[FEM: DAMPING CONCEPT]

[Brian Lee]

Next: 31.1.3 Loading Up: 31.1 General Phenomena Previous: 31.1.1 Mass   Contents   Index

Subsections

• 31.1.2.1 Continuous Damping
• 31.1.2.2 Rayleigh Damping
• 31.1.2.3 Structural Damping

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31.1.2 Damping

^31.1 Either consistent or lumped damping matrices can be used in structural dynamic analysis. In practice the presence of damping reduces the steady-state response and damps out the transient response. A modal analysis [§31.2.1] assumes the application of proportional viscous damping and that the damping matrix C satisfies the orthogonality condition. Modal damping can be employed for this and the magnitude of the damping has to be specified as a percentage of the critical damping factor

iTC j = 2

(31.4)

where is the natural angular frequency and is the damping ratio. The critical damping factor is

ccrit = 2

(31.5)

Where k is the generalized stiffness _i^TK _j and m is the generalized mass _i^TM _j .

31.1.2.1 Continuous Damping

When a form of continuous damping, like viscous dashpots, is included in the finite element model then the direct solution method [§31.2.2], or the direct time integration procedure [§31.4] must be used. Discrete places of damping are obtained by specifying properties for spring elements [§6.2].

31.1.2.2 Rayleigh Damping

Applying a nonmodal solution technique, it is necessary to eval‎uate the damping matrix C explicitly and usually viscous damping effects can be included by assumption of Rayleigh damping which is of the form

C = aM + bK

(31.6)

where a and b are constants to be determined from given damping ratios.

31.1.2.3 Structural Damping

Another type of damping, frequently employed in dynamic analysis, is structural damping also called hysteretic damping. This type of damping is proportional to displacement but in-phase to velocity of a harmonically oscillating system. In that way the equation of motion is expressed as

M + K 1 + i u = f(t)

(31.7)

where is the structural damping factor. Equations like (31.7) can only be solved in the frequency domain.

Next: 8.6.2 Load-Time Diagrams Up: 8.6 Transient Nonlinear Dynamic Previous: 8.6 Transient Nonlinear Dynamic   Contents   Index

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8.6.1 Rayleigh Damping Coefficients

In the transient dynamic analysis, it is necessary to apply Rayleigh damping. Rayleigh damping is characterized by the two constants a and b, whereas the damping coefficient of the concrete is 0.05. So now the question pops up on how to determine the coefficients a and b for given . If a modal analysis would be used, the damping coefficients and for the two lowest frequencies and will equal if

a	= 2	b	= 2	(8.21)
where
	=		=	(8.22)

See the text book of Bathe [1] for this. The values are greater than , the difference increasing for higher i indices. The eigenvalue analysis did show that = 6.308 and = 6.817 so that a = 0.3275 and b = 0.0076.