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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 research aims to explore the free vibration behavior of functionally graded porous (FGP) plates resting on a Kerr-type elastic foundation. This investigation employs an innovative trigonometric shear deformation (ITSD) theory with five variables. The study encompasses various plate configurations, including homogeneous FGP plates, hard-core FGP sandwich plates, and soft-core FGP sandwich plates with both regular and irregular pore structures. The ITSD theory naturally addresses shear stress concerns at the outer surfaces, while also considering the thickness stretching effect, all without the need for correction factors. To formulate the governing equations for the free vibration of such plates on an elastic foundation, Hamilton’s principle is employed. The Navier double trigonometric series approach is then utilized to solve this problem. To validate the plate theory and methodologies used in this work, a comparative analysis is conducted with existing studies. Additionally, comprehensive parametric simulations are employed to examine the impact of different factors, such as geometric properties, material characteristics, sandwich schemes, and parameters of the Kerr-type foundation, on the dimensionless natural frequencies of simply supported rectangular FGP plates.

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.

The blades of helicopters and wind turbines undergo pitch motion in hygrothermal environments. Few works have studied the effect of this movement on blade frequencies. This paper focuses on the bending characteristics of a pitching thin-walled beam, including the effects of hygrothermal environments, rotor speeds, and composite materials. The harmonic pitch function is introduced into the dynamics model of a rotating beam. The method of multiple scales and the Galerkin method are used to solve these equations. Results indicate that the pitch amplitude A dramatically influences natural frequencies. Conclusions are obtained: (1) The flapping frequency varies periodically with pitch amplitude A, and monotonically changes every 2/$\pi$. The pitch amplitude A significantly impacts frequencies, while the pitch frequency ${\omega }_p$ and pitch phase C don’t influence frequencies. (2) An increase in temperature or humidity will decrease the flapping frequencies of a pitching beam. (3) The ply angle, coupled with the dynamic pitching angle, dramatically influences the flapping frequencies of the composite beam. The larger the ply angle, the greater the effect of pitch amplitude A on frequencies.

This study primarily examines a badminton court that is modeled as a rectangular plate enabled by graphene-origami (G-Ori) metamaterials. The model’s dynamic performance is examined in a thermal environment to comprehend the effects of temperature variations. To enhance accuracy, a new quasi-three-dimensional shear and normal trigonometric hyperbolic theory is employed, elucidating the kinematic interactions of the structure. This new hypothesis accommodates transverse normal strain. Differential motion equations are derived from Hamilton’s principle and subsequently solved analytically using Fourier series functions. The primary objective of the study is to investigate the impact of different conditions on the natural frequencies of the model. The findings demonstrate that the incorporation of G-Ori enhances the rigidity of the structure and, as a result, its natural frequencies. Conversely, when the temperature increases, the frequencies diminish.

The free in-plane vibration of a shallow circular arch with uniform cross-section is investigated by taking into account axial extension, shear deformation and rotatory inertia effects. The exact solution of the governing differential equations is obtained by the initial value method. By employing the same solution procedure, the solutions are also given for the other cases, in which each effect is considered alone, as well as no effect. The frequency coefficients are obtained for the lowest five vibration modes of arches with five combinations of classical boundary conditions, and various slenderness ratios and opening angles. The results show that the shear deformation and rotatory inertia effects are also very important as well as the axial extension effect, even if a slender shallow arch is considered. The discrepancies among the results of the five cases decrease, when opening angle increases for a constant radius and slenderness ratio. The effects of the boundary conditions and the slenderness ratio of the arch are investigated. The discrepancies among the results of the cases become much more important in higher modes. The mode shapes of a shallow arch are obtained for each case.

The free vibration problem of a cylindrical shell with an oblique end is considered. A theoretical solution based on the Sanders–Budiansky linear shell theory, and the differential quadrature method, is presented. The surface of the shell is first developed onto a plane, and the resulting irregular domain is then mapped, using blending functions, onto a square parent domain. The analysis is finally carried out in the parent domain. Two solutions are derived, using either trigonometric or polynomial trial functions in the circumferential direction of the domain. Convergence, validation and parametric studies are carried out. Results from the two solutions are compared with each other and with finite element results. The paper ends with an appropriate set of conclusions.

The displacement components are expressed by trigonometric series representation through the plate thickness to develop a two-dimensional theory. This trigonometric shear deformation plate theory is used to perform free-vibration analysis of a simply supported functionally graded thick plate. Lamé's coefficients and density for the material of the plate are assumed to vary in the thickness direction only. Effects of rotatory inertia are considered in the present theory and the vibration natural frequencies are investigated. The results obtained from this theory are compared with those obtained from a 3D elasticity analysis and various equivalent theories that are available. A detailed analysis is carried out to study the various natural frequencies of functionally graded material plates. The influences of the transverse shear deformation, plate aspect ratio, side-to-thickness ratio and volume fraction distributions are investigated.

This paper is concerned with the vibration problem of initially stressed micro/nano-beams. The vibration problem is formulated on the basis of Eringen's nonlocal elasticity theory and the Timoshenko beam theory. The small scale effect is taken into consideration in the former theory while the effects of initial stress, transverse shear deformation and rotary inertia are accounted for in the latter theory. The governing equations and the boundary conditions are derived using the principle of virtual work. These equations are solved analytically for the vibration frequencies of micro/nano-beams with different initial stress values and boundary conditions. The effect of the initial stress on the fundamental frequency and vibration mode shape of the beam is investigated. The solutions obtained provide a better representation of the vibration behavior of initially stressed micro/nano-beams which are stubby and short, since the effects of small scale, transverse shear deformation and rotary inertia are significant and cannot be neglected.

This paper is concerned with the study of the oscillatory behavior of an electrostatically actuated microcantilever gyroscope with a proof mass attached to its free end. In mathematical modeling, the effects of different nonlinearities such as electrostatic forces, fringing field, inertial terms and geometric nonlinearities are considered. The microgyroscope is subjected to bending oscillations around the static deflection coupled with base rotation. The primary oscillation is generated in drive direction of the microgyroscope by a pair of DC and AC voltages on the tip mass. The secondary oscillation occurring in the sense direction is induced by the Coriolis coupling caused by the input angular rate of the beam along its axis. The input angular rotation can be measured by sensing the oscillation tuned to another DC voltage of the proof mass. First, a system of nonlinear equations governing the flexural–flexural motion of electrostatically actuated microbeam gyroscopes subjected to input rotations is derived by the extended Hamilton principle. The oscillatory behavior of the microgyroscopes subjected to DC voltages in both directions is then analytically investigated. Finally, the effects of the geometric parameters, base rotation and fringing field on the natural frequencies of the system are assessed.

The vibration characteristics of an axially moving vertical plate immersed in fluid and subjected to a pretension are investigated, with a special consideration to natural frequencies, complex mode functions and critical speeds of the system. The classical thin plate theory is adopted for the formulation of the governing equation of motion of the vibrating plates. The effects of free surface waves, compressibility and viscidity of the fluid are neglected in the analysis. The velocity potential and Bernoulli’s equation are used to describe the fluid pressure acting on the moving plate. The effect of fluid on the vibrations of the plate may be regarded as equivalent to an added mass on the plate. The formulation of added mass is obtained from kinematic boundary conditions of the plate–fluid interfaces. The effects of some system parameters such as the moving speed, stiffness ratios, location and aspect ratios of the plate and the fluid-plate density ratios on the above-mentioned vibration characteristics of the plate–fluid system are investigated in detail. Various different boundary conditions are considered in the study.

Flapwise flexural vibration of rotating beams has been extensively studied since the 1970s. Existing methods for solving the aforementioned vibration problem range from the conventional finite element method to variable-order finite element method, Frobenius method, differential transformation method and dynamic stiffness method (DSM). Although various approximation methods are available, most of these methods are based on perturbation or discretization of the governing equation, often leading to tedious calculations. This paper re-examines flapwise flexural vibration of rotating beams using the method of variational iteration, which is relatively new and capable of providing accurate solutions for eigenvalue problems. The extracted natural frequencies and mode shapes for sample rotating beams with various rotational speeds and hub radii are compared with existing results that were published in the open literature.

The present study investigates the dynamical behavior of lattice plates, including both bending and shear interactions. The exact natural frequencies of this lattice plate are calculated for simply supported boundary conditions. These exact solutions are compared with some continuous nonlocal plate solutions that account for some scale effects due to the lattice spacing. Two continualized and one phenomenological nonlocal UflyandMindlin plate models that take into account both the rotary inertia and the shear effects are developed for capturing the small length scale effect of microstructured (or lattice) thick plates by associating the small length scale coefficient introduced in the nonlocal approach to some length scale coefficients given in a Taylor or a rational series expansion. The nonlocal phenomenological model constitutes the stress gradient Eringen’s model applied at the plate scale. The continualization process constructs continuous equation from the one of the discrete lattice models. The governing partial differential equations are solved in displacement for each nonlocal plate model. An exact analytical vibration solution is obtained for the natural frequencies of the simply supported rectangular nonlocal plate. As expected, it is found that the continualized models lead to a constant small length scale coefficient, whereas for the phenomenological nonlocal approaches, the coefficient, calibrated with respect to the element size of the microstructured plate, is structure-dependent. Moreover, comparing the natural frequencies of the continuous models with the exact discrete one, it is concluded that the continualized models provide much more accurate results than the nonlocal Uflyand–Mindlin plate models.

This paper is concerned with the flexural vibration analysis of rotating Timoshenko beams by using the variational iteration method (VIM). Accurate natural frequencies and mode shapes of rotating Timoshenko beams under various rotation speeds and rotary inertia are obtained. The VIM solutions are verified by comparing with some existing results in the literature as well as validated from a comparison study with experimentally measured ones. High accuracy and efficiency of VIM are demonstrated by the use of only a small number of iteration steps required for convergence of the first to the tenth mode frequencies of rotating Timoshenko beam.

This paper is concerned with the free vibration of a rotating tapered Timoshenko beam with preset and pre-twist angles. The power series method is used to obtain the frequencies and complex modes of the structure. The rotating velocity related terms are re-classified into three types, namely, static centrifugal terms, dynamic centrifugal terms and the gyroscopic terms. This reclassification provides clearer descriptions of the varying frequencies with respect to the rotating velocity. The gyroscopic coupling among different directions are discussed. The overall contour of the complex modal vibrations is recorded and investigated by time series snapshots of neutral line motions and tip end cross-section motions.

This study focuses on developing and implementing Mamdani hybrid fuzzy logic inference system (FIS) for transverse crack detection and fault diagnosis in a woven fiber laminated glass/epoxy composite beam using different vibration modes of natural frequencies. The shifting of vibration is attributed to the implication of cracks. These vibration signatures are fuzzified through hybrid fuzzy sets (triangular, trapezoidal, Gaussian) and scaled to crack location and depth using the fuzzy rules and defuzzification process. The vibration signatures are recorded using ABAQUS finite element (FE) simulation software for a fixed beam and are fed as input parameters to the developed FIS for computing the desired outputs. The realization for crack depth and position is experimentally verified through a Fast Fourier Transform (FFT) analyzer. The experimental results with simulated data show that fuzzy logic application detects crack positions and depth accurately at different levels. It is concluded that the hybrid FIS bears a close resemblance to the experimental analysis and also stands out as an effective method for crack detection in LCB over other standalone methods. The current method can be used as a cost-effective non-destructive technique for health monitoring and fault diagnosis of composite beam structures in any practical field.

A dual-cable parallel winding hoisting system (DCPWHS) is a feasible system for ultra-deep vertical shaft hoists. In this system, two hoisting cables are winded on two separate drums, and their lower ends are connected to a conveyance. Two guiding cables are installed in the shaft to provide guidance for the conveyance; thereby, transverse vibrations of the cables and the conveyance interact with each other. A mathematical model of the DCPWHS with flexible guides that constitute two guiding cables is presented to analyze its coupled vibrations in out-of-plane transverse and torsional directions. Lagrange’s equations with constraints in combination with the continuous vibration theory are used to derive equations of motion of the system, and coupling relations among hoisting cables, guiding cables, and the conveyance are described by geometric matching conditions. Numerical simulation results are in good agreement with those calculated by MSC.ADAMS. Natural frequencies (NFs) of the stationary system at different positions of the conveyance are also obtained, and a graphic approach is presented to distinguish and extract hoisting cable transverse NFs and guiding cable NFs from system NFs. Numerical results show that transverse vibrations of hoisting cables cause the conveyance to vibrate in both transverse and torsional directions. However, due to restrictions of guiding cables, the rotation of the conveyance is small and negligible. The influence of the guiding cable tension on the dynamic response of the system is also investigated, which can be used for selection of the guiding cable tension in the design of the DCPWHS.

This paper presents modeling and free vibration analysis of variable stiffness system for the truncated sandwich conical shell made of porous aluminum foam core with variable thickness and carbon fiber face sheets under the simply supported boundary condition. The thickness of the core layer varies along the longitudinal direction. Five different types of porosity distribution of the aluminum foam core, which contains Type-X, Type-O, Type-U, Type-V and Type-$\mathrm{\Lambda }$ along the direction of thickness, are considered. Considering the effect of thermal environment, we derive the nonlinear dynamic equations based on first-order shear deformation theory and Hamilton’s principle, and obtain the natural frequencies of the system by employing the Galerkin method. The comparison and validation are conducted by contrast with the determined results of the literature. The influences of porosity distribution pattern, porosity coefficient, the total number of layers, temperature increment, semi-vertex angle, the exponent of thickness function, the minimum radius-thickness and length-thickness ratio of the core layer on the natural frequencies, modal and mode shapes are studied comprehensively.

In this paper, a three-dimensional dynamic model of a dual-cable winding hoisting system is presented to simulate and analyze its coupled vibrations. The equations of motion of this system are derived based on a substructure method and Lagrange’s equations of the first kind, in which the longitudinal–torsional coupled mechanical characteristics of the hoisting cables are considered. Longitudinal and transverse natural frequencies of the system are obtained and studied, and its dynamic responses due to some assumed displacement excitations are calculated. Numerical results agree well with those from ADAMS simulation, and the results have shown that longitudinal and transverse resonances are inevitable, but the transverse resonance has little effect on the conveyance; Slow longitudinal excitations should be avoided, thus avoiding the first-order longitudinal resonance since the system longitudinal vibration is dominated by its first-order mode; the torque in the hoisting cable is mainly induced by its tension under longitudinal–torsional coupled mechanical characteristics of the cable, and a torque release device is suggested to protect the hoisting cable from torsion failure; Torsional and transverse vibrations of the conveyance are been well restricted by the guiding cables even when the guiding cable pre-tension is zero. The results of this paper can be used to improve the parameter design of the ultra-deep shaft hoisting systems.

Functionally-graded materials (FGMs) have great potential in many industry areas for the development of novel acoustic devices such as sensors, electromechanical transducers, actuators and filters. The study of the propagation of elastic waves is a primordial step for a number of such applications. In this study, the stiffness matrix method and the Stroh formalism with the formulation of Ingebrigsten and Tonning were used to establish the relationship between the stress and displacement from the top to the bottom of the fictive multilayer. A power-law inhomogeneity distribution is introduced in the mechanical tensor of the magneto-electro-elastic (MEE) composite. The obtained results indicate that the introduction of heterogeneity has a great influence on nondimensional frequency and modal shape. It is also found that the frequency vibration decreases with the increase in gradient coefficient $\alpha$. Furthermore, the metallization of the free surface (vanishing of electrical and magnetic potential) highly decreases the stress, especially in the median of the plate.