This paper introduces a general and flexible design method for the inverse modal optimization of undamped vibrating systems, i.e., for the computation of mass and stiffness linear modifications ensuring the desired system eigenstructure. The technique is suitable for the design of new systems or the optimization of the existing ones and can handle several design requirements and constraints. Paramount strengths of the method are its capability to modify an arbitrary number of parameters and assigned vibration modes, as well as the possibility of dealing with mass and stiffness matrices with arbitrary topologies. To this purpose, the modification problem is formulated as a constrained inverse eigenvalue problem and then solved within the frame of convex optimization. The effectiveness of the method is assessed by applying it to two different test cases. In particular, the second investigation deals with a meaningful mechanical design application: the optimization of a system recalling an industrial vibratory feeder. The results highlight the generality of the method and its capability to ensure the achievement of the prescribed eigenstructure.

The design of vibratory feeders ensuring effective product feeding is particularly demanding since the flexibility of the device components severely affects the dynamic response. Traditional design approaches are based on expensive and time-consuming trial and error iterations, which include building and testing tentative devices. In order to overcome these limitations, this paper proposes a novel systematic approach for designing linear vibratory feeders, based on an inverse dynamic structural modification approach. After synthesising a reduced-order model, the design is cast as an inverse eigenvalue problem and solved numerically through the minimization of a convex quadratic function. Constraints on the design variables, i.e. inertial and elastic parameters, are represented through a convex domain and included in the optimization problem. The numerical and experimental results obtained prove the effectiveness of the method and its ease of application, that make is suitable for both the design of new feeders and the optimization of existing one

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