Rayleigh damping

Figure 9 Mean out-crossing computation for iinear eiastic structures subjected to white noise from at rest initiai conditions (a) SDOF system (T = 0.3 i s, = 0. i0),and (b) 3-DOF steel buiiding modei (Ti =0.38 s,l2 =0. i 3 s.Ts = 0.09 s, i = 3 =0.02, Rayleigh damping).

The damping effects can be represented using the Rayleigh damping and the final equations of motion can be obtained in terms of relative displacements. 

Trombetti, T, Silvestri, S. (2006). On the modal damping ratios of shear-type stmctures equipped with Rayleigh damping systems. Journal of Sound and Vibration, 292(1-2), 21-58. doi 10.1016/j. jsv.2005.07.023... 

Hall, J. F. (2006). Problems encountered from the use (or misuse) of Rayleigh damping. Earthquake and Structural Dynamics, 35, 522-545. 

Trombetti, T., Silvestri, S. (2007). Novel schemes for inserting seismic dampers in shear type systems based upon the mass proportional component of Rayleigh damping matrix. Journal of Sound and idbration, 502(3), 486-526. doi 10.1016/j.jsv.2006.11.030... 

In order to solve Eq. 10, it is necessary to define the damping model. The so-called Rayleigh damping model is often assumed at the level of the structure. In this case, the damping matrix is expressed as a linear combination of the mass and stiffness matrices ... 

An extended Rayleigh damping matrix, called Caughey damping matrix, can be computed from... 

In the above analysis, one step was tacitly skipped. When adopting the Rayleigh damping expression of Eq. 21, the unknown constants ao and ai need to be determined for the damping matrix to be fully described. Indicatively this can be done as follows ... 

Here, 5u, u, v, a, and F indicate the incremental displacement, displacement, velocity, acceleration, and external force vectors, respectively. M, C, and K indicate mass, damping, and stiffness matrices, respectively, dt and n indicate time increment and time step number, respectively. The Rayleigh damping matrix is used for C the element damping matrix C" is calculated using the element mass matrix and the element stiffiiess matrix K" as... 

Rayleigh damping is used to model the inherent damping of the building with a 5 % damping ratio for the first and second modes. 

Rayleigh damping instead of a viscoelastic model Empirical relations (no simulation) for Northern California... 

For damped systems, the above decoupling is feasible only if the damping matrix c is orthogonal, otherwise Eq. 6 does not apply. Usually c is derived as a combination of k and m, i.e., c = aok -I- aim, where ao, ai are constants. This form of damping is known as Rayleigh damping. Another approach would be to form the damping matrix so that every mode is damped with C = 2 co . This is achieved with the aid of the formula (Chopra 2000) ... 

The most commonly used formulation for evaluation of the viscous damping coefficient (c) is the Rayleigh damping (model). The viscous damping coefficient can be evaluated as... 

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