Piecewise-smooth nonlinear dynamics system caused by dryfriction is becoming hot problems in mechanics with the development ofscience and technology. The study of nonlinear dynamics including dryfriction systems has made many progresses. Because of the complexity ofequations, many researches were based on phase-plane orbit analysis andnumerical analysis and experimental research. In this paper, a mathematicalmodel of self-excited vibration caused by dry friction between two elasticstructures was established using the Chinese cultural relic dragon washbasinas an example. An approximate analytical solution of the piecewise-smoothnonlinear dynamics systems of multi-degrees-of-freedom induced by dryfriction was derived by means of averaging method. According to theapproximate analytical solution, the curves of relation between swing andrubbing velocity of hands, the relation between swing and natural frequencyof hands and the relation between phase angle and rubbing velocity of handswere obtained. The vibration mechanism of the water droplets spurtingphenomenon of the Chinese cultural relic dragon washbasin is furtherexplained. The results not only enhanced the precision but also explainedqualitatively the whole kinematic process. If the parameters of the systemin the design were changed, the design could be optimized according to therelated curves, which supplied the theoretical basis foridentifying parameter and analysis and research of steady region of thiskind of nonlinear vibration systems. Furthermore, the results are inexcellentagreement with that of the numerical solution, so that an efficient andcredible analytical method to investigate piecewise-smooth nonlinear systemsof multi-degrees-of-freedom was given in this paper.