||Although many theories have been proposed in order to determine the effective linear elastodynamic properties of composite media, a complete theory in the case with resonance included is yet to be found. A new approach of linear elastodynamic homogenization based on requirement of measurement constrain and separation of scales is developed. This theory exploits the phenomenon of identical surface response properties of two different structured materials with one of them as homogeneous under certain constrained measurement (such as limited resolution or narrow band excitation frequency), and the homogeneous one contains certain large characteristic length as separated scale. The homogeneous one has been defined as the reconstructed homogenized correspondence from procedure of homogenization, and its properties are just the effective properties pursued. This definition is naturally valid for the local resonance conditions, and with aid of components expression of the Green function in its eigenstates representation, the effect of resonance is clearly provided. Therefore, formal expressions of effective parameters such as the mass density, the elastic modulus and thus the dispersion relationship could be obtained as functions of materials eigenstates and eigenfrequencies. A one dimensional problem of weighted elastic string has been investigated extensively and the existence of negative mass density, negative modulus, evanescent wave and negative refraction have been confirmed. Transmission problem has also been investigated to make a comparison with experiments, and three examples of 3 layers coated spheres, nonuniform membrane and Helmholtz resonator which were reported in previous literatures have been calculated with results well agreed with experiment.