Linear vibration problems involving harmonic excitation of discrete and continuous systems are solved by using the classical perturbation technique. The perturbation parameter is proportional to a mass and the square of the excitation frequency. The power series solution for the displacement of some point in the system is converted to the quotient of two polynomials by the use of continued fractions. The eigenvalues (natural frequencies) of the problem are calculated by finding the roots of the denominator polynomial. The situation wherein a quantity which cannot vanish at any frequency can be found is treated as a special case.

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