Engineering Transactions,

**9**, 3, pp. 425-441, 1961### Zginanie, Stateczność i Drgania Prętów o Zmiennym Przekroju

This is a formally accurate solution of the differential equation (1.1) of vibration of a bar with variable flexural rigidity B(x), variable mass u(x), resting on a Winklerian foundation with variable foundation modulus K(x), compressed by a variable axial force N(x) and subjected to a load q(x) cos ωt.

The ends being simply supported, the deflection curve is assumed in the form of a Fourier sine series (1.2). All variable coefficients of the differential equation are expanded in Fourier cosine series. By means of two auxiliary equations (2.3) and (2.6), the left-hand side of Eq. (1.1) is reduced to a Fourier sine series. Equating the coefficients of this series 1O those of the expansion q(x) = […], we obtain an infinite system of equations (3.8), for the coefficients of the series (1.2). The system (3.8) is reduced to the canonical form by means of the auxiliary equation (2.7). The form of the system of equations (3.8) is very simple even in the most general case of variability of all the coefficients of the differential equation (1.1).

The solution obtained for a simply supported bar is generalized to the problem of bending, buckling and vibration of bars resting on an elastic foundation and having both ends free or one end free, the other being simply supported. It is not difficult to introduce concentrated masses or elastic point supports. The paper is illustrated by a number of numerical examples, each result being confronted with another solution.

The ends being simply supported, the deflection curve is assumed in the form of a Fourier sine series (1.2). All variable coefficients of the differential equation are expanded in Fourier cosine series. By means of two auxiliary equations (2.3) and (2.6), the left-hand side of Eq. (1.1) is reduced to a Fourier sine series. Equating the coefficients of this series 1O those of the expansion q(x) = […], we obtain an infinite system of equations (3.8), for the coefficients of the series (1.2). The system (3.8) is reduced to the canonical form by means of the auxiliary equation (2.7). The form of the system of equations (3.8) is very simple even in the most general case of variability of all the coefficients of the differential equation (1.1).

The solution obtained for a simply supported bar is generalized to the problem of bending, buckling and vibration of bars resting on an elastic foundation and having both ends free or one end free, the other being simply supported. It is not difficult to introduce concentrated masses or elastic point supports. The paper is illustrated by a number of numerical examples, each result being confronted with another solution.

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Copyright © Polish Academy of Sciences & Institute of Fundamental Technological Research (IPPT PAN).

#### References

S. TIMOSHENKO, Elastic Stability, 1936.

S. TIMOSHENKO, Strength of Materials, Part 2, 1941.

A. KACNER, Bending of Plates with Variable Thickness, Arch. Mech. Stos., 3, 13(1961), 393-417.