Transcendental Characteristic Equation and Finite Element Verification of Disk–Frustum Membrane Vibrations

Abstract This work presents an exact analytical–numerical investigation of the small-amplitude, in-plane tension vibrations of a composite membrane formed by a flat, simply supported circular disk seamlessly bonded to a conical frustum. Neglecting bending stiffness reduces the problem to two coupled Sturm–Liouville systems: integer-order Bessel functions describe the disk, while fractional-order Bessel combinations describe the frustum. Enforcing continuity of transverse displacement and radial membrane force at the disk–cone junction produces a compact transcendental determinant whose real roots define the natural frequencies. The determinant is solved with the robust Brent bracketing algorithm, guaranteeing convergence without derivative evaluations. Mode shapes are reconstructed from the closed-form basis functions. Validation is provided by a high-resolution finite element model, employing second-order membrane elements and adaptive mesh refinement. For a fixed geometry, the analytical solutions and finite element model results are nearly one-to-one. The resulting characteristic equation, root-finding framework, and design charts furnish fast, reliable tools for engineers and applied mathematicians working on lightweight acoustic diaphragms, inflatable reflectors, and micro-pressure sensors—extending classical shell vibration theory to its pure-membrane counterpart with rigorous analytical–numerical agreement. In such systems, the disk–frustum configuration offers a compact means of tailoring stiffness and sensitivity through geometry rather than material change, allowing designers to tune resonance frequencies and suppress unwanted modes, or enhance compliance for acoustic response.