- the lower then density, the higher the natural vibration frequency.

Using the foam as the core of a sandwich panel is even more effective because the flexural stiffness, at constant mass, rises even faster as the density of the core is reduced.

Material damping

All materials dissipate some energy during cyclic deformation, through intrinsic material damping and hysteresis. Damping becomes important when a component is subject to input excitation at or near its resonant frequencies.

There are several ways to characterize material damping. Here we use the loss coefficient rç which is a dimensionless number, defined in terms of energy dissipation as follows. If a material is loaded elastically to a stress amax (see

Figure 12.3) it stores elastic strain energy per unit volume, U; in a complete loading cycle it dissipates AU, shaded in Figure 12.3, where r<ymax 1 a 2 r

Figure 12.3 The loss coefficient 77 measures the fractional energy dissipated in a stress-strain cycle

The loss coefficient 77 is the energy loss per radian divided by the maximum elastic strain energy (or the total vibrational energy):

In general the value of 7 depends on the frequency of cycling, the temperature and the amplitude of the applied stress or strain.

Other measures of damping include the proportional energy loss per cycle D = AU/U, the damping ratio £, the logarithmic decrement U, the loss angle ^, and the quality factor Q. When damping is small (7 < 0.01) and the system is excited near to resonance, these measures are related by

They are no longer equivalent when damping is large. Broad-band inputs

If the dominant driving frequency is equal to then from equation (12.6) 1

and the response is minimized by maximizing the material index:

More generally, the input x is described by a mean square (power) spectral density:

where S0, &>0 and k are constants, and k typically has a value greater than 2. It can be shown (Cebon and Ashby, 1994) that the material index to be maximized in order to minimise the response to x is

The selection to maximize Md can be performed by plotting a materials selection chart with log(v) on the x-axis and log(Mw) on the y-axis, as shown in Figure 12.4. The selection lines have slope 1/(1 — k). The materials which lie farthest above a selection line are the best choice.

Loss coefficient log n

Figure 12.4 Schematic diagram of a materials selection chart for minimizing the RMS displacement of a component subject to an input with spectral density S0 (m/m0 )—k

Loss coefficient log n

Figure 12.4 Schematic diagram of a materials selection chart for minimizing the RMS displacement of a component subject to an input with spectral density S0 (m/m0 )—k

If k d 0, the spectrum of input displacement is flat, corresponding to a 'white noise' input, but this is unrealistic because it implies infinite power input to the system. If k D 2, the spectrum of the input velocity is flat (or white), which just gives finite power; for this case the selection line on Figure 12.4 has a slope of —1. For larger values of k, the input becomes more concentrated at low frequencies, and the selection line is less steep. If k = 1, the selection line becomes horizontal and the selection task becomes one of choosing materials with the highest value of &>i, exactly as for the undamped case.

Figure 12.5 shows data for metal foams. Metal foam panels and sandwich panels with metfoam cores have attractive values of Mfd, because of their high flexural stiffness and their relatively high damping capacity. The Alporas range of foams offers particularly good performance. For comparison, aluminum alloys have a values of Mu in the range 1.6-1.8 and damping coefficients, 77, in the range 10—4 — 2 x 10—3 in the same units as those of the figure.

Material index Mu vs. Damping coefficient

Vibes 1 CMS, MFA 14-4-98; Metfoam DB

Fraunhofer (0.551)""^ Alulight (0.95)

Alulight (0.

Duocel (0.235)

' Duocel

' Duocel

Material index Md - n


Figure 12.5 A selection chart for vibration management. The axes are the index. Mu = E1 /3 /p and the damping coefficient Md = ^. All the materials shown are metal foams, and all have better performance, measured by the index Md than the solid metals from which they are made

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