Dynamics
Free vibration: Eigen frequencies
Forced vibration: Harmonic load
Spectral analysis: Seismic
Damping
Karman vibration
8 April 2015
Free vibration: Eigen frequencies
SDOF:
met
Natural circular frequency
MDOF:
Free vibration: Eigen frequencies
Eigen frequency in Scia.Esa PT:
• M-orthonormalisation
•Masses are distributed to the nodes of the mesh
Free vibration: Eigen frequencies
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
Forced vibration: Harmonic load
Forced vibration: Harmonic load
Dynamic magnification factor
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
Forced vibration: Harmonic load
Forced vibration: Harmonic load
Resonance
•Frequency ratio: r ~1  small harmonic load, large deformation
•Deformation has a finite value!
• n=0  Y/Ys =1
w = sqrt (k/m)
r = n/w
•Zone1:
w large  f (k)
•Zone 2:
f (damping ratio)
•Zone 3:
w small  f (m)
Applying a demping is not always effective!!
Forced vibration: Harmonic load
Example: Electrical motor
Spectral analysis: Seismic
Ground motions can be replaced by an external
harmonic load with amplitude
Spectral analysis: Seismic
Response spectra
Eurocode 8 : Elastic response spectrum Se:
Spectral analysis: Seismic
MDOF-systems
Set of uncoupled differential equations: U = Z.Q
With the solution:
And maximal displacements:
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
Spectral analysis: Seismic
Example: horizontal spectrum
Elastic response spectrum
Design spectrum Sd
Spectral analysis: Seismic
Ground type
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
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
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
Spectral analysis: Seismic
Modal combination methods
• used to calculate the response R (displacements,
velocities, acceleration, …)
•Uncoupled differential equations Combine to a global
response Rtot
Spectrale analyse: Seismisch
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
Spectral analysis: Seismic
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, ...
Spectral analysis: Seismic
New functions:
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.
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
h=
10
 0,7
5x
•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)
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, ...
Damping
Or:
Or:
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
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
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)
Demping
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)
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!
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
Vortex Shedding: Karman vibratie