibration of delaminated thin composite plates, Composites Part A: Applied Science and Manufacturing, Vol.30, No. 8, 1999, pp. 989-995
[62] Hou, J. and Jeronimidis, G., Bending stiffness of composite plates with delamination, Composites Part A: applied science and manufacturing, Vol.31, No. 2, 2000, pp. 121-132
[63] Ananda Babu, A. and Vasudevan, R., Vibration analysis of rotating delaminated non-uniform composite plates, Aerospace Science and Technology, Vol.60, 2017, pp. 172-182
[64] Hirwani, C., Sahoo, S., and Panda, S. Effect of delamination on vibration behaviour of woven Glass/Epoxy composite plate-An experimental study. in IOP Conference Series: Materials Science and Engineering. 2016. IOP Publishing.
[65] Mohanty, J., Sahu, S., and Parhi, P., Numerical and experimental study on free vibration of delaminated woven fiber glass/epoxy composite plates, International Journal of Structural Stability and Dynamics, Vol.12, No. 02, 2012, pp. 377-394
[66] Luo, H., Hanagud, S., Luo, H., and Hanagud, S. Delaminated beam nonlinear dynamic response calculation and visualization. in 38th Structures, Structural Dynamics, and Materials Conference. 1997.
[67] Luo, H., Hanagud, S., Luo, H., and Hanagud, S. PVDF sensor and its applications in delamination response detection. in 38th Structures, Structural Dynamics, and Materials Conference. 1997.
[68] Lee, S., Park, T., and Voyiadjis, G.Z., Vibration analysis of multi-delaminated beams, Composites Part B: Engineering, Vol.34, No. 7, 2003, pp. 647-659
[69] Zhu, J.-F., Gu, Y., and Tong, L., Formulation of reference surface element and its applications in dynamic analysis of delaminated composite beams, Composite Structures, Vol.68, No. 4, 2005, pp. 481-490
[70] Ju, F., Lee, H., and Lee, K., Finite element analysis of free vibration of delaminated composite plates, Composites engineering, Vol.5, No. 2, 1995, pp. 195-209
[71] Luo, H. and Hanagud, S., Dynamics of delaminated beams, International Journal of Solids and Structures, Vol.37, No. 10, 2000, pp. 1501-1519
[72] Hammami, M., El Mahi, A., Karra, C., and Haddar, M., Nonlinear behaviour of glass fibre reinforced composites with delamination, Composites Part B: Engineering, Vol.92, 2016, pp. 350-359
[73] Kumar, A. and Shrivastava, R., Free vibration of square laminates with delamination around a central cutout using HSDT, Composite Structures, Vol.70, No. 3, 2005, pp. 317-333
[74] Reddy, J.N. and Phan, N.D., Stability and vibration of isotropic, orthotropic and laminated plates according to a higher-order shear deformation theory, Journal of Sound and Vibration, Vol.98, No. 2, 1985, pp. 157-170
[75] Shu, D. and Della, C.N., Free vibration analysis of composite beams with two non-overlapping delaminations, International Journal of Mechanical Sciences, Vol.46, No. 4, 2004, pp. 509-526
[76] Rout, M., Hota, S.S., and Karmakar, A., Free vibration characteristics of delaminated composite pretwisted stiffened cylindrical shell, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 2017, pp. 0954406216686389
[77] Thornburgh, R. and Chattopadhyay, A. Combined delamination and matrix cracking in adaptive composite laminates. in 44th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference. 2003.
[78] Thornburgh, R. and Chattopadhyay, A., Unified approach to modeling matrix cracking and delamination in laminated composite structures, AIAA journal, Vol.39, No. 1, 2001, pp. 153-160
[79] Sahoo, S.S., Panda, S.K., and Sen, D., Effect of Delamination on Static and Dynamic Behavior of Laminated Composite Plate, AIAA Journal, 2016, pp. 2530-2544
[80] Jafari-Talookolaei, R.-A., Abedi, M., and Hajianmaleki, M., Vibration characteristics of generally laminated composite curved beams with single through-the-width delamination, Composite Structures, Vol.138, 2016, pp. 172-183
[81] Ahmed, K.M., Free vibration of curved sandwich beams by the method of finite elements, Journal of Sound and Vibration, Vol.18, No. 1, 1971, pp. 61-74
[82] Ganesh, S., Kumar, K.S., and Mahato, P.K., Free Vibration Analysis of Delaminated Composite Plates Using Finite Element Method, Procedia Engineering, Vol.144, 2016, pp. 1067-1075
[83] HANAGUD, S. Delaminations in smart composite structures-A parametric study on vibrations. in 31st Structures, Structural Dynamics and Materials Conference. 1990.
[84] Shankar, G., Keshava Kumar, S., and Mahato, P.K., Vibration analysis and control of smart composite plates with delamination and under hygrothermal environment, Thin-Walled Structures, Vol.116, 2017, pp. 53-68
[85] Panda, H., Sahu, S., and Parhi, P., Hygrothermal effects on free vibration of delaminated woven fiber composite plates–Numerical and experimental results, Composite Structures, Vol.96, 2013, pp. 502-513
[86] Della, C.N. and Shu, D., Vibration of delaminated multilayer beams, Composites Part B: Engineering, Vol.37, No. 2–3, 2005, pp. 227-236
[87] Karmakar, A., Roy, H., and Kishimoto, K., Free vibration analysis of delaminated composite pretwisted shells, Aircraft Engineering and Aerospace Technology, Vol.77, No. 6, 2005, pp. 486-490
[88] Hu, N., Fukunaga, H., Kameyama, M., Aramaki, Y., and Chang, F.K., Vibration analysis of delaminated composite beams and plates using a higher-order finite element, International Journal of Mechanical Sciences, Vol.44, No. 7, 2002, pp. 1479-1503
[89] Gadelrab, R.M., the effect of delamination on the natural frequencies of a laminated composite beam, Journal of Sound and Vibration, Vol.197, No. 3, 1996, pp. 283-292
[90] Park, T., Lee, S.Y., and Voyiadjis, G.Z., Finite element vibration analysis of composite skew laminates containing delaminations around quadrilateral cutouts, Composites Part B: Engineering, Vol.40, No. 3, 2009, pp. 225-236
[91] Saravanos, D.A. and Hopkins, D.A., effects of delaminations on the damped dynamic characteristics of composite laminates: analysis and experiments, Journal of Sound and Vibration, Vol.192, No. 5, 1996, pp. 977-993
[92] Pardoen, G.C., Effect of delamination on the natural frequencies of composite laminates, Journal of composite materials, Vol.23, No. 12, 1989, pp. 1200-1215
[93] Lestari, W., Lu, X., and Hanagud, S. Dynamics of a delaminated beam-Effects of nonlinearity. in 41st Structures, Structural Dynamics, and Materials Conference and Exhibit. 2000.
[94] Lu, X., Lestari, W., and Hanagud, S., Nonlinear vibrations of a delaminated beam, Journal of Vibration and Control, Vol.7, No. 6, 2001, pp. 803-831
[2] Sharma, A., Sharma, A.K., Raghav, A.K., and Kumar, V., Vibrational study of square plate with thermal effect and circular variation in density, Romanian Journal of Acoustics and Vibration, Vol.13, No. 2, 2016, pp. 146-152
[3] Muhammad Imran, R.K., Saeed Badshah, Finite Element Analysis to Investigate the Influence of Delamination Size, Stacking Sequence and Boundary Conditions on the Vibration Behavior of Composite Plate, Iranian Journal of Materials Science and Engineering, 2018,
[4] Corum, J., Battiste, R., Liu, K., and Ruggles, M., Basic Properties of Reference Cross ply Carbon-Fiber Composite, Oak Ridge National Laboratory, ORNL/TM-2000, 2000,
[5] Muhammad Imran, R.K., Saeed Badshah, Vibrtaion Analysis of Cracked Composite Laminated Plate, Pakistan Journal of Scientific and Industrial Research Series A: Physical Sciences Vol.61(A), No. 2, 2018,
[6] Imran, M., Pre-stress and Free Vibration Optimization of Composite Ocean Current Turbine Blade, International Journal of Science, Engineering and Innovative Research, Vol.3, 2015, pp. 1-5
[7] Ghiasi, H., Fayazbakhsh, K., Pasini, D., and Lessard, L., Optimum stacking sequence design of composite materials Part II: Variable stiffness design, Composite Structures, Vol.93, No. 1, 2010, pp. 1-13
[8] Cavallo, T., Zappino, E., and Carrera, E., Free-vibration analysis of space vehicle structures made by composite materials, Composite Structures, 2017,
[9] Sajid Riaz, Z.A., Buckling Analysis of an Ocean Current Turbine Blade, International Journal of Science, Engineering and Innovative Research, Vol.2, 2015, pp. 6-11
[10] Liu, Y., Xiao, J., and Shu, D., Free Vibration of Delaminated Beams with an Edge Crack, Procedia Engineering, Vol.75, 2014, pp. 78-82
[11] Abbas, S., Abbas, Z., and Pasha, R.A., Analytical Estimation and Numerical Simulation of Vibration based Piezoelectric Energy Harvesters, International Journal of Science, Engineering and Innovative Research, Vol.13, No. 13, 2016, pp. 1-9
[12] Luo, H. and Hanagud, S. Delamination detection using dynamic characteristics of composite plates. in 36th Structures, Structural Dynamics and Materials Conference. 1995.
[13] Luo, H. and Hanagud, S., Delamination modes in composite plates, Journal of Aerospace Engineering, Vol.9, No. 4, 1996, pp. 106-113
[14] Ghoneam, S.M., Dynamic analysis of open cracked laminated composite beams, Composite Structures, Vol.32, No. 1, 1995, pp. 3-11
[15] Shu, D. and Della, C.N., Vibrations of multiple delaminated beams, Composite Structures, Vol.64, No. 3–4, 2004, pp. 467-477
[16] Parhi, P., Bhattacharyya, S., and Sinha, P., Finite element dynamic analysis of laminated composite plates with multiple delaminations, Journal of reinforced plastics and composites, Vol.19, No. 11, 2000, pp. 863-882
[17] Della, C.N. and Shu, D., Vibration of beams with double delaminations, Journal of Sound and Vibration, Vol.282, No. 3–5, 2005, pp. 919-935
[18] Della, C.N. and Shu, D., Free vibration analysis of composite beams with overlapping delaminations, European Journal of Mechanics - A/Solids, Vol.24, No. 3, 2005, pp. 491-503
[19] Mujumdar, P.M. and Suryanarayan, S., Flexural vibrations of beams with delaminations, Journal of Sound and Vibration, Vol.125, No. 3, 1988, pp. 441-461
[20] Sharma, A., Kumar, V., and Raghav, A.K., Vibrational frequency of circular tapered square plate, Romanian Journal of Acoustics and Vibration, Vol.14, No. 1, 2017, pp. 21
[21] Lestari, W. and Hanagud, S. Health monitoring of structures-Multiple delamination dynamics in composite beams. in 40th Structures, Structural Dynamics, and Materials Conference and Exhibit. 1999.
[22] Chattopadhyay, A., Gu, H., and Dragomir-Daescu, D., Dynamics of delaminated composite plates with piezoelectric actuators, AIAA journal, Vol.37, No. 2, 1999, pp. 248-254
[23] Shen, M.-H. and Grady, J., Free vibrations of delaminated beams, AIAA journal, Vol.30, No. 5, 1992, pp. 1361-1370
[24] Lee, S., Park, T., and Voyiadjis, G.Z., Free vibration analysis of axially compressed laminated composite beam-columns with multiple delaminations, Composites Part B: Engineering, Vol.33, No. 8, 2002, pp. 605-617
[25] Wang, J.T.-S. and Lin, C.-C., Engineering analysis of buckling of delaminated beam plates, Composite structures, Vol.34, No. 4, 1996, pp. 397-407
[26] Kim, H.-Y. and Hwang, W., Effect of debonding on natural frequencies and frequency response functions of honeycomb sandwich beams, Composite Structures, Vol.55, No. 1, 2002, pp. 51-62
[27] Burlayenko, V.N. and Sadowski, T., Influence of skin/core debonding on free vibration behavior of foam and honeycomb cored sandwich plates, International Journal of Non-Linear Mechanics, Vol.45, No. 10, 2010, pp. 959-968
[28] Shu, D., Vibration of sandwich beams with double delaminations, Composites Science and Technology, Vol.54, No. 1, 1995, pp. 101-109
[29] Dey, S. and Karmakar, A., Free vibration analyses of multiple delaminated angle-ply composite conical shells – A finite element approach, Composite Structures, Vol.94, No. 7, 2012, pp. 2188-2196
[30] Marjanović, M. and Vuksanović, D., Layerwise solution of free vibrations and buckling of laminated composite and sandwich plates with embedded delaminations, Composite Structures, Vol.108, 2014, pp. 9-20
[31] Brandinelli, L. and Massabò, R., Free vibrations of delaminated beam-type structures with crack bridging, Composite Structures, Vol.61, No. 1–2, 2003, pp. 129-142
[32] Luo, S.-N., Yi-Ming, F., and Zhi-Yuan, C., Non-linear vibration of composite beams with an arbitrary delamination, Journal of Sound and Vibration, Vol.271, No. 3–5, 2004, pp. 535-545
[33] Chen, D.-L., Dynamic Analysis of Composite Laminated Circular Plate with Circular Delamination, Journal of Mechanics, Vol.32, No. 6, 2016, pp. 683-692
[34] Lee, J., Free vibration analysis of delaminated composite beams, Computers & Structures, Vol.74, No. 2, 2000, pp. 121-129
[35] Wang, J., Liu, Y., and Gibby, J., Vibrations of split beams, Journal of sound and vibration, Vol.84, No. 4, 1982, pp. 491-502
[36] Hirwani, C.K., Patil, R.K., Panda, S.K., Mahapatra, S.S., Mandal, S.K., Srivastava, L., and Buragohain, M.K., Experimental and numerical analysis of free vibration of delaminated curved panel, Aerospace Science and Technology, Vol.54, 2016, pp. 353-370
[37] Nanda, N. and Sahu, S.K., Free vibration analysis of delaminated composite shells using different shell theories, International Journal of Pressure Vessels and Piping, Vol.98, 2012, pp. 111-118
[38] Yam, L.H., Wei, Z., Cheng, L., and Wong, W.O., Numerical analysis of multi-layer composite plates with internal delamination, Computers & Structures, Vol.82, No. 7–8, 2004, pp. 627-637
[39] Lin, D., Ni, R., and Adams, R., Prediction and measurement of the vibrational damping parameters of carbon and glass fibre-reinforced plastics plates, Journal of composite materials, Vol.18, No. 2, 1984, pp. 132-152
[40] Wei, Z., Yam, L.H., and Cheng, L., Detection of internal delamination in multi-layer composites using wavelet packets combined with modal parameter analysis, Composite Structures, Vol.64, No. 3–4, 2004, pp. 377-387
[41] Alnefaie, K., Finite element modeling of composite plates with internal delamination, Composite Structures, Vol.90, No. 1, 2009, pp. 21-27
[42] Krawczuk, M., Ostachowicz, W., and Zak, A., Analysis of natural frequencies of delaminated composite beams based on finite element method, Structural Engineering and Mechanics, Vol.4, No. 3, 1996, pp. 243-255
[43] Krawczuk, M., Ostachowicz, W., and Zak, A., Natural vibration frequencies of delaminated composite beams, Computer Assisted Mechanics and Engineering Sciences, Vol.3, No. 3, 1996, pp. 233-243
[44] Krawczuk, M., Ostachowicz, W., and Zak, A., Dynamics of cracked composite material structures, Computational Mechanics, Vol.20, No. 1, 1997, pp. 79-83
[45] Kim, H.S., Chattopadhyay, A., and Ghoshal, A., Dynamic analysis of composite laminates with multiple delamination using improved layerwise theory, AIAA journal, Vol.41, No. 9, 2003, pp. 1771-1779
[46] Thornburgh, R. and Chattopadhyay, A. Modeling the dynamic effects of delamination in adaptive composite laminate. in 43rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference. 2002.
[47] Ju, F., Lee, H., and Lee, K., Free-vibration analysis of composite beams with multiple delaminations, Composites Engineering, Vol.4, No. 7, 1994, pp. 715-730
[48] Kitipornchai, S., Ke, L.L., Yang, J., and Xiang, Y., Nonlinear vibration of edge cracked functionally graded Timoshenko beams, Journal of Sound and Vibration, Vol.324, No. 3–5, 2009, pp. 962-982
[49] Qu, G.M., Li, Y.Y., Cheng, L., and Wang, B., Vibration analysis of a piezoelectric composite plate with cracks, Composite Structures, Vol.72, No. 1, 2006, pp. 111-118
[50] Chang, T.-P., Hu, C.-Y., and Jane, K.-C., Vibration Analysis of Delaminated Composite Plates Under Axial Load∗, Journal of Structural Mechanics, Vol.26, No. 2, 1998, pp. 195-218
[51] Tenek, L.H., Henneke, E.G., and Gunzburger, M.D., Vibration of delaminated composite plates and some applications to non-destructive testing, Composite Structures, Vol.23, No. 3, 1993, pp. 253-262
[52] Wang, J. and Lin, C.-C. Vibration of beam-plates having multiple delaminations. in 36th Structures, Structural Dynamics and Materials Conference. 1995.
[53] Jafari-Talookolaei, R.-A., Analytical solution for the free vibration characteristics of the rotating composite beams with a delamination, Aerospace Science and Technology, Vol.45, 2015, pp. 346-358
[54] Kulkarni, S.V. and Frederick, D., Frequency as a parameter in delamination problems—a preliminary investigation, Journal of composite materials, Vol.5, No. 1, 1971, pp. 112-117
[55] Hammami, M., El Mahi, A., Karra, C., and Haddar, M., Experimental analysis of the linear and nonlinear behaviour of composites with delaminations, Applied Acoustics, Vol.108, 2016, pp. 31-39
[56] Jian, X., Tzou, H., Lissenden, C., and Penn, L., Damage detection by piezoelectric patches in a free vibration method, Journal of Composite Materials, Vol.31, No. 4, 1997, pp. 345-359
[57] Paolozzi, A. and Peroni, I., Experimental assessment of debonding damage in a carbon-fibre reinforced plastic sandwich panel by frequency variations, Composite Structures, Vol.35, No. 4, 1996, pp. 435-444
[58] Diaz Valdes, S.H. and Soutis, C., Delamination detection in composite laminates from variations of their modal characteristics, Journal of Sound and Vibration, Vol.228, No. 1, 1999, pp. 1-9
[59] Kessler, S.S., Spearing, S.M., Atalla, M.J., Cesnik, C.E.S., and Soutis, C., Damage detection in composite materials using frequency response methods, Composites Part B: Engineering, Vol.33, No. 1, 2002, pp. 87-95
[60] Penn, L., Jump, J., Greenfield, M., and Blandford, G., Use of the free vibration spectrum to detect delamination in thick composites, Journal of Composite Materials, Vol.33, No. 1, 1999, pp. 54-72
[61] Hou, J. and Jeronimidis, G., Vibration of delaminated thin composite plates, Composites Part A: Applied Science and Manufacturing, Vol.30, No. 8, 1999, pp. 989-995
[62] Hou, J. and Jeronimidis, G., Bending stiffness of composite plates with delamination, Composites Part A: applied science and manufacturing, Vol.31, No. 2, 2000, pp. 121-132
[63] Ananda Babu, A. and Vasudevan, R., Vibration analysis of rotating delaminated non-uniform composite plates, Aerospace Science and Technology, Vol.60, 2017, pp. 172-182
[64] Hirwani, C., Sahoo, S., and Panda, S. Effect of delamination on vibration behaviour of woven Glass/Epoxy composite plate-An experimental study. in IOP Conference Series: Materials Science and Engineering. 2016. IOP Publishing.
[65] Mohanty, J., Sahu, S., and Parhi, P., Numerical and experimental study on free vibration of delaminated woven fiber glass/epoxy composite plates, International Journal of Structural Stability and Dynamics, Vol.12, No. 02, 2012, pp. 377-394
[66] Luo, H., Hanagud, S., Luo, H., and Hanagud, S. Delaminated beam nonlinear dynamic response calculation and visualization. in 38th Structures, Structural Dynamics, and Materials Conference. 1997.
[67] Luo, H., Hanagud, S., Luo, H., and Hanagud, S. PVDF sensor and its applications in delamination response detection. in 38th Structures, Structural Dynamics, and Materials Conference. 1997.
[68] Lee, S., Park, T., and Voyiadjis, G.Z., Vibration analysis of multi-delaminated beams, Composites Part B: Engineering, Vol.34, No. 7, 2003, pp. 647-659
[69] Zhu, J.-F., Gu, Y., and Tong, L., Formulation of reference surface element and its applications in dynamic analysis of delaminated composite beams, Composite Structures, Vol.68, No. 4, 2005, pp. 481-490
[70] Ju, F., Lee, H., and Lee, K., Finite element analysis of free vibration of delaminated composite plates, Composites engineering, Vol.5, No. 2, 1995, pp. 195-209
[71] Luo, H. and Hanagud, S., Dynamics of delaminated beams, International Journal of Solids and Structures, Vol.37, No. 10, 2000, pp. 1501-1519
[72] Hammami, M., El Mahi, A., Karra, C., and Haddar, M., Nonlinear behaviour of glass fibre reinforced composites with delamination, Composites Part B: Engineering, Vol.92, 2016, pp. 350-359
[73] Kumar, A. and Shrivastava, R., Free vibration of square laminates with delamination around a central cutout using HSDT, Composite Structures, Vol.70, No. 3, 2005, pp. 317-333
[74] Reddy, J.N. and Phan, N.D., Stability and vibration of isotropic, orthotropic and laminated plates according to a higher-order shear deformation theory, Journal of Sound and Vibration, Vol.98, No. 2, 1985, pp. 157-170
[75] Shu, D. and Della, C.N., Free vibration analysis of composite beams with two non-overlapping delaminations, International Journal of Mechanical Sciences, Vol.46, No. 4, 2004, pp. 509-526
[76] Rout, M., Hota, S.S., and Karmakar, A., Free vibration characteristics of delaminated composite pretwisted stiffened cylindrical shell, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 2017, pp. 0954406216686389
[77] Thornburgh, R. and Chattopadhyay, A. Combined delamination and matrix cracking in adaptive composite laminates. in 44th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference. 2003.
[78] Thornburgh, R. and Chattopadhyay, A., Unified approach to modeling matrix cracking and delamination in laminated composite structures, AIAA journal, Vol.39, No. 1, 2001, pp. 153-160
[79] Sahoo, S.S., Panda, S.K., and Sen, D., Effect of Delamination on Static and Dynamic Behavior of Laminated Composite Plate, AIAA Journal, 2016, pp. 2530-2544
[80] Jafari-Talookolaei, R.-A., Abedi, M., and Hajianmaleki, M., Vibration characteristics of generally laminated composite curved beams with single through-the-width delamination, Composite Structures, Vol.138, 2016, pp. 172-183
[81] Ahmed, K.M., Free vibration of curved sandwich beams by the method of finite elements, Journal of Sound and Vibration, Vol.18, No. 1, 1971, pp. 61-74
[82] Ganesh, S., Kumar, K.S., and Mahato, P.K., Free Vibration Analysis of Delaminated Composite Plates Using Finite Element Method, Procedia Engineering, Vol.144, 2016, pp. 1067-1075
[83] HANAGUD, S. Delaminations in smart composite structures-A parametric study on vibrations. in 31st Structures, Structural Dynamics and Materials Conference. 1990.
[84] Shankar, G., Keshava Kumar, S., and Mahato, P.K., Vibration analysis and control of smart composite plates with delamination and under hygrothermal environment, Thin-Walled Structures, Vol.116, 2017, pp. 53-68
[85] Panda, H., Sahu, S., and Parhi, P., Hygrothermal effects on free vibration of delaminated woven fiber composite plates–Numerical and experimental results, Composite Structures, Vol.96, 2013, pp. 502-513
[86] Della, C.N. and Shu, D., Vibration of delaminated multilayer beams, Composites Part B: Engineering, Vol.37, No. 2–3, 2005, pp. 227-236
[87] Karmakar, A., Roy, H., and Kishimoto, K., Free vibration analysis of delaminated composite pretwisted shells, Aircraft Engineering and Aerospace Technology, Vol.77, No. 6, 2005, pp. 486-490
[88] Hu, N., Fukunaga, H., Kameyama, M., Aramaki, Y., and Chang, F.K., Vibration analysis of delaminated composite beams and plates using a higher-order finite element, International Journal of Mechanical Sciences, Vol.44, No. 7, 2002, pp. 1479-1503
[89] Gadelrab, R.M., the effect of delamination on the natural frequencies of a laminated composite beam, Journal of Sound and Vibration, Vol.197, No. 3, 1996, pp. 283-292
[90] Park, T., Lee, S.Y., and Voyiadjis, G.Z., Finite element vibration analysis of composite skew laminates containing delaminations around quadrilateral cutouts, Composites Part B: Engineering, Vol.40, No. 3, 2009, pp. 225-236
[91] Saravanos, D.A. and Hopkins, D.A., effects of delaminations on the damped dynamic characteristics of composite laminates: analysis and experiments, Journal of Sound and Vibration, Vol.192, No. 5, 1996, pp. 977-993
[92] Pardoen, G.C., Effect of delamination on the natural frequencies of composite laminates, Journal of composite materials, Vol.23, No. 12, 1989, pp. 1200-1215
[93] Lestari, W., Lu, X., and Hanagud, S. Dynamics of a delaminated beam-Effects of nonlinearity. in 41st Structures, Structural Dynamics, and Materials Conference and Exhibit. 2000.
[94] Lu, X., Lestari, W., and Hanagud, S., Nonlinear vibrations of a delaminated beam, Journal of Vibration and Control, Vol.7, No. 6, 2001, pp. 803-831