Exact Free Vibration Research Articles

Exact free vibration study of rectangular Mindlin plates with all-over part-through open cracks

Based on Mindlin plate theory (MPT), a set of exact closed-form characteristic equations incorporating shear deformation and rotary inertia are proposed for the first time to analyze free vibration problem of moderately thick rectangular plates with an arbitrary number of all-over part-through cracks. The proposed rectangular plates have two opposite edges simply supported while six possible combinations of free, simply supported and clamped boundary conditions are taken into account for two other edges. The crack is assumed to be open, non-propagating and perpendicular to two opposite simply supported edges. A continuously distributed line-spring model is used to describe the elastic behavior of an all-over part-through crack. The accuracy of the current approach is investigated through comparing the present exact natural frequencies with those of 3D finite element method obtained by ABAQUS software package. A parametric study is undertaken to show the effect of crack depth, crack location, number of cracks and thickness-to-length ratio on natural frequencies of rectangular moderately thick plates with different boundary conditions in tabular and graphical forms. Finally, the effect of shearing and tearing modes on the modeling of cracks located at the nodal line is shown.

On new symplectic elasticity approach for exact free vibration solutions of rectangular Kirchhoff plates

In the classical approach, it has been common to treat free vibration of rectangular Kirchhoff or thin plates in the Euclidian space using the Lagrange system such as the Timoshenko’s method or Lévy’s method and such methods are the semi-inverse methods. Because of various shortcomings of the classical approach leading to unavailability of analytical solutions in certain basic plate vibration problems, it is now proposed here a new symplectic elasticity approach based on the conservative energy principle and constructed within a new symplectic space. Employing the Hamiltonian variational principle with Legendre’s transformation, exact analytical solutions within the framework of the classical Kirchhoff plate theory are established here by eigenvalue analysis and expansion of eigenfunctions in both perpendicular in-plane directions. Unlike the classical semi-inverse methods where a trial shape function required to satisfy the geometric boundary conditions is pre-determined at the outset, this symplectic approach proceeds without any shape functions and it is rigorously rational to facilitate analytical solutions which are not completely covered by the semi-inverse counterparts. Exact frequency equations for Lévy-type thin plates are presented as a special case. Numerical results are calculated and excellent agreement with the classical solutions is presented. As derivation of the formulation is independent on the assumption of displacement field, the present method is applicable not only for other types of boundary conditions, but also for thick plates based on various higher-order plate theories, as well as buckling, wave propagation, and forced vibration, etc.

Exact free vibration analysis of axially moving viscoelastic plates

In this paper, an exact finite strip method is developed for the free vibration analysis of axially moving viscoelastic plates. Using the differential equation that governs the vibration of plates traveling at a constant axial speed, and utilizing the rheological models to model the viscoelastic behavior of materials, the exact stiffness matrix of a finite strip of plate is extracted in the frequency domain. Terms of this matrix are implicit functions of the eigenvalues of free vibration, in-plane forces, viscoelasticity parameters, axial speed and the geometry of the plate. By assembling the stiffness matrices of these finite strips, the global stiffness matrix of a plate moving on intermediate rollers is obtained, from which the eigenvalues defining the free vibration of the plate are extracted within the domain of complex numbers using a suitable algorithm. These values (that represent the exact frequencies of the plate vibration and damping) can be used as a benchmark to evaluate the accuracy of other numerical techniques. Finally, by presenting a set of numerical examples, the effects of axial speed and viscoelastic parameters on the free vibration of moving plates are examined in this paper.

Exact natural frequencies of structures consisting of two-part beam-mass systems

Using two different, but related approaches, an exact dynamic stiffness matrix for a two-part beam-mass system is developed from the free vibration theory of a Bernoulli-Euler beam. The first approach is based on matrix transformation while the second one is a direct approach in which the kinematical conditions at the interfaces of the two-part beam-mass system are satisfied. Both procedures allow an exact free vibration analysis of structures such as a plane or a space frame, consisting of one or more two-part beam-mass systems. The two-part beam-mass system described in this paper is essentially a structural member consisting of two different beam segments between which there is a rigid mass element that may have rotatory inertia. Numerical checks to show that the two methods generate identical dynamic stiffness matrices were performed for a wide range of frequency values. Once the dynamic stiffness matrix is obtained using any of the two methods, the Wittrick-Williams algorithm is applied to compute the natural frequencies of some frameworks consisting of two-part beam-mass systems. Numerical results are discussed and the paper concludes with some remarks.

Free vibration of transversely isotropic piezoelectric circular cylindrical panels

This paper presents an exact three-dimensional free vibration analysis of a transversely isotropic piezoelectric circular cylindrical panel. The general solution for coupled equations for piezoelectric media that was recently proposed by Ding et al. (Int. J. Solids Struct. 33 (1996) 2283) is employed. By using the variable separation method, three-dimensional exact solutions are obtained under several boundary conditions. Numerical results are finally presented and compared with available data in literature. The results show the non-dimensional frequencies of the piezoelectric panel are bigger than that of the non-piezoelectric one.

FREE VIBRATION ANALYSIS OF NON-CYLINDRICAL COIL SPRINGS BY COMBINED USE OF THE TRANSFER MATRIX AND THE COMPLEMENTARY FUNCTIONS METHODS

The transfer matrix method and the complementary functions method have been employed in the free vibration analysis of helical springs of irregular shape. The rotary inertia effects, the axial and shear deformations and cross-sections of different shapes having double symmetry have been included in the analysis. The formulation has yielded the exact free vibration frequencies of non-cylindrical coil springs. © 1997 John Wiley & Sons, Ltd.

FREE VIBRATION ANALYSIS OF NON‐CYLINDRICAL COIL SPRINGS BY COMBINED USE OF THE TRANSFER MATRIX AND THE COMPLEMENTARY FUNCTIONS METHODS

The transfer matrix method and the complementary functions method have been employed in the free vibration analysis of helical springs of irregular shape. The rotary inertia effects, the axial and shear deformations and cross-sections of different shapes having double symmetry have been included in the analysis. The formulation has yielded the exact free vibration frequencies of non-cylindrical coil springs. © 1997 John Wiley & Sons, Ltd.

The vibration characteristics of thick rectangular multilayered plates

We develop the analysis and numerical calculations for the exact free vibration characteristics of simply-supported, rectangular, thick, multilayered composite plates. We assume that each layer of the composite plate is of arbitrary thickness, is perfectly bonded to adjacent layers, possesses up to orthotropic material symmetry and that its material crystallographic axes are oriented either parallel or perpendicular to the plate's boundaries. Exact formal solutions are obtained for the individual layers which are, in turn, used to relate the field variables at the upper and lower layer surfaces. The solution is carried through by the successive application of appropriate interfacial continuity conditions between adjacent lamina. We facilitate our analysis and its numerical implementation by making use of the matrix transfer technique, thereby rendering the order of the resulting eigenproblem independent of the number of lamina. Numerical results are given in the form of scaled natural frequency versus nondimensionalized mode numbers. The effect of the plate's microstructure on its vibrational characteristics is investigated by examining changes in natural frequency and modal response for selected laminate lay-ups and material combinations.

The unsymmetric formulation for vibration analysis of frames

The unsymmetric formulation is a dynamic model for computing numerically exact free vibration frequencies and modes of structural systems. Based on exact representation of element inertia forces and a frequency independent stiffness matrix, this formulation obtains discretization error-free solutions. Previous studies have shown that in modeling the vibration response of trusses, the unsymmetric formulation is more computationally efficient than the conventional finite element formulation, particularly when accuracy requirements are high. This study extends the development of the unsymmetric formulation to more general vibration problems. The unsymmetric formulation is developed and illustrated here for the vibration analysis of beams and frames with uniform and/or lumped mass distributions. The computational efficiency of the formulation is studied and compared to that of the exact displacement and conventional finite element models. Results from numerical examples suggest that the unsymmetric formulation provides more accurate free vibration modes and frequencies of frames in less computational time than do other dynamic models.