Mechanical Vibration: Where do we Stand?

The basic relationships of the linearized theory of elasticity of a continuous system are reviewed in different notations. The governing equations are expressed in terms of the displacement field, together with the appropriate initial and boundary conditions. The equations of motion of a few structural members are deduced.

This lecture deals with the dynamical behaviour of hemispherical domes and shell panels. The First-order Shear Deformation Theory (FSDT) is used to analyze the above moderately thick structural elements. The treatment is conducted within the theory of linear elasticity, when the material behaviour is assumed to be homogeneous and isotropic. The governing equations of motion, written in terms of internal resultants, are expressed as functions of five kinematic parameters, by using the constitutive and the congruence relationships. The boundary conditions considered are clamped (C), simply supported (S) and free (F) edge. Numerical solutions have been computed by means of the technique known as the Generalized Differential Quadrature (GDQ) Method. These results, which are based upon the FSDT, are compared with the ones obtained using commercial programs such as Ansys, Femap/Nastran, Straus, Pro/Engineer, which also elaborate a three-dimensional analysis.

In the study of thick plates, an accurate description of shear strains is important in order to correctly predict the plate behavior. In the setting of two dimensional theories, improved accuracy with respect to Kirchhoff-Love and Reissner-Mindlin theories can be obtained by introducing higher order terms in the description of the displacement. In this paper, the third order theory proposed by Reddy (1984) for laminated plates is thoroughly evaluated for the isotropic plates. A corresponding finite element model is developed and implemented in a computer code for modal analysis of plates. The results are compared with the three dimensional theory and with the predictions given by traditional two dimensional models.

A continuum three-dimensional Ritz formulation is presented for the free vibration analysis of homogeneous, isotropic, thick rectangular plates. The method is applied to investigate the effects of boundary constraints and thickness ratios on the vibration responses of these plates. The present formulation is based on the linear, three-dimensional, small-strain elasticity theory. The in-plane displacements and deflections are approximated by sets of self generating polynomial shape functions able to satisfy the essential geometric boundary conditions at the outset. Several numerical examples have been computed to demonstrate the accuracy and efficiency of the method. The influence of plate thickness and boundary constraints on the vibratory characteristics of thick plates is also discussed in detail.

In a deformable isotropic infinitely long cylinder a discrete number of propagating guided modes regularly exists in a limited interval of frequency (f) and wavenumber (ξ). The calculation of the guided modes is best done via Helmholtz’s method, where the Bessel functions are used to scale the scalar and wave potentials. Solving the three-dimensional wave equations, leads to displacement and stress componenets in terms of potential to be found. By imposing the stress free boundary conditions for the inner and outer surface of the cylinder, the dispersion equation can be obtained. The dispersion equation shows how the phase velocity, c _p = 2πf/ξ, change with the frequency. The group velocity, i.e. the speed of the propagating guided modes along the cylinder, can be obtained as c _g = ∂(2πf)/∂ξ.

Guided wave is a type of wave propagation in which the waves are guided in plates, rods, pipes or elongated structures such as rails and I-beams. In order to extract the guided wave velocities and wavestructures, for waveguides of arbitrary cross section, a theoretical framework is developed. Here, a semi-analytical finite element (SAFE) method is used for the calculation of the wave propagation characteristics in elastic waveguides immersed in vacuum. The method couples an approximate displacement field over the cross-section of the waveguide and assumes time harmonic representation of the propagating waves along the length of the guide. The Hamilton’s principle and the finite element discretization lead to a discrete weak form of the energy balance equation. The wave propagation problem reduces to a system of algebraic equations, from which the dispersive equation can be obtained. The solution, which depends on both time t and propagation coordinate z, i.e. e ^{i(ξz − ωt)} results in a two-parameter eigensystem. By specifying a real axial wavenumber ξ, the eigenproblem permits real frequencies of propagating modes to be determined. Giving instead real frequency ω, both real and complex axial wavenumbers can be extracted, where real values pertain to propagating modes and the complex ones to the evanescent modes. The method allows us to model a generic cross-section of solid waveguide and it is well suited for computing the phase velocity, the group velocity and the wavestructure or cross-sectional mode shape.

Here, some applications of the Semi-analytical (SAFE) formulation presented in a previous paper of this book (Viola et al., 2005) are shown. In order to emphasize the potentiality of the SAFE method presented, guided wave features are calculated for several isotropic waveguides immersed in vacuum. In particular a plate, a rod and a hollow cylinder are considered. The dispersion results are presented in terms of phase velocity and group velocity along with the wavestructures or cross-sectional mode shapes.

This chapter introduces smart materials suitable for use in vibration related problems. The remaining chapters in this section integrate smart structures into the context of vibration analysis, vibration prevention and structural health monitoring (diagnostics).

This chapter examines several examples of using smart materials for suppressing vibrations in structures. In particular, it is shown that smart materials can greatly increase controllability and observability and can be used for both passive and active control.

This chapter introduces basic concepts from control for suppressing vibrations in structures. These concepts set the background for integrating smart materials into vibration problems for measurement, suppression and monitoring.

This chapter introduces smart materials for problems related to structural health monitoring and diagnostics. Specifically PZT based materials form a natural actuator and sensor for impedance based to structural health monitoring.

This study has two aims. The first aim is to describe a number of wonderful phenomena caused by the action of vibration¹ on nonlinear mechanical systems along with important applications of these effects in technology. The second aim is to present a general mechanical-mathematical approach to the description and investigation of this class of phenomena. We call the approach to be described herein “Vibrational Mechanics”. While it is a new approach, it is based on the classical idea of averaging in the theory of nonlinear oscillations and in the theory of the stability of motion.

The purpose of this part of the book is to provide a review of main linearization methods in analysis of stochastic dynamic systems. Two basic groups of linearization methods namely statistical and equivalent linearization are presented. In particular moment criteria, energy criteria, linearization criteria in the space of power spectral density functions and probability density functions are discussed. Applications of linearization methods to the response analysis and control design for mechanical and structural systems subjected to earthquake excitation or road irregularities are also included.

Problems involving vibration occur in many areas of mechanical, civil and aerospace engineering: wave loading of offshore platforms, cabin noise in aircrafts, earthquake and wind loading of cable stayed bridges and high rise buildings, performance of machine tools — to pick only few examples. Human beings usually regard noise and vibration as uncomfortable. Beside this, an engineering structure can fail due to excessive vibration — the devastating effects of earthquakes on our society is a prime example of this fact. Due to this reasons over the years the aim of the vibration engineers has been to reduce vibration. In order to achieve this in an efficient and economic manner a good understanding of the physics of vibration phenomena in complex engineering structures is needed. In the last few decades, the sophistication of modern design methods together with the development of improved composite structural materials instilled a trend towards lighter structures. At the same time, there is also a constant demand for larger structures, capable of carrying more loads at higher speeds with minimum noise and vibration level as the safety/workability and environmental criteria become more stringent. Unfortunately, these two demands are conflicting and the problem cannot be solved without proper understanding of the vibration phenomena.

This paper is dedicated to derivation of eigenvalues of structures that possess modulus of elasticity and/or material density that vary from point to point. There is a large selection of methods that can deal with such a structures’ vibration spectra. In very rare circumstances one has a possibility to obtain an exact solution, usually in terms of transcendental functions (hypergeometric, Bessel, Lommel, and other special functions).

In other cases, one resorts to powerful numerical methods, like the finite element method; finite difference method; Rayleight-Ritz or Galerkin methods; collocation method and others. In such circumstances, seeking to find a closed-form solution may appear to be a hopeless task.

In 1759, Leonhard Euler was able to derive some closed-form solutions for buckling of non-uniform columns. However, to the best of our knowledge, no closed-form solutions were available for the vibration problems of non-uniform and inhomogeneous structures, until very recently.

In this study some closed-form solutions are reported which we were fortunate to derive over the recent six years. Only a few solutions are discussed, in order to provide the scope of developments. It is hoped that this method, and its generalizations will be further developed, to advance the beautiful world of closed-form solutions, that retain their attractiveness even in the present era of numerical solutions.

Parts 1 and 2 are devoted to the beam vibrations, whereas Parts 3–5 deal with vibrations of circular plates.

In this study four cases of harmonically varying buckling modes are postulated and semi-inverse problems are solved that result in the distributions of the flexural rigidity compatible to the pre-selected modes and to specified axial load distributions. In all cases the closed-form solutions are obtained for the eigenvalue parameter. For comparison the obtained closed form solution is contrasted with an approximate solution based on an appropriate polynomial shape, serving as trial function in an energy method

In the previous work buckling modes has been postulated and semi-inverse problems has been solved that result in the distributions of the flexural rigidity compatible to the pre-selected modes and to specified axial load distributions. In this part II the semi-inverse problem is widen to the dynamic context and it is demonstrated that a harmonic function can serve both as the vibration and the buckling mode of an axially graded beam-column.