Page 30 - Fall 2006

Fig. 13. Typical frequency-temperature dependencies of viscoelastic material properties.
viscoelastomer’s elastic modulus varies strongly with fre- quency and temperature, and where the slope of its modulus is high, it becomes extremely lossy, with loss factors near 1.
The equivalence between the effects of increasing fre- quency and decreasing temperature in a viscoelastomer is called the time-temperature superposition principle, and labo- ratories use this principle to measure the frequency and tem- perature varying dynamic properties of rubbers. Most dynam- ic property characterization tests measure a rubber’s vibration response at a limited set of frequencies, but over a wide range of cold and hot temperatures. The data are shifted in frequen- cy to a common temperature using the time-temperature superposition principle to form so-called ‘master’ curves of elastic moduli and loss factors. Fortunately, many rubber man- ufacturers offer these master curves to their customers (some are available on the internet), so that materials well suited to a given application may be chosen easily.
Prospective CLD users should be cautioned though—it is not uncommon for different testing labs to produce differ- ent master curves of the dynamic properties of the same vis- coelastomer. In my experience, elastic moduli measured by different labs and testing procedures can easily vary by a fac- tor of two. Loss factors, though, are easier to measure, and should be accurate to within 10-20%. The uncertainty in elas- tic modulus, though, can lead to comparable uncertainties in the damping provided to a CLD-treated structure. Other techniques are available to more accurately characterize the master curves,22 but they are quite cumbersome, and best suited to applications where high precision is required.
Bending waves in infinite structures
It is sometimes hard to convince acoustics students that understanding how waves behave in infinite structures is worthwhile. However, the simple formulas that describe the mobilities of infinite structures are tremendously useful, since they represent the mean vibration response of compli- cated, finite structures to force and moment drives. Figure 14 shows an example of our measured glass plate mobilities, along with the drive point mobility of an infinite glass plate, computed using the well-known formula for bending waves in infinite flat plates:1
. (23)
The infinite plate mobility Yinf is purely real (the general finite plate mobility is complex, although we have only looked at plots of the magnitude of mobility), and for the glass plate clearly approximates the mean mobility.
As structural damping increases, the peaks in a finite structure’s mobility become less sharp, until at high damping values and high frequencies, the mobility becomes nearly real, and approaches that of an infinite plate. To physically understand why this is so, consider the waves traveling through a damped plate which are induced by a point drive. As the plate absorbs energy from the vibrations, the wave amplitude decreases as it travels away from the drive. Eventually, the diminished wave strikes a finite boundary, and reflects back toward the source. However, as it travels back, it continues to lose energy to structural damping. The
mobility decreases with increasing loss factor.
The structural damping in the example above is caused
by embedding thin sheets of viscoelastic material (rubber) between two plates of glass, creating a sandwich structure. The technique, called constrained layer damping (CLD) is well known and often applied by noise control engineers con- fronted with vibration problems. As the top and bottom structures (which are much stiffer than the viscoelastic mate- rial) bend, they strain the rubber, which has a high loss fac- tor, and energy is dissipated. Curiously, thinner sheets of vis- coelastomer usually lead to higher loss factors than thick sheets. This is counter-intuitive until the theory behind CLD is well understood. Ungar21 provides a retrospective on the history of CLD which references many early papers describ- ing its theory and examples of how it has been applied.
The number of viscoelastic materials available to noise and vibration control engineers is daunting, and choosing the right material for a given application requires great care. This is because the elastic moduli and loss factors of all viscoelas- tomers depend strongly on both frequency and temperature. This is, in fact, why so many different materials are available, as manufacturers tailor their different rubbers to work well in various conditions (hot, warm, cold) and over many frequen- cy ranges.
Figure 13 shows a typical shear modulus and loss factor curve. At low frequencies (or high temperatures), a viscoelas- tomer is flimsy and rubbery, and not very lossy. At very high frequencies (or very low temperatures), a viscoelastomer is very stiff, or glass-like, and again not very lossy. Within its range of usefulness—mid frequencies and temperatures—a
Fig. 14. Magnitudes of drive point mobilities of 5 mm thick 0.304 m x 0.304 m glass plates with light and heavy damping, compared to infinite plate mobility.
28 Acoustics Today, October 2006