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# Viscous damping

## Viscous damping

(OP)
I'm trying to teach myself a bit more about viscous damping and I had some questions.

I've found a number of references stating that the damping matrix, |C|, is:

|C| = a0 |M| + a1 |K|

Where:

a0 = 4πζ/(T1 + T2)

a1 = T1*T2*ζ/π*(T1 + T2)

And where:
T1 = Period of oscillation of a target mode (usually the first mode)
T2 = Period of oscillation of another target mode
ζ = Viscous damping ratio

Q1) Am I right in thinking that |M| is the mass matrix and |K| is the stiffness matrix? Are these the same matrices from a standard Euler-Bernoulli beam-column 1D 4DOF FE model?

Q2) Also, a few sources I've read state that these values should be lumped at the nodes rather than distributed - is this correct? Distributed values give more representative results, so I'm quite surprised by this 'lumping' if that is the case.

Q3) Can someone explain T1 and T2 to me in more detail? What are the target modes? Is this like 1P and 3P when it comes to wind turbines?

Q4) Is the viscous damping ratio the same damping ratio from soil test data? If not, how is it calculated?

Q5) Finally, am I right in thinking that the |C| matrix will be added to the |M| AND |K| matrices from the remainder of the beam-column formulation?

Thanks in advance for any answers to some or all of the above questions.

Hi

Good Luck

Thomas

### RE: Viscous damping

(OP)
I have searched on the internet and various text books. I asked the questions in the OP because the sources I have read have unfortunately left me with incomplete knowledge on the subject.

Are you (or anyone) able to answer any of those questions please?

### RE: Viscous damping

Before I try and give my opinions, what application are you trying to apply this to?

### RE: Viscous damping

(OP)
Soil mechanics. Though I wouldn't have expected the answers to the questions in the OP are dependent on material type (?).

### RE: Viscous damping

Hi

The two constants you refer to are the mass and the stiffness constants used to define Rayleigh damping.

I Googled a little and found first this: https://www.orcina.com/SoftwareProducts/OrcaFlex/D...

and then this: http://easc.ansys.com/staticassets/ANSYS/staticass...

Hopefully it will help you a bit on the way. Personally I don't use Rayleigh damping if I can avoid it. That is another story but the reason can be found in the first figure in the first link .

Good Luck

Thomas

### RE: Viscous damping

#### Quote (LRJ)

Q1) Am I right in thinking that |M| is the mass matrix and |K| is the stiffness matrix? Are these the same matrices from a standard Euler-Bernoulli beam-column 1D 4DOF FE model?
Yes, make sure the formation you are using does not use the "diagonalized" modal matrices. My reference text uses both |M| and |m|.

#### Quote (LRJ)

Q2) Also, a few sources I've read state that these values should be lumped at the nodes rather than distributed - is this correct? Distributed values give more representative results, so I'm quite surprised by this 'lumping' if that is the case.
It is typical for structures to use lumped masses at the diaphragms with 3 degrees of freedom at each mass lump. It works well enough to give a reasonable behavior of a typical system. Without assuming this the ODE of motion becomes a PDE of motion. Also note, in a lumped mass system, the number of lumps increases the degrees of freedom and as a result the matrix problem becomes larger and more difficult to solve.

#### Quote (LRJ)

Q3) Can someone explain T1 and T2 to me in more detail? What are the target modes? Is this like 1P and 3P when it comes to wind turbines?
I think (after rereading my text) that you would likely want the first and second periods. if there is another mode that dominates the response it would probably do to use those modes instead. I don't know what 1p and 3p are to relate it to.

#### Quote (LRJ)

Q4) Is the viscous damping ratio the same damping ratio from soil test data? If not, how is it calculated?
For building systems, this damping ration is dependent on the type of system. It is difficult to truly quantify. most steel buildings are in the 5-10% range, concrete near 10% and wood in the 15 to 20%. Basically you just have to estimate it. The soil damping will be different from the structure and you can assemble the structure matrix and the soil matrix together to get the overall damping. I have never really done this though.

#### Quote (LRJ)

Q5) Finally, am I right in thinking that the |C| matrix will be added to the |M| AND |K| matrices from the remainder of the beam-column formulation?
Yep, shove that into the equation of motion and use an appropriate technique to solve the Diff EQ.

### RE: Viscous damping

(OP)

#### Quote (RobertHale)

Yes, make sure the formation you are using does not use the "diagonalized" modal matrices. My reference text uses both |M| and |m|.

What do you mean by 'diagonalized' here? Do you mean non-symmetrical or something?

### RE: Viscous damping

Viscous damping is generally used as a mathematical construct to stablise the Diff EQ solving scheme (eg. Newmark Beta) and avoid perpetual motion. Mass-stiffness systems (eg. soil or a building) exert energy through cyclic behaviour, which can be mimicked through including a viscous damping term, however, the actual energy dissipation is not usually viscous but tends to be energy loss to heat or other mechanisms.

Therefore you will find many different methods to model 'viscous damping' such as Rayleigh damping, which is the most common for structural analysis, however, there are others like Wilson-Penzein damping that have other advantages.

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