A generalization of the membrane-plate analogy to non-homogeneous polygonal domains consisting of homogeneous subdomains

Abstract

The well-known membrane-plate analogy that relates the natural frequencies when dealing with polygonal homogeneous domains is herein extended to non-homogeneous systems comprised of homogeneous subdomains. The analogy is generalized and demonstrated and it is shown that certain restrictions among the frequency parameters of the membranes and plates arise. Several examples of membranes and plates with interfaces separating areas with different material properties are numerically solved with different approaches. The subdomains are separated by straight, curved, and closed line interfaces. It is shown that the analogy is verified provided that the restrictions are satisfied. The analogy is first demonstrated and presented as a practical methodology to find the natural frequencies of membranes knowing the corresponding ones of the plates or vice versa. Second, the plate and membrane vibration problems, governed by the bi-Laplacian and Laplacian differential operators, respectively, can be solved without distinction, though under certain conditions, i.e., solve one of them and deduce the other using the analogy. Various numerical examples validate the analogy.