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What

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values

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should

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I

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use

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for

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mass

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-

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and

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stiffness-proportional

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damping?

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Answer:

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Mass

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-

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and

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stiffness-proportional

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damping

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,

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normally

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referred

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to

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as

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Rayleigh

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damping,

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is

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commonly

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used

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in

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nonlinear-dynamic

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analysis.

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Suitability

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for

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an

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incremental

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approach

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to

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numerical

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solution

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merits

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its

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use.

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During

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formulation,

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the

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damping

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matrix

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is

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assumed

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to

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be

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proportional

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to

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the

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mass

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and

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stiffness

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matrices

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as

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follows:

Image Added

where:

  • η is the mass-proportional damping coefficient; and
  • δ is the stiffness-proportional damping coefficient.


Relationships between the modal equations and orthogonality conditions allow this equation to be rewritten as:

Image Added

where:

  • ξ n is the critical-damping ratio; and
  • ω n is the natural frequency ( ω n = 2 π f n ).


Here, it can be seen that the critical-damping ratio varies with natural frequency. The values of η and δ are usually selected, according to engineering judgement, such that the critical-damping ratio is given at two known frequencies. For example, 5% damping ( ξ = 0.05 ) at the first natural frequency of the structure ( ω i = ω 1 ), and at ω j = 188.5 (30 Hz). According to the equation above, the critical-damping ratio will be smaller between these two frequencies, and larger outside.

If the damping ratios ( ξ i and ξ j ) associated with two specific frequencies ( ω i and ω j ), or modes, are known, the two Rayleigh damping factors ( η and δ ) can be evaluated by the solution of a pair of simultaneous equations, given mathematically by:

Image Added

SAP2000 allows users to either specify coefficients η and δ directly, or in terms of the critical-damping ratio either at two different frequencies, f (Hz), or at two different periods, T (sec).

When damping for both frequencies is set to an equal value, the conditions associated with the proportionality factors simplify as follows:

Image Added

References

  • Wilson, E. L. (2004). Static and Dynamic Analysis of Structures (4th ed.). Berkeley, CA: Computers and Structures, Inc.
    Available for purchase on the CSI Products >
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    {link-window:href=http://orders.csiberkeley.com/SearchResults.asp?Cat=2}Books {link-window}

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