  This paper investigates the circuit synthesis problem for a certain
positive-real bicubic (third-order) admittance with a simple pole at the origin
(s = 0) to be realizable as a one-port series-parallel damper-spring-inerter
circuit consisting of at most six elements, where the results can be directly
applied to the design and physical realization of inerter-based control
systems. Necessary and sufficient conditions for such a specific bicubic
admittance to be realizable by a one-port passive series-parallel mechanical
circuit containing at most six elements are derived, and a group of mechanical
circuit configurations covering the whole set of realizability conditions are
presented together with element value expressions. The conditions and element
value expressions are related to the admittance coefficients and the roots of
certain algebraic equations. The circuit synthesis results of this paper are
illustrated by several numerical examples including the control system design
of a train suspension system. Any realization circuit in this paper contains
much fewer passive elements than the ten-element realization circuit by the
well-known Bott-Duffin circuit synthesis approach. The investigations of this
paper can contribute to the theory of circuit synthesis and many other related
fields.
