1. Dynamics Free vibration: Eigen frequencies
Forced vibration: Harmonic load
Spectral analysis: Seismic
Damping
Karman vibration
17 April 2017

2. Free vibration: Eigen frequencies
SDOF:
met
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:
CQC is 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
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