The inverse eigenvalue problem of constructing symmetric positive semidefinite matrix 𝐷 (written as 𝐷≥0) and real-valued skew-symmetric matrix 𝐺 (i.e., 𝐺𝑇=−𝐺) of order 𝑛 for the quadratic pencil 𝑄(𝜆)∶=𝜆2𝑀𝑎+𝜆(𝐷+𝐺)+𝐾𝑎, where 𝑀𝑎>0, 𝐾𝑎≥0 are given analytical mass and stiffness matrices, so that 𝑄(𝜆) has a prescribed subset of eigenvalues and eigenvectors, is considered. Necessary and sufficient conditions under which this quadratic inverse eigenvalue problem is solvable are specified.