A novel scheme for the discrete prediction of high-frequency vibration response: Discrete singular convolution–mode superposition approach

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Abstract

This study introduces a novel scheme for the discrete high-frequency forced vibration analysis based on discrete singular convolution (DSC) and mode superposition (MS) approaches. The accuracy of the DSC–MS is validated for thin beams and plates by comparing with available analytical solutions. The performance of the DSC–MS is evaluated by predicting spatial distribution and discrete frequency spectra of the vibration response of thin plates with two different boundary conditions. The frequency spectra of the time-harmonic excitation forces are in the form of ideal and band-limited white noise so that the natural modes in the frequency band are provoked. The solution exposes high-frequency response behaviour definitely. Therefore, it is hoped with this paper to contribute the studies on the treatment of uncertainties in the high-frequency design applications.

Introduction

In the science of vibro-acoustics, vibration and acoustic problems are classified according to their frequency range as, low-, medium- and high-frequency problems. Since dynamic behaviour of systems changes with regard to the excitation frequency, adaptive approaches are required for reliable solutions. In practice, it is not mentioned about definite boundaries separating frequency ranges from each other due to the fact that they may change from system to system. However, Rabbiolo et al. [1] have put forward an indicator for approximately defining high-frequency thresholds based on “modal overlap count (modal overlap factor)” of simple structures such as beams, plates and acoustical spaces. It is known that modelling high-frequency dynamic systems using deterministic techniques such as finite element method (FEM) and boundary element method (BEM) is numerically expensive. Besides, since the vibro-acoustic response is very sensitive to the changes in system parameters at higher frequencies, some uncertainties are encountered. Therefore, deterministic techniques are feasible only for low-frequency analysis. Albeit hierarchical FEM (p-FEM) has extended the frequency limit of classical FEM and this approach is generally regarded as mid-frequency technique.

The statistical energy analysis (SEA) developed by Lyon and Maidanik in 1962 has proved its validity for high-frequency analysis [2]. However, SEA is based on some pre-assumptions restricting its efficiency and capacity. Therefore, several alternative energy-based techniques have been developed [3], [4], [5], [6], [7], [8], [9], [10], [11]. Among them, energy flow analysis (EFA) and its finite element application, energy finite element method (EFEM) are common approaches in service [3], [4], [5], [6]. However, since all these methods consider average prediction of energy as system variable to describe the response level, they disregard modal information and thus, loose discrete frequency response behaviour of the structure. Specifically, as far as the present authors are aware, any high-frequency method is not present in order to predict frequency response discretely without missing modal information and so being a key for high-frequency uncertainties. Furthermore, there is not any unique method valid for all frequency ranges to perform response analysis.

In the last decade, a novel approach called discrete singular convolution (DSC) has been introduced by Wei [12], [13], [14], [15]. This is a powerful method for the numerical solution of differential equations. The solution technique of the DSC is based on the theory of distribution and wavelets. This approach has been successfully used in various free vibration analyses of isotropic thin simple structures with several boundary conditions [16], [17], [18], [19], [20], [21], [22], [23], [24]. Hou et al. [25] have used DSC–Ritz method for the free vibration analysis of thick plates. Civalek [26], [27], [28] has applied the DSC to the free vibration and buckling analyses of different laminated shells and plates. Seçgin et al. [29] have used the DSC for the free vibration of fiber–metal laminated composite plates. Seçgin and Sarıgül [30] have presented open algorithm of the DSC and have shown the superiority of the DSC over several numerical techniques for the free vibration analysis of symmetrically laminated composite plates. Moreover, for high-frequency free vibration analysis, Wei et al. [31] and Zhao et al. [32] have obtained ten thousands of vibration modes for thin beams and plates. Lim et al. [33] have used DSC–Ritz approach for high-frequency modal analysis of thick shells. Ng et al. [34] have pointed out that the DSC yields more accurate prediction compared to differential quadrature method for the plates vibrating at high frequencies. These successes of the DSC promise that this method would be reliably used for discrete high-frequency response analysis without handling averaged energetic parameters unlike the available high-frequency approaches.

In this regard, the present paper introduces a novel scheme for the discrete high-frequency forced vibration analysis by employing discrete singular convolution (DSC) and mode superposition (MS) approaches. Although at high frequencies thin-structure theories may be hardly satisfied, in order to avoid the additional complexity caused by thick-structure theories, simple physical models were used for the introduction of the present scheme as done in Refs. [31], [32]. The validation of the scheme is realized by the comparisons with the analytical solutions of spatially distributed response of beams and frequency response of infinite plates. Besides, performance and restrictions of DSC–MS approach are discussed by the demonstrations of spatial distribution and frequency spectra of the vibration response for a wide frequency range. The frequency spectra of the time-harmonic point-excitation forces are in the form of ideal and band-limited white noise, so that the natural modes in the considered frequency region are excited. These discrete modes appearing in the response spectra are pleasant signals of recovering the uncertainties of high-frequency applications.

Section snippets

Discrete singular convolution (DSC)

Singular convolution is defined by the theory of distributions. Let T be a distribution and η(t) be an element of the space of test functions. Then, a singular convolution can be given by [12]F(t)=(T*η)(t)=-T(t-x)η(x)dx

Here, the sign * is the convolution operator, F(t) is the convolution of η and T, T(tx) is the singular kernel of the convolution integral. Depending on the form of the kernel T, singular convolution can be applied to different science and engineering problems. Delta kernel

MS technique for plates

MS technique assumes a solution that all system modes discretely contribute to local displacement response. The mathematical foundation of the MS is based on the separation of variables. Bending displacement response of a plate w(x,y,t) can be expressed by the infinite summation of the product of two variables; φp(x,y), the pth natural mode shape of the plate and wp(t), the magnitude of the pth mode [35]:w(x,y,t)=p=1wp(t)φp(x,y)

Eq. (10) can be approximately written in terms of sufficient

Numerical study

This section has been organized as four main parts. The first part is concerned with the high-frequency concept. The second part includes a convergence study for the DSC predictions of thin beams and plates. The third part presents verification study of DSC–MS approach by comparisons of vibration response predictions of thin beams and plates with analytical solutions. The last part concentrates on the performance of the DSC–MS for discrete response analysis of thin plates by presentations of

Conclusion

Available high-frequency approaches are generally based on energy equilibrium between substructures or structural elements. These methods consider average prediction of energy as system variable to describe the response level. Therefore, they disregard modal information and thus, loose discrete response behaviour. This lack of information may cause unrealistic results and leads to unreliable designs.

In the present study, a novel scheme “DSC–MS approach” was introduced for the prediction of

References (42)

  • G.W. Wei

    Vibration analysis by using discrete singular convolution

    Journal of Sound and Vibration

    (2001)
  • G.W. Wei

    A new algorithm for solving some mechanical problems

    Computational Methods in Applied Mechanics and Engineering

    (2001)
  • G.W. Wei et al.

    The determination of natural frequencies of rectangular plates with mixed boundary conditions by discrete singular convolution

    International Journal of Mechanical Sciences

    (2001)
  • Y.B. Zhao et al.

    Plate vibrations under irregular internal supports

    International Journal of Solids and Structures

    (2002)
  • Y.B. Zhao et al.

    DSC analysis of rectangular plates with non-uniform boundary conditions

    Journal of Sound and Vibration

    (2002)
  • S. Zhao et al.

    DSC analysis of free-edged beams by an iteratively matched boundary method

    Journal of Sound and Vibration

    (2005)
  • Ö. Civalek

    Numerical analysis of free vibrations of laminated composite conical and cylindrical shells: Discrete singular convolution (DSC) approach

    Journal of Computational and Applied Mathematics

    (2007)
  • Ö. Civalek

    Free vibration and buckling analyses of composite plates with straight-sided quadrilateral domain based on DSC approach

    Finite Elements in Analysis and Design

    (2007)
  • Ö. Civalek

    A parametric study of the free vibration analysis of rotating laminated cylindrical shells using the method of discrete singular convolution

    Thin Walled Structures

    (2007)
  • A. Seçgin et al.

    Free vibration analysis of symmetrically laminated thin composite plates by using discrete singular convolution (DSC) approach: algorithm and verification

    Journal of Sound and Vibration

    (2008)
  • G.W. Wei et al.

    A novel approach for the analysis of high-frequency vibrations

    Journal of Sound and Vibration

    (2002)
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