M M S S V V 海洋大學力學聲響振動實驗室 MSVLAB HRE NTOU 1 Plate vibration Part III

M M S S V V 海洋大學力學聲響振動實驗室 MSVLAB HRE NTOU 2 Outlines 5. Conclusions 2. Plate vibration 1. Introduction 3. Derivation by Circulants 4. SVD updating terms

M M S S V V 海洋大學力學聲響振動實驗室 MSVLAB HRE NTOU 3 Outlines 5. Conclusions 2. Plate vibration 1. Introduction 3. Derivation by Circulants 4. SVD updating terms

M M S S V V 海洋大學力學聲響振動實驗室 MSVLAB HRE NTOU 4 Vibration of plates Governing Equation: is the lateral displacement is the frequency parameter is the biharmonic operator is the domain of the thin plates is the surface density is the flexural rigidity is the angle frequency is the plates thickness is the Young’s modulus is the Poisson ration

M M S S V V 海洋大學力學聲響振動實驗室 MSVLAB HRE NTOU 5 Field representation using RBF sjsj To match B.C.Field representation xixi s1s1 s2s2 s3s3 s4s4 r i1 r ij s 2N s1s1 s2s2 sjsj xixi s3s3 s4s4 r ij x1x1 x2x2 x3x3 x4x4 s 2N

M M S S V V 海洋大學力學聲響振動實驗室 MSVLAB HRE NTOU 6 Data bank of RBF Radial basis function (RBFs) Mesh method Globally-supported RBFs DRBEM Nardini Brebbia 1982 DRBEM Nardini Brebbia 1982 Method of particular integral Ahmad & Banerjee 1986 Method of particular integral Ahmad & Banerjee 1986 Meshless method Globally-supported RBFs Compactly-supported RBFs C.S. Chen Compactly-supported RBFs C.S. Chen Volume potential Method of fundamental solution Method of fundamental solution (Potential theory) The present method Imaginary-part fundamental solution Equivalence Polyzos & Beskos 1994 Equivalence Polyzos & Beskos 1994

M M S S V V 海洋大學力學聲響振動實驗室 MSVLAB HRE NTOU 7 Outlines 5. Conclusions 2. Plate vibration 1. Introduction 3. Derivation by Circulants 4. SVD updating terms

M M S S V V 海洋大學力學聲響振動實驗室 MSVLAB HRE NTOU 8 Displacement field of plate vibration 2N is the number of boundary nodes s is the source point x is the collocation point  j and  j are the unknown densities P and Q can be obtained from either two combinations of U, , M and V

M M S S V V 海洋大學力學聲響振動實驗室 MSVLAB HRE NTOU 9 Four kernels Imaginary-part fundamental solution

M M S S V V 海洋大學力學聲響振動實驗室 MSVLAB HRE NTOU 10 Four kernels

M M S S V V 海洋大學力學聲響振動實驗室 MSVLAB HRE NTOU 11 Operators

M M S S V V 海洋大學力學聲響振動實驗室 MSVLAB HRE NTOU 12 Slope, Moment and Shear Slope Moment Shear

M M S S V V 海洋大學力學聲響振動實驗室 MSVLAB HRE NTOU 13 B.C.True eigenequation Clamped plate u(x)=0  (x)=0 Simply- supported plate u(x)=0 m(x)=0 Free plate m(x)=0 v(x)=0 True eigenequation of three cases

M M S S V V 海洋大學力學聲響振動實驗室 MSVLAB HRE NTOU 14 Clamped plate Nontrivial

M M S S V V 海洋大學力學聲響振動實驗室 MSVLAB HRE NTOU 15 Simply-supported plate Nontrivial

M M S S V V 海洋大學力學聲響振動實驗室 MSVLAB HRE NTOU 16 Free plate Nontrivial

M M S S V V 海洋大學力學聲響振動實驗室 MSVLAB HRE NTOU 17 Governing Equation ▽ 4 u(x) = λ 4 u(x), x  Ω Find interior modesGame over Flow chart of the present method for clamped case by using U and  kernels Off-Diagonal term Direct computation L’Hôpital’s rule Invariant method Diagonal term Constructing the four influence matrices [U], [Θ], [U  ] and [Θ  ] SVD technique [SM] = Φ s ΣΨ s T Plot det[SM] or plot σ 1 versusλ.Find eigenvaluesλ Construction of [SM] Find form  s

M M S S V V 海洋大學力學聲響振動實驗室 MSVLAB HRE NTOU 18 Case 1: Circular clamped plate using the present method

M M S S V V 海洋大學力學聲響振動實驗室 MSVLAB HRE NTOU 19 Case 2: Circular simply-supported plate using the present method

M M S S V V 海洋大學力學聲響振動實驗室 MSVLAB HRE NTOU 20 Case 3: Circular free plate using the present method

M M S S V V 海洋大學力學聲響振動實驗室 MSVLAB HRE NTOU 21 Mode 1 (λ =2.221)

M M S S V V 海洋大學力學聲響振動實驗室 MSVLAB HRE NTOU 22 Mode 2 (λ=3.196)

M M S S V V 海洋大學力學聲響振動實驗室 MSVLAB HRE NTOU 23 Mode 3 (λ =3.728)

M M S S V V 海洋大學力學聲響振動實驗室 MSVLAB HRE NTOU 24 Outlines 5. Conclusions 2. Plate vibration 1. Introduction 3. Derivation by Circulants 4. SVD updating terms

M M S S V V 海洋大學力學聲響振動實驗室 MSVLAB HRE NTOU 25 The degenerate kernels for interior and exterior problems: Degenerate kernels for circular case X Y Exterior problem X Y Interior problem Blue: field points Red: source points

M M S S V V 海洋大學力學聲響振動實驗室 MSVLAB HRE NTOU 26 Circulants

M M S S V V 海洋大學力學聲響振動實驗室 MSVLAB HRE NTOU 27 Circulants

M M S S V V 海洋大學力學聲響振動實驗室 MSVLAB HRE NTOU 28 Circulants

M M S S V V 海洋大學力學聲響振動實驗室 MSVLAB HRE NTOU 29 The eigenvalues of matrices

M M S S V V 海洋大學力學聲響振動實驗室 MSVLAB HRE NTOU 30 Relationships Diagonal matrix Diagonal element

M M S S V V 海洋大學力學聲響振動實驗室 MSVLAB HRE NTOU 31 Determinant (for clamped)

M M S S V V 海洋大學力學聲響振動實驗室 MSVLAB HRE NTOU 32 Eigenequation for clamped boundary

M M S S V V 海洋大學力學聲響振動實驗室 MSVLAB HRE NTOU 33 Determinant (for simply-supported)

M M S S V V 海洋大學力學聲響振動實驗室 MSVLAB HRE NTOU 34 Relationships

M M S S V V 海洋大學力學聲響振動實驗室 MSVLAB HRE NTOU 35 Eigenequation for simply-supported boundary

M M S S V V 海洋大學力學聲響振動實驗室 MSVLAB HRE NTOU 36 Determinant (for simply-supported)

M M S S V V 海洋大學力學聲響振動實驗室 MSVLAB HRE NTOU 37 Relationships

M M S S V V 海洋大學力學聲響振動實驗室 MSVLAB HRE NTOU 38 Eigenequation for free boundary

M M S S V V 海洋大學力學聲響振動實驗室 MSVLAB HRE NTOU 39 Comparisons of the NDIF and present method KangPresent method Base Clamped plate Simply- supported plate Treatment Net approach Dual formulation with SVD updating

M M S S V V 海洋大學力學聲響振動實驗室 MSVLAB HRE NTOU 40 Circular clamped plate using different methods

M M S S V V 海洋大學力學聲響振動實驗室 MSVLAB HRE NTOU 41 Comparisons of Leissa and present method Leissa (Kitahara) Present method Clamped plate Simply- supported plate Free plate J

M M S S V V 海洋大學力學聲響振動實驗室 MSVLAB HRE NTOU 42 Outlines 5. Conclusions 2. Plate vibration 1. Introduction 3. Derivation by Circulants 4. SVD updating terms

M M S S V V 海洋大學力學聲響振動實驗室 MSVLAB HRE NTOU 43 SVD updating terms (clamped case)

M M S S V V 海洋大學力學聲響振動實驗室 MSVLAB HRE NTOU 44 SVD updating terms (clamped case)

M M S S V V 海洋大學力學聲響振動實驗室 MSVLAB HRE NTOU 45 SVD updating terms (clamped case) Based on the least squares

M M S S V V 海洋大學力學聲響振動實驗室 MSVLAB HRE NTOU 46 SVD updating terms (clamped case) The only possibility for zero determinant of [D] at the same time for the same. The common term is True eigenequation

M M S S V V 海洋大學力學聲響振動實驗室 MSVLAB HRE NTOU 47 Outlines 5. Conclusions 2. Plate vibration 1. Introduction 3. Derivation by Circulants 4. SVD updating terms

M M S S V V 海洋大學力學聲響振動實驗室 MSVLAB HRE NTOU 48 Conclusions True eigenequation can be extract out by using SVD updating term Spurious eigenequation only depends on the adopted kernel function, while the true eigenequation is relevant to the specified boundary condition. Since any two combinations of the four types of potentials, six options( ) were considered.

M M S S V V 海洋大學力學聲響振動實驗室 MSVLAB HRE NTOU 49 J. T. Chen, M. H. Chang, I. L. Chung and Y. C. Cheng, 2002, Comments on eigenmode analysis of arbitrarily shaped two- dimensional cavities by the method of point matching, J. Acoust. Soc. Amer., Vol.111, No.1, pp (SCI and EI) J. T. Chen, M. H. Chang, K. H. Chen and S. R. Lin, 2002, Boundary collocation method with meshless concept for acoustic eigenanalysis of two-dimensional cavities using radial basis function, Journal of Sound and Vibration, Vol.257, No.4, pp (SCI and EI) J. T. Chen, M. H. Chang, K. H. Chen, I. L. Chen, 2002,Boundary collocation method for acoustic eigenanalysis of three- dimensional cavities using radial basis function, Computational Mechanics, Vol.29, pp (SCI and EI)Boundary collocation method for acoustic eigenanalysis of three- dimensional cavities using radial basis function J. T. Chen, S. R. Kuo, K. H. Chen and Y. C. Cheng, 2000, Comments on "vibration analysis of arbitrary shaped membranes using non-dimensional dynamic influence function“, Journal of Sound and Vibration, Vol.235, No.1, pp (SCI and EI) J. T. Chen, I. L. Chen, K. H. Chen and Y. T. Lee, 2003, Comments on “Free vibration analysis of arbitrarily shaped plates with clamped edges using wave-type function.”, Journal of Sound and Vibration, Vol.262, pp (SCI and EI) J. T. Chen, I. L. Chen, K. H. Chen, Y. T. Yeh and Y. T. Lee, 2003, A meshless method for free vibration analysis of arbitrarily shaped plates with clamped boundaries using radial basis function, Engineering Analysis with Boundary Elements, Accepted. (SCI and EI)

M M S S V V 海洋大學力學聲響振動實驗室 MSVLAB HRE NTOU 50 歡迎參觀海洋大學力學聲響振動實驗室 烘焙雞及捎來伊妹兒

M M S S V V 海洋大學力學聲響振動實驗室 MSVLAB HRE NTOU 51 Journal of the Chinese Institute of Engineers Call for Papers Special Issue on Meshless Methods If you are interested, please obser to the following schedule ： 1. Abstract due ： Mar. 31, Full paper due ： Jun. 30, Initial acceptance/rejection ： Sep. 30, Final, modified manuscript due ： Dec. 31, Editing deadline ： Mar. 31, 2004 Guest Editors ： Professor D. L. Young, Professor C. S. Chen, Professor A. H. –D. Cheng, Professor H. -K. Hong, Professor J. T. Chen

M M S S V V 海洋大學力學聲響振動實驗室 MSVLAB HRE NTOU 52 國際邊界元素法電子期刊 Betech 2003 Meshless work shop ICOME 2003 第七屆邊界元素法會議

M M S S V V 海洋大學力學聲響振動實驗室 MSVLAB HRE NTOU 53 Thanks for your kind attention There is nothing more practical than the right theory. Whether the theory is right or not depends on the experiment