1. Waves and Finite Elements Brian Mace ISVR, University of Southampton brm@isvr.soton.ac.uk With acknowledgements to: Y Waki, E Manconi, L Hinke, D Duhamel, M Brennan

2. Background  Waves in structures  Useful approach especially at higher frequencies  Structure borne sound, energy methods, SEA, waveguide structures etc  Wave properties  Wavenumbers, dispersion relations, group velocity etc  Refection and transmission  Forced response, acoustic transmission etc  How to calculate wave properties?  Simple structures: analytical solutions  Complex structures?  Numerical methods, spectral elements, spectral finite elements  Wave/finite element (WFE) method

3. Overview: the WFE approach  The WFE method  Periodic structure theory applied to a homogeneous structure  Mesh small segment of structure  Use conventional FE and commercial FE package  Post-process mass and stiffness matrices  WFE for 1-dimensional waveguides  Beams and rods  Plate strips, isotropic laminated plates, tyres  WFE for 2-dimensional structure  Plates  Orthotropic laminated plates  Cylinders, sandwich cylinders

4. WFE Method for 1-Dimensional Waveguide Structures

5. 1-Dimensional Structural Waveguides  Uniform in one direction (x-axis)  Cross-section may be 0, 1 or 2-dimensional  A wave propagates as x Beam of arbitrary section Plate strip Isotropic laminate Tyre

6. General approach  FEA of short section of waveguide  Find mass and stiffness matrices M, K  Include damping (C, K complex)  Equations of motion:  Periodicity condition:  x phase change attenuation

7. The eigenvalue problem  Can write this in several ways, e.g. project equations of motion onto DOFs q L  In terms of partitions of  Given  2, find  Numerical issues:  Numerical conditioning  FE discretisation rules (6 elements/wavelength)  Periodic structure effects

8. Applications to 1-Dimensional Waveguide Structures

9. Example: laminated panel FE mesh  17 rectangular plane-strain elements  2 DOFs per node (x,y displacements)  36 DOFs per cross-section, 36 wave pairs  1mm 0.6mm 15mm x y aluminium skins viscoelastic core

10. Wavenumbers of propagating waves Many wave modes Dispersive Complex cut-off effects

11. Cut-off effects around 8kHz 7600770078007900800081008200 10 10 0 1 2 3 a b c e g d f Wavenumber (m -1 ) Frequency (Hz) 7810Hz: Wave pair d,e cut-off with same finite, non-zero wavenumber. Mode e has positive phase velocity, negative group velocity: a Michael Jackson “Moonwalk” wave 8016Hz: Wave pair b,e cut off 8039: Wave pair f,g cut off.

12. Waves in a plate strip  Strip of width 2b  Waves propagate in x-direction  WFE method requires segment of width  x y y = b y = -b qRfRqRfR  qLfLqLfL

13. Free edges 1  Analytical solution for dispersion equation  Transcendental  Real, imaginary and complex solutions  WFE model  ANSYS 7.1, SHELL63 elements (4 nodes, 6 DOFs per node)  90 elements across strip  273 reduced DOFs 1. Y Waki et al

14. Propagating waves S0S0 A0A0 A1A1 S1S1 S2S2 A2A2 S3S3 A3A3  Symmetric (S) and antisymmetric (A) wave modes of order n  Cut-off frequencies  n = 0 modes cut-on at  = 0: “beam bending” and “twisting” modes  WFE results used as initial estimates for numerical solutions to analytical dispersion equation Non-dimensional: frequency wavenumber Symmetric Antisymmetric Analytical solutions

15. Complex wavenumbers, symmetric modes  Very complicated behaviour  Wavenumbers either imaginary or complex conjugate pairs  Many bifurcations  Bifurcation from a pair of imaginary wavenumbers to a complex conjugate pair  Bifurcation from a complex conjugate pair to a pair of imaginary wavenumbers  Bifurcation from imaginary to real (cut-on of S 2 propagating wave) 02468101214161820 -5 -4 -3 -2 0 1  - | Im (  ) | | Re (  ) | S0S0 S2S2 S 1,2 S 0,1 S 0,2 S 2,3 S3S3 S 0,3 S 2,3 S 0,3 S3S3 S2S2 S1S1 S1S1 S0S0 Analytical solutions

16.  Tyre/road noise important to 2kHz  Full FEA impractical  The tyre  Complicated construction: steel and textile fibres, several different rubbers  Frequency dependent stiffness  Internal pressure and in-plane tension Application to tyre vibration 1 1. Y Waki et al, InterNoise 2007, ICA 2007

17. The FE model  Bridgestone 195/65R15, smooth tread  ANSYS 7.1,  SOLID46 elements  Segment of angle 1.8º modelled  324 DOFs  Frequency dependence can be included without re-meshing Fixed to rim (R=0.21 m) 0.2 m Belt radius 0.32 m Belt width = 0.18 m

18. A1A1 S1S1 A2A2 TS1TS1 S2S2 A3A3 S3S3 TA2TA2 TS2TS2 TS1TS1 Mode S1: tread bounce Free wave propagation  Complicated behaviour  Real, imaginary and complex wavenumbers:  Antisymettric (A)  Symmetric (S)  Transverse (T) Mode A1: side-to- side motion of tread Mode A2 Mode S2  Curve veering  Cut-on at non-zero wavenumber  Group and phase velocities of opposite sign (Michael Jackson “Moonwalk” waves)

19. Forced response: input mobility at tread centre  Response = sum of waves  Sensitive to exact distribution of load  70 wave modes kept  3 frequency regions: I Below cut-on of S1 mode (stiffness dominated) II Like a beam on flexible foundation III Non-modal – asymptotes to elastic half-space Phase III III Point response Distributed over circle of radius 11.8mm (as in experiment)  Experiment  Freely supported  Force applied through rigid disc Experiment Distributed over circle of radius 11.8mm (as in experiment)

20. Forced response: simulated motion 93Hz 114Hz 145Hz 192Hz 336Hz 374Hz 1206Hz 180Hz

21. Wave Finite Element Method for 2-Dimensional Structures

22. The WFE approach Continuous, uniform 2-D structure Wave propagates as (wavenumbers may be real, imaginary or complex) x z y x y LxLx LyLy FE model of small rectangular segment (internal DOFs reduced) (edge nodes no problem)

23. WFE formulation  FEA of small segment  Use conventional FE package  Typically just 1 rectangular element  Equations of motion:  Periodicity conditions:  Propagation constants and wavenumbers: q1q1 q3q3 q2q2 q4q4 x y DOFs K, M = stiffness and mass matrices  FEA of small segment  Use conventional FE package  Typically just 1 rectangular element  Equations of motion:

24. Eigenvalue problem in 2D  Apply periodicity conditions, write equations of motion in terms of DOFs of node 1 only:  = (reduced) dynamic stiffness matrix of segment  Various forms:  Given x, y, find  2  Given x,  2 find y  Given  2 and direction of propagation, find x, y  Numerical issues:  Conditioning generally not a problem  FE discretisation errors, spatially periodic

25.  FE model of cylinder using piecewise-flat elements  Coordinate axes rotated at right side of element  Wave motion:  Helical waves: k x L x takes any value  Closed cylinder: circumferential order n = 0, 1, 2…, k x L x = n  Application to cylinders 1 Segment of a cylinderFlat shell elements x y   x y 1 Manconi and Mace

26. Applications of WFE Method to Cylindrical Shells

27. Isotropic cylinder  First 3 branches of n = 0, 1 circumferential modes  Cut-off frequencies  Ring frequency:  Above ring frequency ~ asymptotes to a flat plate Frequency/ring frequency n = 0 branches n = 1 branches

28. Comparison with Flugge theory  Steel cylinder, h/R=0.1  Agreement good _____ WFE; + theory n = 3n = 2 n = 1

29. Orthotropic cylindrical sandwich shell  Two glass/epoxy skins with a foam core  Cylindrical, 1m radius  Similar to that analysed by Heron [20] 4 sheets of glass/epoxy, +45/-45/-45/+454 mm ROHACELL foam core 10 mm 4 sheets of glass/epoxy, +45/-45/-45/+454 mm

30. Orthotropic cylindrical sandwich shell  First 3 branches of n = 0, 1 circumferential modes  Ring frequency ~ 617Hz (cut-on of n=0, branch 3)  Very complex behaviour below ring frequency  Veering  Cut-on of propagating waves with non-zero wavenumber  Phase and group velocities of opposite sign  Existence of 2 or more values of k y for same branch at same frequency n = 1 branches n = 0 branches

31. Fluid-filled cylindrical pipes  Cylinder modelled with solid elements  Fluid modelled by acoustic elements  Example:  Steel, water-filled cylinder  ANSYS: 2 x SOLID45 and 20 x FLUID30 elements  30 DOFs after reduction Solid elements Fluid elements

32. Water-filled steel pipe  Each curve associated with wave motion that is predominantly  Bending  Torsion  Axial or  Fluid wave n = 0 n = 1 n = 2 n = 3 Fluid wave Axial wave

33. Concluding remarks  WFE method:  Post-process small, conventional FE model  Periodicity condition, eigenvalue problem  Various examples presented:  1-dimensional waveguide structures  2-dimensional structure  Propagating and attenuating waves (real or complex wavenumbers)  Free and forced response  Results found for negligible computational cost  WFE uses full power of commercial FE packages, element libraries etc