In this study, dynamic behaviors of planar mechanisms with elastic linkages are investigated. For this purpose, slider-crank mechanisms which are widely used in many fields of industry are chosen. Flexible coupler of the mechanism is considered as a pin jointed beam under the effect of elastic oscillations in transverse direction. Euler-Bernoulli beam theory is considered to obtain dynamic responses of the elastic link. Lumped parameters approach is used to model the flexible links. Since, the assumption of small deflections is made, linear and continuous form of the elastic curve equation is written for each lumped masses on the beam to derive the equations of motion of the system. Derived set of nonlinear partial differential equations are reduced to ordinary differential equations by applying finite difference method. Finally, a symbolic mathematical program which gives the dynamic responses of the system is developed to solve the equations of motion. The results obtained from the developed program are tested and verified by the results available in the literature. Elastic deflection results are obtained for different parameters such as mass ratio and length ratio of the links of the mechanisms. The effects of the aforementioned parameters on dynamic response are found and presented in graphical forms.