Transverse vibration of an uniform Euler–Bernoulli beam under linearly varying axial force

This paper is concerned with the transverse vibration of uniform Euler–Bernoulli beams under linearly varying fully tensile, partly tensile or fully compressive axial force distribution. The system parameters are the constant part of the axial force and the constant of proportionality of the varying part. The mode shape differential equation is linear with variable coefficients. The general solution is derived and expressed as the super-position of four independent power series solution functions. The frequency equations of the 16 combinations of classical boundary conditions are listed. The first three frequency parameters are tabulated for example combinations of the system parameters. Increase in the values of one or both of the system parameters ‘stiffens’ the system and results in increase in the frequency parameter. If one or both system parameters are negative, combinations exist for which a frequency parameter is zero. This is the Euler buckling condition i.e., onset of dynamic instability. Example combinations of the system parameters when buckling occurs are tabulated. The results tabulated may be used to judge frequencies, buckling parameter combinations obtained by other methods.