On periodic solutions close to rectilinear normal vibration modes

Manevich, L. I.; Михлин, Ю. В.

doi:10.1016/0021-8928(72)90032-9

Cited by 33 publications

(40 citation statements)

References 3 publications

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“…The beam is divided into a varying number (2,4,8,16) of elements and the equations of motion of the discretized system are obtained. The results of the analysis will demonstrate that the linear part of the problem must be modeled more accurately when significant nonlinearity exists than would traditionally be necessary for a purely linear system.…”

Section: Finite Element Modelmentioning

confidence: 99%

“…In his works, Rosenberg considers the existence and stability of normal modes for a system of n masses interconnected by nonlinear symmetric springs and having n degrees of freedom. Subsequently, determination of nonlinear normal modes has been accomplished via an asymptotic approach [8], energy approaches [9], the method of multiple scales [10], and Poincaré maps [11]. Shaw and Pierre [2] generalize the definition of normal modes of nonlinear systems to include non-conservative, gyroscopic, and continuous systems.…”

Section: Introductionmentioning

confidence: 99%

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On the Convergence of Nonlinear Modes of a Finite Element Model

Balagangadhar¹,

Slater

2007

Shock and Vibration

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Abstract. Convergence of finite element models is generally realized via observation of mesh independence. In linear systems invariance of linear modes to further mesh refinement is often used to assess mesh independence. These linear models are, however, often coupled with nonlinear elements such as CFD models, nonlinear control systems, or joint dynamics. The introduction of a single nonlinear element can significantly alter the degree of mesh refinement necessary for sufficient model accuracy. Application of nonlinear modal analysis [1,2] illustrates that using linear modal convergence as a measure of mesh quality in the presence of nonlinearities is inadequate. The convergence of the nonlinear normal modes of a simply supported beam modeled using finite elements is examined. A comparison is made to the solution of Boivin, Pierre, and Shaw [3]. Both methods suffer from the need for convergence in power series approximations. However, the finite element modeling method introduces the additional concern of mesh independence, even when the meshing the linear part of the model unless p-type elements are used [4]. The importance of moving to a finite element approach for nonlinear modal analysis is the ability to solve problems of a more complex geometry for which no closed form solution exists. This case study demonstrates that a finite element model solution converges nearly as well as a continuous solution, and presents rough guidelines for the number of expansion terms and elements needed for various levels of solution accuracy. It also demonstrates that modal convergence occurs significantly more slowly in the nonlinear model than in the corresponding linear model. This illustrates that convergence of linear modes may be an inadequate measure of mesh independence when even a small part of a model is nonlinear.

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Section: Finite Element Modelmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On the Convergence of Nonlinear Modes of a Finite Element Model

Balagangadhar¹,

Slater

2007

Shock and Vibration

View full text Add to dashboard Cite

show abstract

“…In order to determine the trajectories of normal vibrations (2), the following relationships can be used [4,8]:…”

Section: Normal Vibrations In Lyapunov Systemsmentioning

confidence: 99%

“…For some particular cases curvilinear trajectories were defined by Rosenberg and Kuo [6] and by Rand [8]. In the paper by Manevich and Mikhlin [8] the power series method was proposed for the construction of above mentioned trajectories.…”

Section: Introductionmentioning

confidence: 99%