The partial quadratic eigenvalue assignment problem (PQEVAP) is to shift a few undesired eigenvalues of a damped vibrating system to suitably chosen locations, while leaving the remaining eigenvalues and corresponding eigenvectors unchanged. In this paper, an algorithm for solving PQEVAPs and the minimum norm PQEVAP (MNPQEVAP) using acceleration and velocity feedback is proposed. It is shown that solving the PQEVAP here is transformed into solving an eigenvalue assignment of a linear system of a much lower order. Furthermore, the MNPQEVAP here can be efficiently solved by a gradient-based unconstrained optimization method with the derived gradient formula. This algorithm works directly on the second-order system model, and requires the knowledge of only the open-loop eigenvalues to be replaced and their corresponding eigenvectors. Lastly, through two numerical examples, the results of solving the MNPQEVAP under two different combined feedback signals, velocity and displacement signals, and acceleration and velocity signals, are compared from two points of view, i.e. the F-norms of their feedback matrices and the active control energy required from the actuators.