"Reproduced" damping in Ansys

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Original address:http://www.cnblogs.com/ylhome/archive/2009/08/26/1554195.html

The Ansys Dynamics analysis provides various damping forms, how are these damping calculated in the analysis, and what are the effects on the analysis? This article will do some explaining about this.

A First, it is clear that the damping defined in the full method and the modal superposition method is different . Because the former uses node coordinates, the latter uses the overall coordinates.

1. In complete modal analysis, harmonic analysis and transient analysis, the vibration equation is:

The damping matrix is the sum of the following damping forms:

α is the constant mass damping (α damping) (Alphad β is the constant stiffness damping (beta damping) (Beta command)
ξ is the constant damping ratio, and F is the current Βj is the constant stiffness matrix of the J material coefficients (mp,damp command)
[C] is

Where: [C] = structure damping matrix
α= mass matrix multiplier (input on Alphad command)
[M] = structure mass matrix
β= stiffness Matrix multiplier (input on Betad command)
βc = variable stiffness matrix multiplier (see equation 15–23)
[K] = structure stiffness matrix
Nm = number of materials with damp or DMPR input
= stiffness Matrix multiplier for material J (input as damp on MP command)
= constant (frequency-independent) stiffness matrix coefficient for material J (input as DMPR on MP command)
ω= Circular excitation frequency
Kj = portion of structure stiffness matrix based on material J
Ne = number of elements with specified damping
Ck = element Damping Matrix
cξ= frequency-dependent damping matrix (see equation 15–21)

2. The harmonic analysis of the modal superposition method, and the analysis of the transient analysis, the equations of dynamic solution are:

Each modal produces an effective damping ratio ξid instead of creating a damping matrix

α is constant mass damping
β is constant stiffness damping
ξ as constant damping ratio
The constant damping ratio of the ΞMI as the first mode
Ξj damping coefficient for J-Material
Ejs for the strain energy of the J material, Ansys is calculated from {f}t[k]{f}.

Two For spectral analysis, damping is only included in the modal combination and is not considered when calculating modal coefficients. When the modal superposition method is used, the material damping is added to the extended mode, so that the user must include material damping (MP,DAMP) and calculation of the element stress (Mxpand command) prior to the modal analysis.

Three The modal superposition method supports the use of QR damping, but the user must be aware that, although the modal combination method, damping is included in modal analysis, damping should be calculated using the complete damping matrix [C] above. If the modal extraction method (Mopt,qrdamp) using the QR damping is used, and any form of damping is specified in the pre-processing or modal analysis, then ANSYS will ignore the damping when the modal overlay is performed.

Four It is important to understand that Mp,damp has different roles in different situations.

In a complete analysis, material damping represents a stiffness matrix multiplier of the material, which is similar to viscous damping (linear with frequency, but for all materials). Therefore, in this case, for a single-DOF structure, the material damping value is equal to Ξ/πf or c/k. If there are multiple materials, then the damping matrix is simply the βj of the material multiplied by the stiffness matrix of the corresponding material:

However, when using the modal superposition method, the material damping value is similar to the structural damping (independent of frequency), that is, the material damping value is equal to the ξ of the single degree of freedom System. If there are multiple materials, use the modal Strain Energy method (MSE) to calculate the effective damping ratio of the system:

In other words, a valid constant material damping will be used for all modal calculations.

Five The following table lists the damping that can be used in different analysis types.

§3.8.3 Damping

Damping is present in most systems, and damping should be specified in the Dynamics analysis. There are five forms of damping that can be specified in the ANSYS program:

· Aplha and Beta damping (Rayleigh damping)

• Material-related damping

• Constant damping ratio

• Vibration Type Damping

• Unit Damping

Only constant resistance ratioes and modal damping are available in the Ansys/professional program. Multiple forms of damping can be specified in the model, and the program forms a damping matrix in accordance with the specified damping sum [C]. The following table lists the types of damping available in different structural analyses.

Damping available for different analysis types

Type of analysis

α , Beta damping

[Alphad,

Betad]

Material-related damping

[Mp,damp]

Constant damping ratio

[Dmprat]

Vibration Type Damping

[MDAMP]

Unit Damping [3]

(COMBIN7, etc.)

Statics Analysis

N/A

N/A

N/A

N/A

N/A

Modal analysis

Unobstructed

NO[5]

NO[5]

NO[5]

No

No

With damping

Yes

Yes

No

No

Yes

Harmonic response Analysis

Total law

Yes

Yes

Yes

No

Yes

Reduction method

Yes

Yes

Yes

No

Yes

Modal Superposition method

YES[6]

yes[4,6]

YES[7]

YES[7]

YES[6]

Transient analysis

Total law

Yes

Yes

No

No

Yes

Reduction method

Yes

Yes

No

No

Yes

Modal Superposition method

YES[6]

yes[4,6]

YES[7]

YES[7]

YES[6]

Spectral analysis

SPRS,MPRS[2]

YES[1]

Yes

Yes

Yes

No

DDAM[2]

YES[1]

Yes

Yes

Yes

No

PSD[2]

Yes

No

Yes

Yes

No

Buckling analysis

N/A

N/A

N/A

N/A

N/A

Sub-structure

Yes

Yes

No

No

Yes

N/A indicates that you cannot use

[1] indicates that only beta damping is available, not α damping

[2] indicates that damping is only used for modal merging and is not used to calculate modal coefficients

[3] means including a super-element damping matrix

[4] means that if the modal expansion is converted into a vibration type damping

[5] indicates that if specified, the program calculates a valid damping ratio for subsequent spectral analysis

[6] means that if the QR damping modal extraction method [Modopt,qrdamp] is used, any damping is specified during pre-processing or modal analysis, but Ansys will ignore any damping when performing modal overlay analysis.

[7] If you use the QR damping modal extraction method [Modopt,qrdamp],dmpart and Mdamp cannot be used.

1. Alpha damping and Beta damping

Alpha damping and beta damping are used to define Rayleigh (Rayleigh) damping constants α and β. The damping matrix is calculated by multiplying these constants by the mass matrix [M] and the stiffness matrix [K].

The commands Alphad and Betad are used to determine the Rayleigh (Rayleigh) damping constants α and β respectively. The values of α and β are usually not directly obtained, but are calculated by the damping ratio of the vibration type. is the ratio of the actual damping and critical damping of a certain mode I. If the intrinsic angular frequency of the modal I is, then α and β satisfy the following relationships:

In many practical problems, alpha damping (or mass damping) can be ignored (α=0). In this case, β can be computed by known and calculated:

Since only one β value can be entered in a load step, the most important frequency of activation in the load step should be selected to calculate the beta value.

In order to determine the α and β values corresponding to a given damping ratio of ξ, the sum of α and β is generally assumed to be approximately constant within a frequency range (see Figure 5). Thus, after a given damping ratio of ξ and a frequency range of ωi~ωj, the solution of two parallel equations can be obtained α and β.

Fig. 5 Rayleigh damping

Alpha damping can lead to undesirable results when any large mass is introduced into the model. A common example is the addition of an arbitrarily large mass on the basis of the structure to facilitate the application of the acceleration spectrum (the acceleration spectrum can be converted to the Weili spectrum using a large mass). The alpha damping coefficient produces very large damping forces in such a system after multiplying the mass matrix, which results in inaccurate spectral inputs and imprecise system responses.

Beta damping and material damping can lead to undesirable results in nonlinear analysis. The two kinds of damping are multiplied by the stiffness matrix, and the stiffness matrix is constantly changing in the nonlinear analysis. The resulting damping changes are sometimes contrary to the actual damping changes in the physical structure. For example, the physical structure of softening due to plastic response usually correspondingly increases the damping, while the ANSYS model with beta damping exhibits a decrease in damping when there is a plastic softening response.

2. material-related damping

Material-related damping allows the use of beta damping as a material property to specify [Mp,damp]. However, it is important to note that in spectral analysis [ANTYPE,SPECTR] The mp,damp is specified and the material-related damping ratio ξ, rather than β. It is also important to note that for multi-material units such as solid46,solid65,shell91 and SHELL99, only one β value can be specified for the unit as a whole, not for each material in the cell. In these cases, β is determined by the element's material pointer (set by the mat command), not by the material that the unit real constant mat points to.

3. constant damping ratio

Constant resistance Ratioes is the simplest method of specifying damping in a structure. It represents the ratio of actual damping to critical damping, which is a small value specified with the dmprat command. Dmprat can only be used for spectral analysis, harmonic response analysis and modal superposition method transient dynamics analysis .

4. Vibration Type damping

The modal damping can be used to specify different damping ratios for different vibration modes. It is specified by the Mdamp command and can only be used for spectral analysis and modal superposition method transient dynamics analysis and harmonic response analysis.

5. Unit Damping

Unit damping is involved in the use of element types with viscous damping characteristics, such as unit COMBIN7, COMBIN14,COMBIN37,COMBIN40, etc.

For a more detailed description of the damping refer to the <<ansys theory reference manual >>.

Damping is one of the characteristics of dynamic analysis, but also an easy to cause confusion in the dynamic analysis, and because it only affects the attenuation of the dynamic response, the error is not easy to detect. The nature and performance of damping are quite complex, and the corresponding models are many. ANSYS provides a powerful and rich damping input, but is also in its power and richness to make it easy for beginners to get confused here describes the implementation of various damping mathematical models in Ansys, and the use of damping functions in Ansys.
1. Proportional damping
The most commonly used is also relatively simple damping is probably Rayleigh damping, also known as proportional damping. It is the preferred choice for most practical power analysis and is sufficient for many practical engineering applications. In Ansys, it is the sum of damping and damping, which is entered with the Alphd and Betad commands, respectively. The ratioes of the total resistance of the known structure is that the equivalent damping ratio of the damping and damping at two frequency points is equal to that of the same, and the approximate damping and damping coefficients can be obtained as input:
(5.1.1)
A fitting formula for calculating the coefficient of proportional damping
The damping and damping coefficients can be obtained by using the equation set (5.1.1), and then input with the Alphd and Betad commands, the damping input can be either full (complete) method analysis or the analysis of shrinkage method and mode superposition method, which is equally effective.
However, although the concept of damping and damping is simple and clear, some possible misunderstandings should be carefully used. First, the damping is related to the quality, mainly affects the low-order mode, and the damping is related to the stiffness, mainly affects the modes quantified type; If a nonlinear transient analysis is required and the stiffness varies greatly, then the use of damping is likely to cause difficulties in convergence; for the same reason, sometimes when using some computational techniques, For example, the high-quality method of traveling wave effect analysis, coupled with the false large artificial mass, then can not use damping. Similarly, when a rigid connection is added to the model, it should also be checked to see if the damping will result in some false calculations.
2. Calculation of damping matrices
There are several ways to enter damping characteristics in Ansys. Let's summarize some of the input damping commands commonly used in structural analysis:
Alphad: Input Damping parameter
Betad: Input Damping parameter
Dmprat: Input full-structure damping ratio
MDAMP: The modal damping ratio of the input to the vibration pattern of each frequency
Mp,damp input material damping corresponding to a material??。
The total damping matrices calculated by Ansys for the input of the above commands [C] are:
(5.1.2)
Ansys formula for calculating damping matrices
where M is the number of material types with damping in the structure, n is the number of unit types with unique damping. The first two are used with the defined Rayleigh damping, the third is the damping matrix corresponding to the full structure damping ratio, the fourth is the material damping, the last one is some unit-specific damping matrices.
3. Viscous damping ratio
Viscous damping is expressed as the resistance of a similar object to movement in a viscous fluid, which is proportional to the velocity.
(5.1.3)
Viscous damping Force
For a single degree of freedom System, C is the viscous damping coefficient, to the multi-degree of freedom system, is the damping matrix [C]. [C] is the most basic form of defining the damping characteristics of the structure, however, for viscous damping, there is very little direct definition of damping matrix [C], damping ratio is the most simple way to define viscous damping. In Ansys, you can define a full-structure damping ratio (Dmprat command) in a structural coordinate system, or you can define a modal damping ratio (Mdamp command) for each mode in modal coordinates. The corresponding modal resistance ratioes for each modal of Ansys is the superposition of the mdamp defined modal damping ratio and the full-structure damping ratio defined by Dmprat.
Both Dmprat and Mdamp are only valid for response spectrum analysis, harmonic analysis and transient analysis using modal superposition methods, and their corresponding damping matrices [C] are damping matrices with different frequencies. When the modal damping ratio is known, the corresponding damping matrix [C] is calculated using the following formula:
(5.1.4)
Damping matrix corresponding to the input modal damping ratio
Which is the I-mode vector, is the corresponding modal frequency.
It is noteworthy that the above formula has only theoretical significance, in the mode superposition is the direct use of the defined mode damping ratio and the full structure damping ratio, no program will use the formula (3) to reverse the damping matrix. (It may be possible to reverse the damping matrix in some programs, but at least ansys does not). So in the transient analysis of the full (complete) integration method, the damping defined by the damping ratio is ignored by the program, so many times we need to use a full-structure damping ratio to do the time of the transient analysis of the complete method, (as some specifications stipulate that some structures can be used 0.005~ 0.05 of the damping ratio to do the analysis), how to do? At this time, a simple method is to use damping and damping to approximate a constant damping ratio.

Fig. 5.1 Using ALPHD and Betad to fit constant damping ratio
Select and, you can use the formula (1) to calculate the input with the Alphd and Betad values.
4. Material Damping
Unlike several other damping, material damping is defined in the material parameters (command: mp,damp), material damping is called hysteresis damping, the most significant feature is not related to the structure response frequency.


Fig. 5.2 Relationship between two damping and frequency
In many literatures it is often written in the form of complex stiffness:. where k is the structural stiffness, called the material damping coefficient (also called structural damping coefficient).
In a single degree of freedom case, when the mass m is simple harmonic, (c is the corresponding viscous damping coefficient), so the corresponding damping ratio is:
(5.1.5)
Relationship between damping coefficient of material and viscous damping ratio
(in Japan's structural damping specification, the attenuation coefficient used to define damping is the damping coefficient of this material.) )
In Ansys, it is the multiplier of the stiffness matrix, and the damping array is weighted by the corresponding stiffness of each material.
(5.1.6)
The formula for calculating damping matrix of material damping by Ansys
It is obvious that the corresponding damping matrix [C] can be diagonal, so it can be used in full (complete) method transient analysis, and also in the mode superposition method analysis. The previous section describes: Ansys in the full integration of the transient analysis, the damping ratio defined by the damping is ignored by the program, in many cases, known as viscous damping damping ratio, but also to do the full method of transient analysis, how to do? One way to do this is to convert the viscous damping ratio into a material damping coefficient and then use the Mp,damp input. The conversion relationship between material damping coefficient and viscous damping ratio is: in the case of single degree of freedom: (c is the viscous damping coefficient).



Table 5.1 material damping coefficients for common materials
Pure aluminum steel lead cast iron
0.00002~0.002 0.001~0.008 0.008~0.014 0.003~0.03

Natural rubber hard Rubber glass concrete
0.1~0.3 1.0 0.0006~0.002 0.01~0.06
The above materials are from: "Structural vibration analysis", c.f. Pirc (the author is not responsible for their use)
The damping of the metal is relatively low, do not know whether this is not a defect of steel structure. Generally, the strength ductility of the metal is low. However, there are exceptions, such as manganese copper alloy strength hardness ductility damping is high, but the corresponding price is very high.
5. Calculation of modal damping ratio
When the modal superposition method is used, the modal damping ratio and the structural resistance ratioes are directly used by ANSYS, and the other damping is calculated by calculating the modal damping ratio of the various damping to calculate the response of each modal. Under a variety of damping inputs, the total modal resistance ratioes of the first modal of the ANSYS program is
(5.1.7)
The formula of calculating modal damping ratio by ansys
The first two are the damping and damping corresponding to the modal damping ratio, the third is the input of the full structure damping ratio, the fourth is the input modal damping ratio, the last one is the material damping coefficient of m material produced by the modal damping ratio. It is the modal strain energy corresponding to the J material, which is used to calculate the structure damping ratio in the Japanese damping specification.
Note
As mentioned earlier, when doing the transient analysis of the full integration method, the damping defined by the damping ratio is ignored by the ANSYS program, so the same model uses the full method and the modal superposition method transient analysis, the ANSYS calculation uses the damping may be different, causes the result also to have the difference.
The following is an ANSYS command flow demonstration of several damping inputs commonly used in structural analysis.
1) Use Mp,damp to enter viscous damping
damprato=0.025! Damping ratio of known viscous damping
Lossmodm=2*damprato! The damping ratio of viscous damping multiplied by 2 is the equivalent of the material damping coefficient (day
The "attenuation coefficient" of this specification)
critfreq=2.6! This is the conversion frequency when viscous damping is equivalent to material damping
mp_betad=damprato/(ACOs ( -1) *critfreq)! Viscous damping associated with frequency
/prep7
Mp,damp,1,mp_betad! Define iscous damping, frequency-related
/solu
Antype,modal
modopt,lanb,1
! To make modal calculations consider the effects of damping, material damping must be used, and material damping must be specified before solving
! Mxpand,,,, Yes, options! The damping ratio input is only useful in the superposition of the obtained vibration pattern.
! Ansys does not calculate damping ratio reduction as damping matrix [C]
mxpand,1,,, Yes
,,,
Sole

2) input material damping with Mp,damp
damprato=0.025
Lossmodm=2*damprato! Material damping coefficient, the book gives the general is LOSSMODM
/prep7
Mp,damp,1,damprato! Constant, if it is known that the material damping coefficient is lossmodm, divide by 2.
/solu
Antype,modal! Using the modal overlay method
modopt,lanb,1
! Important
mxpand,1,,, Yes
,,,,
Sole

3) using Betad input viscous damping (mode superposition method)
! Msup method with Betad
! Betad is damping_ratio/pi*f, een for msup
damprato=0.025! Damping ratio
Lossmodm=2*damprato! Equivalent damping coefficient of material
/prep7
! Mp,damp,1,damprato
betad,damprato/(ACOs (-1) *442)! Note this formula! 442 is the frequency value you given.
/solu
Antype,modal! Modal analysis
modopt,lanb,1
! Important
mxpand,1,,, Yes
Lumpm,on
,,,,
Sole
/solu
Antype,harmic! Harmonic analysis
Hropt, Msup
Hrout, ON, off
Harfrq, Freqbegn, FREQENDG
,,, Sole

4) constant damping ratio of the whole structure defined by the Dmprat (modal superposition method)
! Msup method with Dmprat
! Shows that Dmprat is damping ratio
damprato=0.025! Total structure resistance ratioes is 0.025
Lossmodm=2*damprato
/prep7
!mp,damp,1,damprato
/solu
Antype,modal! First, the decomposition of the damping mode.
Sole
/solu
Antype,harmic
Hropt,msup
Hrout,on,off
Harfrq,freqbegn,freqendg
Nsubst,num_step
kbc,1
Dmprat,damprato! This damping ratio is defined here, the constant
,,,,,, Sole

5) Full transient analysis with mp,damp defined viscous damping
! Viscous damping increases with frequency increase, high frequency attenuation is fast
! Full method with Mp,damp
! Shows that mp,damp with full are damping_ratio/pi*f
! As freq increases, damping is huge
damprato=0.025
Lossmodm=2*damprato
critfreq=480
mp_betad=damprato/(ACOs ( -1) *critfreq)! Note this formula
/prep7
Mp,damp,1,mp_betad

6) Define the total structure constant damping ratio with Dmprat
! Full method with Dmprat
damprato=0.025
Lossmodm=2*damprato
critfreq=480
mp_betad=damprato/(ACOs ( -1) *critfreq)
/prep7
et,1,1
! Mp,damp,1,mp_betad! If the input is in the form of material damping, enter
Dmprat,damprato! Constant damping ratio
/solu
Antype,modal! Vibration type decomposition with damping
modopt,lanb,3
! Important
mxpand,3,,, Yes
Lumpm,on
,,,
Sole
/solu
Antype,harmic
Hropt,full! Full harmonic analysis

6. Unit damping
Many units have unit damping, and unit damping is entered in the relevant cell data. Units with unit damping in Ansys are:
BEAM4, Combin7, Link11, Combin14, Pipe16, Combin37, Fluid38, Combin40, Fluid79, Fluid80, Fluid81, Surf153, Surf154
There are also user-defined cell property matrix Matrix27, which can be defined as a damping matrix, in addition to being defined as mass and stiffness matrices. The unit damping data in the BEAM4 unit has been introduced in the previous two chapters. Here is a brief introduction to the damping data of several units not mentioned earlier.
1) COMBIN14 Unit
Et,4,combin14
r,4,10,0.01,0.02,! 0.01 is the damping coefficient and 0.02 is the nonlinear damping coefficient.

7. Friction damping
The commonly used Coulomb damping models are:
(5.1.8)
Calculation formula of friction force of Coulomb model
The symbol of this resistance is opposite to the velocity of the contact surface relative to the movement, which is independent of the structure movement, and is related to the positive pressure and coefficient of friction on the contact surface, and the static friction coefficient and the dynamic friction coefficient are usually different. In many structural dynamic problems, friction damping is very important, ANSYS has many kinds of units that can simulate friction. However, the analysis with friction is generally a nonlinear analysis. If you do not want to do nonlinear analysis, a linearization approximation is to use the first or previous items of the Fourier series of the friction equation as input for the equal-generation viscous damping. (Examples of friction damping)
8. Other damping functions of ansys
Fluid damping, boundary damping and so on.
(Note: The formula in this article is missing, and it is recommended to refer to the Ansys Help file and the chapter on damping in the book "Structural Dynamics" and Du Siuli "structural dynamics". )

"Reproduced" damping in Ansys

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