Abstract: Utilizing Sylvester's theorem and Fourier seriesapproximations, commonly used in the matrix decomposition, the spectralcomponents for both the free and forced responses of linear parametricallyexcited system are obtained theoretically, and the effects of parametricstability and damping on the spectral amplitudes are discussed in detail.Consequently, the outer excitation resonant condition for parametric systemis found. It is concluded from the theoretical results as follows:parametric excitation leads to multi-frequency response; these spectra areclosely related to the natural frequency, parametric frequency and the outerexcitation frequency (only for the forced response) of the system; thedistribution for these spectra in frequency domain follow certain laws;there are multi-resonances caused by outer excitation in parametricvibration system.