On the equivalent mass-spring parameters and assumed mode of a cantilevered beam with a tip mass

In the Rayleigh-Ritz approach to the mathematical model of a cantilevered beam with a tip mass, the proper selection of basis functions is critical in representing the original system by an equivalent mass-spring system. Although the fundamental bending mode shape of a beam varies for a different tip mass magnitude, the numerical values of 33/140 and 3 have been conventionally employed as those of the normalized dimensionless equivalent mass and spring constants, respectively, which correspondingly yield errors in its calculated natural frequencies. This work firstly proposes a method to evaluate more accurate values of the equivalent mass and spring for a wide range of the tip mass-to-beam mass ratio by direct use of a fundamental mode, and then proposes a new basis function as a linear combination of two polynomials, which represent static deflection shapes of a beam under a tip force and a uniformly distributed force, respectively, yielding natural frequencies fairly close to those by the continuous beam equation.