A novel scheme for the discrete prediction of high-frequency vibration response: Discrete singular convolution–mode superposition approach
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]
Here, the sign * is the convolution operator, F(t) is the convolution of η and T, T(t−x) 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]:
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)
- et al.
Definition of a high-frequency threshold for plates and acoustical spaces
Journal of Sound and Vibration
(2004) - et al.
Energy flow analysis of vibrating beams and plates for discrete random excitations
Journal of Sound and Vibration
(1997) - et al.
Prediction of flow-induced structural vibration and sound radiation using energy flow analysis
Journal of Sound and Vibration
(1999) A wave intensity technique for the analysis of high frequency vibrations
Journal of Sound and Vibration
(1992)A hybrid energy method for predicting high frequency vibrational response of point-loaded plates
Journal of Sound and Vibration
(1997)- et al.
Complex envelope displacement analysis: a quasi-static approach to vibrations
Journal of Sound and Vibration
(1997) A vibro-acoustic model for high frequency analysis
Journal of Sound and Vibration
(1998)- et al.
Prediction of vibrational energy distribution in the thin plate at high-frequency bands by using the ray tracing method
Journal of Sound and Vibration
(2001) Discrete singular convolution for the sine-Gordon equation
Physica D
(2000)Discrete singular convolution for beam analysis
Engineering Structures
(2001)
Vibration analysis by using discrete singular convolution
Journal of Sound and Vibration
A new algorithm for solving some mechanical problems
Computational Methods in Applied Mechanics and Engineering
The determination of natural frequencies of rectangular plates with mixed boundary conditions by discrete singular convolution
International Journal of Mechanical Sciences
Plate vibrations under irregular internal supports
International Journal of Solids and Structures
DSC analysis of rectangular plates with non-uniform boundary conditions
Journal of Sound and Vibration
DSC analysis of free-edged beams by an iteratively matched boundary method
Journal of Sound and Vibration
Numerical analysis of free vibrations of laminated composite conical and cylindrical shells: Discrete singular convolution (DSC) approach
Journal of Computational and Applied Mathematics
Free vibration and buckling analyses of composite plates with straight-sided quadrilateral domain based on DSC approach
Finite Elements in Analysis and Design
A parametric study of the free vibration analysis of rotating laminated cylindrical shells using the method of discrete singular convolution
Thin Walled Structures
Free vibration analysis of symmetrically laminated thin composite plates by using discrete singular convolution (DSC) approach: algorithm and verification
Journal of Sound and Vibration
A novel approach for the analysis of high-frequency vibrations
Journal of Sound and Vibration
Cited by (28)
Analysis of the free vibration of thin rectangular plates with cut-outs using the discrete singular convolution method
2020, Thin-Walled StructuresCitation Excerpt :Abdullah Seçgin et al. analyzed the free vibration and Stochastic vibration of laminated plates by the DSC method [33,34]. Abdullah Seçgin et al. studied the low-to high frequency vibrations of thin plates using discrete singular convolution-mode superposition approach [35,36]. Xinwei Wang et al. [37–41] put forward the Taylor series expansion method and applied it to the solution of the case with free boundary.
Discrete singular convolution method for one-dimensional vibration and acoustics problems with impedance boundaries
2019, Journal of Sound and VibrationCitation Excerpt :Civalek and co-workers have also made great contribution to the development of the DSC especially for the vibration analysis of laminated plates [17–22] and of nano-structures [23–25]. There are also important studies on the application of the DSC to wave propagation problems [26,27] and high frequency vibration analyses [28–30]. However, the DSC suffers from providing accuracy in the modelling of complex boundary conditions when the classical boundary condition implementation is applied.
Thermo-electro-mechanical impedance based structural health monitoring of plates
2014, Composite StructuresCitation Excerpt :Different works are examined for numerical solution quantum problem [24], mechanical vibration [25–27] and Navier–Stokes equation [28]. This algorithm is a good, robustness and efficient method for prediction vibration problem in high frequency [28–33]. Wei [27] investigate three classes of benchmark beam problems, including bending, vibration and buckling using DSC method and conclude that it is much more accurate than the standard local methods, such as the finite element methods.
Geometrically nonlinear dynamic and static analysis of shallow spherical shell resting on two-parameters elastic foundations
2014, International Journal of Pressure Vessels and PipingCitation Excerpt :DSC was proposed to solve linear and nonlinear differential equations by Wei [32], and later it was introduced to solid and fluid mechanics [33–39]. Plate and shells structures have also been modeling via DSC method [40–51]. In general, delta type singular kernels have been performed in the concept of kernels.