2 july 2015 dynamics free vibration: eigen frequencies forced vibration: harmonic load spectral...

April 19, 2023

Dynamics

• Free vibration: Eigen frequencies• Forced vibration: Harmonic load• Spectral analysis: Seismic• Damping• Karman vibration

Free vibration: Eigen frequencies

met

Natural circular frequency

SDOF:

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/2

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!• =0 Y/Ys =1

•Zone1: large f (k)•Zone 2: f (damping ratio)•Zone 3:small f (m) Applying a demping is not always effective!!

sqrt (k/m)

r =

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: SeismicExample: 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: ag/g•Definition of in the load case manager, since the same spectrum can have different values of

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

Value for •‘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

•Damping ratio > 14,3% no more influence•Damping ratio = 5 % •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% < < 85%(0 is not possible, because this will lead to an infinite deformation)

100,7

5

Damping

•Important influence in the case of resonance!

•Structural damping: always presentCaused 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 gives a harmonic solution!

In the most cases, < 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 PTSupport 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 casesSeismic spectrum XSeismic 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