C.P. Filipich

Vibrations of rings of variable cross section

Abstract The present paper deals with the determination of the lower natural frequencies of vibration of rings of variable cross-sectional area using three approximate schemes: • —using polynomial coordinate functions in the angular coordinate in order to approximate the fundamental mode shape • —expanding the ring response in terms of a sinusoidal truncated series • —by means of a finite element algorithm. When using the first two procedures the Ritz method is applied in order to obtain the frequency equation. In general very good agreement is obtained between the eigenvalues predicted by the three approaches.

Transverse vibrations of a stepped beam subject to an axial force and embedded in a non-homogeneous Winkler foundation

Abstract The title problem is tackled using two different approaches: the exact solution applicable in a rather limited amount of practical situations and an approximate methodology which is quite convenient from a practical viewpoint since it requires only approximating the deflection function by means of a polynomial co-ordinate function which contains an arbitrary exponential parameter n . Making use of the classical Rayleigh method one is able to optimize the frequency coefficients under investigation by minimizing them with respect to n .

A note on free flexural vibrations of a non-uniform elliptical ring in its plane

Abstract An approximate solution for the title problem is obtained by using the Rayleigh-Ritz method. Two different types of co-ordinate functions are used: sinusoids, and polynomial expressions which contain an undetermined exponential parameter which allows for minimization of the natural frequency coefficients. Continuous and discontinuous variation of the ring thickness are considered. The analytical predictions are compared with values obtained by means of the finite element method, and good engineering agreement is shown to exist. No claim of originality is made, but it is hoped that present results will be of interest to designers.

A note on vibrations of elliptical plates carrying concentric, concentrated masses

Abstract The present paper deals with the determination of the fundamental frequency of vibration of clamped and simply supported elliptical plates carrying concentric, concentrated masses. Numerical values of the fundamental frequency coefficient are presented as a function of the plate aspect ratio and of the dimensionless parameter concentrated mass/plate mass.

ANALYTICAL AND EXPERIMENTAL INVESTIGATION ON VIBRATING FRAMES CARRYING CONCENTRATED MASSES

Abstract Two solutions for the title problem are presented in this study: (i) an exact approach using the Bernouilli theory of vibrating beams; (ii) a finite element solution using classical beam elements. Excellent agreement is achieved for all cases considered. Experimental results are also presented using a clamped portal frame model specially designed during this investigation, and it is shown that the measured fundamental frequencies are in rather reasonable agreement with the analytical predictions.

In-plane vibrations of a circumferential arch elastically restrained against rotation at one end and with an intermediate support

Abstract The determination of the fundamental frequency of in-plane vibration of the structural system described in the title is tackled using three different approaches: (a) an optimized Rayleigh-Ritz method; (b) a modified Dunkerleys method; (c) the finite element formulation. The effect of a concentrated mass placed at the free end of the arch structure is also taken into account. It is concluded that, in view of the simplicity and accurate results provided by the first approach, one could use it in more complex situations where an exact solution seems out of the question (non-uniform cross section, presence of elastically mounted masses, etc.).

A lower bound for the smallest frequency determined by means of the Rayleigh-Ritz method

Abstract The present paper deals with the derivation of a simple expression which constitutes a valid lower bound with respect to the Rayleigh-Ritz lowest frequency in the case of a vibrating mechanical system. The expression is obtained by diagonalizing the strain energy matrix currently constructed when using the classical Rayleigh-Ritz method. The approach is first employed in a situation where the strain energy matrix is a priori diagonal due to the type of coupled displacement fields present in the system, and it is shown that the calculated lower bounds are acceptable from a practical viewpoint. The methodology is then applied to the determination of lower bounds of natural frequencies in planar structures. It is then concluded that it seems both possible and convenient to construct such a limit leading to simple algebraic expressions which are effective from a designers viewpoint, since the geometric and mechanical parameters which come into play in the dynamic behavior of the system can be easily varied as opposed to the case in which complicated transcendental equations are obtained.

A note on free flexural vibrations of a non-uniform ring with fixed supports

Abstract An approximate solution for the title problem is obtained by using the “optimized” Rayleigh-Ritz method. The fundamental mode shape is approximated by means of polynomial co-ordinate functions which contain two exponential optimization parameters. Since the Rayleigh-Ritz method provides upper bounds, the fundamental frequency coefficient is determined by minimizing the lowest root of the frequency equation with respect to the optimization parameters. Continuous and discontinuous variation of the ring thickness are considered. Results are presented for a circular ring with two opposite, infinitely rigid supports.