Analysis methods to support design for damping

Abstract

Methods are documented for active and passive damping studies with capabilities within NX I-deas and NX Nastran (http://www.ugs.com/products/nx/). The focus is on the methods that can be applied to a wide variety of products and design goals. Project examples are used to illustrate the methods and indicate the level of success obtained from real applications. Specific examples include passive damping for a liquid rocket engine, actuators for spacecraft interface that can suppress spacecraft response to launch vehicle vibration, active ceramic-fiber vibration suppression, constrained and free layer damping as well as the use of a secondary mode to increase effective damping or improve settling time of a manipulator arm.

1 Introduction and background

The goal for designing structures that meet some damping target is to maintain the dynamic response of a structure in a known environment to specified or tolerable limits. When the design driving loads are sine dwell and acoustic, damping becomes a critical issue. The structure must be able to dissipate some of the energy to which it is exposed to avoid large dynamic amplification. This could mean including materials that have significant loss factors or adding discrete energy absorbing elements such as shock absorbers. The process of designing energy dissipation into a structure generally involves the following steps: identify the modes that dominate the response to be controlled; determine the strain energy distribution in the basic structural concept and the important modes; determine potential strain energy in the candidate active or passive energy absorption treatment; add elements to model the damping material or treatment and predict the damping by mode and possibly the effect on the forced response. This approach is based on the understanding [1, 2] that the contribution of each materials or region of material to the loss factor for a structure is proportional to the strain energy in the structure carried by that material region.

For example, in Fig. 1, the results of a forced response analysis of a satellite antenna system (a simplified system has been used for paper figures to protect proprietary information) shows the peak strain energy distribution due to launch loads. The strain energy is concentrated in the connection of the booster to the satellite electronics as highlighted in the figure.

Fig. 1

Strain energy distribution at time of peak response during boost phase

The frequency response functions (FRF) for stress and acceleration, shown in Fig. 2, were obtained by applying a unit input force uniform for all frequencies to show the structural characteristics. In particular this FRF identifies the fifth mode, at 9.7 Hz, as the important mode for the response of interest and the input direction selected. The strain energy distribution in this mode is shown in Fig. 3. Those peak strain energy areas are candidate locations for added damping treatment to control the response.

Fig. 2

Stress and acceleration response shows mode at 9.7 Hz is an important mode excitation in a selected direction

Fig. 3

Strain energy distribution in important mode

Table 1 provides another way to determine the locations most effective for damping treatment application. Regions of the structure are organized into groups; I-deas outputs spreadsheet text files with the percent of total strain energy defined by group in each mode. For the fifth mode this table shows that the strain energy is concentrated in the local attachment between the satellite and the booster.

Table 1 Strain energy per mode by structural group

From both approaches, it is clear that this is the location where we should put damping materials to control the response. In order to design the distribution and type of damping, we need to quantify the modal damping contribution of candidate damping treatments.

Energy loss in a dynamic system can be modeled as a force term in the equation of motion. The damping force is always out of phase with the displacement. This force term can be a viscous term where the force is proportional to velocity as shown in (1) for a single degree of freedom system (or a mode of a system with any number of degrees of freedom expressed as modes),

$$ \begin{aligned}{} & m\ifmmode\expandafter\ddot\else\expandafter\"\fi{x} + c\ifmmode\expandafter\dot\else\expandafter\.\fi{x} + kx = f(t)\,, \\ \end{aligned} $$

(1)

or it can be a stiffness term which is proportional to the material loss factor (structural or hysteretic damping) where the resisting force is proportional to displacement as shown in (2)

$$ \begin{aligned}{} & m\ifmmode\expandafter\ddot\else\expandafter\"\fi{x} + ik{\underbrace \eta _{{{\text{loss}}\;{\text{coeff}}{\text{.}}}}}x + kx = f(t)\,, \\ & m\ifmmode\expandafter\ddot\else\expandafter\"\fi{x} + {\underbrace {k(i\eta + 1)}_{{{\text{complex\_stiffness}}}}}x = f(t)\,. \\ \end{aligned} $$

(2)

With the following definitions for ζ and ω _n ,

$$ \zeta \equiv \frac{c} {{2{\sqrt {km} }}}\,, $$

$$ \omega _{n} \equiv {\sqrt {\frac{k} {m}} }\,. $$

Equation (1) becomes

$$ \ifmmode\expandafter\ddot\else\expandafter\"\fi{x} + 2\varsigma \omega _{n} \ifmmode\expandafter\dot\else\expandafter\.\fi{x} + \omega _{n} ^{2} x = 0\,. $$

In these equations, m is the mass, k is the stiffness, c is the viscous damping coefficient, η is the material loss factor, ζ is the critical damping ratio, x is the single degree of freedom and f(t) is the forcing function.

Two types of loss modeled by these equations involve different physics but have the same peak amplitude at resonance when the following relationship holds:

$$ \eta = 2\zeta \,. $$

Any finite element program assembles a mass matrix and a stiffness matrix from the element definitions and the material properties. If the material properties include the material loss factor or discrete viscous damping elements connecting nodal degrees of freedom, I-deas and NX Nastran will also assemble a physical damping matrix, [C]. The mass and stiffness matrices are used in an eigen-solution to obtain the undamped natural frequencies and mode shape vectors, [ψ]. Using the following triple matrix product, an estimate of the modal damping is provided by I-deas.

$$ {\underbrace {{\left[ {\;\;\;\;\;c\;\;\;\;\;} \right]}}_{{{\text{Modaldamping}}}}} \equiv {\left[ c \right]} = {\left[ \psi \right]}^{{\text{T}}} {\underbrace {{\left[ C \right]}}_{\begin{subarray}{*{20}l} {\text{physical}} \\ {\text{damping}} \\ {\text{matrix}} \end{subarray} }}{\left[ \psi \right]}\,. $$

(3)

Although the modal damping matrix, [c] is not in general diagonal, the diagonal terms provide a modal damping estimate that is listed and can be directly applied for forced response. For light damping, which is mostly material damping, the off-diagonal terms tend to be small compared to the diagonal terms, indicating that this estimate of mode by mode damping is reasonable. I-deas (the internal solver called Model Solution) calculates modal damping twice: once to calculate hysteretic damping (structural damping) and a second time to calculate viscous damping (damper elements). The viscous modal damping values are converted to modal damping factors by dividing by the critical damping value for each mode. (This capability is standard in I-deas but currently requires DMAP implementation in any version of Nastran including NX Nastran).

The off-diagonal terms are used only to provide confidence factors for the diagonal damping values. A confidence factor of 1.0 indicates that the sum of the off-diagonal terms is zero. A confidence factor of 0.5 indicates that the sum of the off-diagonal terms equals the value of the diagonal element.

Thus once the locations of structure with high strain energy in the forced response or the important modes have been identified, the design can start to include loss mechanisms at these locations. What is important is to design the materials with high loss factors into the structure such that they are forced to undergo significant strain. These materials are never stiff compared to the structural materials that carry most of the load. Thus the design must include materials with loss in such a way that stiff parts of the structure force deformation and strain in the materials with high loss. Some candidate treatments are discussed here along with modeling approaches.

One of the most effective damping treatments is to constrain a high loss material between two stiff layers of structural material such that the damping material must deform significantly. The damping material is forced to undergo shear deformation as it is forced to follow the contours of the stiff layers on either side. Thus strain is forced into the damping material which can be much less stiff than the constraining material layers. This constrained layer damping can be modeled with the above methods. As an example two strips of aluminum were modeled with solid elements with a layer of damping material [3M Visco-elastic Damping Polymer: loss factor 0.9, shear modulus 1 MPa, (145 psi)] between these layers. Four layers of parabolic solid elements through the thickness were used for each of the aluminum layers and for the damping material layer. This model was obtained by first building the elements with the thickness much greater and then using the I-deas automatic updating to shrink the thicknesses to the correct values. Although the damping layer elements do not have a good aspect ratio, a successively finer mesh was tried until confidence in the adequacy of the aspect ratio was achieved.

The first three bending modes are shown in Fig. 4 along with the structural damping ratio predicted for each. Experiments with these strips indicated that the bending modes die out in just a few cycles which are consistent with these damping estimates.

Fig. 4

Constrained layer damping modeled in I-deas

In the next section these methods will be illustrated using application examples from projects.

2 Damping treatment examples

ATA Engineering has performed a number of projects in which increased damping was the design goal. NX I-deas and NX Nastran provided strong capabilities for these studies. The following project design goals are described.

• 1. Application of constrained layer damping and semi-active ceramic fibers to spacecraft motion and pointing accuracy.

• 2. Application of damping treatment to the exterior (not constrained layer) of a spacecraft after the design was complete with the intension of having the material weight burned off in flight.

• 3. Use of a secondary mode to reduce settling time due to a step or shock excitation.

2.1 Controlling spacecraft pointing accuracy

For the satellite system shown in Fig. 1, the FRF for the rotation of the antenna, shown in Fig. 5, indicates that the important modes for this response is mode 5.

Fig. 5

Frequency response functions (FRF) for rotation of antenna due to vibration at the base of the spacecraft

For this FRF, baseline damping by mode was obtained by using structural damping of 2% (1% equivalent viscous) for the aluminum and antenna composite material and the springs connecting the three major components; thus the modal damping for this baseline was 2% structural in each mode.

The strain energy table (Table 1) and plot (Fig. 1) show that damping would be effective in the satellite body as well as the connectors in the shell elements and the springs on the satellite at the interface with the booster and antenna. The maximum-principal-stress plot, Fig. 6, for the satellite body at the booster end of the satellite suggests load paths for this structural region in this mode.

Fig. 6

Location of maximum principal stress at location controlling mode 5

Piezoelectric fibers were considered for this region. When these fibers are stretched, they generate electrical energy which can be stored in capacitors and then used to cause the fibers to create a restoring force. A microprocessor can be used to cause a time delay which represents a shift in phase. This phase shift will be different for each mode. In this case, we have identified a single important mode which makes it feasible to use this approach. This effect creates a force much like the structural damping force in (2) above. This type of piezoelectric fiber composite has been used successfully in skis and tennis rackets [3]. It may also be possible to use these fibers as passive dampers [4]. When used passively the electrical energy generated by the fibers is dissipated through resistors and not used for active control.

For the design process, these treatment fibers were included in the finite element model. The fibers were modeled as beams using cross-sectional and material properties from one supplier of these fibers [3]. Figure 7 shows the fibers on the satellite surface next to the booster.

Fig. 7

Ceramic fibers shown as strips on the base of the satellite near the booster attachment

The fibers were given a structural damping factor of 0.5 to represent the best possible efficiency. The resultant modal damping for the first ten modes is provided in Table 2, which shows we have more than doubled the effective structural damping in the important mode 5 from 2 to 5.9%.

Table 2 Effective structural and viscous damping by mode for ceramic fibers, constrained layer and connector shock absorbers

As an alternative approach the top and bottom of the satellite were replaced with aluminum of approximately the same thickness but with a constrained layer of visco-elastic damping polymer. As a first estimate of this effect, the sample problem shown above was used to estimate the material damping. Taking the conservative value of damping found in the first bending mode of 0.16 loss factor (16%), the shells on the top and bottom of the satellite used a material with this structure damping loss factor. Table 2 shows that this results in much higher modal damping in the first five modes with 11.6% in the fifth mode.

Another approach is to use viscous dampers (shock absorber) with 30% critical viscous damping in the spring elements that are used to connect the components. These results are also shown in Table 2. The equivalent viscous damping percentage is twice as effective as structural damping at resonance. The discrete shock absorber approach is also more effective for mode 5.

Thus three design approaches for controlling the first five modes have been demonstrated. Selection of the solution of choice might come from using these approaches in a forced response for transient, random, or sinusoidal excitation. The damping values for all modes computed in this way are automatically provided to the forced response module in I-deas; here, that module was used to predict the transient, random and sinusoidal response of angular displacement at the base of the antenna before and after damping treatment. Figure 8 shows the FRF comparisons which should be the right relative response for sinusoidal loads.

Fig. 8

FRF response provides comparison of three types of damping treatment to baseline

Figure 9 compares the responses for a launch transient for three types of damping and the baseline. Constrained layer damping treatment provides the best settling time of this response, but the ceramic fiber approach shows the lowest peak value. The input transient is shown in Fig. 10.

Fig. 9

Transient response to launch loads compares three types of damping treatment to the baseline

Fig. 10

Input acceleration at the base used to obtain forced angular displacements in Fig. 9

For vertical random vibration applied at the booster, the power spectral density (PSD) response at the base of the antenna was computed for each type of damping treatment; responses are shown in Fig. 11. For comparison, the PSD vibration input is shown as a dotted line in this figure. The plot has been zoomed to the range 2–50 Hz where the response is the greatest. For this environment, the most dramatic decrease in RMS value is provided by the constrained layer.

Fig. 11

Power spectral density (PSD) response compares three types of damping treatment to baseline with input shown as dotted line

In the responses computed above we focused primarily on modes 1 through 6. Modes 7, 8 and 10 are primarily antenna modes. In order to reduce vibration response here, a treatment that could be applied to the composite material is needed. The following discussion about free layer damping may be of interest for these modes and this part of the structure.

2.2 Free layer damping

Free layer damping is sometimes attractive because it can be added after the structure is already built without greatly affecting the design and can be more easily removed. The disadvantage of free layer damping is that it is hard to get a significant amount of strain energy into the damping materials. Most materials with a high loss factor, such as visco-elastic materials, have low stiffness relative to the structural materials. For constrained layer damping, the visco-elastic material is forced to undergo large shear deformation by the constraining layers of structural material. For free layer damping, the deformation comes from bending or stretching of the outer fibers of structural material on which the material is fastened. Thus free layer damping depends on the damping layer having a stiffness that is not trivial in comparison to the base material in order to undergo enough strain to contribute the desired loss factors. Because of the challenge of getting strain energy into the damping material, the analytical prediction of the effective damping becomes even more important. This is so to ensure that the structural concept has a reasonable chance of achieving the desired damping before the design is taken too far or too much detailed modeling is done.

For free layer damping, just as for constrained layer, finite element approaches can be used. For example, Fig. 12 shows a carbon–carbon cone, which is similar to a segment of a nozzle of a launch vehicle design project completed by ATA [5, 6]. At one stage in this launch vehicle project, the nozzle was believed to be over-stressed due to sinusoidal and acoustic loading environments and the very light damping of the baseline carbon–carbon material. The model shown is not the real structure but has similar structural characteristics. Two layers of solid parabolic orthotropic elements are used for the base material; for the damping material, two layers through the thickness of the free layer damping material are used. This damping material was selected such that it would burn off in flight when the nozzle segment becomes hot and when the weight of the nozzle was critical.

Fig. 12

Carbon–carbon cone with free layer damping modeled by two solid elements through the thickness of both the base cone and damping layer

Figure 13 shows the loss factors as percent hysteretic damping due to the damping material as a function of frequency for each mode. The base material was given zero damping such that the effect of the damping treatment could be identified. An equivalent viscous damping between 0.2 and 0.5% was later shown to be appropriate for the base material. Thus the added damping due to the damping treatment of 0.5–1.2% represented a doubling of the damping although it was still relatively lightly damped.

Fig. 13

Predicted hysteretic damping by mode for cone varies from 0.5 to 1.2%

Modeling the damping layer explicitly with solid elements can lead to large models that may be impractical for design studies. Modeling damping using shell models is very difficult to accomplish because of the need to have a layer of structural material next to a layer of damping material with shear transfer across the interface. Very detailed offset shell models are more time consuming and of questionable accuracy compared to the solid element approach described here. Thus a solid model with today’s computing power is perhaps the best way to perform detailed modeling. However, one alternative approach is to determine from a breakout model such as the cone, the damping that can be expected from the detailed multiple layer solids model and then use this loss factor in the simpler representation of the structural component in a global spacecraft model.

For example, in the actual project the nozzle was represented by a single layer of shells representing the carbon–carbon. Two damping layers of shells were added, one with the correct loss factor for membrane and one for the correct loss factor for bending. In considering how to simulate the effect of damping treatment in these shells, it is important to recognize that the effect of the damping layer changes as the deformation in the shells varies between mainly bending to mainly membrane behavior. There is no single-shell material loss factor that can correctly represent the modal damping in a single layer of shells as the shells undergo different types of deformation.

The dependency of loss factor on deformed shapes can be better understood and accounted for by looking at the loss factors predicted by classical solutions which show differences for membrane and bending behavior. In particular, the loss factor in bending, η _b , can be determined by the following expression:

$$ \eta _{b} = \eta _{D} \,\frac{n} {{1 + sn}}{\left[ {\frac{{3 + 6n + 4n^{2} + 2sn^{3} + s^{2} n^{4} }} {{1 + 2sn(2 + 3n + 2sn^{2} ) + s^{2} n^{4} }}} \right]}\,, $$

(4)

from Overst and Frankenfeld [7, 8], where

h :

plate thickness

h _D :

damping layer thickness

n :

h _D /h; non-dimensional thickness

E :

elastic modulus of the base material

E _D :

real part of the complex modulus of the damping material

s :

E _D /E; non-dimensional stiffness

η _D :

damping material loss factor

E _D (1+iη _D ):

complex modulus

For membrane waves, Cremer and Heckle [7, 9] derived the following expression for loss factor, η _m ,

$$ \eta _{m} = \eta _{D} \frac{{sn}} {{1 + sn}} $$

(5)

and warned that it is much harder to damp longitudinal waves than bending waves. Note that loss factors from these formulas are not explicit functions of frequency. The damping can depend on frequency if the loss factor, η _D , or the modulus, E _D , for the damping layer varies with frequency. There almost surely is some dependence on frequency and temperature for these damping material properties; as a result, selecting the damping material and performing the analysis may require some knowledge of the important frequency and certainly the expected temperatures.

To illustrate the difference in damping achievable for bending versus membrane deformation, Fig. 14 depicts the expressions (3) and (4) for loss factor plotted for a range of ratios, s, for elastic moduli and ratios, n, for thickness of damping layer to thickness of base material layer. Note how much lower the effective damping is in membrane deformation.

Fig. 14

Achievable loss factor for structure plotted versus ratio of their elastic modulus for several different thickness ratios shows that free layer damping is much more effective for bending than for membrane deformation

The modeling approach using layers of solids for the damping and base materials should correctly show this behavior which represents how the damping layer properties E _D , η _D , combine with the base materials for different deformations to achieve the effective damping in each type of deformation. Every mode is some combination of bending (out-of-plane) and membrane (in-plane) behavior.

Figure 15 shows results for a detailed solids model of a flat plate made of the same materials that were used in the conical nozzle and the same thicknesses for damping and base material and the same frequency range. For the flat plate, nearly all the modes are mostly bending as opposed to a mix of bending and membrane deformation that is expected from the conical shell. Note the higher damping is predicted for the flat plate.

Fig. 15

Hysteretic damping predicted for flat plate varies from 0.9 to 1.2% with the typical value closer to 1.2%

It is possible to model the nozzle with a single layer of shells and include different material properties for bending and membrane deformation. However, for the prediction of modal damping using the mode shapes, both NX Nastran and I-deas use the membrane material loss factor to scale the total element stiffness matrix, after the bending and membrane stiffnesses have already been combined, no matter what loss factor is entered for bending. Because the membrane damping is normally less, and the lightly damped modes tend to be more important to the response, it may be appropriate to just use the membrane material loss factors to estimate the damping to be expected from free layer damping. An alternative approach is to use a double layer of shells, one each for membrane stiffness and loss factor only, and for bending stiffness and loss factor only.

It is important not to over estimate the damping achievable from free layer damping in the real structure by predicting or measuring the damping for a flat plate with free layer damping treatment. The real structure’s behavior may exhibit much more membrane behavior than a small sample of the material with a free layer of damping. In the actual project involving a launch vehicle nozzle, the first assessment of damping was done on small samples that were modal tested. These samples where excited into bending mode for which the modal damping was determined before and after adding free layer damping. In this particular case, damping evaluation was about 5%. It was later learned from modal testing on the actual nozzle that damping of 0.2–0.5% was more typical for the important modes.

2.3 Damping for position accuracy

In the previous projects the structure being designed was subjected to a stationary or periodic vibration or acoustic environment. If the loading is transient, such as a shock pulse or a step positioning command, damping cannot effectively change the peak response, but it can greatly reduce the time necessary for the free vibration response to settle to an acceptable position error. Figure 16 shows the end of a beam-like structure that is positioned by a step command. The accuracy of the positioning is diminished by the mode of the beam ringing. With added damping this ringing will die out more quickly and achieve the desired accuracy.

Fig. 16

End of a beam-like structure which required very accurate placement when provided a step position command

Damping of the structure’s modes proved challenging: It was difficult to obtain enough damping with free layer or constrained layer damping. However, by adding another spring-mass system within the primary structure, the settling time and thus effective accuracy was improved. The added spring-mass system provided an additional modal degree of freedom. This mode brought improved performance in two ways: the internal mass tended to move out-of-phase with the primary structural motion and directly acted to suppress the primary vibration just as a tuned mass damper does. In this case the mode of the internal motion did not need to be exactly tuned to the fundamental beam motion, since both were excited by the step input. The secondary opportunity for damping was a discrete damping mechanism in the internal spring to further dissipate energy.

This system was modeled in I-deas by adding an additional beam-mass system with springs to maintain the position of the secondary mass within the primary structure. The discrete damper within the beam could also be included in the simulation, along with the effective material damping from the constrained layer damping to obtain an overall effective modal damping. Figure 17 shows the improved settling time when the incremental spring mass system was added to the already damped primary structure.

Fig. 17

Baseline beam end deflection at a given wait time is greatly reduced by adding both a mass damper and constrained layer damping to the beam

3 Conclusions and recommendations

Analytical methods have been illustrated that support the design of damping treatment on various structures such as spacecraft and launch vehicles. These methods can guide the selection of damping approaches, the location of damping treatment and the selection of damping material. The key is the prediction of strain energy, important modes and the estimate of damping on a mode-by-mode basis. Simulation methods included constrained and free layer damping, active ceramic fibers and the use of secondary modes. The difference in bending and membrane behavior deformation of the damping structures was found to be important for the accurate extrapolation of damping measured on samples to damping that can be achieved in an actual structure.