Ricardo Oscar Grossi

A general algorithm for the study of the dynamical behaviour of beams

Abstract This paper presents a simple, accurate and flexible general algorithm for the study of a great number of beams vibration problems. The approach is developed based on the Rayleigh–Ritz method with characteristic orthogonal polynomial shape functions. It allows the inclusion of a number of complicating effects such as varying cross-sections, presence of an arbitrarily placed concentrated mass, ends elastically restrained against rotation and translation and presence of an axial, tensile force. Several cases are treated to show the simplicity and great flexibility of this approach, in the determination of frequencies. To demonstrate the accuracy of the present approach natural frequency coefficients are given for beams, from which comparison results are available. New results are also given for tapered beams with several complicating effects.

Eigenfrequencies of generally restrained beams

We deal with the exact determination of eigenfrequencies of a beam with intermediate elastic constraints and generally restrained ends. It is the purpose of this paper to use the calculus of variations to obtain the equations of motion and the natural boundary conditions, and particularly those at the intermediate constraints. Numerical values for the first five natural frequencies are presented in a tabular form for a wide range of values of the restraint parameters. Several particular cases are presented and some of these cases have been compared with those available in the literature.

Some observations on the application of the Rayleigh–Ritz method

Abstract This paper deals with the applicability of the Rayleigh–Ritz method for the determination of frequency coefficients of beams and plates with elastically restrained edges. Natural frequencies of beams elastically restrained against rotation and translation at both ends and of rectangular isotropic plates with elastic edge restraints are studied by using the Rayleigh–Ritz method along with orthogonal polynomials as co-ordinate functions. It is shown that the approximate satisfaction of boundary conditions introduces additional constraints into the formulation that bring unexpected values in the results. On the other hand, it is shown that there are defects in the approximations when the natural boundary conditions in beams are taken into account. The adequate procedure for constructing the co-ordinate functions to avoid numerical errors is presented.

Vibration frequencies for a beam with a rotational restraint in an adjustable position

Abstract The problem of free vibrations of uniform beams having a combination of clamped, pinned and free ends supports, has been extensively studied by several investigators 1 , 2 , 3 , 4 . Transverse vibration of uniform beams with elastically restrained ends have also been extensively investigated 5 , 6 , 7 , 8 , 9 , 10 . In contrast to the body of information available, there is only a very limited amount of information for uniform beam elastically restrained at some point [11] . The present work deals with the exact and the approximate determination of frequency coefficients of a uniform beam, with one end spring-hinged and a rotational restraint in a variable position. The approximate eigenvalues are obtained by means of the Rayleigh–Schmidt method, and the results obtained were compared with the exact values. It is shown that a one term approximation with various undetermined exponents yields fundamental frequency values which are in good agreement with exact values.

Free Vibrations of a Trapezoidal Plate with an Internal Line Hinge

This paper deals with a general variational formulation for the determination of natural frequencies and mode shapes of free vibrations of laminated thin plates of trapezoidal shape with an internal line hinge restrained against rotation. The analysis was carried out by using the kinematics corresponding to the classical laminated plate theory (CLPT). The eigenvalue problem is obtained by employing a combination of the Ritz method and the Lagrange multipliers method. The domain of the plate is transformed into a rectangular domain in the computational space by using nonorthogonal triangular coordinates and the transverse displacements are approximated with a set of simple polynomials automatically generated and expressed in the triangular coordinates. The developed algorithm allows obtaining approximate analytical solutions for mentioned plate with different geometries, aspect ratio, position of the line hinge, and boundary conditions including translational and rotational elastically restrained edges. It allows studying the influence of the mentioned line on the vibration frequencies and respective mode shapes. The algorithm can easily be programmed and it is numerically stable. Additionally, as a particular case, the results of triangular plates can be easily generated.

Natural frequencies of edge restrained tapered isotropic and orthotropic rectangular plates with a central free hole

Abstract Natural frequencies of tapered orthotropic rectangular plates with a central free hole and edges restrained against rotation and translation are studied by using orthogonal polynomials in the Rayleigh–Ritz method and applying a generalization of the Rayleigh–Schmidt method. The two methods are quite general and can be used to study plates with any combinations of boundary conditions, taper and geometric parameters. To demonstrate the accuracy of the present approach natural frequency coefficients are given for isotropic plates with a central free hole, from which comparison results are available. New results are also given for orthotropic plates with several complicating effects. The studied problems are of interest in several field of engineering, since holes are present in plates due to operational conditions.

How Did Generalized Solutions Arise in Solid Mechanics

In recent decades, engineers and physicists have shown an increasing interest in functional analysis and its applications. As many of these practitioners lack special training in mathematics, they sometimes run into trouble when trying to use the tools of this powerful branch of knowledge. Our purpose is to outline the connection between the traditional ideas of mechanics and the newer mathematical concepts of generalized solution and distribution.