تحليل ارتعاش آزاد غيرخطي ور ق هاي مستطيلي از جنس ماده مدرج تابعي دوجهته

Nonlinear Free Vibration Analysis of Bi-directional Functionally Graded Rectangular Plates

در ﺗﺤﻘﯿﻖ ﺣﺎﺿﺮ، ارﺗﻌﺎش آزاد ﻏﯿﺮﺧﻄﯽ ورق ﻣﺴﺘﻄﯿﻠﯽ دور ﻣﻔﺼﻞ از ﺟﻨﺲ ﻣﻮاد ﻣﺪرج ﺗﺎﺑﻌﯽ دو ﺟﻬﺘﻪ ﺑﺮاي اوﻟﯿﻦ ﺑﺎر ﺑﻪ ﺻﻮرت ﮐﺎﻣﻼ ﺗﺤﻠﯿﻠﯽ ﺑﺮرﺳﯽ ﺷﺪه اﺳﺖ. ﺑﺮاي اﯾﻦ ﻣﻨﻈﻮر ﺑﻪ ﮐﻤﮏ اﺻﻞ ﻫﺎﻣﯿﻠﺘﻮن و رواﺑﻂ ﮐﺮﻧﺶ ﺟﺎﺑﺠﺎﯾﯽ ﻏﯿﺮﺧﻄﯽ ون ﮐﺎرﻣﻦ، ﻣﻌﺎدﻻت دﯾﻔﺮاﻧﺴﯿﻞ ﺟﺰﺋﯽ ﺣﺮﮐﺖ ورق اﺳﺘﺨﺮاج ﺷﺪه اﻧﺪ. ﭘﺲ از آن ﺑﺎ اﻋﻤﺎل روش ﮔﺎﻟﺮﮐﯿﻦ، ﻣﻌﺎدﻻت دﯾﻔﺮاﻧﺴﯿﻞ ﺟﺰﺋﯽ ﻏﯿﺮﺧﻄﯽ ورق ﺑﻪ ﻣﻌﺎدﻻت دﯾﻔﺮاﻧﺴﯿﻞ ﻣﻌﻤﻮﻟﯽ ﻏﯿﺮﺧﻄﯽ واﺑﺴﺘﻪ ﺑﻪ زﻣﺎن ﺗﺒﺪﯾﻞ ﮔﺸﺘﻪ اﻧﺪ. ﺳﭙﺲ ﺑﻪ ﻣﻨﻈﻮر ﻣﺤﺎﺳﺒﻪ ﻓﺮﮐﺎﻧﺲ ﻫﺎي ﻏﯿﺮﺧﻄﯽ ورق، ﻣﻌﺎدﻟﻪ ﻏﯿﺮﺧﻄﯽ ورق ﺑﻪ ﺻﻮرت ﺗﺤﻠﯿﻠﯽ ﺑﺎ اﺳﺘﻔﺎده از روش ﻟﯿﻨﺪﺷﺘﺪ ﭘﻮاﻧﮑﺎره ﺑﻬﺒﻮدﯾﺎﻓﺘﻪ ﺣﻞ ﺷﺪه اﺳﺖ. ﺧﻮاص ﻣﻮاد ورق ﺑﺮاﺳﺎس ﻣﺪل ﺗﻮزﯾﻊ ﺗﻮاﻧﯽ در دو راﺳﺘﺎي ﻃﻮﻟﯽ و ﻋﺮﺿﯽ ورق ﺑﻪ ﺻﻮرت ﭘﯿﻮﺳﺘﻪ ﻣﺘﻐﯿﺮ ﻓﺮض ﺷﺪه اﻧﺪ. درﻧﻬﺎﯾﺖ اﺛﺮ ﺑﺮﺧﯽ ﭘﺎراﻣﺘﺮﻫﺎي ﮐﻠﯿﺪي ﺳﯿﺴﺘﻢ ﻫﻤﭽﻮن، داﻣﻨﻪ ارﺗﻌﺎش، اﻧﺪﯾﺲ ﻣﺎده ﻣﺪرج ﺗﺎﺑﻌﯽ و ﻧﺴﺒﺖ ﻫﻨﺪﺳﯽ ورق روي ﻓﺮﮐﺎﻧﺲ ﻏﯿﺮﺧﻄﯽ ورق ﺑﻪ ﺻﻮرت ﺟﺰﺋﯽ ﻣﻮرد ﺗﺤﻠﯿﻞ ﻗﺮار ﮔﺮﻓﺘﻪ اﻧﺪ. ﺑﺮاي ﺻﺤﻪ ﮔﺬاري ﻣﺴﺎﻟﻪ، ﻧﺘﺎﯾﺞ اﯾﻦ ﭘﮋوﻫﺶ ﺑﺎ ﻧﺘﺎﯾﺞ ادﺑﯿﺎت ﮔﺬﺷﺘﻪ و ﻫﻤﭽﻨﯿﻦ ﺣﻞ ﻋﺪدي ﻣﻘﺎﯾﺴﻪ ﺷﺪه اﺳﺖ و ﺗﻄﺎﺑﻖ ﺑﺴﯿﺎرﺧﻮﺑﯽ ﻣﺸﺎﻫﺪه ﮔﺸﺘﻪ اﺳ

In the present study, nonlinear free vibration analysis of bi-directional functionally graded simply supported rectangular plates is investigated analytically for the first time. For this purpose, with the aid of Hamilton’s principle and von Karman nonlinear strain-displacement relations, the partial differential equations of motion are developed. Afterward, the nonlinear partial differential equations are transformed into the time-dependent nonlinear ordinary differential equations by applying the Galerkin method. The nonlinear equation of motion is then solved analytically by the Modified Lindstedt-Poincare method to determine the the plate nonlinear frequencies. The volume fraction distribution is assumed to be continuously graded in both the length and width directions of the plate. Finally, the effects of some system parameters such as the vibration amplitude, FG indexes and aspect ratio on the nonlinear frequency are discussed in detail. To validate the analysis, the results of this paper are compared with both the published data and numerical method, and good agreements are found.