# Damping

The stiffness damping in version 960 is completely reformulated. Even though you may provide a 960 COEF value which is consistent with the 950 BETA value, that is,...

`COEF = BETA * (w/2)`

...the 950 stiffness damping and the 960 stiffness damping will not be exactly equivalent. The 960 formulation of stiffness damping provides an APPROXIMATE fraction of critical damping in the high frequency domain. The details of the formulation are proprietary. The change in formulation was prompted by the frequent occurences of instability when using the old formulation.

In rev. 3510 (or higher) of v. 970, the old 950-style stiffness damping formulation is available as an option and is invoked by setting the COEF parameter to a negative value. The parameter is then interpreted as a BETA value as documented in the v. 950 User's Manual.

As an alternative in version 970 is a frequency-independent damping option for a range of frequencies and a set of parts (`*DAMPING_FREQUENCY_RANGE`

). This technique was developed by Richard Sturt of Arup. It allows LS-DYNA to tackle vibration prediction problems properly - including vehicle NVH time-history analysis as well as certain classes of seismic problems and civil/structural vibration problems. The key points of `*DAMPING_FREQUENCY_RANGE`

are:

- Use for low amounts of damping only, e.g. up to 1% or 2%
- The damping treatment slightly reduces the stiffness of the response - that's because the applied damping force lags slightly behind the "theoretically correct" damping force, due to the need to evaluate frequency content.
- The frequency range specified by the user should ideally be no more than a factor of 30 between highest and lowest. Damping is still achieved outside the frequency range but the amount of damping reduces.

Mass damping in LS-DYNA, which includes *`DAMPING_GLOBAL`

and `*DAMPING_PART_MASS`

, is intended to damp low-frequency structural modes but it has the added effect of damping rigid body modes. Thus parts that undergo significant rigid body motion should be excluded from mass damping (or the mass damping should be turned off during the time the part undergoes rigid body motion). The critical mass damping coefficient is `4*pi/T`

where T is the period of the mode targeted for damping (usually the lowest frequency (fundamental) mode). The period can be determined from an eigenvalue analysis or estimated from results of an undamped transient analysis. In version 970, `*DAMPING_RELATIVE`

provides a means to invoke mass damping which is relative to the motion of a particular rigid body.

Damping is completely optional. If the user decides to use mass damping, a damping coefficient less than the critical damping coefficient is suggested. A value of 10% of critical damping, or `0.4*pi/T`

is fairly typical. You can choose to damp all parts using the same damping coefficient (`*DAMPING_GLOBAL`

) or, to tailor the damping to the individual response characteristics of each part, you can assign a different damping coefficient to each part (`*DAMPING_PART_MASS`

). In either case, the damping coefficient can vary with time (useful to turn damping off or on in the middle of a simulation).

I know of no shortcut to producing good agreement with the observed loss in a test. I can only suggest good judgement and a trial-and-error approach (making a series of runs) in order to tune the numerical damping.

Rayleigh damping does not require diagonal mass and stiffness matricies. The only requirement is that the damping matrix be expressed as a linear combination of the mass and stiffness matricies:

`C = alpha*M + beta*K`

It is possible that the customer is thinking about mode superposition analysis using eigenmodes. If you express the displacement field as a linear combination of eigenmodes, then the reduced mass and stiffness matricies become diagonal. If Rayleigh damping is applied, then the damping matrix will also be diagonal. But this is a special case in linear analysis. (We are now adding this mode superposition capability using `*PART_MODES`

.)

The LS-DYNA implementation of Rayleigh damping for standard, nonlinear analysis is done at the element level, as you stated. This is done for numerical convenience, since in the explicit method we don't need to form the stiffness matrix K. Instead, we compute internal forces by simply integrating stresses over the element area. The Rayleigh damping terms are implemented as corrections to these stresses.

Both mass and stiffness damping are implemented for implicit transient analysis.

Ref: Bathe and Wilson "Numerical Methods in Finite Element Analysis", Prentice-Hall, 1976, p 339