Transverse Vibration Analysis of an Arbitrarily-shaped Membrane by the Weak-form Quadrature Element Method

Size: px

Start display at page:

Download "Transverse Vibration Analysis of an Arbitrarily-shaped Membrane by the Weak-form Quadrature Element Method"

Brandon Lee
5 years ago
Views:

1 COMPUTATIOAL MECHAICS ISCM7, July 3-August, 7, Being, China 7 Tsinghua University Press & Springer Transverse Vibration Analysis of an Arbitrarily-shaped Membrane by the Weak-form Quadrature Element Method H. Zhong*, M. Gao Department of Civil Engineering, Tsinghua University, Being 84, China hzz@tsinghua.edu.cn Abstract The recently proposed weak-form quadrature element method (QEM) is applied to transverse vibration analysis of membranes. It differs considerably from the strong-form quadrature element method where the differential equations are tackled directly. The variational description of the governing Helmholtz equations is used to establish discrete equations. The membrane is partitioned into a few large quadrilateral quadrature elements. In each element, the integrands involved in the variational description of the problem are approximated by high order interpolations at Gauss-Lobatto sampling points and the derivatives in the integrands are approximated using differential quadrature analog at the same sampling points. Two examples are studied to demonstrate the present quadrature element method. In the first example, a benchmark problem is examined to validate the convergence of the present method. Computed results in the transverse vibration analysis of a square membrane are compared with the analytical solution. In the second example, transverse vibrations of circular membranes with an eccentric cutout are considered for various eccentricity ratios. Comparison with other available results is made and good agreement is reached. It is shown that the weak-form quadrature element method is highly efficient and applicable to vibration analysis of arbitrarily-shaped membranes. Key words: differential quadrature, Helmholtz equation, membrane, quadrature element method, vibration ITRODUCTIO The finite element method is characterized by low order piecewise approximation. It typically uses large number of degrees-of-freedom and therefore often results in large computational effort in practice. Membrane structures are widely used in a variety of industrial areas. Finite element analysis of membrane structures can be performed in a routine manner. onetheless, research of development of new numerical methods has been going on for years in pursuit of high efficiency and versatility. The differential quadrature method, introduced first by Bellman and Casti in 97 [], has been shown to be powerful numerical tool []. Since it was first used in structural analysis in 988 [3], significant progress on the development of the differential quadrature method has been made. It has been successfully applied to a variety of structural problems. However, the differential quadrature method is aimed at solution of differential equations. As a result, it is generally restricted to problems with simple geometry and those whose governing equations are known and well defined. To make the differential quadrature more useful and competitive in the computational mechanics, a weak-form quadrature element method (QEM) was proposed recently by the first author[4]. It differs considerably from the strong-form quadrature element method [5, 6] where the differential equations are tackled directly. Variational principles are used to formulate the weak-form quadrature elements. The derivatives in the variational principle are approximated using differential quadrature analog and numerical integration is adopted to evaluate the integrals involved in the variational process. Thus, weak-form quadrature elements enjoy the advantages of weak-form solution technique as the finite element method and preserve the high approximation efficiency of the differential quadrature method. 9

2 In this paper, the concept of the weak-form quadrature element is applied to transverse vibration analysis of arbitrarily-shaped membranes. Two examples are given to demonstrate the quadrature element for vibration analysis of membranes. In the first example, transverse vibrations of a square membrane are studied to verify the convergence of the quadrature element method. In the second example, transverse vibrations of circular membranes with an eccentric cutout are investigated for various eccentricity ratios. All computational results are compared with other available solutions and good agreement is reached. It is shown that the proposed weak-form quadrature element is highly efficient and applicable to transverse vibration analysis of arbitrarily-shaped membranes. QUADRATURE ELEMET AALYSIS 4 3 4(,) η 3(,) y ξ x (, ) (, ) Figure : An arbitrarily-shaped membrane and the standard square for a quadrature element Consider an arbitrarily-shaped membrane that is divided into a number of large quadrilaterals as shown in Figure. For an arbitrary quadrilateral domain, a quadrature element, which is feasible for transformation onto the standard computational domain (see also Figure ), the following parametric point transformation is introduced x= x( ξη, ) = Si( ξη, ) xi ξη,, () y = y( ξη, ) = Si( ξη, ) yi where x and y are space variables defined on the physical domain of the problem, ξ and η are the variables on the standard square domain. The transformation functions Si ( ξ, η) are herein constructed based on the idea of serendipity so that only points on the boundary of the element are needed in the transformation [7]. It is noted that the blending function transformation [8], which is widely used in p-version finite elements [9], is another alternative as long as parametric expressions of the curved side are available. Clearly,, i = j; Si( ξ j, ηj) = δ =, i j. With the chain rule of differentiation, x y ξ ξ ξ x J J x x = = [ J ]. x y J J = η η η y y y Solving for the derivatives with respect to x and y, one obtains x ξ = y η [ J ], in which the inverse Jacobian matrix is given as follows () (3) (4)

3 y y η ξ J J b = = = η ξ [ J ], J x x J J J where x y x y J = JJ JJ = ξ η η ξ (5) (6) is the determinant of the Jacobian matrix in (3). Clearly, one must make sure that the sign of J remains unchanged in the element to avoid nullifying the transformation. The rule of thumb for selecting the nodes and geometric shapes for transformation can refer to many finite element books such as []. Denote the deflection of the membrane as w. The nodal displacement vector of a quadrature element is written as = w w w ξ η ( et ) d, where ξ, η are the numbers of nodes in the two directions in the standard square. Usually, it is convenient to choose = =. (8) ξ η The kinetic energy of the element is given by T = ρhw dxdy, (9) Ω where ρ and h are the density and the thickness of the membrane, respectively; the single dot represents differentiation with respect to time variable. The primary procedure in quadrature element analysis is the approximation of the integrand in the first place. An expedient way to approximate the kinetic energy is to introduce the following interpolation ξ η = i( ) j( )( ) i= j= ρhw l ξ l η ρhw with l i ξ ξ, ξ = = ξ. k j k=, k iξi ξk k=, k jξ j ξk k ( ξ) l ( ξ) The Lagrange interpolation functions are chosen herein in the two directions ξ and η for simplicity and efficiency. As a result, the kinetic energy reads, T T = dm d where the element mass matrix is given as + + ( ρhj) l( ξ) l( η) dξdη =. + + ( ρhj) l ( ξ) l ( η) dξdη ξ η ξ η M For uniform and homogeneous membranes, the density and the thickness of the membrane are constants and therefore the diagonal mass matrix in (3) can be simplified. On the other hand, the potential energy in a quadrature element is given as (7) () () () (3)

4 T T U = + + ε Eεdxdy A dξdη, A, = = ε E ε J (4) where T w w ε =. x y (5) The material behavior matrix of an isotropic uniform membrane is given as E = S, (6) where S is the uniform tension in the membrane. With the inverse of the Jacobian matrix in Eq. (3), slopes are expressed as ε = bε, (7) where an auxiliary vector is given as T w w ε =. ξ η (8) Thus, the potential energy density in terms of the auxiliary vector reads T T T A = ε Eε J = ε bebε J. (9) Again, the Lagrange interpolation functions are used in the two directions to approximate the potential energy density in Eq.(4) in terms of its values at all nodes generated in the domain, i.e. ξ η i ξ j η i= j= A l ( ) l ( ) A, = () where the potential energy at a node in the standard square is expressed as T T A = ε be bε J. () With differential quadrature analog of derivatives, the auxiliary vector at a node is expressed as T ξ η ( ξ) ( η) = Cik wkj Cjk wik = k= k= ε Bd, () where the weighting coefficients for the first order derivatives are obtained using the formula first developed by Quan and Chang [] i j=, j i i k k j j=, j k ( ξ) ( ξ) ii = C j=, j i ( ξ ξ ) ( ξ ) Cik =, i, k =,,...,, and i k; ( ξ ξ ) ( ξ ξ ) C In the same manner,. j (3)

5 j i ( η ) i=, i j Cjk = j k = j k C j k k i i=, i k ( η) ( η) jj = Cji i=, i j ( η η ),,,,...,, and ; ( η η ) ( η η ). Since the membrane theory is C continuous, weighting coefficients for higher order derivatives are not needed. Details of the differential quadrature method can be referred to [3,]. To preserve the high order approximation characteristic of the present method, non-uniform node patterns are used. The Lobatto non-uniform node distribution in the normalized range [3] is ξ =,..., ξi,, ξ =, i=,...,, (5) ξi being the (i-) th zero of dp ( ) ξ =, dξ where P ( ) ξ is the (-)th order Legendre polynomial. The same node distribution is used in the η direction. Then, the potential energy at a node reads T T T T A = d BbDb J Bd = d Dd, (7) where D B b D b B J (8) T T =. Substitution of Eq.() into Eq.(4), along with Eq.(7), yields T U = + A dξdη =, where the stiffness matrix is given as d K d (9) ξ η + + K = D li ( ξ ) lj ( η) dξdη. (3) i= j= Apparently, K is symmetric. Ideally, the numerical integration rule in its expression uses the same node pattern as that for approximation of derivatives and achieves the highest accuracy. With numerical integration rule, the stiffness matrix in Eq. (3) reads ξ η K = WW i jd, (3) i= j= where Wi and W j are the weight coefficients for a numerical integration rule. For Gauss-Lobatto rule, l i ξ dξ = Wl i + Wli + Wkli ξk + R l i k = ( ) ( ) () ( ) [ ], (3) where R[ li] is the error that is of order (-) and W = W =, ( ) Wk =, k =,...,. ( )[ P ( ξ )] k (4) (6) (33) 3

6 In an analogous manner, the mass matrix in Eq.(3) is written as ( ρhj) WW M. (34) = ( ρ hj) W W ξ η ξ η Thus, the Lagrange function of the entire membrane is written as T T L= U T = dkd dmd, (35) where K, M and d are the global stiffness matrix, the global mass matrix and the global displacement vector, respectively, and e),. (36) U = U T = T Application of the Hamilton principle to the system yields the global dynamic equilibrium equations Kd + Md =, (37) where the double dot represents the second order differentiation with respect to time variable. When displacement vector d is harmonic in time with circular frequency ω, i.e. d= d exp( iωt), Eq. (37) turns into ( ω ) K M d= which can be readily cast into a standard eigenvalue problem. After imposition of essential boundary conditions, the natural frequencies and the corresponding modes shapes are extracted from the eigen-equations using numerical methods such as the QR algorithm [4]. UMERICAL EXAMPLES. Example one The convergence of quadrature element analysis of membrane vibrations is examined by considering a square membrane. Assume that the side length of the square is a. The commonly used non-dimensional frequency parameter is introduced, i.e. ω = ωa ρh S. (39) The boundary condition of the membrane requires the enforcement of Dirichlet condition at the periphery of the membrane, i.e. w =. (4) The exact solution of the non-dimensional frequency [5] is ω = π m + n, (4) where m and n are the numbers of half-wave of the mode shape in the x and y directions, respectively. (38) The entire square membrane is modeled using one quadrature element with the same number of nodes in the two directions. Figure displays the convergence of three selected frequencies, ω, ω 4 and ω of a square membrane. It is found that the convergence of the quadrature element solution is not monotonic. This is believed to be ascribed to the prorating scheme adopted during the construction of mass matrix. The relative error of the fundamental frequency is less than.4% when is increased to 5. With only = 8, the relative error of the th frequency is less than.%, highlighting the rapid convergence rate of the quadrature element solution. 4

7 Figure:Convergence of frequencies of a square membrane. Example To illustrate the ability of the quadrature element method to cope with vibration analysis of arbitrarily-shaped membranes, a circular membrane with an eccentric circular cutout is considered in this example. Analytical solutions of the problem were given by agaya [6] and Lin [7] using series expansion technique. In order to compare with the available results, a non-dimensional frequency parameter similar to that given in Eq. (39) is defined in this example, i.e. ω = ωr ρhs, (4) where R is the radius of the outer circle of the membrane. The radius of the inner circle and the eccentricity are denoted as R and e. To exploit the symmetry of the membrane, only half of the membrane is dealt with in quadrature element analysis, which can be modeled using one quadrature element as shown in Figure 3. The following blending function transformation + η ξ η ξ η x = R cos π + R cos π e + η ξ η ξ y = R sin π + R sin π is used herein to map the quadrature element onto the standard square. Membranes with two radius ratios and various eccentricity ratios are considered. The frequencies of the first five symmetric and anti-symmetric mode shapes are presented in Table. A relatively fine grid, = 5, is used in all computational cases of this example to ensure high accuracy of the results. Comparison with other available results in [6] and [7] is also made. In majority cases, the present predictions agree very well with those results given in [6] and [7]. For some cases where discrepancies appear, it is believed that the relevant results given in [6] and [7] suffer from calculating error as pointed out in [4]. y (43) R 4 R 3 e x Figure 3: A circular membrane with an eccentric circular cutout 5

8 Table on-dimensional frequencies of the first five symmetric and anti-symmetric modes of circular membrane with an eccentric circular cutout R R er R R er Symmetric modes Present [6] [7] Present [6] [7] Present [6] [7] Present [6] [7] Present [6] [7] Present [6] [7] Present [6] [7] Present [6] [7] Anti-symmetric modes Present [6] [7] Present [6] [7] Present [6] [7] Present [6] [7] Present [6] [7] Present [6] [7] Present [6] [7] Present [6] [7]

9 It is seen that the frequency of the membrane increases with the increase of the radius ratio given the same eccentricity ratio. It is also seen that the frequencies of the first symmetric and anti-symmetric modes decreases with the increase of eccentricity ratio. This observation differs from the results reported in Lin s work [7] where the first two frequencies of symmetric and anti-symmetric modes were reported to exhibit in the same manner. For other symmetric and anti-symmetric modes, the values of frequencies increase first before a peak value is reached with the increase of the eccentricity. Further increase of the eccentricity ratio leads to the decline of these frequencies. The rise-fall phenomenon of these frequencies was also reported in [7]. COCLUDIG REMARKS Vibration analysis of membranes has been performed using the weak-form quadrature element method. Results have been compared with available analytical solutions and good agreement has been reached. It has been shown that the quadrature element method enjoys rapid convergence rate. Although the full stiffness matrix in one quadarture element is a disadvantage, this is more or less offset by the gain in the diagonal element mass matrix resulting from the prorating scheme. It is shown that the quadrature element method is a promising numerical tool and further investigation is under way to extend the present method to other computational problems. Acknowledgements The support of Tsinghua Fundamental Research Foundation JCxx569 is gratefully acknowledged. REFERECES. Bellman RE, Casti J. Differential quadrature and long-term integration. J. Math. Anal. Appl., 97; 34: Bert CW, Malik M. Differential quadrature method in computational mechanics: a review. Appl. Mech. Rev., 996; 49: Bert CW, Jang SK, Striz AG. Two new approximate methods for analyzing free vibration of structural components, AIAA J., 988; 6: Zhong H, Yu T. Flexural vibration analysis of an eccentric annular Mindlin plate. Arch. Appl. Mech., 7; 77: Striz AG, Chen W, Bert CW. Static analysis of structures by the quadrature element method (QEM), Int. J. Solids Struct., 994; 3: Zhong H, He Y. Solution of Poisson and Laplace equations by quadrilateral quadrature element, Inter. J. Solids Struct., 998; 35(): Zhong H, He Y. A note on incorporation of domain decomposition into the differential quadrature method, Commun. umer. Meth. Engng., 3; 9(4), Gordon WJ. Blending function methods of bivariate and multivariate interpolation and approximation. SIAM J. umer. Analysis, 97; 8: Szabó BA, Babuška I. Finite Element Analysis, John Wiley & Sons, ew York, 99.. Zienkiewicz OC, Taylor RL. The Finite Element Method, 5th edn. Oxford: Butterworth Heinemann,.. Quan JR, Chang CT. ew insights in solving distributed system equations by the quadrature method- I. Analysis. Comput. Chem. Engng., 989; 3: Shu C. Differential Quadrature and Its Application in Engineering. Springer-Verlag, London,. 3. Davis PI, Rabinowitz P. Methods of umerical Integration, nd edn. Academic Press, Orlando, Press WH, Flannery BP, Teukolsky SA, Vetterling WT. umerical Recipes The Art of Scientific Computing. Cambridge University Press, Cambridge, Weaver WJr, Timoshenko SP, Young DH. Vibration Problems in Engineering, 5th Edn, John Wiley and Sons, ew York, 99. 7

10 6. Lin WH. Free transverse vibrations of uniform circular plates and membranes with eccentric holes. J. Sound Vib., 98; 8(3): agaya K. Transverse vibration of a plate having an eccentric inner boundary. ASME J. Appl. Mech., 977; 44:

Vibration of Thin Beams by PIM and RPIM methods. *B. Kanber¹, and O. M. Tufik 1

APCOM & ISCM -4 th December, 23, Singapore Vibration of Thin Beams by PIM and RPIM methods *B. Kanber¹, and O. M. Tufik Mechanical Engineering Department, University of Gaziantep, Turkey. *Corresponding