Vibration monitoring, notably in the fields of civil, mechanical and aeronautical engineering, aims at detecting damages at an early stage, in general by using output-only vibration measurements under ambient excitation. In this paper, a new method is proposed for the detection of small changes in the eigenstructure of such systems. The main idea is to transform the multiplicative eigenstructure change detection problem to an additive one, by means of perturbation analysis based on the assumption of small eigenstructure changes. Another transformation then further simplifies the detection problem into the framework of a linear regression subject to additive white Gaussian noises, leading to a numerically efficient solution of the considered problem. Compared to existing methods, it has the advantages of focusing on chosen system parameters and efficiently addressing random uncertainties. A numerical example of a simulated mechanical structure and a lab experiment on a beam, each with the detection of different damages, are reported.