Bounded natural frequencies of completely free rectangular plates

Mochida, Yusuke; Ilanko, Sinniah

doi:10.1016/j.jsv.2007.10.022

Cited by 19 publications

(8 citation statements)

References 10 publications

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“…The figures also show that the proportion of higher modes contributing to the transient response of the plates with aspect ratios 1.5 and 2.0 are larger than that of a square plate. It is known that both the SM and the R-R method give upper bounds for the natural frequencies of completely free plates [1,4] and lowest values would be considered as benchmarks. The SM and the R-R method using the degenerated free-free beam functions give lower values of natural frequencies than those given by the R-R method using the ordinary free-free beam functions for the completely free plates.…”

Section: Resultsmentioning

confidence: 99%

“…It seems appropriate at this stage to treat the transient responses obtained using the natural frequencies and modes given by the SM as benchmark results for the response of the plates with all edges simply supported when being suddenly released from all the supports after being subject to an initial displacement corresponding to that due to a uniformly distributed load. The first few participating modes, transient coefficients, corresponding natural frequency parameters, λ 2 [5,6] and modal displacements at the centre, the point of 0.75a, 0.75b and the corner are given in Table 1. We believe the numerical results would serve as benchmarks for transient response of plates.…”

Section: Resultsmentioning

confidence: 99%

“…This relaxation can be achieved by simply floating θ in Eqs (4) and (5). The θ is replaced by a coefficient, which will be determined during the usual minimisation procedure in the R-R method.…”

Section: Natural Frequencies and Modes Of Vibrationmentioning

confidence: 99%

“…An excellent review of the literature relating to vibration analysis of plates was published by Leissa [1]. Among the methods utilised for vibration analysis, it is known that Gorman's Superposition method is very efficient and accurate for a range of geometric shapes and in many cases the results from the Superposition method are the benchmarks for the natural frequencies [2][3][4][5][6]. In a recent publication, the Superposition Method was shown to be applicable for the determination of steady state response of plates [7].…”

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confidence: 99%

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Transient vibration analysis of a completely free plate using modes obtained by Gorman's superposition method

Mochida

Ilanko

2010

Journal of Sound and Vibration

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This paper shows that the transient response of a plate undergoing flexural vibration can be calculated accurately and efficiently using the natural frequencies and modes obtained from the Superposition Method. The response of a completely free plate is used to demonstrate this. The case considered is one where all supports of a simply supported thin rectangular plate under self weight are suddenly removed. The resulting motion consists of a combination of the natural modes of a completely free plate. The modal superposition method is used for determining the transient response, and the natural frequencies and mode shapes of the plates used are obtained by Gorman's ω radian frequency of vibration

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Section: Resultsmentioning

confidence: 99%

Section: Resultsmentioning

confidence: 99%

Section: Natural Frequencies and Modes Of Vibrationmentioning

confidence: 99%

mentioning

confidence: 99%

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Transient vibration analysis of a completely free plate using modes obtained by Gorman's superposition method

Mochida

Ilanko

2010

Journal of Sound and Vibration

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“…Exact solutions to the vibration of rectangular plates are only available for plates with two opposite edges subject to simply supported or clamped conditions. Otherwise, they are analysed by using approximate methods [31]. Nevertheless, it is possible to obtain a very good approximation of the eigenvalues and eigenvectors associated to a free rectangular plate, and express the displacement u solution of (5.3) in semi-analytical form.…”

Section: Semi-analytical Solution For Free Impact Of a Rectangular Panelmentioning

confidence: 99%

Energy conservative finite element semi-discretization for vibro-impacts of plates on rigid obstacles

2016

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Our purpose is to describe and compare some families of fully discretized approximations and their properties, in the case of vibro-impact of plates between rigid obstacles with non-penetration Signorini's conditions. To this aim, the dynamical Kirchhoff-Love plate model is considered and an extension to plates of the singular dynamic method, introduced by Renard and previously adapted to beams by Pozzolini and Salaün, is described. A particular emphasis is given in the use of an adapted Newmark scheme in which intervene a discrete restitution coefficient. Finally, various numerical results are presented and energy conservation capabilities of several numerical schemes are investigated and discussed.Mathematics Subject Classification. 35L85, 65M12, 74H15, 74H45.

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Upper and lower bounds for natural frequencies: A property of the smoothed finite element methods

Zhang

Liu

2010

Numerical Meth Engineering

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SUMMARYNode-based smoothed finite element method (NS-FEM) using triangular type of elements has been found capable to produce upper bound solutions (to the exact solutions) for force driving static solid mechanics problems due to its monotonic 'soft' behavior. This paper aims to formulate an NS-FEM for lower bounds of the natural frequencies for free vibration problems. To make the NS-FEM temporally stable, an -FEM is devised by combining the compatible and smoothed strain fields in a partition of unity fashion controlled by ∈[0, 1], so that both the properties of stiff FEM and the monotonically soft NS-FEM models can be properly combined for a desired purpose. For temporally stabilizing NS-FEM, is chosen small so that it acts like a 'regularization parameter' making the NS-FEM stable, but still with sufficient softness ensuring lower bounds for natural frequency solution. Our numerical studies demonstrate that (1) using a proper , the spurious non-zero energy modes can be removed and the NS-FEM becomes temporally stable; (2) the stabilized NS-FEM becomes a general approach for solids to obtain lower bounds to the exact natural frequencies over the whole spectrum; (3) -FEM can even be tuned for obtaining nearly exact natural frequencies.

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Bounded natural frequencies of completely free rectangular plates

Cited by 19 publications

References 10 publications

Transient vibration analysis of a completely free plate using modes obtained by Gorman's superposition method

Transient vibration analysis of a completely free plate using modes obtained by Gorman's superposition method

Energy conservative finite element semi-discretization for vibro-impacts of plates on rigid obstacles

Upper and lower bounds for natural frequencies: A property of the smoothed finite element methods

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