Abstract. 1 Introduction

Transcription

1 A numerical approach for the free vibration of isotropic triangular plates J.P. Arenas Institute ofacoustics, Facultad de Ciencias de la Ingenieria, Universidad Austral de Chile, PO Box 567, Valdivia, Chile Abstract The final aim of this paper is to solve the classic differential equation for the behaviour of the free flexural vibration of a thin, triangular, isotropic and homogeneous plate subject to a classic boundary condition by means of a discrete relaxation of a rectangular mesh. Modified Numerow-Cowling algorithm is applied to get the tridimensional shapes modes. The characteristic frequencies are calculated by means of a collocation method and their results are compared to the Rayleigh-Ritz method. Results for a thin aluminium plate of triangular shape for different aspect ratios are shown. 1 Introduction Any vibrating surface, especially a thin plate radiates sound due to the interaction between the solid surface and the air particles that surround it. The frequency of the radiated sound goes in direct relation with the bending waves propagated on the plate. In a similar way, an acoustic wave that hits a plate will propagate flexural waves proportional to the incident sound. It is difficult to get a plate thin enough which possesses a small mass per unit area and yet stiff enough to have a high-frequency fundamental. However, plate diaphragms have been successfully used on miniature condenser microphones. Similarly, a certain type of acoustic compliances are used together with thin plates, which have a useful frequency range restrained to the regions below the lowest frequency of resonance. A great number of theoretical studies designed to find the solutions to the classical equation that describes the behaviour of the thin plates have been reported. Nevertheless, the theoretical studies are restricted to only a few of the coordinated systems, expressing theflexuralwaves in terms of a suitable sum of parallel waves[l] and studying the properties of the integrals. These can be very

2 20 Computational Acoustics difficult to evaluate, due to the big amount of numerical integrations needed to find the shape modes, specially if the plates have irregular forms as in triangular shapes. The problem of the free vibration of triangular plates is of practical importance and it has produced a big number of works in the area. Different methods have been utilized to get both the frequencies and the modes with the clamped boundary condition. Cox & Klein[2] used the collocation method, which consists of satisfying a given differential equation, or set of equations, at a finite number of points. They change the classical equation expressing it in skew coordinates. The finite element method has been employed by other authors like Mirza & Bijlani[3] and Utjes et. al.[4], who demostrate the analytical and computational complexity of the problem. Gorman[5] reports a modified superposition method for the problem of right triangular plates. In Kim & Dickinson's formidable works [6,7] a simple, straightforward Rayleigh-Ritz analysis is applied for triangular plates having any combination of the classical free, simply supported or clamped boundary conditions. In these works, the problem has been transformed into an eigenvalue equation and the tabulated results are expressed as a frequency parameter. In this report, a discrete points algorithm will be used to numerically solve the classical differential equation through relaxation. 2 Differential equation discretization The classical equation describing the free flexural waves behaviour in a plate thin, isotropic and homogeneous is, according to Timoshenko & Woinowsky- Krieger[8]andGraff[9]: where Y is the deflection of the vibration, E is the Young' s moduli, 8 is the plate thickness, p is the material density, and v is the Poisson ratio. Usually, the quantity E8^/12(l-v^) is represented by B, the bending stiffness. The difficulty to solve this equation lays in the operator V^, which can be separated for polar coordinates, that is to say, for circular plates. In the reference [8], an analytical technique to separate the operator in two differential equations is shown, but it is necessary to calculate the bending and twisting moments. If in equation (1), a small amplitude, simple harmonic vibration is supposed, we can consider *F= (x,y)exp{jcot}, where co is the radian natural frequency. Thus, the equation will be: To discretizase the coordinates in the x-y plane, a rectangular mesh was used, so as not to have first and second class points, that is to say points inside or outside the physical limits of the triangular plate. The width of x will be h and for y, it will be k. The fourth order derivatives were discretized through

3 Computational Acoustics 21 Bickley's given tables[10], while the operator with the crossed derivatives was obtained, according to the Numerow-Cowling method[ll], expanding in Taylor's series the functions Y(x+nh,y+nk) where n varies between -2 and +2. Operating with these series, the following discrete version for Y(x,y) will be used: (3) where y is co^p5/b. 3 Numerical Method The former equation can be easily transformed into a computer code. If the x axis is divided into N intervals of h width and the y axis into M intervals of k width, a discretization of the triangular plate surface is obtained. Now, placing the coordinates origin in a corner of the triangle's base, h and k can be obtained through the base and height dimensions, as shown in figure 1. The discretized version allows to calculate the value of Y in point (x,y) starting from the adyacent values, so the relaxation method is used, that means that random data is assigned to all the inner surface points ( not the borders ) and later on the surface is swept a number of times until the solution is convergent. ry 2 -a/2 Figure 1: Description of the reference systems used. The $ variable is defined for the collocation method, while h and k define the relaxation mesh.

4 22 Computational Acoustics The contours must be calculated starting with the boundary conditions. It is convenient to improve the solution starting with simultaneous border corrections, like the one suggested by Liebmann[12]. According to Gorman[5], since any triangular plate can be descomposed into two right triangular shapes, it was numerically experimented to test the method with plates bearing this shape. The eigenfrequencies are calculated by collocation in skew coordinates according to the Cox & Klein method for plates with the classic clamped boundary condition[2], this is, Y(x,y)=0 over the boundary and 3 /3n=0, where n is the normal direction to a boundary. 4 Numerical results and discussion In order to compare the results obtained through Cox & Klein collocation method with the Rayleigh-Ritz analysis used by Kim & Dickinson, it was necessary to change the skew coordinates to a cartesian coordinates system. It was usual to define an aspect ratio, which will be b/a in this paper. The computer codes were worked on in TurboPascal and for the recurrences, double precision was used. The fundamental frequencies obtained for different aspect ratios of isoceles and right aluminium triangular plates ( E = 72x10^ N/nf, 8=0.001 m, p=2,700 Kg/nP, v=0.34 ) are shown in figure 2. A relative concordance can be seen with Rayleigh-Ritz analysis, particularly for aspect ratios between 1 and 2. The differences in the other orders may be due to the sensibility of the collocation method when choosing the skew coordinates origin. In figure 3, thefirstsix mode frequencies for a right triangular plate of aspect ratio equal to 1, compared to the classic squared and circular plates of equivalent surfaces, are shown for design purposes. The relaxation of the algorithm described in the equation (3) allowed to estimate the shape modes and nodal patterns for the right triangular plate of aspect 1. Figure 4 shows the results for thefirstthree modes, including the flapping mode. The amplitudes are only referential and were used for visualization purposes. The results of the nodal lines agrees with the ones obtained by other authors. It can be seen that in the second mode, the nodal line splits the plate into 2 equal parts, while a third mode a fourth of a circle is produced with its center in the vertex of an angle. 5 Conclusions The frequencies of the vibration modes for triangular plates can be obtained through the previously described methodology with a certain grade of accuracy. Similarly, relaxation is possible when a simultaneous correction of the boundary conditions is developed until a convergent value is found. This will allow to tridimensionally visualize the shape modes and nodal lines of the flexural vibration, responsible of the acoustic radiation This can be useful way to calculate the modal behaviour for design purposes in structures radiating sound.

5 Computational Acoustics 23 Isosceles by collocation A Isosceles by Rayleigh-Ritz 18 Right by collocation _ D Right by Rayleigh-Ritz _ Hz ,0 0,8 1, , ,8 5, aspect ratio Figure 2: Fundamental frequency of clamped isosceles and right triangular aluminium plates for different aspect ratios b/a D O A squared circular triangular Mode number Figure 3: Comparison of frequency modes for a right triangular aluminium plate of aspect ratio equal to 1 and the classic squared and circular of equivalent surfaces.

6 24 Computational Acoustics Figure 4: Mode shapes for the free flexural vibration of a right triangular plate of aspect ratio equal to 1.

7 References Computational Acoustics Morse, P.M. Vibration and Sound, Acoust. Soc. of America, New York, Cox, H.L. & Klein, B Fundamentalfrequenciesof clamped triangular plates, Journal of Acoust. Soc. Am., 1955, 27(2), Mirza, S. & Bijlani, M. Vibration of triangular plates, American Institute of Aeronautics and Astronautics Journal, 1983,21, Utjes, J.C., Ercoli, L, Laura, PA A & Santos, R.D. Transverse vibrations of right triangular plates having the hypotenuse free, Journal of Sound and Vibration, 1985, 102, Gorman, D.J. A modified superposition method for the free vibration analysis of right triangular plates, Journal of Sound and Vibration, 1987, 112, Kim, C S & Dickinson, S. M Thefreeflexural vibration ofrighttriangular isotropic and orthotropic plates, Journal of Sound and Vibration, 1990, 141, Kim, C.S. & Dickinson, S. M. Thefreeflexuralvibration of isotropic and orthotropic general triangular shaped plates, Journal of Sound and Vibration, 1992, 152, Timoshenko, S. & Woinowsky-Krieger, S. Theory of plates and shells, McGraw-Hill Kogakusha, New York, Graff, K.F. Wave motion in elastic solids, Dover Publ., New York, Bickley, W.G. Formulae for numerical differentiation, Math. Gaz., 1941, 25, Koonin, S. Computational Physics, Benjamin-Cummings, New York, Demidowitsch, B.P., Maron, LA. & Schuwalowa, E.S. Numerical Methods ofanalysis, Paraninfo, Madrid, 1980.