en Optimal design Optimal design پژوهشي Research The mass matrix formulation is very important to achieve a high-convergent model in structural dynamics. This study calculates the optimum mass matrix for in-plane free vibrations of the plane problems. In fact, the parameterized mass and stiffness for a rectangular element are formulated by the template approach. By using perturbation theory and sensitivity analysis, the changes of the natural frequencies are obtained as a function of the free parameter variations. Based on the natural frequencies, the objective function is established. Through an optimization process, the optimum values for template-free parameters are determined. Findings are used to calculate the plane problems’ natural frequencies. Some structural analyses and comparative studies with the other schemes are performed. Base on the obtained results, the efficiencies and high-convergence properties of the optimal element are demonstrated by numerical examples. eigenvalue, optimum mass, perturbation, plane vibration, sensitivity, template 249 269 http://ijoce.iust.ac.ir/browse.php?a_code=A-10-66-301&slc_lang=en&sid=1 M. Rezaiee Pajand 180031947532846001736 180031947532846001736 Yes N. Baghiee 180031947532846001737 180031947532846001737 No