On non-linear vibration analyses of continuous systems with quadratic and cubic non-linearities
- 作者：Akira Abe
- 关键词：Non-linear vibration of continuous system ; Quadratic and cubic non-linearities ; Method of multiple scales ; Galerkin's procedure ; Shooting method ; Finite difference method
- 刊名：International Journal of Non-Linear Mechanics
- 出版年：2006
- 期刊代码：87_00207462
- 类别：et
- 出版时间：October 2006
- 卷：41
- 期：8
- 页码：873-879
- 文件大小：522 K
摘要
This paper examines the validity of non-linear vibration analyses of continuous systems with quadratic and cubic non-linearities. As an example, we treat a hinged–hinged Euler–Bernoulli beam resting on a non-linear elastic foundation with distributed quadratic and cubic non-linearities, and investigate the primary none; color:black" href="/science?_ob=MathURL&_method=retrieve&_udi=B6TJ2-4KSD4C7-1&_mathId=mml42&_user=10&_cdi=5298&_rdoc=2&_acct=C000050221&_version=1&_userid=10&md5=c2b64442818dc9da0afb40d77dfd2582" title="Click to view the MathML source">(Ω≈ωn) and subharmonic none; color:black" href="/science?_ob=MathURL&_method=retrieve&_udi=B6TJ2-4KSD4C7-1&_mathId=mml43&_user=10&_cdi=5298&_rdoc=2&_acct=C000050221&_version=1&_userid=10&md5=cac762498819e1172b0270fa59de8de6" title="Click to view the MathML source">(Ω≈2ωn) resonances, in which none; color:black" href="/science?_ob=MathURL&_method=retrieve&_udi=B6TJ2-4KSD4C7-1&_mathId=mml44&_user=10&_cdi=5298&_rdoc=2&_acct=C000050221&_version=1&_userid=10&md5=7a9ee203e846dcd008f826e634d01ce1" title="Click to view the MathML source">Ω and none; color:black" href="/science?_ob=MathURL&_method=retrieve&_udi=B6TJ2-4KSD4C7-1&_mathId=mml45&_user=10&_cdi=5298&_rdoc=2&_acct=C000050221&_version=1&_userid=10&md5=bf37a99165ed94d984cc84bfbfdaebc4" title="Click to view the MathML source">ωn are the driving and natural frequencies, respectively. The steady-state responses are found by using two different approaches. In the first approach, the method of multiple scales is applied directly to the governing equation that is a non-linear partial differential equation. In the second approach, we discretize the governing equation by using Galerkin's procedure, and then apply the shooting method to the obtained ordinary differential equations. In order to check the validity of the solutions obtained by the two approaches, they are compared with the solutions obtained numerically by the finite difference method.