Abstract: s: The complete analytical methods for random earthquake response and equivalent damping of arbitrary MDOF dissipation structures with supporting braces and general integral model viscoelastic dampers in series are studied systematically. The structural response equations are firstly established. Then, by using the modal analytical method, the dynamic integral-differential model earthquake response equations are established. And then, by using the MDOF stochastic averaging method, the structural stochastic averaging Itôequations are established, the analytical formulas of the drift vector and the diffusion matrix of averaging Itôequations are given, the analytical solutions of the transient joint probability density function for every structural modal amplitude and phase, for every structural modal displacement and velocity are derived, the analytical solutions of transient mean-square values of various structural modal displacements and velocities are established. By the equivalent criterion of which all random response characteristics are identical in the meaning of MDOF stochastic averaging analysis, the analytic formulas of every modal equivalent damping ratio are established, which make it possible for structures to use response spectrum technique. Lastly, by CQC and SRSS methods, the analytic formulas for the random earthquake response of foregoing MDOF dissipation structures are given, the analytical formulas of the foregoing random response characteristics and every structural modal equivalent damping ratio for MDOF dissipation structures with supporting braces and general Maxwell model dampers and general differential model dampers in series are given, it is verified that the proposed methods are effective through accuracy in comparison with the Modal Strain Energy method for some typical problems, so the complete solutions of random earthquake response characteristics and equivalent damping for arbitrary MDOF dissipation structures with supporting braces and arbitrary linear viscous and viscoelastic dampers in series are achieved.