Analysis of a linear one-dimensional active noise control system by means of block diagrams and transfer functions

In arriving at the ideal filter transfer function for an active noise control system in a duct, the effect of the auxiliary sources (generally loudspeakers) on the waves generated by the primary source has invariably been neglected in the existing literature, implying a rigid wall or infinite impedance. The present paper presents a fairly general analysis of a linear one-dimensional noise control system by means of block diagrams and transfer functions. It takes into account the passive as well as active role of a terminal primary source, wall-mounted auxiliary source, open duct radiation impedance, and the effects of mean flow and damping. It is proved that the pressure generated by a source against a load impedance can be looked upon as a sum of two pressure waves, one generated by the source against an anechoic termination and the other by reflecting the rearward wave (incident on the source) off the passive source impedance. Application of this concept is illustrated for both the types of sources. A concise closed-form expression for the ideal filter transfer function is thus derived and discussed. Finally, the dynamics of an adaptive noise control system is discussed briefly, relating its standing-wave variables and transfer functions with those of the progressive-wave model presented here.

Dynamics of Cricket Sound Production

The clever designs of natural transducers are a great source of inspiration for man-made systems. At small length scales, there are many transducers in nature that we are now beginning to understand and learn from. Here, we present an example of such a transducer that is used by field crickets to produce their characteristic song. This transducer uses two distinct components-a file of discrete teeth and a plectrum that engages intermittently to produce a series of impulses forming the loading, and an approximately triangular membrane, called the harp, that acts as a resonator and vibrates in response to the impulse-train loading. The file-and-plectrum act as a frequency multiplier taking the low wing beat frequency as the input and converting it into an impulse-train of sufficiently high frequency close to the resonant frequency of the harp. The forced vibration response results in beats producing the characteristic sound of the cricket song. With careful measurements of the harp geometry and experimental measurements of its mechanical properties (Young's modulus determined from nanoindentation tests), we construct a finite element (FE) model of the harp and carry out modal analysis to determine its natural frequency. We fine tune the model with appropriate elastic boundary conditions to match the natural frequency of the harp of a particular species-Gryllus bimaculatus. We model impulsive loading based on a loading scheme reported in literature and predict the transient response of the harp. We show that the harp indeed produces beats and its frequency content matches closely that of the recorded song. Subsequently, we use our FE model to show that the natural design is quite robust to perturbations in the file. The characteristic song frequency produced is unaffected by variations in the spacing of file-teeth and even by larger gaps. Based on the understanding of how this natural transducer works, one can design and fabricate efficient microscale acoustic devices such as microelectromechanical systems (MEMS) loudspeakers.