# Damping coefficients by experiments and the application to transient FE analyses of cable trays

Damping entails significant effects in transient analyses, and it is a mistake to ignore it to reach a conservative solution. It is also possible that the analysis delivers no meaningful results. For steel structures, a dimensionless damping coefficient of 1% of the critical damping is widely accepted; however, for structures consisting of several materials, damping coefficients may be higher and estimating them reliably is very important. Nevesbu has written a paper in which the case of damping in a cable pack supported by a steel tray is studied. On Wednesday, 12 June, our specialists Jack Reijmers and Alessandro Zambon will present this paper at the NAFEMS congress in Staffordshire, UK.

## Retrieving realistic damping coefficients

To retrieve realistic damping coefficients, free-vibration signals were acquired using a steel beam without and with cables attached to it. These registrations were made through a basic smartphone’s app, which uses the device’s own acceleration sensor. The logarithmic decrements calculated from the oscillation signals resulted in different dimensionless damping coefficients for different numbers of cables supported by the steel beam.

The calculated damping coefficient for the beam without cables resulted to be consistent with the usual 1% value. Five configurations were tested, ranging from zero to twenty cables attached to the beam. The resulting damping coefficients showed an increase from 1% (without cables) to 3% (with twenty cables). These test results were applied to an FE model of a ladder-type tray carrying cable packs. This configuration was excited with a shock pulse; then, the transient response was investigated for different cases from nearly zero damping up to a 3% level. With negligible damping, the analysis did not converge; with the damping coefficients found in the experiment, realistic numerical results were found instead.

Hence, it is shown that valuable information could be obtained through a simple experimental setup. Although widely accepted values for damping are available, structures with several material components can easily be tested, and the outcome gives a better input for transient FE analyses.

## Undamped transient analysis

The effect of damping is illustrated by the analysis of a cable tray subjected to a shock load. Figure 1(a) shows the model of the cable tray under consideration. This cable tray is supported by spring-damper elements, and one of the design requirements is the natural frequency of the resilient support. A low frequency value, characteristic for spring-mounted equipment, results in considerable motion amplitudes that require space around this equipment. To reduce the motion after a shock load a more rigid support is desired, and this is realized by a minimum natural frequency of 50 [Hz].

The mass attached to one support and the required frequency gives the stiffness to be applied. One spring element in Figure 1(a) carries a mass of 31.7 [kg] resulting in a spring stiffness of 3.129·106 [N/m]. For the damping coefficient an arbitrary value of 1 [N·s/m] is applied. The cable tray itself is undamped.

The shock pulse is presented in Figure 1(b) showing a loading time of 30 [ms].

*Figure 1a (left): FE model cable tray. Figure 1b (right): Shock pulse*

The displacement of the cable tray is shown in Figure 2. The expected free vibration after 30 [ms] shows an unstable character and at 81 [ms] the analysis stops with an excessive deflection. So far, the effect of damping is neglected. The cable tray has no structural damping, and the input for the spring-damper element has an arbitrary value of C=1.0 [N∙s/m], which is small enough to make the effects of damping on the system’s response negligible.

The critical damping determined through the mass and stiffness properties is given by:

Hence, this gives the following dimensionless damping coefficient:

With a general accepted value for steel structures of ξ≈0.01, it looks therefore clear that the response shown in Figure 2 is unstable as it does not converge.

*Figure 2: Response of the cable tray with nearly undamped setting*

## Realistic damping values

As stated above, for the steel structure a dimensionless damping coefficient of ξ≈0.01 is realistic. However, the cable pack will increase the overall damping. The effect of the added cables on the tray was demonstrated through a simple test setup, as shown in Figure 3.

*Figure 3: Test setup for the free vibration experiment*

A steel beam was fixed to the desk with clamps, and at its free end a smartphone was fastened. The app Physics Toolbox by Vieyra Software was installed and used to record oscillation signals (Android version). This app has the option to export the time-domain registration to Excel.

Recordings were made with: (i) the steel beam only; (ii) different bundles of cables attached to the beam. The experiments considered the beam with 2, 8, 14, and 20 cables. The oscillations were produced by imparting an initial vertical displacement to the beam from its equilibrium position. Figure 4 shows the registration of the configuration using 20 cables.

The motion signal in Figure 4 offers the possibility to determine the corresponding logarithmic decrement, and consequently the dimensionless damping coefficient.

These experiments produce different damping coefficients of the beam-cables systems as the number of cables attached to the mean varies.

*Figure 4: Registration of free vibration of the beam carrying 20 cables*

## Transient analysis with damping coefficient = 0.03

The damping coefficients determined from the experiments are applied in the transient analyses, and the result by using damping coefficient = 0.03 is shown in Figure 5. With a bundle of 20 cables, the displacement after the shock pulse (t = 30 [ms]) is negligible if compared to the motion during the loading.

*Figure 5: Damped transient analysis*

The experiment illustrated in Figure 4 may not be representative for an actual cable tray. However, it clearly indicates the increasing effect of damping as the number of cables increases. It can be concluded that the dimensionless character of the damping coefficient represents a significant variable influencing the outcomes of transient analyses for such structures.