Constrained substructure approach to optimal strain energy analysis

The chief tool for design of viscoelastic-based damping treatments over the past 20 years has been the modal strain energy (MSS) approach. This approa ch to damping design traditionally has involved a practitioner to vary plac ement and stiffness of add-on elements using experience and trial and error so as to maximize the add-on element share of system MSE in modes of inter est. Ira this paper we develop a new technique for maximizing strain energy as a function of stiffness for add-on structural elements modeled as rank r perturbations to the original stiffness matrix. The technique is based on a constrained substructure approach allowing us to parameterize strain ene rgy in terms of the eigenvalues of the perturbed structure. An optimality c ondition is derived that relates the input-output response at the attachmen t location of the add-on elements to the maximum achievable strain energy. A realizability, condition is also derived which indicates whether or not t he optimal solution is achievable with passive structural elements. This me thod has applications in the design of structural treatments for controllin g sound and vibration and promises an efficient means of determining the li mits of performance of passive structural treatments. An advantage of our a pproach over existing methods is that the maximum achievable strain energy fraction in the add-on elements is directly computable with the realizabili ty condition then indicating whether the optimal solution is achievable.