On the Zeros, Initial Undershoot, and Relative Degree of Collinear Lumped-Parameter Structures

This paper considers collinear lumped-parameter structures where each mass in the structure has a single degree of freedom. Specifically, we analyze the zeros and relative degree of the single-input, single-output (SISO) transfer function from the force applied to an arbitrary mass to the position, velocity, or acceleration of another mass. In particular, we show that every SISO force-to-motion transfer function of a collinear lumped-parameter structure has no positive (real open-right-half-plane) zeros. In addition, every SISO force-to-position transfer function of a spring-connected collinear lumped-parameter structure has no non-negative (real closed-right-half-plane) zeros. As a consequence, the step response does not exhibit initial undershoot. In addition, we derive an expression for the relative degree of SISO force-to-position transfer functions. The formula depends on the placement of springs and dashpots, but is independent of the values of the spring constants and damping coefficients. Next, we consider the special case of serially connected collinear lumped-parameter structures. In this case, we show that every SISO force-to-position transfer function of a serially connected collinear lumped-parameter structure is minimum phase, that is, has no closed-right-half-plane zeros. The proofs of these results rely heavily on graph-theoretic techniques.