Mechanical bandpass filter design with lumped element prototypes

A method is presented which provides means for the design of mechanical filters by the comparison of whole "pi" sections of the filter with the corresponding sections of a simple electric prototype filter. This replaces the well-known equivalences for each filter component and reduces the accumulation of eventual comparison errors. The results are the computation of broader passbands with more accuracy and more precise narrow-band filter design. This is achieved by first splitting the shunt arms of a mechanical filter ladder in two one-ports, providing a succession of symmetrical "pi" sections, formed by two half-resonators and a coupling element in between. The calculated chain matrices of these sections which are established including more or less mechanical phenomenons of the elements, compatible with the desired approximation and bandwidth. Second, a low-pass filter is synthesized with the imposed filter characteristics, using well-known insertion loss theory. This filter is put into a form which results in a succession of symmetrical "pi" sections as well. The elements of the chain matrices of the mechanical filter sections are compared with those of the transformed electrical prototype; the comparison of the matrix coefficients yield a frequency mapping function, applicable in the same way as in the design of electrical broadband filters. The degree of approximation for each bandwidth is immediately found by comparison of the plotted matrix coefficients. Extensional, torsional, and flexural vibrations are dealt with and some commonly used combinations are calculated with their basic matrices for demonstration purposes.