MSC Damping 1999

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1999 Nordic MSC Users Conference, Gothenburg 1(14)

Mechanical damping simulation in MSC.Nastran

Claes R Fred, Peter Andrn and Tommy Falk

Ingemansson Automotive ABBox 276, S-401 24 Gothenburg, Sweden

[email protected]

IntroductionDesign of structural damping has the potential to reduce dynamic problems by orders of

magnitude. However, the structural damping treatment efficiency changes with frequency

and temperature. Therefore, the damping treatment must be carefully tuned with respectto the application. Use of MSC.Nastran is an integral part in this design process.

Why is damping important?Structural deflection is governed by the structural stiffness in static analysis. This is no

longer the case in dynamic analysis where the dynamic response shifts between being

stiffness-, mass- or damping- controlled in the frequency regions below, above and near

resonance, respectively. The example in Figure 1 shows that the force that is applied (1 N)

can be amplified at resonance whereas amplification is not present in the mass- and

stiffness- controlled frequency regions. The load amplification that occurs at resonance (100

times) is clearly visible in the example and not unrealistic for typical engineering structures.

The force balance of the single degree of freedom system in Figure 1 is

[ ] FXKiM =++ 2 , (1)where the mass force is2MX, the spring force isKXand the damping force is iX.

Examination of the force balance in equation (1) shows that the spring force dominates at

frequencies below 0 and the mass force dominates at frequencies above 0, Figure 1.

Figure 1. The force balance of equation (1). The largest forces in the system appear at

resonance. The stiffness force (green) dominates below resonance while the mass force(blue) dominates above resonance. The damper force (red) peaks at resonance and is of

little influence in the mass- and stiffness- controlled regions. Note that the forces in the

example are caused by an exciting force of 1 N at all frequencies, i.e. the peak force

magnitude of the damper force at resonance.

Stiffness forceMass force

Dam er force

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Resonance occur when the mass and spring forces cancel, i.e. when2MX+KX= 0,

which gives that the resonance frequency is

M

K

=0 . (3)

The force balance

FXi = (2)

shows hat the vibration response is damping controlled at resonance. Note that theresonant response is only reduced by less excitation and/or increased damping.

The vibration amplitude at resonance is

0

FX = . (4)

The largest vibration amplitudes are normally found at resonance when excitation is

broadband noise or tonal that seeps across resonance. Half of the energy is located near

resonance, i.e. within the half power bandwidth

0=f . (5)

Combination of equation (4) and equation (5) shows that a small damping value leads to

large displacement amplitudes and a narrow half power bandwidth. Similarly, a high

damping value leads to small vibration amplitudes and that the structural energy is

distributed over a wide frequency bandwidth.

Damping a forgotten design parameter?Equation (4) hold, too a good approximation, also for multi-modal structures that areuniformly damped. The structural response scales linearly with the damping magnitude.

In simple terms, a factor of 10 higher damping can reduce the structural stress and/ordeflection by a factor of 10.

As a coarse rule of thumb, built-up plate structures with connections that involve friction

and air pumping in narrow gaps (riveted seams, bolted connections, linings etc.) tend to

have damping values about 2% to 0.5%. Structures with rigid connections like line (weld

seams, solid connections etc) tend to have damping values about 1% to 0.1%. A structure

with a well designed damping treatment tend to have damping values in the range 40% to

10%.

Therefore, a designed structural damping treatment can lead to a reduction in stressand/or deflection that ranges from 400 times down to 5 times! Few, if any, design

parameters can match this design change potential in dynamic analysis.

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How does damping change with temperature and frequency?Temperature is usually considered to be the most important environmental factor that

affect the properties of damping materials. The variation of Youngs modulus,E, and loss

factor, , with temperature and constant frequency and strain amplitude are typically inthe form shown in Figure 2. Three regions can be observed.

The first region is the so-called glassyregion, where the material takes on its maximumvalue for the storage modulus,EorG, while having extremely low values for the loss

factor, . The modulus in the glassy region changes slowly with temperature, while the

loss factor increases significantly with increasing temperature.

The second, transition region, is characterized by having a Youngs modulus that

decreases rapidly with increasing temperature, while the loss factor takes on itsmaximum.

The third is the rubbery region where both the modulus and the loss factor usually take

on somewhat low values and vary slowly with temperature [2].

There is also a fourth region, not shown in Figure 2, after the rubbery region at very high

temperature, the so-called flow region. In this region the material become soft with high

loss factor, but very low Youngs modulus. For most rubber like materials, this fourth

region does not exist [2]. For an overview of the temperature effects for modulus and loss

factor see Table 1.

Figure 2. Variation of material properties with temperature and frequency [3].

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Table 1. Behavior of damping in different temperature regions with constant frequency.Region Young s modulus Loss factor

Glassy High, slowly decreasing Low, slowly increasing

Transition Decreasing Maximum value region

Rubbery Stable low Stable low

Flow Negligible Slightly increasing

The material behavior illustrated in Figure 2 shows the archetypical behavior. All rubber-

like materials have different specific properties, characterized mainly by various levels of

the storage modulus and loss factor within each temperature region and the location of

each region.

Typical values of the storage modulus could be as high as 108 kPa in the glassy region

and as low as 10 kPa in the rubbery region [2]. The loss factor in the glassy region isusually below 10-2 or 10-3, whereas it can reach values over 2,5 in the transition region.

Typical loss factor values in the rubbery region are usually between 0.1 or 0.3 for manymaterials, depending on their composition.

To exemplify, the viscoelastic damping material Swedac DG-V4 shows a typical

behavior of temperature effects for loss factor and shear modulus in the transition region,

see Figure 3(a,b). The graphs have been calculated and plotted by use of Ingemanssons

in-house computer program, DCP (Damping Calculation Program) and our database with

damping materials.

If you study Figure 2 you can see that the curve with constant temperature is the mirror

image of the one with constant frequency. However, note that it takes several decades offrequency to reflect the same change of behavior with a few degrees of temperature [1].

Figure 4 shows how Youngs modulus and loss factor changes for different temperatures.

One of the effects which frequency has on the damping properties is the fact that modulus

always increases with increasing frequency. This increase is rather small in both the

glassy (temperatures T-1 and T-2) and rubbery regions (temperatures T1 and T2) while it

takes on its greatest rate of change in the transition region (temperature To), Figure 4(a).

Figure 4(b) shows that the loss factor, , increases with increasing frequency in the

rubbery region (T1and T2), while it takes on its maximum value in the transition region

(To), and then decreases with increasing frequency in the glassy region (T-1 andT-2).

The viscoelastic damping material Swedac DG-V4 shows a typical behavior of frequency

effects for shear modulus and loss factor in the transition region, see Figure 5(a,b). Note

that the shape of the shear modulus in Figure 5(a) is almost a mirror image of the one inFigure 3(a).

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0.00

0.25

0.50

0.75

1.00

1.25

Single Viscoelastic Material

ShearModulus[Pa][e7

]

Tem erature Celcius

100 [Hz]

1.25

1.50

1.75

2.00

2.25

2.50


Lossfactor

Figure 3. A) Shear modulus for viscoelastic damping glue, Swedac DG-V4, at 100 Hz in

the transition region. B) Loss factor for viscoelastic damping glue, Swedac DG-V4, at

100 Hz in the transition region.

Figure 4. Variation of the (A) storage modulus and (B) loss factor with frequency for

different temperature [2].

0.0

2.5

5.0

7.5

10.0


ShearModulus[Pa][e6]

Frequency [Hz]

50 [Celsius]

1.75

2.00

2.25

2.50


Lossfactor

Frequency [Hz]

50 [Celsius]

Figure 5. A) Shear modulus for viscoelastic damping glue, Swedac DG-V4, at 50o C. B)Loss factor for viscoelastic damping glue, Swedac DG-V4, at 50o C.

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What does a damping treatment look like?Figure 6(a-c) shows three typical structural damping arrangements.

The principle of creating a high structural loss factor in FLDT is that the dampingmaterial should take up strain energy, Figure 6(a). The strain energy in the free damping

layer can only be large if the damping material is stiff, if the bonding between the

damping layer and the structure is strong and the damping material is located at a

distance from the neutral bending axis. An advantage with FLDT is that it adds stiffness

to the structure - the disadvantage is the weight added.

The principle of creating a high structural loss factor in a CLDT and MCLDT is that the

damping layer should take up shear energy, Figure 6(b,c). The largest shear energy is

found along the neutral axis of the composite. Again, the structural damping layer can

only be effective if the bonding between the damping layer and the constraining

structures is sufficiently strong. It is easier to load the damping layer in CLDT becausethe shear angle across the damping material is the largest when the damping materials

shear stiffness is negligible. An advantage with CLDT is that it can be designed withoutadding weight the cost is that stiffness is reduced. The stiffness reduction depends on

how the CLDT is stacked and the stiffness of the damping material.

The MCLDT composite stack is primarily used when damping is required over a wide

temperature and/or frequency range. Each damping layer is then tuned for optimal

performance in a certain temperature and frequency domain.

Damping treatments with the highest loss factors are often created with CLDT or

MCLDT.

(A) (B)

(C)

Figure 6. Surface damping treatment from [1]. A) Free Layer Damping Technique

(FLDT). B) Constrained Layer Damping Technique (CLDT). C) Multiple Constrained

Layer Damping Technique (MCLDT).

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How do I design a damping application?The structural damping application can be designed at various levels of sophistication.

Material levelThe simplest approach is, naturally, to find a material with high damping in the required

temperature and frequency interval. The Ingemansson, in-house, DCP program can

search the database and produce a list over the materials that produce a certain structural

damping factor magnitude that is larger than a specified value. Note that the fact that a

damping material has a high loss factor does by no means guarantee that the structure

will become effectively damped.

Archetypical level

Data for the cross section dimension and material is specified and the damping is

thereafter computed as a function of frequency for the composite cross section. The

advantage is that this calculation is very fast and that it gives a first rough design of thedamping application. It is also a very good tool for design of damping application at high

frequency.

Generic levelThe damping application can be simulated for a generic structure like a flat plate or a

straight beam and the shift in eigenfrequency with temperature be computed. The generic

level is restricted to simple shapes for the beam cross section and straight structures.

Component level

Analysis of a damped component requires use of a FE model. The damping treatment can

be applied over the whole surface or at parts of the component. The position and shape of

the damping treatment can be optimised for the partially covered component.

System levelA damped component increases also the damping of the global structure. The increase in

damping depends on whether the component participates in the dynamic load path or not.

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How do I calculate damping in my FE model?Eigenvalue analysis

The modal damping can be approximated with the use of the Modal Strain Energy (MSE)

method. The structure is divided intoNregions with different damping. The modaldamping is

total

N

j

jj

eE

E=

=1

mod

, (6)

where , the structures total potential energy isEtotal, the loss factor and strain energy of

partj of the structure are j andEj, respectively.

The MSE method is best suited for cases where it is clear whether strain or shear is the

primary loss mechanism. Use of equation (6) should be changed to shear energy for CLDT

and is applicable only for flat structures. The MSE can be developed into a more complicatedform for curved or stiffened structures where the strain and shear loss factors are treated as

separate regions. The computational advantage in the MSE method is that it uses real modes.

The SOL 107 and the EIGC option in MSC.Nastran can be used to compute complexmodes. The modal damping is then directly computed in the eigenfrequency analysis. An

advantage in using complex modes a better representation of the energy flow between

parts. The disadvantage is the larger computational effort involved.

Forced response analysis

The structural damping can be derived from the point Frequency Response Function

(FRF). The modal eigenfrequency and damping is determined from a circle fit on the

computed FRF or by automatic pole fitting in modal analysis software. The FRF can be

computed with the direct or the modal approach in MSC.Nastran. Note that a non-

orthogonal modal basis should be used in the case that the modal approach is used andthe frequency effect is strong.

A version of the above mentioned forced response analyses is to compute the active part

of the structural power input and the total strain energy of the structure. The Power Input

Method (PIM) shows that the structures loss factor is

( )tot

total

E

Fvreal

2

*

= , (7)

where the complex conjugate is *, the vibration velocity is v, and the angular frequency is

A version of the PIM would be to compute the strain energy of only a portion of the

structure. The loss factor would then represent the parts structural damping.

Free decay techniquesIn principle, the structural damping can be computed from the free decay when the

structure is excited by an impulse. However, this option would lead to poorcomputational economy from the many calculation steps involved in capturing the time

history of the decay.

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How do I model damping in MSC.Nastran?There are several ways to model damping treatments in MSC.Nastran. However, none of

them is really convenient for the user since the use of frequency and temperature

dependent material properties and a non-orthogonal modal base is not in the mainstreamof MSC.Nastran use. As mentioned above, the temperature dependence is the most

important factor to cope with.

For CLDT and MCLDT, the structural damping treatment can be modelled with spring

elements that connect between the constraining layers. This technique can be applied

with MPCs and CELAS1 elements or with PBUSH elements. This technique is time

consuming for the simple reason that it necessitates detailed modelling of the springs

with alignment into local coordinate systems.

FLDT, CLDT and MCLDT can be modelled with the use of solid elements for the

damping material layer. The damping layer should have two or more elements across thecross section. This technique is therefore restricted when large models are involved but is

otherwise a relatively straightforward process.

The most economical way of modelling the damping layer is to use shell elements. Itwould be tempting to use MSC.Nastrans functionality for composite materials.

Unfortunately, MSC.Nastrans PCOMP (version 70.5) does not offer much advantage

since the PCOMP stack does not compute the composite damping and because a

PSHELL entry that reference the PCOMP stack does only handle a single damping value.

A well designed damping application should be differently damped for bending and

membrane (in-plane) motion. In particular, a CLDT application should have a large

difference in damping between bending and membrane motion. As mentioned above, thePSHELL entry does only reference a single damping value. A work around is therefore

needed.

The way to model damping in MSC.Nastran is therefore to import bending and

membrane damping with the use of two separate PSHELL entries. The elements are

duplicated in areas at which damping material is applied and the PSHELL entries for

bending refers to one of the element sets while the membrane entry refers to the other

element set.

Equivalent frequency dependent parameters (Youngs modulus, density and damping) forthe composites bending and the membrane materials must be computed outside of

MSC.Nastran. The composites Poisson ratio and the composite thickness are values thatare specified and the other parameters are computed with respect to these values.

In our case, data are automatically output at each temperature and frequency by

Ingemanssons in-house program DCP. One advantage with DCP is that the composites

damping factor is computed taking into account also the material damping of the

constraining layers. The equations involved are lengthy and, thus not given here.

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ExamplesEigenvalue analysis of an oil sump

The simplified oil sump structure is shown in Figure 7. The sump edges were clamped.

This example was chosen because it is customary to use CDLT for this type of enginecomponent.

The oil sump was first meshed and the elements thereafter duplicated. The materials that

were used were divided into two separate PSHELL entries in order to get the correct

damping loss factors for the bending and the membrane deformations. The CLDT stack

consists from a 1mm and a 2 mm thick steel sheet and a 0.1 mm thick damping material

layer that is bonded between the steel sheets. The composite loss factor and equivalent

stiffness is computed in DCP for the bending and membrane motions.

The objective with the simulation was to compute the variation in damping and

eigenfrequency with temperature. SOL107 and the complex Lanczos eigenmode solverwere used to compute the data.

The work flow was set up and automatically executed in LMS Optimus, Figure 8(a,b).

The normalised difference between the guessed frequency that was used to extract thematerial data and the computed eigenfrequency was used as the optimisation criterion in

the optimisation loop. The cases that were computed were chosen with the help of the

adjustable full factorial Design of Experiments (DoE) method where the step length was

controlled for the input variables. The step length on the input variables was controlled

such that data was computed at 5 different temperatures and for 10 modes, i.e. 50

simulation runs were made. The optimisation loop did typically require 3 to 6 iterations

to converge on a sufficiently small difference between the computed frequencies for the

material and the eigenmode, i.e. 150 to 300 computation runs in MSC.Nastran wereautomatically executed by LMS Optimus.

The data computed was thereafter curve fitted with a 5th order polynome which involves

cross terms between the input variables (temperature and mode number). In this way a

mathematical model, i.e. a Response Surface Model (RSM), that relates the data output

variables (eigenfrequency and damping) to the input variables could be created. The

RSM was thereafter used to plot the results. The RSM can be used for optimisation

purposes as well, e.g. to find the temperature that produces the best overall damping.

Figure 9(a) shows the variation in eigenfrequency with temperature for the ten firstmodes. There is a shift in eigenfrequency for the modes of about 30%. Figure 9(b) shows

the variation in loss factor for modes 1 (deep blue), 5 (magenta) and 10 (light blue). Theovershoot at high and low temperatures is an effect from the polynomial function. The

peak damping magnitude is quite similar for all of the modes, which can be expected

from the fact that the damping material is uniformly distributed over the oil sump. The

damping for mode 10 can be seen to be located into a narrower temperature interval than

is the case for modes 1 and 5. Figure 10(a,b) shows 3D plots that gives and overview of

the eigenfrequency and damping variations with frequency.

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The RSM is an efficient mathematical model that can be used to generate data at many

points. It is therefore possible also to gain some understanding on the robustness of the

design. Figure 11 shows the expected variation in eigenfrequency in the case that there is

a variation in temperature. As can be seen in the plot, the shift in eigenfrequency is quitelarge because the damping material is in the transition region between the glassy and the

rubbery region.

Figure 7. FE model of the simplified oil sump.

Figure 8. The workflow used to compute the data for the oil sump. A) Master loop that

executes the optimisation loop shown in subfigure B, i.e. Optimus is used to drive

Optimus. B) The slave loop in which the optimisation takes place. Data is collected from

the database with DCP and MSC.Nastran is used to compute the complex

eigenfrequencies and their damping values. The workflow shown in this subfigure needs

use of some commands from the LMS Optimus syntax and is used in batch mode.

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Figure 9. Data derived from calculations the oil sump. A) The variation in

eigenfrequency with temperature (T) for mode 1 to 10. B) The variation in damping withtemperature (T).

Figure 10. A 3D overview of the data that is generated by the RSM. A) The variation ineigenfrequency (fmode) with temperature (T) and mode number (Mode_no). B) The

variation in damping (DAMPING) with temperature (T) and Modenumber (Mode_no).

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Figure 11. The expected variation in eigenfrequency when the temperature is uncertain.

The mean temperature is 55 C and is determined with 90% probability (3sigma) to vary

within 12 C. The eigenfrequency of mode 6 will then with 90% probability lie in the

frequency region 232 Hz to 304 Hz, with the most probable frequency located at 262 Hz.

Forced response analysis and use of PIM for a simple cup

A simple cup that consists from two materials is shown in Figure 12. The bottom of the

cup consists from a damped material while the rest of the cup is made from ordinary

steel. The bottom plate is flat and the analysis was simplified to use only the bending

material for both the bending and the membrane materials (= an early case ofDCP/Nastran application).

The forced response is calculated at one frequency and the material parameters arethereafter updated for the next frequency. The workflow is simpler for this case because

the optimisation loop in the example from above is not needed, Figure 13.

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Figure 12. FE model of the simple cup. The bottom of the cup is damped. A) The

excitation position. B) Alternate view.

Figure 13. A) The workflow that is used in the PIM analysis. B) The end result of the

PIM analysis. The variation in damping (DesETA) with temperature (t) and frequency

(freq).

References

[1] Nashif A.D, Design of damped structures. Anatrol Corporation, Ohio (1990).

[2] Nashif, Jones, Henderson, Vibration damping. John Wiley & Sons, USA (1985).

[3] Garibaldi, Onah, Viscoelastic material damping technology. Becchis Osiride,

Torino (1996).

DCP

Nastran

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