The localization factor, which was first developed in solid state physics [1][2][3][4][5][6], has often been used to quantify vibration localization in mono-coupled nearly periodic structures [7][8][9][10][11][12]. The localization factor is the average exponential decay rate of the vibration amplitudes, measured from one substructure to the next. For multi-coupled structures, a traditional localization factor cannot be found, but one may compute the Lyapunov exponents of the system wave transfer matrix [4,8]. The Lyapunov exponents are analogous to the localization factor, since they provide a measure of the spatial amplitude decay rates for the multiple wave types. In fact, for the mono-coupled case, the largest Lyapunov exponent is equivalent to the localization factor [13,14]. Lyapunov exponents may be calculated numerically using an efficient algorithm developed by Wolf et al. [15].
Spatial wave decay in a nearly periodic structure, however, is not necessarily due to disorder. The wave decay can be caused by other mechanisms, such as off-resonance or damping. We are thus posed with the problem that, although we can quantify the decay rate, we do not know if this value truly indicates localization. Indeed, a localization factor or Lyapunov exponent is sometimes greatest in frequency regions where there is, in fact, little localization. The crucial distinction between localization due to disorder and wave decay caused by other mechanisms is that localization is a confinement, rather than a dissipation or attenuation, of the vibration energy. This energy confinement can lead to higher response amplitudes in a disordered system than would be found in its ordered counterpart. It is therefore of practical interest to identify the frequency regions in which localization is the primary source of decay.
In reference [16], Lyapunov exponents were shown to provide a valuable tool for analyzing wave decay in multi-coupled systems. Here, the work of reference [16] is extended: we propose that localization may be predicted by calculating the statistics of the Lyapunov exponents. Since a numerical Lyapunov exponent is actually an average of the values computed at each iteration of the algorithm, by considering only the mean we have discarded valuable information. The standard deviation of the iterate values can be used as well. If the decay is due mostly to damping or off-resonance, then each iterate value will be approximately the same, and the standard deviation will be small relative to the mean. If the decay is due to disorder, then the iterate values will vary significantly, and the larger standard deviation indicates localization. Thus, by considering both the mean and standard deviation, we can systematically identify frequency ranges where localization occurs.
Very few studies have examined the statistics of Lyapunov exponents. Cha and Morganti [17] investigated the mean, and probability density of the localization factor for the mono-coupled system considered here. However, they used this information to make correct inferences about the rate of exponential decay of a typical system, while we use the standard deviation for a different purpose. Cusumano and Lin [18] calculated the 0022-460X/97/210151