Relations between various values of the internal friction (tgdelta, Q(-1), Q(-1*), and Lambda/pi) measured by free decay and forced vibration are analyzed systemically based on a fundamental mechanical model in this paper. Additionally, relations between various natural frequencies, such as vibration frequency of free decay omega(FD), displacement-resonant frequency of forced vibration omega(d), and velocity-resonant frequency of forced vibration omega(0) are calculated. Moreover, measurement of natural frequencies of a copper specimen of 99.9% purity has been made to demonstrate the relation between the measured natural frequencies of the system by forced vibration and free decay. These results are of importance for not only more accurate measurement of the elastic modulus of materials but also the data conversion between different internal friction measurements.