1. M M S S V V 0 Free vibration analysis of a circular plate with multiple circular holes by using the multipole Trefftz method Wei-Ming Lee Department of Mechanical Engineering, China Institute of Technology, Taipei, Taiwan 2009 年 06 月 17 日國立海洋大學

2. M M S S V V 1 Outlines 4. Concluding remarks 3. Illustrated examples 2. Methods of solution 1. Introduction

3. M M S S V V 2 Outlines 4. Concluding remarks 3. Illustrated examples 2. Methods of solution 1. Introduction

4. M M S S V V 3 Intrduction Circular holes can reduce the weight of the whole structure or to increase the range of inspection. These holes usually cause the change of natural frequency as well as the decrease of load carrying capacity.. Over the past few decades, most of the researches have focused on the analytical solutions for natural frequencies of the circular or annular plates.

5. M M S S V V 4 Laura et al. determined the natural frequencies of circular plate with an eccentric hole by using the Rayleigh-Ritz variational method. Lee et al. proposed a semi-analytical approach to the free vibration analysis of a circular plate with multiple holes by using the indirect and direct boundary integral method. Spurious eigenvalues occur when using BEM or BIEM.

6. M M S S V V 5 The Trefftz method was first presented by Trefftz in 1926 and is categorized as the boundary-type solution such as BEM or BIEM. The Trefftz formulation is regular and free of the problem of improper boundary integrals. The concept of multipole method to solve multiply- connected domain problems was firstly devised by Zaviska. The multipole Trefftz method was proposed to solve plate problems with the multiply-connected domain in an analytical way.

7. M M S S V V 6 Outlines 4. Concluding remarks 3. Illustrated examples 2. Methods of solution 1. Introduction

8. M M S S V V 7 Free vibration of plate Governing Equation: w(x)w(x) ω is the angular frequency D is the flexural rigidity h is the plates thickness E is the Young’s modulus μ is the Poisson’s ratio ρ is the surface density w is the out-of-plane displacement is the frequency parameter is the biharmonic operator is the domain of the thin plates

9. M M S S V V 8 Problem Statement Problem statement for an eigenproblem of a circular plate with multiple circular holes

10. M M S S V V 9 The integral representation for the plate problem The solution of free vibration in the polar coordinate is The Bessel equation The modified Bessel equation

11. M M S S V V 10 The solution for where is defined by

12. M M S S V V 11 The slope, moment and effective shear slope Moment Effective shear

13. M M S S V V 12 Analytical derivations for the eigensolution The lateral displacement by the multipole expansion

14. M M S S V V 13 The Graf's addition theorem

15. M M S S V V 14

16. M M S S V V 15 The addition theorem The displacement field near the circular boundary B 0

17. M M S S V V 16 where

18. M M S S V V 17 The field of bending moment, m(x), near the circular boundary B p (p=1,…,H)

19. M M S S V V 18 The moment operator is defined as The effective shear operator is defined as

20. M M S S V V 19 The field of effective shear, v(x), near the circular boundary B p (p=1,…,H)

21. M M S S V V 20 For an outer clamped circular plate (u = θ = 0) containing multiple circular holes with the free edge (m = v = 0) A coupled infinite system of simultaneous linear algebraic equations A (H+1)(2M+1) system of equations+ the direct-searching scheme by SVD m=0, ±1, ±2, …., ±M

22. M M S S V V 21 Outlines 4. Concluding remarks 3. Illustrated examples 2. Methods of solution 1. Introduction

23. M M S S V V 22 Case 1: A circular plate with an eccentric hole Geometric data: R 0 =1m R 1 =0.4m e=0.5m thickness=0.002m Boundary condition: Inner circle : free Outer circle: clamped

24. M M S S V V 23 Natural frequency parameter versus the number of coefficients of the multipole representation

25. M M S S V V 24 The minimum singular value versus the frequency parameter

26. M M S S V V 25 The former seven frequency parameters, mode types and mode shapes

27. M M S S V V 26 Case 2: A circular plate with three holes Geometric data: R 0 =1m R 1 =0.4m R 2 =0.2m R 3 =0.2m O 0 =(0.0,0.0) O 1 =(0.5,0.0) O 2 =(-0.3,0.4) O 3 =(-0.3,-0.4) thickness=0.002m Boundary condition: Inner circles: free Outer circle: clamped

28. M M S S V V 27 Natural frequency parameter versus the number of coefficients of the multipole representation

29. M M S S V V 28 The minimum singular value versus the frequency parameter

30. M M S S V V 29 The former six natural frequency parameters and mode shapes

31. M M S S V V 30 Outlines 4. Concluding remarks 3. Illustrated examples 2. Methods of solution 1. Introduction

32. M M S S V V 31 Concluding remarks The multipole Trefftz method has successively derived an analytical model for a circular plate containing multiple circular holes. An exact eigensolution can be derived from a coupled infinite system of simultaneous linear algebraic equations. No spurious eigenvalue occurs in the present formulation. Numerical results show good accuracy and fast rate of convergence thanks to the analytical approach. 1. 2. 3. 4. 5. The proposed results match well with those provided by the FEM using many elements to obtain acceptable data for comparison.

33. M M S S V V 32 Thanks for your kind attention The End