Effects Of Rotary Inertia Research Articles

Wavenumber dynamic stiffness formulation for exact dispersion analysis of moderately thick symmetric cross-ply laminated plate built-up waveguides

A wavenumber dynamic stiffness (WDS) formulation for exact dispersion analysis of moderately thick symmetric cross-ply laminated plate built-up waveguides is presented. The effects of shear deformation and rotatory inertia are considered in the formulation by adopting the first order shear deformation theory. The elemental WDS matrix is derived from the exact general solutions of the governing differential equations and can be assembled to model complex plate built-up waveguides using the assembly procedure similar to that of the finite element method. The nonlinear eigen-solution technique, the Wittrick–Williams (WW) algorithm, is employed to compute dispersion curves from the obtained WDS matrices. To enhance computational efficiency, the J0 count problem of the WW algorithm is addressed by developing explicit expressions for the J0 term. This eliminates the need for mesh refinement over the entire frequency/wavenumber range. The accuracy and efficiency of the proposed method are demonstrated through comparisons with analytical solutions based on the classical laminated plate theory and the first order shear deformation theory, as well as numerical results obtained from the wave and finite element method.

In-Plane and Out-of-Plane Bending Vibration Analysis of the Laminated Composite Beams Using Higher-Order Theories

In this paper, higher-order shear deformation theories for a thorough analysis of the in-plane and out-of-plane vibrational characteristics of laminated composite beams have been presented. Through the introduction of new displacement fields and the consideration of rotary inertia and Poisson's effect, the kinetic and potential energies of the beams have been derived. This formulation, displaying significant generality, accommodates arbitrary stacking sequences. Utilizing the finite element method, a new element has been presented for calculating the beam's vibrational characteristics. Featuring three nodes, each with seven degrees of freedom, this higher-order element provides a detailed representation of complex behaviors. Mass and stiffness matrices have been derived using the energy method and apply boundary conditions through the penalty approach. The results exhibit a high degree of consistency and alignment with those obtained from the 3D commercial software ANSYS, validating the accuracy and reliability of the proposed methodology for structural analysis. This comprehensive approach contributes to advancing the understanding and modeling of laminated composite beams in diverse engineering applications. The effects of different parameters on the in-plane and out-of-plane vibration analysis of laminated composite beams have been investigated in detail.

Guided wave propagation in a high-speed rotating concrete shell reinforced by advanced nanocomposites

This paper investigates the propagation of guided waves in a high-speed rotating concrete shell reinforced with graphene oxide powders (GOP) using higher-order shear deformation theory (HSDT). The study addresses the complexities introduced by initial hoop tension and provides a comprehensive analytical solution procedure. Concrete shells are widely used in various engineering applications due to their robustness and durability; however, their mechanical performance can be significantly enhanced by the incorporation of GOP, which improves the material properties, such as tensile strength and modulus of elasticity. The analysis begins with the formulation of the governing equations based on HSDT, which accurately captures the effects of transverse shear deformation and rotary inertia, crucial for high-speed rotating structures. The presence of initial hoop tension, resulting from the centrifugal forces in the rotating shell, is integrated into the model to reflect real-world conditions. The modified equations are then solved analytically to obtain the dispersion relations for the guided wave modes. The results reveal that the inclusion of GOP significantly alters the wave propagation characteristics, enhancing the shell’s ability to transmit high-frequency waves with minimal attenuation. The initial hoop tension is found to shift the frequencies of the guided wave modes, indicating a coupling effect between the mechanical loading and wave propagation. This study provides valuable insights into the dynamic behavior of reinforced concrete shells in high-speed rotational environments, offering a potential pathway for the design of advanced structural components in aerospace, automotive, and civil engineering applications.

Hygro-Thermo-Magneto-Elastic Vibration of Multidirectional Graded Porous Nanobeams with Axial Motion by Considering Rotary Inertia and Thickness Effects

Scale-dependent vibration and stability of three-dimensional functionally graded (FG) porous Rayleigh nanobeams with axial motion in varying environmental conditions are investigated by considering scale effects in the thickness orientation. It is supposed that the effective material characteristics of the nanobeam in the longitudinal, width, and thickness orientations are graded according to exponential and power-law functions. Different patterns involving uniform, non-uniform, and logarithmic distributions are considered to model the porosity impacts. Based on the nonlocal strain gradient Rayleigh beam model and incorporating thickness effects, the dynamical equation of the system is derived by considering linear, parabolic, and sinusoidal elastic foundations. The Galerkin discretization technique and Laplace transform are adopted to solve the eigenvalue problem and accomplish the stability analysis. Several comparative studies are performed with the available data in the open literature to validate the solution approach and results. Also, detailed parametric investigations are accomplished to interpret the importance of scale factor, rotary inertia factor, porosity characteristics, magneto-hygro-thermal loads, foundation parameters, gradient indices, and boundary conditions on the dynamical behavior, critical divergence, and flutter axial speeds. The results revealed that, contrary to the effect of the material gradient in thickness orientation, the longitudinal and width gradation of materials can lead to the increment of vibrational frequencies. The outcomes declared that the instability threshold improves significantly by considering the scale effects along the thickness orientation. Also, it is demonstrated that when the middle surface of the cross-section has the highest porosity, the stability regions expand by increasing the porosity coefficient. The present research outcomes can offer valuable insights into the optimum design of high-speed moving multi-directional graded porous nanosystems.

Free Vibration of Ogival Circular Arch

This study aims to analyze the free vibration of an ogival arch in which two symmetrical arched shapes are discontinuously met at the mid-arc. Differential equations governing the free vibration of an ogival circular arch were derived, where the rotatory inertia and shear deformation effects are included. The governing equations were numerically solved to calculate natural frequencies and mode shapes. In numerical solution methods, the symmetric and anti-symmetric boundary conditions at the mid-arc were focused rather than the boundary conditions of supported end due to the discontinuity of the mid-arc. For the first time, the free vibration problem for discontinuous arches, such as ogival arches, is solved in this study. Calculation results of this study for natural frequencies are compared well with those of the finite element method. The effects of various arch parameters on natural frequencies were highlighted and discussed in detail.

Critical velocities of a three-layer composite tube incorporating the rotary inertia and material anisotropy

Critical velocities of a three-layer composite tube subjected to a uniform internal pressure moving at a constant velocity are obtained in closed-form expressions. A Love–Kirchhoff thin shell model including the rotary inertia and material anisotropy effects is used in the formulation. The composite tube is made of three perfectly bonded cylindrical layers of dissimilar materials, each of which can be orthotropic, transversely isotropic, cubic or isotropic. Closed-form formulas for the critical velocities are first derived for the general case by incorporating the effects of material anisotropy, rotary inertia and radial stress. Specific formulas are then obtained for composite tubes without the rotary inertia effect and/or the radial stress effect and with various types of material symmetry for each layer as special cases. It is also shown that the current model for three-layer tubes can be reduced to those for single- and two-layer tubes. To illustrate the newly derived formulas, an example is provided for a composite tube consisting of an isotropic inner layer, an orthotropic core, and an isotropic outer layer. All four critical velocities of the composite tube are computed using the new closed-form formulas. Three values of the lowest critical velocity of the three-layer composite tube are analytically obtained from three sets of the new formulas, which agree well with the value computationally determined by others.

Dynamic analysis and stability of variable length vertical pipe containing fluid using Timoshenko’s beam theory

In this article, the dynamics of pipes containing fluid is investigated according to Timoshenko’s beam theory. The problem’s equations were extracted with the help of Hamilton’s principle, and then their responses were obtained using Galerkin’s numerical approach. The modeling of the vertical pipe containing fluid was done using Timoshenko’s beam theory. It is prominent to mention that the considered vertical pipe containing fluid has variable length. The results of this analysis will be compared with previous researches, which were according to Euler–Bernoulli theory. Also, the gravity effect on the pipe stability will be analyzed. Finally, by examining the impact of rotational inertia and shear deformation separately, it was determined that the effect of shear deformation on the pipes’ dynamic behavior containing fluid will be more than the effect of rotational inertia.

Natural Characteristics of a Marine Two-Stage Tandem Hybrid Planetary System

This study focuses on a marine two-stage tandem hybrid planetary system. Natural frequencies and vibration modes are determined using a translational–torsional coupled dynamic model. Based on the motion characteristics of the transmission system, free vibration is categorized into three typical modes. The parameter sensitivity of natural frequencies is computed, and the effects of structural parameters such as unequally spaced planet, mesh stiffness, planet mass and rotational inertia on the natural frequencies are analyzed. Utilizing the coupling factor, the mode transition criterion for the natural frequencies response to parameters is formulated. The results demonstrate that the vibration modes of the two-stage tandem hybrid planetary system can be classified as the fixed-axis train vibration mode, the differential train vibration mode, and the coupled vibration mode. Unequally spaced planet primarily disrupts vibration modes without significantly affecting natural frequencies. In contrast, the effects of mesh stiffness, planet mass and rotational inertia on the modes are opposite to those of unequally spaced planets. The validity of the parameter sensitivity and mode transition criterion is substantiated through illustrative examples.

Vibration behavior prediction of submerged nanobeams with axially traveling supports: numerical, analytical, and machine learning approaches

This work surveyed the vibration behavior and stability analysis of an underwater moving flexible nanosize beam, implementing numerical and analytical methods as well as tree-based machine learning (ML) algorithms. Dynamic modeling is performed based on the nonlocal stress-strain gradient theory (NSSGT), incorporating variable environmental conditions, surface energy, and rotational inertia effects. The natural frequencies and instability threshold are computed numerically. The exact closed-form mathematical expression for the critical towing velocity of the nanosize beam is determined analytically. Also, decision tree regression and least-squares boosting tree (LSBT) regression algorithms are exploited to predict stability boundaries and the fundamental vibration frequency, and their efficiency is surveyed accordingly. Comparative studies and parametric investigations are conducted. The impressions of geometry, added mass coefficient, scale ratio parameter, surface layer characteristics, fluid mass ratio, humidity, temperature rise, and external magnetic field intensity on the nanosystem dynamics are examined and illuminated. It is detected that the results of the analytical method and ML-based approaches are consistent with those reported by the numerical technique and literature. The results asserted that the proposed ML algorithms have acceptable performance and excellent effectiveness in computational time and cost. It is comprehended that the dynamic features of submerged traveling nanobeams drastically depend on the rotational inertia effects and thickness-dependent scale effects. The stability of the immersed movable nanoscale beams is perceived to improve by increasing the scale ratio parameter and the added mass coefficient. From the perspective of the optimal design of engineering instrumentation tools, the outcomes of the current article can be applied as an inclusive benchmark.

Free vibration analysis of tall buildings using a generalized continuous model: simplified approach to soil-structure interaction

Currently, due to the simplifications imposed on continuous models, there is an absence of a generalized approximate solution to investigate all possible dynamic behaviors of a tall building. In an attempt to address this deficiency, this paper proposes a new generalized continuous model and its solution methods to generalize all types of behavior that may be encountered in structural systems and tall buildings. The continuous model results from a composite beam that serially couples the classical sandwich beam and a pure shear beam. Unlike existing continuous models, the serial coupling, inclusion of new kinematic fields, and inclusion of rotational inertia allow for the resolution of complex interaction among different types of bending and shear behaviors. Specifically, it is shown how local shear behavior simultaneously interacts with global bending behavior, global shear, and local bending. Two solutions to the dynamic problem are presented: a closed-form analytical solution method for structures with uniformity along their height, and a numerical method that combines the advantages of the continuous method and the transfer matrix method to resolve structural and geometric variability in height. The transfer matrix is analytically solved by maintaining a fixed 6 × 6 matrix range that is independent of the number of levels of the structure, drastically reducing computational cost by eliminating the classical need to calculate the inverse of the zero matrix. Numerical applications investigate the effect of rotational inertia and local shear on the accuracy of the proposed results compared to results from the finite element method. To this end, a parametric analysis of 1560 different cases covering a wide range of typical behaviors in individual structural systems and tall buildings is conducted. The results exhibit promising accuracy, showing that the qualities of the proposed continuous model significantly improve precision compared to classical continuous models and therefore justify its practical application by academics and practicing engineers. Furthermore, the proposed mathematical approach is easily applicable in various domains of composite materials engineering.

Modeling and analysis of a macro-fiber piezoelectric bimorph energy harvester operating in d33-Mode using Timoshenko theory

Piezoelectric bimorph cantilevered beams are frequently employed in energy harvesting applications. The use of macro-fiber composites, which combine piezoelectric fibers with rectangular cross-sections and interdigitated electrodes, has significantly enhanced their performance by enabling operation in the 33-mode of piezoelectricity, thereby increasing conversion efficiency. In this article, the Timoshenko beam theory is applied to overcome the limitations of the Euler-Bernoulli theory in modeling transverse vibrations of harvesters with low slenderness ratios (length to thickness ratios). The mixture rule formulation is applied to determine the equivalent properties of the macro-fiber composite (MFC) structure. The electromechanical properties of a representative volume element (RVE) bounded by two successive interdigitated electrodes are coupled to the overall electro-elastic dynamic model of the structure using the Timoshenko theory. Frequency response functions for the power and tip vibration displacement of the MFC bimorph beam are determined from the model. Experimental data from the literature are used to validate the improved results predicted by the model developed in this article. A comparative analysis between the electromechanical responses predicted using a model based on Timoshenko and a model based on Euler-Bernoulli theory is conducted showing agreement, for harvesters with large slenderness ratios and discrepancies of up to 30% for low slenderness ratios. It has also been shown that the model based on Euler-Bernoulli theory overestimates both the mechanical and the electrical responses for low slenderness ratios as the shear and rotary inertia effects are not negligible in this case.

Free Vıbratıon of a Carbon Nanotube-Reınforced Nanowıre/Nanobeam wıth Movable Ends

ObjectiveThe purpose of this work is to investigate the size-dependent free vibrations of carbon nanotube-reinforced nanowires/nanobeams with movable ends. For this purpose, two movable end conditions are considered for carbon nanotube-reinforced nanowires/nanobeams with different carbon nanotube distributions. The size effect is addressed with the modified couple stress theory, which includes a material length scale parameter.MethodsIn this study, a solution approach based on the Fourier sine series and Stokes' transformation is used. With the help of this approach, both rigid and movable end conditions can be investigated. Firstly, equation sets consisting of infinite series and movable end parameters are derived. Then, eigenvalue problems are obtained for the free vibration of carbon nanotube-reinforced nanowires/nanobeams. The eigenvalues of these problems give the frequencies of the carbon nanotube-reinforced nanowires/nanobeams with movable ends.ConclusionsThe eigenvalue problems obtained in this study can be used to examine both rigid and movable end conditions. The accuracy of the problems obtained for solution is proven by various comparison studies. Then, a series of analyses are carried out for nanowire/nanobeam reinforced with carbon nanotubes, including both the size effect and the rotary inertia effect. When the frequencies of nanowire/nanobeams reinforced with carbon nanotubes are examined, it is understood that the material length scale parameter, carbon nanotube distribution, carbon nanotube volume fraction, rotational and lateral spring parameters can cause significant changes in free vibration.

Multimodal vibration damping of a three-dimensional circular ring coupled to analogous piezoelectric networks

Analogous piezoelectric networks have been shown to be effective for multimodal vibration attenuation in structures, including beams, plates, and rings. Previous studies for rings have only accounted for in-plane transverse vibration attenuation and disregarded the out-of-plane vibration modes. Furthermore, these previous numerical models and experiments have only been studied on thin rings, which ignore the effects of shear deformation and rotary inertia. As a result, these networks are not suitable for attenuating vibrations in thick rings. This study enhances on the previous electrical networks, considering both shear deformation and rotary inertia for both in-plane and out-of-plane vibrations. A new passive network topology is developed for the out-of-plane dynamics of thick rings, and the existing passive analogous network of in-plane vibration of thin rings is enhanced by considering the effects of shear deformation and rotary inertia. Combined, these new networks are capable of multimodal vibration damping of a thick ring in three-dimensions, encompassing primarily six types of vibration modes: the inextensional bending modes, the extensional modes, the thickness-shear modes, the coupled twist-bending modes, the torsional modes, and the transverse thickness-shear modes. By using piezoelectric elements to couple two separate analogous passive electrical networks derived from both the in-plane and out-of-plane governing equations of a ring and optimizing the internal resistance in each unit cell, it becomes possible to replicate the dynamics and effectively attenuate different types of vibration modes. This study serves as a theoretical foundation for implementations of passive vibration attenuation in ring structures.

An analytical model for the analysis of vibration and energy flow in a clamped stiffened plate using integral transform technique

Based on Kirchhoff thin plate and Mindlin thick plate theories, the vibration and energy flow characteristics of clamped stiffened plate are studied by using the analytical model constructed by finite integral transform method. The results show that the energy flow characteristics of the stiffened plate at the beam/plate coupling interface depend on the position of the rib in the vibration modes of the plate. The effects of shear deformation and rotatory inertia on the energy flow across the beam/plate coupling interface of the stiffened plate are further investigated. It is found that the inclusion of rotatory inertia of the beam and plate in the model only affects the energy flow component controlled by the moment coupling but not that controlled by the shear force coupling. Whilst the inclusion of the shear deformation of the beam and plate mainly causes a decreased amplitude of the energy flow for the mode group where the beam is located away from both the nodal and antinodal lines of modes, in addition to the shear deformation of the plate which also leads to an increased amplitude of the energy flow component controlled by the shear force coupling for the mode group where the beam locates at the antinodal line of modes. The understanding of energy flow characteristics of the stiffened plate at the beam/plate interface is essential to effectively control the noise and vibration problems of structures such as transformer tanks and machine covers.

An analytical investigation on nonlinear vibrations and stability of Timoshenko pipes conveying two-phase flow

The primary objective of this study is to conduct a comprehensive investigation on the nonlinear free vibration and stability analysis of the multilayer graphene reinforced pipe conveying liquid–gas two–phase flow based on the Timoshenko beam model. The Halpin–Tsai micromechanics model is employed which provides a means to evaluate and analyze the mechanical behavior and performance of the multilayer FG reinforced pipe. According to Von–Kármán strain relations and applying the Hamilton's principle, the governing equations of motion are derived. The Galerkin technique is operated to convert the partial differential equation (PDE) to the ordinary differential equation (ODE). The homotopy analysis method is employed to solve the fourth order nonlinear differential equation. Closed–form expressions for the nth nonlinear frequency and time history as well as the critical flow velocity based on the Timoshenko theory are achieved. The results demonstrate that neglecting the effects of rotary inertia and shear deformation may cause some errors by the increment of the initial amplitude. Furthermore, the critical flow velocity, depends not only on the mechanical and geometrical properties of the pipe but also on the initial mode shape excitation. The results indicate that by the increment of flow velocity, liquid density and slenderness ratio, the nonlinear frequency decreases, on the other hand, by the increment of the initial amplitude, the nonlinear frequency increases.

Grey wolf optimizer based-natural frequency optimization of laminated composite shells carrying distributed mass

In this study, a grey wolf optimizer is proposed to find the optimum stacking sequence to maximize the fundamental frequency of shells. The vibration analysis of the shells is done using first-order shear deformation theory and a nine-node isoparametric element with five degrees of freedom per node. A mass lumping scheme with rotary inertia is considered in the finite element formulation to include the rotary inertia effect. The design variable considered is the fiber angle orientation of the shells. The present approaches are compared with previously published results using layerwise optimization theory, the Ritz method, and classical lamination theory. A comparison study is also performed, comparing the GWO-based finite element model against six different metaheuristic approaches, and it demonstrates that the Grey Wolf Optimizer consistently outperforms these competing algorithms. The optimum fiber angle orientation of shells carrying distributed mass for various boundary conditions and shell types is analyzed for the first time, to the best of the author’s knowledge. This novel discovery will benefit future studies.

Dynamical behavior of a spinning tapered Rayleigh beam subjected to an axial load

As a first attempt, the vibration and stability analysis of a spinning tapered beam subjected to an axial load are investigated. The radius of the cross-section of the beam is assumed to vary exponentially in the longitudinal direction. The tapered beam is modeled in Rayleigh beam assumption with rotary inertia, eccentricity, and gyroscopic effects. Based on the Hamilton principle, the governing equations of motion of the spinning tapered beam with an axial force are obtained. Also, a Galerkin discretization scheme is employed to obtain the natural frequency and critical divergence rotating speeds of the system. As a result, the instability thresholds and stability evolution of the system are acquired. Furthermore, a detailed parametric study is conducted to evaluate the effects of the slenderness ratio, taper parameter, axial load, and eccentricity ratio on the dynamical behavior of the system. The results show that the natural frequency and critical divergence rotating speed of the tapered beam are larger than those of the circular-section beam. In addition, the axial load, taper parameter, and eccentricity are the key factors that influence the stability of the system.

Rotary inertia effect on dynamic analysis of embedded FG porous nanobeams under deformable boundary conditions with the effect of neutral axis

Rotary inertia effect on dynamic analysis of embedded FG porous nanobeams under deformable boundary conditions with the effect of neutral axis

Vibration of cylindrical shells: Design criteria for transition from shell modes to beam modes

The dynamic behavior of the cylindrical shell can be predicted by more simplified beam models for a wide range of applications. The present paper deals with finding design conditions in which the cylindrical shell performs like a beam. Employing the Hamilton’s principle, the governing equations are obtained using both Donnell-Mushtari and Flugge shell theories and the analytical solution is obtained for long cylinders with simply-supported boundary conditions at both ends. Then, by equalizing the shell and beam vibration frequencies, the shell-to-beam transition conditions are obtained for both theories. To account for the effects of shear distortion and rotatory inertia of the shell, the finite element method is applied to find the best transition conditions with less approximating assumptions. Finally, the effects of boundary conditions on the transition parameters as well as the frequency response are studied. The obtained conditions simply define that if the shell can be assumed as a beam in any specific geometrical and material conditions.

The effects of shear deformation and rotary inertia on the electrical analogs of beams and plates for multimodal piezoelectric damping

AbstractAnalogous electrical networks were previously derived from the Euler–Bernoulli and Kirchhoff–Love theories to represent beams and plates, respectively, for use in multimodal structural vibration damping. However, these networks do not account for shear deformations or rotary inertia, which can result in suboptimal vibration damping performance when used on moderately thick beams and plates. In this paper, we investigate the incorporation of shear deformation and rotary inertia using Timoshenko–Ehrenfest beam theory and Mindlin–Reissner plate theory to develop improved electrical networks that can more accurately represent thick beams and plates. Our findings suggest that the inclusion of shear deformation and rotary inertia can significantly improve the frequency coherence of the electrical networks and multimodal vibration damping for thicker structures. The electrical analogs presented here are of use for various applications, especially to conveniently design complex circuit topologies in fields spanning from vibration attenuation to energy harvesting.