The Non-Linear Free Vibration of Fully Clamped Rectangular Plates: Second Non-Linear Mode for Various Plate Aspect Ratios

Kadiri, M. El; Benamar, Rhali; White, Robert G.

doi:10.1006/jsvi.1999.2410

Cited by 34 publications

(37 citation statements)

References 17 publications

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“…This method does not require a very good initial estimate of the solution or a step procedure, similar to other methods described for beams and rectangular plates. [20][21][22][23] It was adopted here to ensure rapid convergence when varying the amplitude, which allowed solutions to be obtained with a quite reasonable number of iterations. The fundamental nonlinear mode shape was calculated in the neighborhood of the linear solution corresponding to a small numerical value of the coefficient a r0 (r 0 = 1) of the basic function w * r0 .…”

Section: Iterative Methods Of Solutionsmentioning

confidence: 99%

“…Such an assumption has been made when calculating the first two nonlinear mode shapes of circular plates and fully clamped rectangular plates. 16,20,21,23 For the first nonlinear mode shape, the range of validity of this assumption has been discussed in the light of the experimental and numerical results obtained for the nonlinear frequency-amplitude dependence and the nonlinear bending stress estimates obtained at large vibration amplitude. 20,22 In order to examine the effects of large vibration amplitudes on the membrane stress patterns for clamped circular plates, the contribution of the in-plane displacement U should be taken into account in the membrane strain expression.…”

Section: Vibration Analysismentioning

confidence: 99%

“…Previous studies [20][21][22][23] have shown that the nonlinear mode shapes of beam-and plate-like structures are amplitude dependent. This effect is illustrated in the present case in Fig.…”

Section: Amplitude Dependence Of the First Nonlinear Axisymmetric Modmentioning

confidence: 99%

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Geometrically Nonlinear Free Axisymmetric Vibrations Analysis of Thin Circular Functionally Graded Plates Using Iterative and Explicit Analytical Solution

Kaak¹,

Bikri²,

Benamar³

2016

IJAV

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This paper deals with nonlinear free axisymmetric vibrations of functionally graded (FG) thin circular plates whose properties vary in thickness. The inhomogeneity of the plate is characterized by a power law variation of the Young's modulus and mass density of the material along the thickness direction, whereas Poisson's ratio is assumed to be constant. The theoretical model is based on Hamilton's principle and spectral analysis using a basis of admissible Bessel's functions to yield the frequencies of the circular plates under clamped boundary conditions on the basis of the classical plate theory. The large vibration amplitudes problem, reduced to a set of nonlinear algebraic equations, is solved numerically. The nonlinear to linear frequency ratios are presented for various values of the volume fraction index n showing a hardening type nonlinearity. The distribution of the radial bending stresses associated to the nonlinear mode shape is also given for various vibration amplitudes and compared with those predicted by the linear theory. Then, explicit analytical solutions are presented, based on the semi-analytical model previously developed by El Kadiri et al. for beams and rectangular plates. This model allows direct and easy calculation for the first nonlinear axisymmetric mode shape with its associated nonlinear frequencies and nonlinear bending stresses of FG circular plates, which are expected to be very useful in engineering applications and in further analytical developments. An excellent agreement is found with the results obtained by the iterative method.

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Section: Iterative Methods Of Solutionsmentioning

confidence: 99%

Section: Vibration Analysismentioning

confidence: 99%

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Geometrically Nonlinear Free Axisymmetric Vibrations Analysis of Thin Circular Functionally Graded Plates Using Iterative and Explicit Analytical Solution

Kaak¹,

Bikri²,

Benamar³

2016

IJAV

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“…Additionally, if sound transmission through the plates is concerned (what is the subject of this paper), the primary paths are also nonlinear. The nonlinearities can be caused due to vibrations of clamped plates (El Kadiri et al, 1999;Saha et al, 2005) and also by actuators (i.e. d33 effect of MFC patches) (Stuebner et al, 2009).…”

Section: Nonlinear Feedforward Controlmentioning

confidence: 99%

Hammerstein Nonlinear Active Noise Control with the Filtered-Error LMS Algorithm

Mazur¹,

Pawełczyk²

2013

Archives of Acoustics

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Active Noise Control (ANC) of noise transmitted through a vibrating plate causes many problems not observed in classical ANC using loudspeakers. They are mainly due to vibrations of a not ideally clamped plate and use of nonlinear actuators, like MFC patches. In case of noise transmission though a plate, nonlinerities exist in both primary and secondary paths.Existence of nonlinerities in the system may degrade performance of a linear feedforward control system usually used for ANC. The performance degradation is especially visible for simple deterministic noise, such as tonal noise, where very high reduction is expected. Linear feedforward systems in such cases are unable to cope with higher harmonics generated by the nonlinearities. Moreover, nonlinearities, if not properly tackled with, may cause divergence of an adaptive control system.In this paper a feedforward ANC system reducing sound transmitted through a vibrating plate is presented. The ANC system uses nonlinear control filters to suppress negative effects of nonlinearies in the system. Filtered-error LMS algorithm, found more suitable than usually used Filtered-reference LMS algorithm, is employed for updating parameters of the nonlinear filters. The control system is experimentally verified and obtained results are discussed.

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“…It is true that the concept of a non-linear mode shape is not absolutely clear and universally accepted like its linear equivalent. However, it is becoming quite familiar in the literature (see references [4,5,7,9,21], Nayfeh's paper [22] and his references [1, 2, 9, 11, 14}16, 18}20]), is very useful for the qualitative understanding of the non-linear behaviour, and is expected to play an important role in the development of the &&non-linear modal analysis theory'' mentioned in reference [21].…”

mentioning

confidence: 98%

Authors' Reply

Moussaoui

Benamar²,

White

2001

Journal of Sound and Vibration

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The Non-Linear Free Vibration of Fully Clamped Rectangular Plates: Second Non-Linear Mode for Various Plate Aspect Ratios

Cited by 34 publications

References 17 publications

Geometrically Nonlinear Free Axisymmetric Vibrations Analysis of Thin Circular Functionally Graded Plates Using Iterative and Explicit Analytical Solution

Geometrically Nonlinear Free Axisymmetric Vibrations Analysis of Thin Circular Functionally Graded Plates Using Iterative and Explicit Analytical Solution

Hammerstein Nonlinear Active Noise Control with the Filtered-Error LMS Algorithm

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