In this work, the radial basis function approximations are used to improve the accuracy of meshfree Galerkin method. The method is applied to the free vibration problems of non-rotating and rotating Euler-Bernoulli beams. The stiffness and mass matrices are derived by using conventional methods. In this meshfree method, only six nodes are considered within a single sub-domain. The parameters are varied for different approximations; the results are obtained with different approximations and found accurate. Two new basis function have been developed which are relatively accurate than conventional basis function: the first new basis function is obtained by multiplication of linear function to radial basis function and second new basis function is obtained by multiplying cubuic radial basis function to Gaussian radial basis function. The first few modes show same result that is available in literature using finite element method and higher modes are found very accurate as well. The result are found to be more accurate for first three modes of non-rotating and rotating Euler-Bernoulli beams where the cantilever beam boundary conditions are used; the first three modes do not change with the change in the parameter c of radial basis function.