Dynamics Free vibration Eigen frequencies Forced vibration Harmonic
Dynamics • Free vibration: Eigen frequencies • Forced vibration: Harmonic load • Spectral analysis: Seismic • Damping • Karman vibration 11/28/2020
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/2 x 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 • Zone 1: 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 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 ag. R • Ground acceleration: f(ag. R) • 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 > 15 m, . . .
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 • 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:
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 in Scia. Esa PT • Possible on 1 D-elements, 2 D-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 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 (1 D/2 D) element, a damping can be specified • If this is not done: default value Material default (vb. For S 235) 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 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
- Slides: 38