Computing natural frequencies and mode shapes of a non-uniform circular membrane

https://doi.org/10.1016/j.mechrescom.2020.103553Get rights and content

Highlights

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    Novel techniques to compute natural frequencies and mode shapes of an inhomogeneous circular membrane.

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    Solution of an open problem.

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    Validated the novel techniques by comparison with results from finite element analysis.

Abstract

This paper presents algorithms to compute natural frequencies and mode shapes of a circular membrane with concentric or eccentric non-uniformity as found in many applications such as Indian musical drums. Algorithm for concentric non-uniformity is based on the theory of linear time varying systems whereas that for eccentric non-uniformity is developed by expressing each eigenfunction as a linear combination of eigenfunctions of the corresponding circular membrane with concentric non-uniformity. Numerical results from these approaches are corroborated by comparison with those from a finite element method and existing results in the literature.

Introduction

Vibration of non-uniform circular membranes has been an important topic of research since the publication of experimental results by Raman [1]. For a uniform circular membrane, natural frequencies and mode shapes are obtained analytically [2]. But, numerical approaches are used for non-uniform circular members [3], [4], [5], [6], [7], [8], [9], [10], [11], [12]. There are two types of non-uniformity: concentric (Fig. 1) and eccentric (Fig. 2) spatial variation of mass-density; for example, Indian musical drums [1,[3], [4], [5], [6], [7]], known as Tabla. The right hand Tabla is called “Dayan” and is composed of a circular membrane with a concentric circular dark spot as shown in Fig. 1. The left hand Tabla is called “Bayan” and is also composed of a circular membrane but with an eccentric circular dark spot as shown in Fig. 2. These dark spots are due to “Syahi”, which is a tuning paste composed of flour, water and iron fillings [13], which is applied to make the drum a harmonic instrument; i.e, all natural frequencies are integral multiples of the lowest frequency [1]. Effect of “Syahi” is that it alters the mass of the membrane per unit area at places where it is applied. Then the governing partial differential equation of motion is the Helmholtz equation with a non-uniform mass density. Ramakrishna and Sondhi [3] modeled the Dayan Tabla (circular membrane with concentric non-uniformity) as a composite membrane, and used the classic Bessel functions based solution [2] in each region and matched the solutions at the boundary of each concentric membrane. This approach is analytically cumbersome and has also been used by Gaudet et al. [6], Laura et al. [8] and Rossit et al. [9]. Spence and Horgan [10] used the integral equation method, and found the bounds of natural frequencies of a circular membrane with stepped mass density. Wang [11] found the exact solution for the mass density inversely proportional to the square of radial location. Jabareen and Eisenberger [12] obtained exact power series solutions for polynomial variations in the mass density.

For eccentric variation in the mass density (Fig. 2), the dependence of membrane deflection on polar coordinates r and θ can not be separated. Ramakrishna [4] modeled the Bayan Tabla as a circular membrane for the syahi region alone, and modeled the boundary conditions for this circular membrane as an equivalent, circumferential spring support. First, he found the frequencies for constant stiffness around circumference. He then used a perturbation approach to include circumferentially varying spring stiffnesses and computed frequencies. Sarojini and Rahman [5] developed a variational method using a bipolar coordinate system. These methods [4,5] are cumbersome and do not yield accurate results [7]. Sathej and Adhikari [7] developed a mathematical expression for non-uniform mass density due to Syahi in each Tabla, and then used Fourier-Chebyshev spectral collocation technique to compute natural frequencies and mode shapes. This technique is analytically complex and it is not straightforward to develop a computer program based on this algorithm. Outside the context of Indian musical drum, a paper dealing with eccentric non-uniformity of the mass distribution has not been found by the author of this paper.

In this paper, natural frequencies and mode shapes of a circular membrane with concentric mass non-uniformity are calculated using the novel technique recently developed by the author of this paper [14] utilizing the theory of linear time-varying systems [15]. This technique [14] yields an “almost” closed-form solution of an one-dimensional non-uniform continuous structure because it only requires specification of spatial variations of parameters and the error is guaranteed to be within pre-specified bound at each discrete spatial points. With the concentric patch as shown in Fig. 1, the dependence of membrane deflection on coordinates r and θ can be separated. The θ -equation is a standard second-order differential equation for which solutions are represented as cosine and sine functions. The r-equation turns out to be a second order differential equation with coefficients which are functions of r. Here, the concept developed in [14] is applied to solve this r-equation, and natural frequencies and mode shapes are computed in a straightforward manner without any need to match boundary conditions as required in earlier papers [3,6,8,9]. The advantage of this approach is that it is applicable to any arbitrary variation in the mass density along a radial direction. Also, this method is not analytically complex like Fourier-Chebyshev method [7]. In the earlier paper [14], structures are one-dimensional and represented in cartesian coordinates. This paper is the first application of the new technique in polar coordinates.

For the eccentric mass non-uniformity where the separation of variables is not possible, the solution is expressed as a linear combination of all modes of the circular membrane with the corresponding concentric mass non-uniformity in this paper as they represent a complete set of eigenvectors. Considering only a finite number of modes, an algorithm is developed to compute the natural frequencies and mode shapes of the circular membrane. To the best of author's knowledge, this method has not been used to compute natural frequencies and mode shapes of a circular membrane with eccentric inhomogeneity in the mass distribution. This approach has been used to develop a reduced-order model of a mistuned bladed disk [16], where the cyclic symmetry is lost because of variations in modal properties of blades, known as mistuning. In fact, like a mistuned bladed disk, the cyclic symmetry is lost in the circular membrane because of eccentricity of mass distribution. Again, compared to Fourier-Chebyshev method, this method is very simple to implement. And, there is no need to find the equivalent stiffness at the boundary of circular Syahi region as required by Ramakrishna [4].

First, the governing partial differential equation of motion and associated boundary conditions are presented. Next, new algorithms to compute natural frequencies and mode shapes of circular membranes with concentric and eccentric inhomogeneities are presented. Then, numerical results are presented for the mass density of the circular membrane with Syahi, developed by Sathej and Adhikari [7]. Lastly, Matlab PDE toolbox [17] is used to obtain natural frequencies and mode shapes via Finite Element Method (FEM) for comparison with those from techniques developed in this paper. Results from new algorithms are also compared to those by Sathej and Adhikari [7].

Section snippets

Formulation

Let the radius of a circular membrane be ℓ. The non-dimensional radial coordinate r is defined asr=rrℓwhere rr is the dimensional radial coordinate. Then the time-independent partial differential equation for membrane deflection w(r, θ)with frequency ω is given by [3], [4], [5], [6], [7], [8], [9], [10], [11], [12]∇2w(r,θ)+αρ¯(r,θ)w(r,θ)=0where∇2=∂2∂r2+1r∂∂r+1r2∂2∂θ2α=ℓ2ρ0ω2σρ¯(r,θ)=ρ(r,θ)ρ0

Here, σ is the tension in the membrane and is assumed be a constant. The non-dimensional mass per unit

Numerical results

Numerical results are generated for the following representation of the mass per unit area [7]:ρ¯(r,θ)=1+(β2−1)2[1−tanh(χ(r,θ)−κξ)]whereχ(r,θ)=(rcosθ−ε)2+(rsinθ)2and ε is the eccentricity as shown in Fig. 2.

When β = 1, ρ¯(r,θ)=1 and it represents the case of a constant mass per unit area. Parameters ξ and κ represent smoothness in the variation of ρ¯(r,θ), and ratio of radii of “Syahi” and complete membrane, respectively.

Case I: Concentric Non-Uniformity, Dayan Tabla (ε = 0)

For Dayan Tabla,

Conclusions

First, an efficient computational technique has been developed to compute natural frequencies and mode shapes of a circular membrane with concentric and symmetric non-uniformity in mass distribution by solving the radial differential equation using the theory of linear time-varying systems [14,15]. Then, vibration of a circular membrane with an eccentric and asymmetric non-uniformity in mass distribution is represented as a linear combination of eigenmodes of the corresponding circular membrane

Declaration of Competing Interests

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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