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In this work, a new analytic solution for vibrations of shallow shells is presented. The equations of motion consist of three coupled partial differential equations. Solutions to such complex coupled equations were only available for the Navier and Levy cases of boundary conditions, which is a small part of the scope of the problem. A superposition of two solutions enables to satisfy both the equations of motion and any combination of boundary conditions. For isotropic square shallow shell, there are 9316 different combinations of support conditions. For isotropic rectangular shell or square orthotropic shell, the number is 18496. These numbers apply for a single type of curvature and aspect ratio. For all these a general solution is derived. The functions for the solution are obtained by using carefully chosen series that solve the coupled partial differential equations of motion for in-plane and out-of-plane deformations for all possible combinations of edge conditions. The number of terms in the series is taken such that convergence is assured to the number of digits as shown. Examples of the new solutions are given and compared with available solutions in the open literature.

This paper presents a new analytical method for the solution of the vibration frequencies of rectangular plates with cutouts. The cutouts may be internal or bordering the edges of the plate. In this work, a newly derived exact solution for the vibrations of rectangular plates for all the possible combinations of boundary conditions is extended for the solution of the vibrations of rectangular plates with cutouts. The problem is modeled as an assembly of plates. The continuation requirements along the connecting edges are enforced as part of the solution. The accuracy of the solution is studied through comparisons with published results from other methods.

We present extensive results of computer simulations on vibrations in one- and two-dimensional lattices with quartic anharmonicity. The existence of localized mode in ordered lattice is confirmed but the long term stability is questionable. Rather our results indicate an ergodic behavior at large times when several modes are present. Concerning disordered systems we have found that anharmonicity leads to a surprising new phenomenon of anomalous diffusion and thus to the possibility of anharmonically induced transport.

The vibrational behavior of double-walled carbon nanotubes is studied by the use of the molecular structural and cylindrical shell models. The spring elements are employed to model the van der Waals interaction. The effects of different parameters such as geometry, chirality, atomic structure and end constraint on the vibration of nanotubes are investigated. Besides, the results of two aforementioned approaches are compared. It is indicated that by increasing the nanotube side length and radius, the computationally efficient cylindrical shell model gives rational results.

The vibration during external cylindrical grinding is caused by many factors such as the rigidity of the technology system, machining modes, machining materials, cooling mode, etc. This paper employed a Taguchi method to design experiments and evaluate the influence of machining mode parameters and workpiece material hardness on the vibrations when machining some types of alloy steel in external cylindrical grinding process. The influence of machining conditions on the vibrations was investigated. Besides, the mathematical models of vibration amplitudes were also modeled. The achieved results can be used to control the vibrations through machining conditions to improve the surface quality of the product.

The published work on analytical ("mathematical") and computer-aided, primarily finite-element-analysis (FEA) based, predictive modeling of the dynamic response of electronic systems to shocks and vibrations is reviewed. While understanding the physics of and the ability to predict the response of an electronic structure to dynamic loading has been always of significant importance in military, avionic, aeronautic, automotive and maritime electronics, during the last decade this problem has become especially important also in commercial, and, particularly, in portable electronics in connection with accelerated testing of various surface mount technology (SMT) systems on the board level. The emphasis of the review is on the nonlinear shock-excited vibrations of flexible printed circuit boards (PCBs) experiencing shock loading applied to their support contours during drop tests. At the end of the review we provide, as a suitable and useful illustration, the exact solution to a highly nonlinear problem of the dynamic response of a "flexible-and-heavy" PCB to an impact load applied to its support contour during drop testing.

Complex vibrations of an Euler–Bernoulli beam with different types of nonlinearities are considered. An arbitrary beam clamping is considered, and deflection constraints (point barriers) are introduced in some beam points along its length. The influence of a constraint, as well as of the amplitude and frequency of excitation on the vibrations is analyzed. Scenarios of transition to chaos owing to the introduced nonlinearities are reported.

A closed cylindrical shell with circular cross-section having constant stiffness and density and subjected to sign changeable loading and embedded into a temperature field is analyzed. Both Bubnov–Galerkin (with a higher approximation) and Fourier methods are applied to solve the derived nonlinear nondimensional partial differential equations. Among others, the novel scenario of transition from shell harmonic to chaotic vibrations via the collapse of quasi-periodic vibrations with one independent frequency and Hopf bifurcation is detected, illustrated and discussed. In addition, it is shown how for various intensities of the temperature field (including its absence) the increase of the loading yields qualitative changes in the investigated shell dynamics, and how chaotic zones are transmitted into periodic ones and vice versa.

The problem considered in this paper is one of damped vibrations of the beam in transportation. The dynamical analysis of systems in motion is a very well-known issue, but many detailed cases have not been published yet. The considered case is applied with the double-sided fixed beams placed on a rotational disk. The disk treated as a rigid one is rotated with a constant angular velocity in the stationary reference frame. An application of vibrations of the beam is transformed from a local to global reference frame, damping forces and forces arising from the rotational motion of a local reference center are also taken into consideration. The beam is assumed as a homogenous one with symmetric cross-sections. This work is the dynamical analysis of such a type of vibrations, which is a mathematical one based on considering transient response and dynamical flexibility of this type of systems.

In this paper, the free vibration problem of a system consisting of a three-member type telescopic boom of the laboratory crane, the hydraulic cylinder of crane radius change and the elastic support system has been presented. The model has been formulated in order to represent the vibrations in the lifting plane of the laboratory crane. The transverse vibrations of the boom, longitudinal and transverse vibrations of the hydraulic cylinder have been considered in the coupled model of the system taking into account the elasticity of the support system. In the computational model, the analyzed system has been replaced by the discrete-continuous model. The free vibration problem of the analyzed system is formulated and solved on the basis of Lagrange multiplier formalism. The experimental modal analysis is done and the genetic algorithm worked out with the aim of identifying spring constants of the support system elements. The analysis of the influence of the hydraulic cylinder and the length of the telescopic boom on the free vibration frequencies is carried out on the basis of the identified vibration model of the considered system. The examples of the experimental and numerical results are presented.

This paper presents our study of dynamics of fractal solids. Concepts of fractal continuum and time had been used in definitions of a fractal body deformation and motion, formulation of conservation of mass, balance of momentum, and constitutive relationships. A linearized model, which was written in terms of fractal time and spatial derivatives, has been employed to study the elastic vibrations of fractal circular cylinders. Fractal differential equations of torsional, longitudinal and transverse fractal wave equations have been obtained and solution properties such as size and time dependence have been revealed.

A new approach called the ''Variational Theory of Complex Rays'' (VTCR) is being developed in order to calculate the vibrations of slightly damped elastic structures in the medium-frequency range. Here, the emphasis is put on the extension of this theory to analysis across a range of frequencies. Numerical examples show the capability of the VTCR to predict the vibrational response of a structure in a frequency range.

Vibrations induced by machinery and traffic have become of increasing concern in the last years, for example, when constructing buildings near railway lines. In this paper we will present a model designed to predict the vibration level in the ground. Since in practice it is nearly impossible to determine exact material parameters for soil layers, we use a model with a stochastic shear modulus G. Under moderate assumptions G can be split with the Karhunen–Loeve expansion into a mean value G₀ and a stochastic part G_stoch. Using a combination method of finite elements, Fourier transformation and Polynomial Chaos, it is possible to transform the partial differential equation describing the system into a matrix-vector formulation Kx = b which can be split into a deterministic and a stochastic part (K₀ + K_s) x = b analog to the shear modulus. To keep the dimensions of the matrices involved with this system small, we use a Neumann-like iteration to solve it. Finally, results for a small example are presented.

Using Taguchi design of experiments (DoE), experiments were conducted with 3 factors and 3 levels. The factors were the depth of cut, spindle speed, and feed. The responses were surface roughness, flank wear, material removal rate, and spindle vibration along $x$ ($V_x$), $y$ ($V_y$), and $z$ ($V_z$) axis. To convert the multi-response optimization problem into a single response optimization problem, the technique for order of preference by similarity to ideal solution (TOPSIS) was applied. The S/N of the closeness coefficients from TOPSIS was calculated and optimum machining conditions were obtained. Further, analysis of variance (ANOVA) was performed to verify which input parameter significantly affects the output responses. From TOPSIS optimization, the responses like surface roughness and flank wear were decreased by 0.99% and 2.55%. The vibration in $x$, $y$, and $z$-axis decreased by 3.84%, 16.87% and 12.48% respectively. This optimization significantly enhances the machining characteristics.

The exact vibration frequencies of continuous beams with internal releases are found using the dynamic stiffness method. Two types of releases are considered: hinge and sliding discontinuities. First, the exact dynamic stiffness matrix for a beam element with a release is derived and then used in the assembly of the structure dynamic stiffness matrix. The natural frequencies are found as the values of frequency that make this matrix singular. Then the mode shapes are found exactly. Examples are given for continuous beams with different releases.

The effect of a constant axial load either tensile or compressive on the natural frequencies of uniform multi-span beams is presented. The proposed solution is based on the exact dynamic stiffness matrix for the member, for any desired precision. The natural frequencies are found as the frequencies that cause the determinant of the stiffness matrix to become zero. The influence of the axial load and variation of span ratios on the vibration frequencies for beams with up to 4 spans are studied, and presented in graphical forms.

A thin circular plate with a free or movable boundary, and weakened along an internal concentric circle, is considered. The frequencies of the circular plate varying with the radius of the weakened circle and the extent of the weakening represented by an elastic rotational restraint are determined. The results for the first four modes are tabulated. For the free plate, the internal weakening only slightly decreases the fundamental frequency of the plate, which is advantageous. For the plate with a movable edge, the internal weakening greatly decreases the fundamental frequency, especially for the purely hinged case.

Higher-order multiple scales methods have been developed for a strong nonlinear differential equation modeling the nonlinear transverse dynamic behavior of piezoelectric–elastic–piezoelectric sandwich beams. The proportional and derivative potential feedback controls via sensor and actuator layers are used. Two higher-orders multiple scales methods are developed leading to accurate approximate time-response and frequency and phase-amplitude relationships. As the classical multiple scales method disregards the quadratic nonlinear terms, the developed higher-order multiple scales methods overcame this drawback and lead to more accurate results. Static and dynamic stability criteria are elaborated and tested. Critical bifurcation points and peak amplitudes are characterized in accordance with the excitation amplitude and the feedback parameters.

The solutions for the vibrations of cracked thin plates are obtained by the Ritz method with admissible functions. Based on the classical plate theory, the basis functions comprising polynomials and crack functions are adopted to generate the admissible functions by the moving least-squares approach for a set of nodes randomly distributed in the domain. The crack functions account for the singular behaviors of stress resultants at crack tip(s), which are discontinuous in displacement and slope across the crack. The present solutions are validated through convergence tests of frequencies and by comparison with the published results for simply-supported cracked rectangular plates. The solutions are further employed to determine the natural frequencies of cantilevered skewed rhombic and isosceles triangular plates and completely free circular plates, each with a crack of varying length, location and orientation. The numerical results are tabulated and some corresponding mode shapes are also presented, by means of nodal patterns. Most of the results shown here are new to the literature.

The well-known vibration model of axially moving beam is considered. Both axial moving speed and axial force are assumed to vary harmonically. The Method of Multiple Time Scales (a perturbation method) is used. The natural vibrations of beam are considered for different values of beam parameters. Resonances are obtained for seven different conditions. Solvability conditions for each resonance case are found analytically. Effects of transport velocity, axial force, rigidity and damping are discussed. Stability analysis are obtained for principal parametric resonances. Stable and unstable regions are obtained regarding velocity and force effects separately and together.