Lekhnitskiiʹs classic formula serving as an exact mode shape of simply supported polar orthotropic inhomogeneous circular plates

The semi-inverse problem of free vibration of simply supported inhomogeneous polar orthotropic plates is studied. The mode shape is postulated in the form of the classic formula by Lekhnitskii, namely, the static deflection of the associated uniform polar orthotropic circular plate under uniform loading. The ratios of the circumferential rigidity to the radial rigidity are identified as integer numbers, so that the candidate mode shapes constitute the polynomial functions. The flexural rigidities themselves are also represented by polynomials. The semi-inverse problem of identifying the coefficients of the flexural rigidities is solved analytically. It appears remarkable, that for numerous cases the simple method developed in this study provides novel closed form solutions for the design of the polar orthotropic circular plates with pre-specified mode shapes.