Research Article
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Vibration response analysis of an aircraft structure in terms of crashworthiness by using differential transformation method

Year 2025, Early View, 1 - 1

Aysun Soysal , İbrahim Ozkol , Erol Uzal

https://doi.org/10.35378/gujs.1383428

Abstract

Due to importance of certification procedure and safety of occupants, crashworthiness is one of the most principal elements to be considered in the design and production of civil and military aircrafts. In the component level, the crashworthiness of an aircraft structure is significantly affected by many factors including the cross-section of the aircraft structure, boundary conditions of the aircraft structure, and applied crush loading to the aircraft structure. The aim of this study is to contribute to the literature on improving the design of aircraft structures in order to increase the energy absorption properties of the aircraft structures and therefore support the crashworthiness capability of the structures by performing vibration analysis. In this context, after deriving the governing equations of an aircraft structure exposed to an axial crush loading, three different applications are conducted to investigate the effects of the axial crush loading on the dynamic characteristics of the aircraft structure. The findings of the study concluded that vibration characteristics of an aircraft structure subjected to an axial crush loading are affected by the boundary conditions of the structure, material of the structure, cross-section of the structure, magnitude of the axial load applied to the structure, and direction of the axial load applied to the structure. In addition, the findings showed that the response of the structure under ultimate axial crush loading varies depending on the geometriy of the structure, material of the structure and the direction of the ultimate axial crush loading applied to the structure.

Keywords

Crush loading Vibration analysis Energy principle Differential transform method

References

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  • [5] Wang, T., Wang, L., Wang, C., Zou, X., “Crashworthiness analysis and multi-objective optimization of a commercial vehicle frame: A mixed meta-modeling-based method,” Advances in Mechanical Engineering, 10(5), (2018). DOI: 10.1177/1687814018778480.
  • [6] GuidaGuida, M., Marulo, F., Abrate, S., “Advances in crash dynamics for aircraft safety,” Progress in Aerospace Sciences, 98: 106–123, (2018). DOI: 10.1016/j.paerosci.2018.03.008.
  • [7] Xue, P., Ding, M.L., Qiao, C.F., Yu, T.X., “Crashworthiness Study of a Civil Aircraft Fuselage Section,” Latin American Journal of Solids and Structures, 11: 1615–1627, (2014).
  • [8] Guida, M., Lamanna, G., Marulo, F., Caputo, F., “Review on the design of an aircraft crashworthy passenger seat,” Progress in Aerospace Sciences, 129, (2022).
  • [9] Garofano, A., Sellitto, A., Acanfora, V., Di Caprio, F., Riccio, A., “On the effectiveness of double-double design on crashworthiness of fuselage barrel,” Aerospace Science and Technology, 140, (2023). DOI: 10.1016/j.ast.2023.108479.
  • [10] Mou, H., Chen, Y., Feng, Z., Liu, H., “Damage and energy absorption behavior of CFRP/aluminum hybrid open-section thin-walled columns subjected to quasi-static loading,” Thin-Walled Structures, 197, (2024). DOI: 10.1016/j.tws.2024.111593.
  • [11] Li, Q., Xiao, Q., Liu, X., Cheng, X., Mao, C., Hu, H., “Hierarchical double-hat beam for axial crashworthiness,” Aerospace Science and Technology, 141, (2023). DOI: 10.1016/j.ast.2023.108515.
  • [12] Tang, T., Zhang, W., Yin, H., Wang, H., “Crushing analysis of thin-walled beams with various section geometries under lateral impact,” Thin-Walled Structures, 102: 43–57, (2016). DOI: 10.1016/j.tws.2016.01.017.
  • [13] Fanthorpe, C., Soban, D., Price, M., Butterfield, J., “Developing a capability function relating aircraft systems cost overruns to aircraft design parameters,” in 11th AIAA Aviation Technology, Integration, and Operations (ATIO) Conference, including the AIAA Balloon Systems Conference and 19th AIAA Lighter-Than-Air Technology Conference, (2011). DOI: 10.2514/6.2011-6837.
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  • [15] Liang, H., Liu, B., Pu, Y., Sun, H., Wang, D., “Crashworthiness analysis of variable thickness CFRP/Al hybrid multi-cell tube,” International Journal of Mechanical Sciences, 266, (2024). DOI: 10.1016/j.ijmecsci.2024.108959.
  • [16] Xue, P., Wang, L., Qiao, C.F., “Crashworthiness Study on Fuselage Section and Struts under Cabin Floor,” International Journal of Protective Structures, 4(2): 515-525, (2011). DOI: 10.1260/2041-4196.2.4.515
  • [17] Ren, Y., Xiang, J., “A comparative study of the crashworthiness of civil aircraft with different strut configurations,” International Journal of Crashworthiness, 15(3): 321–330, (2010). DOI: 10.1080/13588260903343823.
  • [18] Ren, Y., Xiang, J., “Influences of Geometrical factors on the crashworthiness of open shells,” in In 51st AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, 1–10, (2010).
  • [19] Ren, Y., Xiang, J., “Energy absorption structures design of civil aircraft to improve crashworthiness,” Aeronautical Journal, 118(1202): 383–398, (2014). DOI: 10.1017/S0001924000009180.
  • [20] Patil, S., Pangavhane, D., “Crashworthiness analysis and multiobjective optimization for variable thickness square thin-wall columns under axial loading,” Materials Today: Proceedings, 77: 860–870, (2023). DOI: 10.1016/j.matpr.2022.11.506.
  • [21] Moas, E., Boitnott, R.L., Griffin, H., “An Analytical and Experimental Investigation of the Response of the Curved, Composite Frame/Skin Specimens,” Journal of the American Holicopter Society, 39(3): 58–66, (1994).
  • [22] Woodson, M.B., Johnson, E.R., Haftka, R.T., “Optimal design of composite fuselage frames for crashworthiness,” International Journal of Crashworthiness, 1(4): 369–380, (1996). DOI: 10.1533/cras.1996.0027.
  • [23] Collins, J.S., Johnson, E.R., “Static and Dynamic Response of Graphite-Epoxy Curved Frames,” Journal of Composite Materials, 26(6): 792–803, (1992). DOI: https://doi.org/10.1177/002199839202600602.
  • [24] Perez, J.G., Johnson, E.R., Boitnott, R.L., “Design and test of semicircular composite frames optimized for crashworthiness,” in Collection of Technical Papers- AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, AIAA, 27–38, (1998). DOI: 10.2514/6.1998-1703.
  • [25] Zhang, C., Binienda, W.K., Horvat, F.E., Wang, W., “Application of numerical methods for crashworthiness investigation of a large aircraft wing impact with a tree,” Mathematical & Computational Forestry & Natural Resource Science, 5(1), 71–85, (2013).
  • [26] Xie, C., Wang, D., Zong, L., Kong, D., “Crashworthiness analysis and multi-objective optimization of spatial lattice structure under dynamic compression,” International Journal of Impact Engineering, 180(104713), 1-15, (2023).
  • [27] Bildik, N., Konuralp, A., Bek, F.O., Küçükarslan, S., “Solution of different type of the partial differential equation by differential transform method and Adomian’s decomposition method,” Applied Mathematics and Computation, 172(1): 551–567, (2006).
  • [28] Odiba Odibat, Z.M., Bertelle, C., Aziz-Alaoui, M.A., Duchamp, G.H.E., “A multi-step differential transform method and application to non-chaotic or chaotic systems,” Computers and Mathematics with Applications, 59(4): 1462–1472, (2010). DOI: 10.1016/j.camwa.2009.11.005.
  • [29] Yaghoobi, H., Torabi, M., “The application of differential transformation method to nonlinear equations arising in heat transfer,” International Communications in Heat and Mass Transfer, 38(6): 815–820, (2011). DOI: 10.1016/j.icheatmasstransfer.2011.03.025.
  • [30] Li, J., Shen, R., Hua, H., Jin, X., “Bending-torsional coupled dynamic response of axially loaded composite Timosenko thin-walled beam with closed cross-section,” Composite Structures, 64(1): 23–35, (2004). DOI: 10.1016/S0263-8223(03)00210-1.
  • [31] Chemartin, L., Lalande, P., Chazottes, A., Elias, P.Q. Delalondre, B.G., Lago, F., “Direct Effects of Lightning on Aircraft Structure: Analysis of the Thermal, Electrical and Mechanical Constraints,” Aerospace Lab, 5: 1–15, (2012), [Online]. Available: https://hal.science/hal-01184416.
  • [32] Soysal, A., Ozkol, I., Uzal, E., “An Analytical-Based Lightning-Induced Damage Model for an Aircraft Wing Exposed to Pressure Loading of Lightning,” Mathematical Problems in Engineering, 2024: 1–17, (2024). DOI: 10.1155/2024/8313135.
  • [33] Hashemi, S.M., Richard, M.J., “Free vibrational analysis of axially loaded bending-torsion coupled beams: a dynamic fnite element,” Computers & Structures, 77: 711–724, (2000). [Online]. Available: www.elsevier.com/locate/compstruc.
  • [34] Dursun, T., Soutis, C., “Recent developments in advanced aircraft aluminium alloys,” Materials & Design, 56: 862–871, (2014). DOI: 10.1016/j.matdes.2013.12.002.
  • [35] Zhou, J.K., “Differential transformation and its applications for electrical circuits”, Wuhan: Huazhong University Press, (1986).
  • [36] Soysal, A., Özkol, İ., Uzal, E., “Flexural-torsional-coupled vibration analysis of Euler-Bernoulli beam by using the differential transform method,” Academic Perspective Procedia, 5(3): 26–33, (2022). DOI: 10.33793/acperpro.05.03.642.
  • [37] Banerjee, J.R., Guot, S., Howson, W.P., “Exact Dynamic Stiffness Matrix of a Bending-Torsion Coupled Beam Including Warping,” Computers & Structures, 59(4):613–621, (1996).
  • [38] Jun, L., Rongying, S., Hongxing, H., Xianding, J., “Coupled bending and torsional vibration of axially loaded Bernoulli-Euler beams including warping effects,” Applied Acoustics, 65(2): 153–170, (2004). DOI: 10.1016/j.apacoust.2003.07.006.
  • [39] Banerjee, J.R., Fisher, S.A., “Coupled bending–torsional dynamic stiffness matrix for axially loaded beam elements,” International Journal for Numerical Methods in Engineering, 33(4): 739–751, (1992). DOI: 10.1002/nme.1620330405.
  • [40] Banerjee, J.R., “Coupled bending-torsional dynamic stiffness matrix for beam elements,” International Journal for Numerical Methods in Engineering, 28: 1283–1298, (1989).
  • [41] Eslimy-Isfahany, S.H.R., Banerjee, J.R., “Use of generalized mass in the interpretation of dynamic response of bending-torsion coupled beams,” Journal of Sound and Vibration, 238(2): 295–308, (2000). DOI: 10.1006/jsvi.2000.3160.
  • [42] Larsen, C.E., Raju, I.S., “Moving aerospace structural design practice to a load and resistance factor approach,” in 57th AIAA/ASCE/AHS/ASC Structures, Structural, Dynamics, and Material Conference, 230, (2016).

Year 2025, Early View, 1 - 1

Aysun Soysal , İbrahim Ozkol , Erol Uzal

https://doi.org/10.35378/gujs.1383428

Abstract

References

  • [1] Grote, M., Williams, I., Preston, J., “Direct carbon dioxide emissions from civil aircraft,” Atmospheric Environment, 95: 214–224, (2014). DOI: 10.1016/j.atmosenv.2014.06.042.
  • [2] Prabandari, A.P., Puteri, E.A.P., “Standardization of Private Aircraft: Implications of Climate Change,” in IOP Conference Series: Earth and Environmental Science, IOP Publishing, 012028, (2023). DOI: 10.1088/1755-1315/1270/1/012028.
  • [3] Anand, S., Alderliesten, R., Castro, S.G., “Low-fidelity crashworthiness assessment of unconventional aircraft: Modeling of plastic bending,” AIAA SCITECH 2024, 0833, (2024).
  • [4] Kurtaran, H., Eskandarian, A., Marzougui, D., Bedewi, N.E., “Crashworthiness design optimization using successive response surface approximations,” Computational mechanics, 29(4–5): 409–421, (2002). DOI: 10.1007/s00466-002-0351-x.
  • [5] Wang, T., Wang, L., Wang, C., Zou, X., “Crashworthiness analysis and multi-objective optimization of a commercial vehicle frame: A mixed meta-modeling-based method,” Advances in Mechanical Engineering, 10(5), (2018). DOI: 10.1177/1687814018778480.
  • [6] GuidaGuida, M., Marulo, F., Abrate, S., “Advances in crash dynamics for aircraft safety,” Progress in Aerospace Sciences, 98: 106–123, (2018). DOI: 10.1016/j.paerosci.2018.03.008.
  • [7] Xue, P., Ding, M.L., Qiao, C.F., Yu, T.X., “Crashworthiness Study of a Civil Aircraft Fuselage Section,” Latin American Journal of Solids and Structures, 11: 1615–1627, (2014).
  • [8] Guida, M., Lamanna, G., Marulo, F., Caputo, F., “Review on the design of an aircraft crashworthy passenger seat,” Progress in Aerospace Sciences, 129, (2022).
  • [9] Garofano, A., Sellitto, A., Acanfora, V., Di Caprio, F., Riccio, A., “On the effectiveness of double-double design on crashworthiness of fuselage barrel,” Aerospace Science and Technology, 140, (2023). DOI: 10.1016/j.ast.2023.108479.
  • [10] Mou, H., Chen, Y., Feng, Z., Liu, H., “Damage and energy absorption behavior of CFRP/aluminum hybrid open-section thin-walled columns subjected to quasi-static loading,” Thin-Walled Structures, 197, (2024). DOI: 10.1016/j.tws.2024.111593.
  • [11] Li, Q., Xiao, Q., Liu, X., Cheng, X., Mao, C., Hu, H., “Hierarchical double-hat beam for axial crashworthiness,” Aerospace Science and Technology, 141, (2023). DOI: 10.1016/j.ast.2023.108515.
  • [12] Tang, T., Zhang, W., Yin, H., Wang, H., “Crushing analysis of thin-walled beams with various section geometries under lateral impact,” Thin-Walled Structures, 102: 43–57, (2016). DOI: 10.1016/j.tws.2016.01.017.
  • [13] Fanthorpe, C., Soban, D., Price, M., Butterfield, J., “Developing a capability function relating aircraft systems cost overruns to aircraft design parameters,” in 11th AIAA Aviation Technology, Integration, and Operations (ATIO) Conference, including the AIAA Balloon Systems Conference and 19th AIAA Lighter-Than-Air Technology Conference, (2011). DOI: 10.2514/6.2011-6837.
  • [14] Lobitz, L., Hahn, A., Vogt, D., Luplow, T., Michalowski, P., Heimbs, S., Garnweitner, Gl., “Conceptual Challenges for Crashworthy Battery-Electric Commercial Aircraft – A Review,” in AIAA SCITECH 2024 Forum, Reston, Virginia: American Institute of Aeronautics and Astronautics, (2024). DOI: 10.2514/6.2024-2156.
  • [15] Liang, H., Liu, B., Pu, Y., Sun, H., Wang, D., “Crashworthiness analysis of variable thickness CFRP/Al hybrid multi-cell tube,” International Journal of Mechanical Sciences, 266, (2024). DOI: 10.1016/j.ijmecsci.2024.108959.
  • [16] Xue, P., Wang, L., Qiao, C.F., “Crashworthiness Study on Fuselage Section and Struts under Cabin Floor,” International Journal of Protective Structures, 4(2): 515-525, (2011). DOI: 10.1260/2041-4196.2.4.515
  • [17] Ren, Y., Xiang, J., “A comparative study of the crashworthiness of civil aircraft with different strut configurations,” International Journal of Crashworthiness, 15(3): 321–330, (2010). DOI: 10.1080/13588260903343823.
  • [18] Ren, Y., Xiang, J., “Influences of Geometrical factors on the crashworthiness of open shells,” in In 51st AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, 1–10, (2010).
  • [19] Ren, Y., Xiang, J., “Energy absorption structures design of civil aircraft to improve crashworthiness,” Aeronautical Journal, 118(1202): 383–398, (2014). DOI: 10.1017/S0001924000009180.
  • [20] Patil, S., Pangavhane, D., “Crashworthiness analysis and multiobjective optimization for variable thickness square thin-wall columns under axial loading,” Materials Today: Proceedings, 77: 860–870, (2023). DOI: 10.1016/j.matpr.2022.11.506.
  • [21] Moas, E., Boitnott, R.L., Griffin, H., “An Analytical and Experimental Investigation of the Response of the Curved, Composite Frame/Skin Specimens,” Journal of the American Holicopter Society, 39(3): 58–66, (1994).
  • [22] Woodson, M.B., Johnson, E.R., Haftka, R.T., “Optimal design of composite fuselage frames for crashworthiness,” International Journal of Crashworthiness, 1(4): 369–380, (1996). DOI: 10.1533/cras.1996.0027.
  • [23] Collins, J.S., Johnson, E.R., “Static and Dynamic Response of Graphite-Epoxy Curved Frames,” Journal of Composite Materials, 26(6): 792–803, (1992). DOI: https://doi.org/10.1177/002199839202600602.
  • [24] Perez, J.G., Johnson, E.R., Boitnott, R.L., “Design and test of semicircular composite frames optimized for crashworthiness,” in Collection of Technical Papers- AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, AIAA, 27–38, (1998). DOI: 10.2514/6.1998-1703.
  • [25] Zhang, C., Binienda, W.K., Horvat, F.E., Wang, W., “Application of numerical methods for crashworthiness investigation of a large aircraft wing impact with a tree,” Mathematical & Computational Forestry & Natural Resource Science, 5(1), 71–85, (2013).
  • [26] Xie, C., Wang, D., Zong, L., Kong, D., “Crashworthiness analysis and multi-objective optimization of spatial lattice structure under dynamic compression,” International Journal of Impact Engineering, 180(104713), 1-15, (2023).
  • [27] Bildik, N., Konuralp, A., Bek, F.O., Küçükarslan, S., “Solution of different type of the partial differential equation by differential transform method and Adomian’s decomposition method,” Applied Mathematics and Computation, 172(1): 551–567, (2006).
  • [28] Odiba Odibat, Z.M., Bertelle, C., Aziz-Alaoui, M.A., Duchamp, G.H.E., “A multi-step differential transform method and application to non-chaotic or chaotic systems,” Computers and Mathematics with Applications, 59(4): 1462–1472, (2010). DOI: 10.1016/j.camwa.2009.11.005.
  • [29] Yaghoobi, H., Torabi, M., “The application of differential transformation method to nonlinear equations arising in heat transfer,” International Communications in Heat and Mass Transfer, 38(6): 815–820, (2011). DOI: 10.1016/j.icheatmasstransfer.2011.03.025.
  • [30] Li, J., Shen, R., Hua, H., Jin, X., “Bending-torsional coupled dynamic response of axially loaded composite Timosenko thin-walled beam with closed cross-section,” Composite Structures, 64(1): 23–35, (2004). DOI: 10.1016/S0263-8223(03)00210-1.
  • [31] Chemartin, L., Lalande, P., Chazottes, A., Elias, P.Q. Delalondre, B.G., Lago, F., “Direct Effects of Lightning on Aircraft Structure: Analysis of the Thermal, Electrical and Mechanical Constraints,” Aerospace Lab, 5: 1–15, (2012), [Online]. Available: https://hal.science/hal-01184416.
  • [32] Soysal, A., Ozkol, I., Uzal, E., “An Analytical-Based Lightning-Induced Damage Model for an Aircraft Wing Exposed to Pressure Loading of Lightning,” Mathematical Problems in Engineering, 2024: 1–17, (2024). DOI: 10.1155/2024/8313135.
  • [33] Hashemi, S.M., Richard, M.J., “Free vibrational analysis of axially loaded bending-torsion coupled beams: a dynamic fnite element,” Computers & Structures, 77: 711–724, (2000). [Online]. Available: www.elsevier.com/locate/compstruc.
  • [34] Dursun, T., Soutis, C., “Recent developments in advanced aircraft aluminium alloys,” Materials & Design, 56: 862–871, (2014). DOI: 10.1016/j.matdes.2013.12.002.
  • [35] Zhou, J.K., “Differential transformation and its applications for electrical circuits”, Wuhan: Huazhong University Press, (1986).
  • [36] Soysal, A., Özkol, İ., Uzal, E., “Flexural-torsional-coupled vibration analysis of Euler-Bernoulli beam by using the differential transform method,” Academic Perspective Procedia, 5(3): 26–33, (2022). DOI: 10.33793/acperpro.05.03.642.
  • [37] Banerjee, J.R., Guot, S., Howson, W.P., “Exact Dynamic Stiffness Matrix of a Bending-Torsion Coupled Beam Including Warping,” Computers & Structures, 59(4):613–621, (1996).
  • [38] Jun, L., Rongying, S., Hongxing, H., Xianding, J., “Coupled bending and torsional vibration of axially loaded Bernoulli-Euler beams including warping effects,” Applied Acoustics, 65(2): 153–170, (2004). DOI: 10.1016/j.apacoust.2003.07.006.
  • [39] Banerjee, J.R., Fisher, S.A., “Coupled bending–torsional dynamic stiffness matrix for axially loaded beam elements,” International Journal for Numerical Methods in Engineering, 33(4): 739–751, (1992). DOI: 10.1002/nme.1620330405.
  • [40] Banerjee, J.R., “Coupled bending-torsional dynamic stiffness matrix for beam elements,” International Journal for Numerical Methods in Engineering, 28: 1283–1298, (1989).
  • [41] Eslimy-Isfahany, S.H.R., Banerjee, J.R., “Use of generalized mass in the interpretation of dynamic response of bending-torsion coupled beams,” Journal of Sound and Vibration, 238(2): 295–308, (2000). DOI: 10.1006/jsvi.2000.3160.
  • [42] Larsen, C.E., Raju, I.S., “Moving aerospace structural design practice to a load and resistance factor approach,” in 57th AIAA/ASCE/AHS/ASC Structures, Structural, Dynamics, and Material Conference, 230, (2016).
There are 42 citations in total.

Details

Primary Language English
Subjects Theoretical and Applied Mechanics in Mathematics
Journal Section Research Article
Authors

Aysun Soysal 0000-0003-0708-4529

İbrahim Ozkol 0000-0002-9300-9092

Erol Uzal 0000-0003-0008-1376

Early Pub Date September 23, 2024
Publication Date
Submission Date October 30, 2023
Acceptance Date April 26, 2024
Published in Issue Year 2025 Early View

Cite

APA Soysal, A., Ozkol, İ., & Uzal, E. (2024). Vibration response analysis of an aircraft structure in terms of crashworthiness by using differential transformation method. Gazi University Journal of Science1-1. https://doi.org/10.35378/gujs.1383428
AMA Soysal A, Ozkol İ, Uzal E. Vibration response analysis of an aircraft structure in terms of crashworthiness by using differential transformation method. Gazi University Journal of Science. Published online September 1, 2024:1-1. doi:10.35378/gujs.1383428
Chicago Soysal, Aysun, İbrahim Ozkol, and Erol Uzal. “Vibration Response Analysis of an Aircraft Structure in Terms of Crashworthiness by Using Differential Transformation Method”. Gazi University Journal of Science, September (September 2024), 1-1. https://doi.org/10.35378/gujs.1383428.
EndNote Soysal A, Ozkol İ, Uzal E (September 1, 2024) Vibration response analysis of an aircraft structure in terms of crashworthiness by using differential transformation method. Gazi University Journal of Science 1–1.
IEEE A. Soysal, İ. Ozkol, and E. Uzal, “Vibration response analysis of an aircraft structure in terms of crashworthiness by using differential transformation method”, Gazi University Journal of Science, pp. 1–1, September 2024, doi: 10.35378/gujs.1383428.
ISNAD Soysal, Aysun et al. “Vibration Response Analysis of an Aircraft Structure in Terms of Crashworthiness by Using Differential Transformation Method”. Gazi University Journal of Science. September 2024. 1-1. https://doi.org/10.35378/gujs.1383428.
JAMA Soysal A, Ozkol İ, Uzal E. Vibration response analysis of an aircraft structure in terms of crashworthiness by using differential transformation method. Gazi University Journal of Science. 2024;:1–1.
MLA Soysal, Aysun et al. “Vibration Response Analysis of an Aircraft Structure in Terms of Crashworthiness by Using Differential Transformation Method”. Gazi University Journal of Science, 2024, pp. 1-1, doi:10.35378/gujs.1383428.
Vancouver Soysal A, Ozkol İ, Uzal E. Vibration response analysis of an aircraft structure in terms of crashworthiness by using differential transformation method. Gazi University Journal of Science. 2024:1-.