Abstract: Based on the boundary conditions of transversevibration of non-uniform beams, second order inexplicit-recursivedifferential equations with variable coefficients about deflection andbending moment are obtained from the fourth order differential vibrationequation with variable coefficients by the method of order reduction. Bythe methodof finite difference, the numerical calculation method and its precision for the natural frequency of the transverse vibration of the simply supportednon-uniform beams are studied. Examples of theoretical analysis and orthogonalcalculation show that the numerical calculation algorithm is very simple, andits precision depends on the number of calculation steps and thevertical variationrates of the gradually changed cross-section and is independent of width andlength of beams; the precision of the numerical calculation can be estimatedaccording to a given length or the number of calculation steps and thereasonable length or the number of calculation steps can be determinedby a given precision requirement.