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.

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