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Page 1: ANSYS Damping

Date October 21, 2000 Memo Number STI:001021ASubject Tips and TricksTips and TricksTips and TricksTips and Tricks: : : : Structural DampingStructural DampingStructural DampingStructural DampingKeywords Beta damping: Damping Ratio: Mode-Superposition Method: Full Method

1.1.1.1. Introduction:Introduction:Introduction:Introduction:Damping is required in many dynamic applications, yet because there are many ways to include

it, there is often confusion regarding the implementation of damping in ANSYS. This memo hopes toprovide a general summary of the representation of damping available in ANSYS.

2.2.2.2. Background Discussion on Damping:Background Discussion on Damping:Background Discussion on Damping:Background Discussion on Damping:Damping results in energy loss in any dynamic system, which results in decay of amplitude of

motion. Damping can be considered (a) in different forms (e.g., structural, viscous, or Coulomb) (b)for single or multiple DOF systems, (c) depending on whether the oscillations are harmonic orgeneral. Although physical damping behavior is quite complicated, the mathematical representationtends to be simplified and is dependent on whether nodal or generalized coordinates are used (i.e.,full vs. mode-superposition methods).

3.3.3.3. Nodal vs. Generalized Coordinates:Nodal vs. Generalized Coordinates:Nodal vs. Generalized Coordinates:Nodal vs. Generalized Coordinates:Before continuing with the discussion, it may be useful to first define nodal coordinates and

generalized coordinates.Nodal coordinates1 are always used in FEA, where the displacements at each node are solved

for, such as in the basic equation:

{ } [ ]{ } [ ]{ } [ ]{ }xKxCxMF ++= &&&This is also known in ANSYS as the full method when solving dynamic equations. Modal (DAMP,QRDAMP), harmonic, and transient analyses may be solved for with this method, including anynonlinearities which may be present. Damping is defined through the damping parameter [C].

Generalized coordinates2 can be used in FEA, where the response is assumed to be comprisedof a linear combination of the eigenvectors of the system (mode-superposition method). Hence, theuse of generalized coordinates necessitates performing a modal analysis first to obtain n number ofmode shapes (eigenvectors). Then, the response {x} is assumed to be a linear combination of the nnumber of mode shapes {φ} by solving for the mode coefficients y:

{ } [ ]{ } [ ]{ } [ ]{ }

{ } { }

{ } { } { } [ ]{ }{ } { } [ ]{ }{ } { } [ ]{ }{ }iiT

iiiT

iiiT

iT

i

n

iii

yKyCyMF

yx

xKxCxMF

φφφφφφφ

φ

++=

=

++=

∑=

&&&&

&&&

1

Due to the use of undamped eigenvectors and use of modal damping only, the equations areuncoupled and can be simplified further as:3

{ } { } { } { } { }iiiiiiT

i yyyF 22 ωξωφ ++= &&&&The attractiveness of using the mode-superposition method (i.e., solving in terms of generalizedcoordinates) is that the equations are uncoupled, and only n equations are solved for (where n isusually much less than the total number of nodal DOF), resulting in efficiency of solution. However,because it is using a linear combination of modes, only linear behavior is allowed for modal(QRDAMP)4, harmonic, or transient analyses. Damping is specified through a dimensionlessparameter called the critical damping ratio ξ, which is actual damping over critical damping c/ccr.

1 “Nodal coordinates” is sometimes referred to as “discrete coordinates”.2 The author may sometimes refer to “generalized coordinates” as “modal coordinates,” although because of the similarity between theterms “modal” and “nodal”, the author will intend on using “generalized coordinates” throughout this memo.3 For details on the mode-superposition method, see to Ch. 15.11 “Mode Superposition Method” in the ANSYS 5.6 Theory Manual.4 See Memo: STI68:001014 “CSI Tip of the Week: QR Damped Eigenvalue Extraction Method” for more details on QRDAMP. Notethat QRDAMP is actually two solutions at once: full method to get undamped eigenvectors, then modal method to get dampedeigenvalues.

Page 2: ANSYS Damping

4.4.4.4. Different Types of Damping:Different Types of Damping:Different Types of Damping:Different Types of Damping:Regarding the different types of damping, these are usually categorized into viscous, structural,

and Coulomb damping.

Viscous damping usually arises in cases where a system vibrations in a fluid, such that thedamping force is proportional to velocity by a constant “c”. This is expressed as:

xcF dv &=

The equation can also be written assuming harmonic motion as:

( )xicF

xeiceFd

v

titidv

ωω ωω

=

=

which indicates that viscous damping is an imaginary term linearly proportional to frequency.

Structural damping (also known as solid damping) is due to internal friction of the material orof entire system. The former is a characteristic of the material.5 The latter is due to energy loss atstructural joints, for example. In literature, the representation of structural damping assumesharmonic motion and is written as:

igkxFigkxeeF

ds

titids

=

= ωω

Unlike viscous damping, structural damping is usually assumed to be constant with respect tofrequency. Also, note that because of this independence on frequency, one can view this as animaginary term to be an imaginary elastic force. Hence, one can refer to complex stiffness (orcomplex moduli)6 as k(1+ig) where g is the structural damping factor. (When used in nodalcoordinates, “structural damping” changes and becomes equivalent to viscous damping, where “gk”is the same value as “c”, although this is not really “structural damping” anymore.)

Coulomb damping is due to frictional effects of the sliding of two dry surfaces. Coulombdamping is not dependent on the displacement or velocity but rather on the normal force FN and thecoefficient of friction µk:

Nkd

c FF µ=Since the Coulomb damping force opposes motion, the sign is the opposite of that of the velocity, sofor harmonic oscillations, the sign keeps changing for each half-period. Also, this is usually includedwith contact elements with a distinction often being made between static and kinetic coefficients offriction, resulting in nonlinear behavior.

There are also other types of damping not discussed in detail here. Negative damping is theaddition of energy into the system rather than its dissipation. Some consider plasticity and othermaterial nonlinearities as sources of damping because of the fact that energy is lost in the system.

5 Although often known as “material damping” when referring to energy loss due to internal friction of the material (microscopic ormacroscopic material behavior), the author will use this term only in the context of ANSYS usage, defined later.6 Some consider the complex moduli as another way of viewing viscoelasticity in the frequency domain, although one often refersspecifically to shear behavior (as in the case of incompressible elastomers). In either case, the real term is called the dynamic (orshear) storage and the imaginary term is the dynamic (or shear) loss, and the ratio of the imaginary to real modulus is tan(delta)where delta is the difference in phase between input strain and output stress.

Page 3: ANSYS Damping

It is instructive to note that viscous and structuraldamping are the same only at a given frequency. Forexample, if one were to plot damping force vs. frequency underconstant displacement harmonic motion for both types ofdamping, the graph would look as shown on right. Only at agiven value ω (red) will the two values intersect. At thisfrequency, g·k=c·ω.

As noted earlier, the damping ratio ξ is defined as c/ccr,where ccr is the critical damping ratio. At resonance,ω=sqrt(k/m), so this means that:

221

2g

kmkmgk

mc

cccr

=

===

ωξ

Another useful thing to note when harmonic oscillations occur is the behavior of the force vs.deflection curve. The damping force Fd is an ellipse, as illustrated in the bottom-left figure. Forlinear elastic materials, we expect the elastic force Fk to be a straight line. When combining theeffect of Fd and Fk, however, we get a more interesting response, as shown on the bottom-right figurein blue.7 The difference is due to the fact that there is a phase lag in the force response. The areaenclosed by the blue hysteresis loop is the energy dissipated per cycle due to damping.

Because of this hysteresis loop due to damping, structural damping is also known as hysteresisdamping.8 However, it is important to note that both viscous and structural damping result in asimilar response with a hysteresis loop under harmonic oscillations, and this is not limited to juststructural damping. The difference is that the energy dissipated per cycle for structural damping isindependent of frequency, whereas it linearly increases with frequency for the case of viscousdamping.

Other forms of damping may not produce a hysteresis curve as an ellipse, as the damping energyloss needs to be a quadratic function of the amplitude (or strain), which, as mentioned earlier, is truefor viscous and structural damping.

There are two attached files, “hyst_msup.inp” and “hyst_full.inp” which illustrate this for asingle DOF system at a given frequency. The former uses structural damping, the latter uses viscousdamping, and both produce ellipical hysteresis loops.

7 Note that the line specified by Fk (purple) does not define the axis of the ellipse Fk+Fd (blue) in the figure.8 For examples of the use of the term “hysteresis damping”, refer to pages M1-22, M1-24, and M1-25 in the ANSYS 5.6 DynamicsSeminar. For reasons described in the text above (coupled with the fact that sometimes users confuse the term “hysteresis” withnonlinear stress-strain behavior, such as in metal plasticity under cyclic loading), the author will not use this terminology.

Page 4: ANSYS Damping

5.5.5.5. Characterization of Damping for Single DOF Systems:Characterization of Damping for Single DOF Systems:Characterization of Damping for Single DOF Systems:Characterization of Damping for Single DOF Systems:There are various ways to measure or to characterize damping. It is important to note that the

following methods assume a single DOF system, so its extension to a multiple DOF system (as in thecase with FEA) needs to be done with some care, as will be discussed shortly.

The half-power bandwidth (or 3 dB bandwidth)9 ∆ω isdefined as the difference of the two half-power points ω1 and ω2,as shown on the graph on the right. The half-power points ω1

and ω2 surround the resonance value ωn, and these half-powerpoints are frequencies where the response is 1/√2 or 0.707 of itspeak value.10 The power is proportional to the square of theresponse, so the half-power is defined as 1/√2 of the peakresponse. Hence, the frequencies at which this occurs are calledthe half-power points. For lightly damped structures, the half-power bandwidth can be related to the damping ratio with thefollowing approximation:

ξωω n2=∆

The quality factor Q provides information on the sharpness of resonance. Q is determined bycomputing the ratio of the resonant frequency ωn with the half-power bandwidth ∆ω:

ξωωω

ωω

21

12

=−

=∆

= nnQ

The loss factor η is also sometimes used, and this is defined as the ratio of damping energy perradian to strain energy, i.e., the loss factor is equal to the inverse of the quality factor (η=1/Q).

The logarithmic decrement δ is measured not in thefrequency domain but in the time domain. If the rate of decayof a free oscillation is measured, the ratio of two successiveamplitudes is the logarithmic decrement:

( ) ( ) dnt

tdn

dn

n

ee

exx τξωδ τξω

τξω

ξω

==

=

= −

+−

lnlnln1

1

2

1

where τd is the damped period. The damped period can bereplaced by 2π/ωn·sqrt(1-ξ2), which leads to the equation:

πξξ

πξξωπξωτξωδ 2

12

12

22≈

−=

−==

n

ndn

The last expression holds for light damping (small values of ξ).11

Because of the fact that common characterizations of damping such as the above – half-powerbandwidth, quality factor, loss factor, and logarithmic decrement – are for single degree-of-freedomsystems, the use of these in ANSYS should be done with care. A harmonic (or, in the case oflogarithmic decrement, transient) analysis should be carried out with enough resolution around thefrequency of interest12 to verify that the damping captures the response at a given location to what isexpected. Because multiple modes are usually excited, the damping due to one mode will affect theother, so validation of appropriate damping values (discussed next) can be performed in this fashion.These are just recommendations, of course, and it is the user’s responsibility to understand wherehis/her damping values came from and how they relate to the FE model. Also, there are other waysof characterizing damping not discussed here, although the more common ones have been covered.

9 Recall that dB is defined as 10 log (P/Pref). If P is half the power of Pref, then this results in -3 dB.10 ‘Peak response’ refers to a result such as max (total) displacement at resonance, depending on how it was measured.11 Note that on page M1-28 of the ANSYS 5.6 Dynamics Seminar, the table of conversion is missing a factor of 2 for log decrement ∆.12 A general guideline: in harmonic, use 10-15 substeps in the half-power bandwidth; in transient, use 20-30 substeps in a period

Page 5: ANSYS Damping

6.6.6.6. Defining Damping Values in ANSYS for Full vs. Mode-Superposition Methods:Defining Damping Values in ANSYS for Full vs. Mode-Superposition Methods:Defining Damping Values in ANSYS for Full vs. Mode-Superposition Methods:Defining Damping Values in ANSYS for Full vs. Mode-Superposition Methods:The previous sections have discussed damping in general with little emphasis on the ANSYS

implementation of damping. In this section, attention is drawn to different methods to account forthis affect in ANSYS.

The first important thing to note is how damping is defined in the full method compared withthe mode superposition method. Because the former deals with nodal coordinates and the latterwith generalized coordinates, accounting for damping is different between the two.

As noted in Section 3, in the full method of modal13, harmonic, or transient analyses, theequation of motion is:

{ } [ ]{ } [ ]{ } [ ]{ }xKxCxMF ++= &&&The damping matrix [C] is formed from the following components:14

[ ] [ ] [ ] [ ] [ ] [ ]∑∑==

++

++=N

kk

M

jjj CKK

fKMC

11β

πξβα

• α is the constant mass matrix multiplier for alpha damping (ALPHAD command)• β is the constant stiffness matrix multiplier for beta damping (BETAD command)• ξ is the constant damping ratio, and f is the current frequency (DMPRAT command)15

• βj is the constant stiffness matrix multiplier for material j (MP,DAMP command)• [Ck] is the element damping matrix for supported element types (ET and TYPE commands)

On the other hand, in the mode superposition method for harmonic, transient, or spectrumanalyses, the equation solved for has also been covered in Section 3:

{ } { } { } { } { }iiiiiiT

i yyyF 22 ωξωφ ++= &&&&

Instead of creating a damping matrix [C], an effective damping ratio ξi

d is created for each mode i:16

=

=+++

+

= M

j

sj

M

j

sjj

mii

i

di

E

E

1

1

22

ξξξβω

ωαξ

• α is the inversely-related damping parameter for alpha damping (ALPHAD command)17

• β is the linearly increasing damping parameter for beta damping (BETAD command)• ξ is the constant damping ratio (DMPRAT command)• ξmi is the damping ratio specified for mode i (MDAMP command)• ξj is the damping ratio specified for material j (MP,DAMP command)18

• Ej

s is the strain energy for material j, calculated by ANSYS as ½{φj}T[Kj]{φj}

For spectrum analyses, damping is included not in the calculation of mode coefficients but in modecombination only. Also, in the case of mode-superposition method, material-dependent damping isadded in the expansion of modes, so the user must include material-dependent damping (MP,DAMP)and request element stress calculations (MXPAND) before running the modal analysis. Lastly, at 5.7,mode superposition methods will support the use of QRDAMP, but the user should know thatalthough it is a mode-superposition method, damping is included in the modal analysis phase(QRDAMP), so the full method damping equation [C] above should be used.

Table 5-8 “Damping for Different Analysis Types” of the ANSYS 5.6 Structural Analysis Guideprovides a summary of when different forms are damping are available for different analysis types.

13 This is the DAMP or QRDAMP eigenvalue extraction methods.14 This is similar to Equation 15.3-1 of the ANSYS 5.6 Theory Manual.15 This term is not available in damped modal or full transient analyses.16 This is similar to Equation 15.11-22 and 17.7-1 of the ANSYS 5.6 Theory Manual.17 No alpha damping permitted for certain types of spectrum analysis (SPRS, MPRS, DDAM)18 No material-dependent damping ratio is permitted for certain types of spectrum analysis (PSD)

Page 6: ANSYS Damping

7.7.7.7. Description of Individual Damping Input:Description of Individual Damping Input:Description of Individual Damping Input:Description of Individual Damping Input:It may be instructive to describe how the different damping

input in ANSYS affects the response. The damping ratio ξ will beused as the reference point. Recall that damping ratio is the ratioof damping to critical damping at a given frequency (mode). Ifone were to plot the various types of damping input as a functionof damping ratio vs. frequency, the graph would look as shownon the right.

Damping is always cumulative in both full and modalmethods, so this should always be kept in mind.

The constant damping ratio, specified by DMPRAT, isconstant for each frequency, as shown in the solid dark green lineon right.

Modal damping, defined by MDAMP, is specified for each frequency. This is frequency-dependent, so it is arbitrary and not shown on the graph above. This is added in addition to theconstant damping ratio DMPRAT (if defined).

Element damping, defined by the element types noted in Chapter 15.3 of the ANSYS 5.6 TheoryManual, is also not necessarily predefined and is dependent on the element, so it is not shown in thegraph of damping ratio vs. frequency above.

Alpha damping (specified via ALPHAD, shown in dark red in the graph) results in a dampingratio which is inversely related to frequency. Because of this, alpha damping affects low frequencies.Alpha damping is not similar to any of the damping types discussed in Section 4.

Beta damping (specified via BETAD, shown in dark blue in the graph) provides a damping ratiothat is linearly related to frequency. Consequently, it tends to affect higher-frequency content. Thisis similar to viscous damping, discussed earlier, where β·k=c for a single DOF system.

Rayleigh damping is the use of a combination of alphaand beta damping. Because the full method for transientanalysis does not support any form of a constant dampingratio, users can determine values of α and β which providean approximately constant damping ratio in a given range,as shown in the figure on the left.19 This can be done bysolving the two equations below for α and β.

22222

2

1

1

βωωαβω

ωαξ +=+=

Rayleigh damping arises because it is a form of proportional damping ([C] is proportional to [K] and[M] in this method), which makes it easier to deal with numerically.

Beta damping, without alpha damping, is sometimes used in linear dynamics. In nonlineartransient applications, because beta damping is proportional to the stiffness matrix which cansometimes change drastically, it is not preferred in these situations.

While beta damping affects higher-frequencies, alpha damping affects lower ones. This meansthat alpha damping should not be used in the “large mass method” commonly used in linear dynamicapplications, as the large mass would create an artificially high damping force. However, by thesame token, alpha damping is sometimes used in the ‘slow dynamics’ approach of nonlinear transientproblems to help deal with possible rigid-body motion by damping out this behavior.

19 Graph copied from Figure 5.6 “Rayleigh Damping” in the ANSYS 5.6 Structural Analysis Guide

Frequency

Dam

ping

Rat

io

DMPRATBETADALPHADMP,DAMP (FULL)MP,DAMP (MSUP)

Page 7: ANSYS Damping

It is extremely important to note that MP,DAMP means different things, depending on theanalysis method used.

In the full method, material-dependent damping values represent a stiffness matrix multiplierfor that material (dotted dark blue line), similar to viscous damping but per material. Hence, in thiscase, the value for MP,DAMP will be equal to ξ/π·f or to c/k for a single DOF system. When multiplematerials are present, the damping matrix [C] simply applies each value of βj to the portion of thestiffness matrix associated with that given material j:

[ ] [ ]∑=

=M

jjjDAMPMP KC

1, β

In the mode-superposition method, however, material-dependent damping values indicate thedamping ratio for that material (dotted dark green line), similar to structural damping. This meansthat the value supplied via MP,DAMP will be equal to ξ or to g/2 for a single DOF system. Whenmultiple materials are present, the Modal Strain Energy Method (MSE) is used to calculate an‘effective’ damping ratio for the system, as shown below:

=

== M

j

sj

M

j

sjj

DAMPMPi

E

E

1

1,

ξξ

This means that an ‘effective’ constant damping ratio is calculated for all modes.

Because the last point of the difference in behavior of MP,DAMP is often a point of confusion,several input files are attached of a simple LINK1 model, which may hopefully provide some insight:

• sdof_full.inp is a single DOF model using full method and MP,DAMP. This is a form ofviscous damping.

• sdof_msup.inp is a single DOF model using mode-superposition method and MP,DAMP.This is a form of structural damping.

• sdof_betad.inp is a single DOF model using mode-superposition method and BETAD.This is a form of viscous damping, as noted above.

• sdof_dmprat.inp is a single DOF model using mode-superposition method and DMPRAT.This is a constant damping ratio.

• mdof_full.inp is a 3 DOF model using full method and MP,DAMP. This is a form ofviscous damping, so higher modes are affected more by the damping.

• mdof_msup.inp is a 3 DOF model using mode-superposition method and MP,DAMP. This isa form of structural damping, so the damping ratio is constant.

• mdof_dmprat.inp is a 3 DOF model using full method and DMPRAT. This is a constantdamping ratio, which is also applicable for full harmonic analyses.

One important thing to note is that for the single DOF model, the forces due to the mass, damping,and stiffness terms are added in /POST26 and compared with the applied force. For the full method,these two values are always equal, as expected. In the mode superposition method, however, asdamping is increased, the slight inaccuracy becomes more noticeable. This is due to the fact thatsome differences are introduced by using undamped modal coordinates in the mode-superpositionphase.

Page 8: ANSYS Damping

8.8.8.8. Conclusion:Conclusion:Conclusion:Conclusion:This memo hoped to review some background information on the different types of damping

(viscous, structural, and Coulomb) and its characterization in single DOF systems (half-powerbandwidth, quality factor, loss factor, and log decrement).

The implementation of damping models in ANSYS (ALPHAD, BETAD, MDAMP, DMPRAT) andtheir differences were also covered, including their applicability in full and modal methods.

Lastly, the difference in behavior of MP,DAMP in full and modal methods was emphasized,indicating that in modal methods, MP,DAMP provides a similar response to complex modulus forstructural damping. Some input files were provided to illustrate some of the points.

In the future, the author hopes to cover the use of damping elements (rotordynamics, surfaceeffect elements, discrete lumped-parameter elements, and fluid elements) and the QR-Dampeigenvalue extraction method introduced at ANSYS 5.6.

9.9.9.9. References:References:References:References:• ANSYS Theory Manual, Version 5.6• Hurty, W.C. and Rubinstein, M.F., “Dynamics of Structures”, Prentice Hall, 1964• Thomson, W.T. and Dahleh, M.D., “Theory of Vibration with Applications”, 5th ed., Prentice Hall,

1993

__________________________Sheldon Imaokahttp://ansys.net/ansys/