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Dynamics

Dynamics. Free vibration: Eigen frequencies Forced vibration: Harmonic load Spectral analysis: Seismic Damping Karman vibration. Free vibration: Eigen frequencies. met. SDOF:. Natural circular frequency. MDOF:. Free vibration: Eigen frequencies. Eigen frequency in Scia.Esa PT:

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Dynamics

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  1. Dynamics Free vibration: Eigen frequencies Forced vibration: Harmonic load Spectral analysis: Seismic Damping Karman vibration

  2. Free vibration: Eigen frequencies met SDOF: Natural circular frequency MDOF:

  3. Free vibration: Eigen frequencies • Eigen frequency in Scia.Esa PT: • M-orthonormalisation • Masses are distributed to the nodes of the mesh

  4. Free vibration: Eigen frequencies

  5. Free vibration: Eigen frequencies • Remarks: • Self weight is automatically taken into account • ‘Create masses from load case’!! • Project data: Acceleration of gravity  9.81 m/s^2 • Mass remains unchanged after adapting the related load case • Only generation of the vertical load component

  6. Forced vibration: Harmonic load

  7. Forced vibration: Harmonic load Dynamic magnification factor

  8. Forced vibration: Harmonic load • Y/Ys : large if r ~1  resonance! • Small harmonic load huge deformation • No infinite deformation, but a limit value: Ys/2x • Harmonic load in Scia.Esa PT: • Parameters: • Forcing frequency • Logarithmic decrement • Nodal force: moment or force • Value of the forcing frequency is valid for each load in a load case • Linear calculation: static results are multiplied with • the dynamic magnification factor • Results of the harmonic load case: take both directions into account  • Envelope combination

  9. Forced vibration: Harmonic load

  10. Forced vibration: Harmonic load • Resonance • Frequency ratio: r ~1  small harmonic load, large deformation • Deformation has a finite value! • n=0  Y/Ys =1 • Zone1: • w large  f (k) • Zone 2: • f (damping ratio) • Zone 3: • w small  f (m) • Applying a demping is not always effective!! w = sqrt (k/m) r = n/w

  11. Forced vibration: Harmonic load Example: Electrical motor

  12. Spectral analysis: Seismic Ground motions can be replaced by an external harmonic load with amplitude

  13. Spectral analysis: Seismic Response spectra Eurocode 8 : Elastic response spectrum Se:

  14. Spectral analysis: Seismic MDOF-systems Set of uncoupled differential equations: U = Z.Q With the solution: And maximal displacements:

  15. Spectral analysis: Seismic • Seismic load case in Scia.Esa PT • Same procedure as with the free vibration, • extended with the properties of the seismic load case. • Instead of the vibration of the ground because of an earthquake  • Applying of forces on the static structure so that a linear calculation • can be performed • Linear calculation + included the calculation of the free vibration • Elastic response spectrum Se is reduced to a design spectrum • Sd with parameters: •  Ground type • Ground acceleration • Behaviour factor • Damping

  16. Spectral analysis: Seismic Example: horizontal spectrum Elastic response spectrum Design spectrum Sd

  17. Spectral analysis: Seismic Ground type

  18. Spectral analysis: Seismic • Ground acceleration • Seismic hazard is constant in each zone • Performance is described by the peak ground acceleration agR • Ground acceleration: f(agR) • Mostly, use of acceleration coefficient: a = ag/g • Definition of a in the load case manager, • since the same spectrum can have different values of a

  19. Spectral analysis: Seismic • Behaviour factor q • To avoid inelastic behaviour during the design • Ductile behaviour is taken into account •  Reducing the response spectrum with q • Favourable: large q, but the system has to possess this ductility

  20. Spectral analysis: Seismic • Damping • Standard: 5% • If we have a value different from this one  correction factor h • Value for b • ‘lower bound factor’ for the horizontal spectrum • Advised: 0,2 • Type 1 & 2 • Introducing both spectra

  21. Spectral analysis: Seismic • Modal combination methods • used to calculate the response R (displacements, velocities, acceleration, …) • Uncoupled differential equations Combine to a global response Rtot

  22. Spectrale analyse: Seismisch

  23. Spectral analysis: Seismic • Conclusions: • CQCis based on the modal frequency and modal damping • For CQC, mostly the same damping ratio is used for all modes • CQC is going to take into account the correlations between the different modes. • SRSS if Tf < = 90% Ti

  24. Spectral analysis: Seismic

  25. Spectral analysis: Seismic • 90%-rule: • Take into account as much modes till 90% of the mass is in vibration • In certain cases, also the vertical component has to be taken into account • If avg > 25% + other conditions as: span > 15m, ...

  26. Spectral analysis: Seismic New functions:

  27. Spectral analysis: Seismic • New functions: • Participation mass only: user has to consider 90% rule • Missing mass: Scia.Esa PT creates automatically extra masses until 100% is reached in each direction. • Effective mass is regarded in each direction for each mode. • Residual mass: Scia.Esa PT creates automatically extra masses until 100% is reached in each direction. • Effective mass is regarded in each node in each direction for each mode.

  28. Damping • Standard in Scia.Esa PT: damping ratio is equal to 5% • If there is a deviation: • Spectrum is corrected with the damping coefficient h • Damping ratio > 14,3%  no more influence • Damping ratio = 5 %  h = 0 • Meaning: • Spectral accelerations are augmented because the damping is lower then the standard value, • With other words, there is less damping in the system • 0,0016% < x < 85% • (0 is not possible, because this will lead to an infinite deformation)

  29. Damping • Important influence in the case of resonance! • Structural damping: always present • Caused by hysteresis of the material: • Tranfer of little quantitie energy into warmth augmented by friction • Aerodynamic damping, ...

  30. Damping Or: Or:

  31. Damping • Critical damping: • System becomes in equilibrium without vibration in the shortest possible period Only x < 1 gives a harmonic solution! In the most cases, x < 0,02

  32. Damping • Damping in Scia.Esa PT • Possible on 1D-elements, 2D-elements and on supports • Substructures with different damping properties: • 3 types of damping: • Rayleigh Damping (proportional damping) • Stiffness-weighted Damping (most used!) Or Only valid if resultant damping values < 20% of the critical

  33. Damping Damping in Scia.Esa PT • Support damping (on flexible nodal supports) • quasi not possible: solution: • Replacing support by a beam with the same stiffness • Summation possible of 5.17 + 5.18 • Not every support needs to have a damping, only the flexible • On every (1D/2D) element, a damping can be specified • If this is not done: default value • Material default (vb. For S235) • Global default (demper setup)

  34. Demping

  35. Damping Remark: Following Eurocode: • In Scia.Esa PT: • Creating of 2 load cases • Seismic spectrum X • Seismic spectrum Y Both load cases in a load group of the type ‘Together’ and ‘Accidental’ Combining of load cases in develope combinations (with resp. Coefficient 1 and 0,3)

  36. Vortex Shedding: Karman vibration • At a critical wind velocity: flow lines break away at some points • and vortices are formed • Rising of forces perpendicular on the wind direction • Resultant pressure difference  Formation of a harmonic varying lateral load • with the same frequency as the ‘vortex shedding’ • If the frequency of the vortex shedding ~frequency of the structure •  resonance!

  37. Vortex Shedding: Karman vibration Karman vibration in Scia.Esa PT • Following Czech code • Only influence between a maximal and a minimal wind velocity • Dependant on the number of Reynolds • Introducing sufficient geometrical nodes to the structure! • Solutions: • Special ribs on the surface: reduce the Karman-effect • Applying of damping on the system

  38. Vortex Shedding: Karman vibratie

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