On use of the Dirac delta function in the vibration analysis of elastic structures

The Dirac delta function has often been employed to represent the amplitude of concentrated harmonic forces in the analysis of vibration of elastic structures such as beams and plates. It is known that this function, as represented by a truncated Fourier series, does not provide a true representation of a concentrated force, nevertheless, it is frequently employed and good convergence is usually, though not always, encountered in solutions thereby obtained. In this paper, the nature of the function is discussed and for illustrative purposes it is used to obtain series solutions for some selected beam and plate free vibration problems. In some cases problems are chosen for which exact solutions are already obtainable by analytical means. This permits powerful checks to be made on rates of convergence experienced when the series solutions are investigated. Rates of convergence are discussed in detail and it is explained why convergence is to be expected when analyzing certain families of problems when employing this function and a lack of convergence is to be expected in others.