This paper introduces a metric called degree of shapeability to determine whether or not it is possible to assign a set of desired eigenstates to a particular linear dynamic system. Specifying an eigenstate, which consists of an eigenvalue and its associated eigenvector, can be essential in designing certain vibrating structures and undulating mechanisms. In general it may be desirable to specify more than one eigenstate. The number of eigenstates that can be specified for a particular linear system is not always immediately evident, however, because of physical and kinematic constraints, as well as actuator placement. To quantify the potential for active feedback and/or passive parameter tuning to influence a system’s eigenstates, we identify a metric, the degree of shapeability, that relates the structure of the linear system to the number of eigenstates that a designer may fully define. We investigate the effect of system equality and inequality constraints on the ability to specify eigenstates for linear dynamic systems.

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