The eigenvalues and the first and second-order eigenvalue sensitivities of a uniform Euler–Bernoulli beam supported by the standard linear solid model for viscoelastic solids are studied in detail. A method is proposed that yields the approximate eigenvalues and allows the formulation of a frequency equation that can be used to obtain approximate eigenvalue sensitivities. The eigenvalue sensitivities are further exploited to solve for the perturbed eigenvalues due to system modifications, using both a first- and second-order Taylor series expansion. The proposed method is easy to formulate, systematic to apply, and simple to code. Numerical experiments consisting of various beams supported by a single or multiple viscoelastic solids validated the proposed scheme and showed that the approximate eigenvalues and their sensitivities closely track the exact results.