Mode Dispersion Curves

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1 Mode Dispersion Curves Fluid-Filled Pipe using FEM George Grigoropoulos Civil Engineer, MSc. Department of Civil and Environmental Engineering Hong Kong University of Science and Technology

2 outline wave quantities FEM model mode identification rigid pipe deformable pipe pipe-only 2

3 wave quantities angular frequency 2 2 f (rad/s) where f 1 frequency (s ), T period (s) spatial frequency analogous to the angular frequency, the spatial frequency is defined as: 2 k (rad/m) wavenumber phase velocity where for a given wave and mode, the phase velocity is defined by: c f = (m/s) k wavelength (m) group velocity is associated with the propagation velocity of a group of waves with similar frequency: c g d = (m/s) dk 3

4 outline wave quantities FEM model automated mode identification rigid pipe deformable pipe pipe-only 4

5 acrylic pipe geometric properties length: 2 m diameter: mm pipe thickness: 4.83 mm element size min max FEM model material properties i. water density: 1000 kg/m 3 bulk modulus: 2.25 GPa c w (m/s) ii. acrylic density: 1190 kg/m 3 E: GPa Poisson s ratio: 0.39 the minimum element dimension can be calculated using the formula: c w Lmax m (n n f min 10 is recommended) element type Abaqus acoustic element, AC3D8 for water shell element S4R is used for the pipe eigensolver boundary conditions Lanczos algorithm c L (m/s) number of eigenvalues in the order of thousands 5

6 outline wave quantities FEM model mode identification rigid pipe deformable pipe pipe-only 6

7 path definition (1) description mode identification mode identification is quintessential in plotting the dispersion curves; a mode has a distinct deformation pattern paths on the system are defined; waveform mode classification (m,n) =(wavelengths in the circumferential path, zero crossings in the radial path) circumferential radial longitudinal path phase velocity: c f = λ freq 7

8 outline wave quantities FEM model mode identification rigid pipe deformable pipe pipe-only 8

9 rigid pipe mode (0,0) or planar mode the last mode shape in 0-40 khz Hz 54 wavelengths wavelength: λ = m 9

10 rigid pipe mode (0,1) the last mode shape in 0-40 khz Hz 22 wavelengths wavelength: λ = m 10

11 rigid pipe mode (1,0) the last mode shape in 0-40 khz Hz 49 wavelengths wavelength: λ = m 11

12 rigid pipe mode (2,0) the last mode shape in 0-40 khz Hz 37.5 wavelengths wavelength: λ = m 12

13 Phase Velocity (m/s) rigid pipe dispersion curves reference comparison Mode (0,0) Mode (0,1) Mode (1,0) Mode (2,0) Reference (0,0) Reference (0,1) Reference (1,0) Reference (2,0) Dispersion Curves - Rigid Pipe Reference (1,0) Mode (1,0) Mode (0,1) Reference (0,1) Mode (2,0) Reference (2,0) 2000 Mode (0,0) 1000 Reference (0,0) Frequency (Hz.) 13

14 outline wave quantities FEM model mode identification rigid pipe deformable pipe pipe-only 14

15 deformable pipe: 0-10 khz mode (0,0)* - water domain >1 peaks between consecutive ZCs cut-on frequency: Hz wavelength: λ = 1.33 m 15

16 deformable pipe: 0-10 khz mode (0,0)* - structure domain pressure peaks and translation peaks have opposite signs wavelength: λ = 1.33 m 16

17 deformable pipe: 0-10 khz mode (2,0) water domain 2 wavelengths along the circumference cut-on frequency: Hz wavelength: λ = 2 m 17

18 deformable pipe: 0-10 khz mode (2,0) structure domain cylindrical R translations shown wavelength: λ = 2 m 18

19 Phase Velocity (m/s) deformable pipe: 0-10 khz dispersion curves Dispersion Curves - Frequency Range: 0-10 khz Mode (0,0)* Mode (0,0) Mode (1,0)* Mode (1,0) Mode (2,0) Mode (3,0) Mode (4,0) Mode (5,0) Torsional Frequency (Hz.) 19

20 Phase Velocity (m/s) deformable pipe: 0-10 khz dispersion curves comparison with Derek s Model Frequency Range: 0-10 khz - Comparison Mode (0,0) Mode (0,0)* Mode (1,0)* Derek's (0,0) Derek's (0,1) Mode (1,0)* 3000 Derek's (1,0) Derek's (1,0) 2000 Derek's (0,1) 1000 Mode (0,0)* Mode (0,0) 0 Derek's (0,0) Frequency (Hz.) 20

21 deformable pipe: khz mode (1,0)** - water domain the last mode shape in khz Hz 18 wavelengths wavelength: λ = m 21

22 deformable pipe: khz mode (1,0)** - structure domain the last mode shape in khz wavelength: λ = m 22

23 Phase Velocity (m/s) deformable pipe: khz dispersion curves 7000 Dispersion Curves - Frequency Range: khz Frequency (Hz.) Mode (0,1) Mode (0,0)* Mode (0,0) Mode (1,0)* Mode (1,0)** Mode (1,0) Mode (2,0) Mode (2,0)* Mode (3,0) Mode (3,0)* Mode (4,0) Mode (5,0) Mode (6,0) Mode (7,0) Torsional 23

24 deformable pipe: khz mode (0,1) water domain the last mode shape in khz Hz 30.5 wavelengths wavelength: λ = m 24

25 deformable pipe: khz mode (0,1) structure domain the last mode shape in khz wavelength: λ = m 25

26 Phase Velocity (m/s) deformable pipe: khz dispersion curves 7000 Dispersion Curves - Frequency Range: khz Frequency (Hz.) Mode (0,1) Mode (0,0) Mode (1,0)** Mode (1,0) Mode (2,0)* Mode (2,0)** Mode (2,0) Mode (3,0) Mode (3,0)* Mode (4,0) Mode (4,0)* Mode (5,0) Mode (6,0) Mode (7,0) Mode (8,0) Torsional 26

27 outline wave quantities FEM model mode identification rigid pipe deformable pipe pipe-only 27

28 pipe-only: 0-10 khz mode (1,0)* the last mode shape in 0-10 khz 28

29 Phase Velocity (m/s) pipe-only: 0-10 khz comparison with the coupled system Dispersion Curves - Mode (1,0)/(1,0)* khz Pipe-only (1,0)* (LF) Pipe-only (1,0)* (HF) Pipe-only (1,0) Coupled (1,0)* (LF) Coupled (1,0)* (HF) Coupled (1,0) Frequency (Hz.) 29

30 pipe-only: khz mode (6,0) the last mode shape in khz 30

31 Phase Velocity (m/s) pipe-only: 0-10 khz comparison with the coupled system Dispersion Curves - Mode (6,0) khz Pipe-only (6,0) Coupled (6,0) Frequency (Hz.) 31

32 thank you 32

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