The smoothed finite element method(S-FEM)has been recently developed as an effective solver for solid mechanics problems.The S-FEM uses different smoothed domains(and hence strain fields)and can have various models.The smoothing domains can be cell-based(CS-FEM),edge-based(ES-FEM)and node-based(NS-FEM),and these S-FEM models have useful properties.This paper represents a unique approach to compute the lower bounds of eigenvalues of elasto-dynamic problems,making use of the important softening effects of the node-based smoothed finite element method(NS-FEM).We first use the NS-FEM,standard FEM and the analytic approach to compute the eigenvalues of transverse free vibration in a stationary rectangular membrane.It is found that eigenvalues by NS-FEM is always smaller than those by FEM and the analytic method.However,NS-FEM produces spurious unphysical modes simultaneously,due to its overly soft behavior.A technique is then proposed to remove these modes by analyzing vibration modes of the eigenvectors.It is observed that they have excessively large wave numbers that are not related to the waves in the rectangular membrane,but to the spatial discretization,and therefore they can be removed.The final results of NS-FEM are the correct and lower bounds.The proposed NS-FEM procedure offers an available practical computational means to effectively compute the lower bounds of eigenvalues in solid mechanics problems for the first time.