FINITE ELEMENT MODELLING OF TRAVELING WAVES By Ashwin Ajith Chandran H00114624 Year 4, B.Eng (Hons.) Mechanical Engineering Supervisor : Prof. Andrew. J. Moore 1
Transcript
1. FINITE ELEMENT MODELLING OF TRAVELING WAVES By Ashwin Ajith
Chandran H00114624 Year 4, B.Eng (Hons.) Mechanical Engineering
Supervisor : Prof. Andrew. J. Moore 1
2. PROJECT BREAKDOWN Natural frequencies of an edge clamped
circular plate Natural frequencies of a centre clamped circular
plate Dual frequency excitation traveling wave (model) FE
Animations 2
3. INTRODUCTION Natural Frequency Natural frequency is the
frequency at which a system tends to oscillate in the absence of
any driving or damping force. 3
4. THEORY ( ) = 2 f (Hz) = 2 Where, D = 3 12(12) And, Flexural
rigidity, D = 6.5462 Radius, R =0.07m Density, = 2700 kg/m3
Thickness, h = 0.001m Youngs Modulus, E = 70 GPa Poissons ratio, =
0.33 Base Mesh 4
5. OBJECT 5 Aluminium Disc
6. STANDING WAVES 6
7. STANDING WAVE 3D DISC 7 Fundamental mode 2 Diametral nodes
and 1 circumferential node 1 circumferential node and 1 diametral
node
9. TRAVELING WAVE EQUATION w(, t) = A(K ) cos( t) + B(K ) sin(
t) (1) Where, K = Spatial Wave number A(K ) and B(K ) are functions
that are both position dependent A(K ) = A1 cos(K ) + A2 sin(K )
(2) B(K ) = B1 cos(K ) + B2 sin(K ) (3) 9
10. TRAVELING WAVE EQUATION Substituting (2) and (3) in (1) w(,
t) = 1 2 (( A1 + B2 ) cos( t K ) + ( B1 A2 ) sin( t K ) + ( A1 B2 )
cos( t + K ) + ( B1 + A2 ) sin( t + K ) ) 10
11. TRAVELING WAVE EQUATION The terms ( t K ) and ( t + K )
determine the positive or negative direction of the traveling wave,
w+ = 1 2 (( A1 + B2 ) cos( t K ) + ( B1 A2 ) sin( t K )) w = 1 2 ((
A1 B2 ) cos( t + K ) + ( B1 + A2 ) sin( t + K ) ) 11
12. SIMULATIONS Case 1 : Edge Clamped Disc Case 2 : Centre and
Unclamped Case 3 : Traveling Wave 12
17. UNCLAMPED AND CENTRE CLAMPED DISCS FE time history
animation comparison and error % Unclamped Centre Clamped f (hz)
n=0 n=1 n=2 n=3 s=0 0 0 0.926469 0.526134 s=1 0.830351 0.67474
0.96337 -0.60419 s=2 0.532026 -0.39209 -0.72923 0.225907 f (hz) n=0
n=1 n=2 n=3 s=0 0 0 -0.62808 -0.45037 s=1 0.788995 0.665105
0.626816 -0.74246 s=2 0.870545 -0.75544 -0.89656 0.495936 17
18. UNCLAMPED AND CENTRE CLAMPED DISCS Eigen frequency chart
comparison Unclamped Disc Centre Clamped Disc 18
19. EIGEN FREQUENCY CONVERGENCE GRAPHS 0 3000 6000 9000 0 10000
20000 30000 40000 50000 60000 70000 EigenFrequency Number of
elements Edge Clamped Disc(m=2,n=0) Frequency(m=2,n=0) 0 2000 4000
6000 0 10000 20000 30000 40000 50000 60000 70000 EigenFrequency
Number of elements Unclamped/Centre Clamped Disc (m=2,n=0)
Frequency (m=2,n=0) 19
20. STEADY STATE MODAL ANALYSIS Modal Dynamics Applying
Constant (1N) Steady State Dynamics, Modal Frequency range 0 Hz
10000 Hz New load, concentrated force, continuous 20
21. STEADY STATE MODAL ANALYSIS FE Animations 21
22. TRAVELING WAVE ANIMATION 22
23. CHALLENGES Hourglassing A spurious deformation mode of a
Finite Element Mesh, resulting from the excitation of zero-energy
degrees of freedom. Caused due to inability to resist deformation
where no stiffness in the mode Resolved by mesh refinement 23
24. FURTHER SCOPE Structural Intensity Injected power
Vibrational energy flow Power flow 24