Volterra series

The damage detection problem in mechanical systems, using vibration measurements, is commonly called Structural Health Monitoring (SHM). Many tools are able to detect damages by changes in the vibration pattern, mainly, when damages induce nonlinear behavior. However, a more difficult problem is to detect structural variation associated with damage, when the mechanical system has nonlinear behavior even in the reference condition. In these cases, more sophisticated methods are required to detect if the changes in the response are based on some structural variation or changes in the vibration regime, because both can generate nonlinearities. Among the many ways to solve this problem, the use of the Volterra series has several favorable points, because they are a generalization of the linear convolution, allowing the separation of linear and nonlinear contributions by input filtering through the Volterra kernels. On the other hand, the presence of uncertainties in mechanical systems, due to noise, geometric imperfections, manufacturing irregularities, environmental conditions, and others, can also change the responses, becoming more difficult the damage detection procedure. An approach based on a stochastic version of Volterra series is proposed to be used in the detection of a breathing crack in a beam vibrating in a nonlinear regime of motion, even in reference condition (without crack). The system uncertainties are simulated by the variation imposed in the linear stiffness and damping coefficient. The results show, that the nonlinear analysis done, considering the high order Volterra kernels, allows the approach to detect the crack with a small propagation and probability confidence, even in the presence of uncertainties.

The emulation of nonlinearities of audio devices can be achieved by means of a nonlinear convolution method, which is based on a particular case of the Volterra series, called the Diagonal Volterra series. The Volterra kernels characterize the
nonlinear audio device being tested, dependent upon the level of the signal that passes through the device. In this paper, a method is presented that approximates Volterra kernels in a "continuous" range of levels by means of an interpolation procedure of an achieved number of measurements. In order to obtain the best emulation for real signals, several parameters were tested. An extension of the
previous Diagonal Volterra model is also developed to an arbitrary order of nonlinearities in order to allow better emulation of harmonics of low and medium frequencies; a particular case for the experiments is presented.