Solution For Free Vibration Research Articles

Free vibration analysis of rectangular plates resting on an elastic foundation

Abstract Rectangular plate on elastic foundations is a common structural form in aerospace fields, while there still lacks a general analytical method for free vibration analysis of this kind of structure. Thus, the extended separation-of-variable method is used to obtain the analytical free vibration solutions for plates with all kinds of homogeneous boundary conditions. The mode functions are assumed to be in separation-of-variable form, and the natural frequencies of two-direction mode functions are mathematically independent. Then, the free vibration solutions are obtained according to the two-direction boundary conditions. The correctness of the present results is verified via numerical comparisons. Parameter study shows that the elastic foundation significantly affects the flexural vibration behaviors of rectangular plates, and the influence is dependent on boundary conditions and mode orders of rectangular plates.

New benchmark free vibration solutions of passive constrained layer damping beams by the symplectic method

New benchmark free vibration solutions of passive constrained layer damping beams by the symplectic method

Free vibration of doubly-curved panels reinforced by carbon nanotubes: New analytic solutions under non-Lévy-type boundary conditions

Existing analytic solutions for the free vibration of functionally graded carbon nanotube reinforced doubly-curved panels primarily address the cases with two parallel simply supported boundaries, known as Lévy-type boundary conditions (BCs). However, doubly-curved panels with non-Lévy-type BCs are more commonly encountered in practical engineering applications, yet their analytic solutions are rarely available due to significant mathematical challenges. This gap motivates us to develop new analytic free vibration solutions under these more complex BCs. The nanocomposites’ material properties are first computed according to the rule of mixture. The Hamiltonian-system governing equation for the free vibration of doubly-curved panels is then formulated from the Donnell-Mushtari theory, and is solved by adopting the analytic symplectic superposition method. The obtained analytic solutions are derived without requiring predefined solution forms, and have been thoroughly validated by comparison with the results from the finite element method. By utilizing the accurate analytic solutions, the effects of aspect ratios, BCs, types of CNT distributions, and volume fractions of CNT on the free vibration behaviors are further analyzed. The present solution procedure and the resulting analytic solutions are expected to be useful for dynamic modeling of composite shell panels, supporting both future research and practical applications.

New Analytical Solutions for Free Vibration of Cracked Functionally Graded Rectangular Plates

In aerospace engineering, the existence of cracks highly alters the mechanical behaviors of structural components. As a new class of key structural components, functionally graded material (FGM) rectangular plates with an interior/side crack are focused on in this study. New analytical free vibration solutions are put forward for such cracked FGM plates. First, a cracked plate is decomposed into several subplates with the necessary continuity conditions to address the physical discontinuity issue caused by the crack. The challenge then becomes deriving analytical free vibration solutions for each subplate. To accomplish this task, a symplectic superposition method (SSM) is developed to deal with FGM rectangular plates with various boundary conditions, including the complex continuity conditions. The SSM eliminates the need for a predefined solution form, and its solution procedure, including variable separation and eigen expansion, is mathematically rigorous. Afterward, the analytical solutions obtained via the SSM are integrated to serve as the final free vibration solution of the cracked FGM plate. Comprehensive results based on the obtained analytical solutions, including natural frequencies and vibration modes, are presented and validated. The solutions are further used to investigate the effects of some important parameters, such as crack length and crack location.

Analytic analysis of free vibration problem of the plate with a rectangular cutout using symplectic superposition method combined with domain decomposition technique

Plates with rectangular cutouts is widely seen in the field of engineering structures. Therefore, it is crucial to examine analytical solutions for free vibration (FV) of these structures. Despite the existence of approximate/numerical methods, analytical solutions are lacking in the literature. In this study, we employ the symplectic superposition method to effectively analyze the FV problems encountered in plates with rectangular cutouts while integrating the domain decomposition technique. To address the issue of irregular geometry, the rectangular cutout plate is divided into multiple sub-plates. By dividing the problem into multiple sub-problems and solving them separately using variable separation and symplectic eigen expansion, we obtain analytical solutions. Finally, we combine the sub-problem solutions to resolve the initial issue. This solution method can be considered a logical, analytical, and systematic approach as it starts with the fundamental governing equation and is derived without assuming forms of solutions. The study presents a comprehensive set of numerical results that include mode shapes (MSs) and natural frequencies (NFs). The results are rigorously validated using the finite element method (FEM) and relevant literature. The symplectic superposition method demonstrates excellent convergence and precise accuracy, making it suitable for analytically modeling more complex mechanical problems of plates.

Symplectic analytical solutions for free vibration of elastically line-hinged orthotropic rectangular plates with rotationally restrained edges

The elastically line-hinged orthotropic rectangular plates with rotationally restrained edges are commonly used in engineering applications such as deployable structures. However, it is intractable to obtain the analytical solutions for the free vibration problems of such structures owing to the challenges in processing the high-order partial differential equations. Here, we make a first attempt to deal with such issues within the Hamilton system-based symplectic space. The problem of a plate is transformed into the symplectic space from the Euclidean space, and the subplates that may be analyzed analytically by the symplectic superposition method are then obtained with the division of the entire plate. The complex boundary and connection forms are achieved by enforcing mechanical quantities with undetermined expansion coefficients on the edges of the subplates. By integrating the solutions of the subplates, the final solution of an elastically line-hinged orthotropic rectangular plate with rotationally restrained edges is accessible. The proposed solution scheme is performed with rational derivations, with no requirement for pre-defined solution forms. Comprehensive results with validations under various boundary and hinge connection cases are presented. Moreover, detailed parametric investigations are conducted, which could facilitate the engineering design of deployable structures.

Approximate analytical solutions of nonlinear vibration and bifurcation of dielectric elastomer balloon

Dielectric elastomers (DE), a novel class of actuators, are increasingly employed in fields such as soft robotics and vibration control. Most existing studies investigating the dynamic characteristics of DE structures neglect the viscoelasticity and strain-stiffening effects of materials, with only a few offering approximate analytical solutions. This paper establishes the governing equation for the dynamics of a DE balloon, utilizing the principle of virtual work and considering material viscoelasticity and strain-stiffening effects through linear damping and the Gent hyperelastic model, respectively. The nonlinear governing equations are simplified using the Chebyshev polynomial approximation. The multiple scales method is applied to derive approximate solutions for the structure's free vibration under static voltage excitation and parametrically excited vibration under harmonic voltage excitation. The accuracy of these approximate analytical solutions is validated by comparison with numerical solutions. Additionally, the paper analyzes stability and bifurcation phenomena through time-domain response curves, amplitude-frequency curves, and phase trajectories. Furthermore, the influence of the stretching limit, voltage, and damping on the nonlinear vibration of the balloon structure is studied, providing a valuable theoretical foundation for the future design and utilization of dielectric elastomers.

Parametric Analysis of Free Vibration of Functionally Graded Porous Sandwich Rectangular Plates Resting on Elastic Foundation.

Based on the three-dimensional elasticity theory, the free vibration of functionally graded porous (FGP) sandwich rectangular plates is studied, and a unified solution for free vibration of the plates is proposed in this study. The arbitrary boundary conditions of FGP sandwich rectangular plates are simulated by using the Rayleigh-Ritz method combined with artificial spring theory. The calculation performances of the unified solution for FGP sandwich rectangular plates such as convergence speed and computational efficiency are compared extensively under different displacement functions. In addition, three kinds of elastic foundation (Winkler/Pasternak/Kerr foundations) and three porosity distributions are considered. Some benchmark results and accurate values for the free vibration of FGP sandwich rectangular plates resting on elastic foundations are given. Finally, the effects of diverse structural parameters, elastic foundations with different parameters, and boundary conditions on the free vibration of the FGP sandwich rectangular plates are analyzed.

New exact free vibration solution of two-dimensional decagonal quasicrystal cylindrical shells in the symplectic framework

An accurate free vibration analysis of two-dimensional decagonal quasicrystal (QC) cylindrical shells is performed under the framework of symplectic mechanics. Unlike the classical Lagrangain system, the governing equations are reduced into a set of lower order ordinary differential equations, and therefore exact solutions of displacements, rotation angles, internal forces and bending moments in the phonon and phason fields are simultaneously obtained and expressed in terms of symplectic vectors. The present results are compared with existing literature and finite element solutions, and excellent agreements are observed. Furthermore, the effects of key influencing factors on the free vibration characteristics are revealed also.

Vibration Analysis of Porous Functionally Graded Material Truncated Conical Shells in Axial Motion

In this study, a vibration equation for axially moving truncated conical thin shells made of functionally gradient materials with uniformly and non-uniformly distributed pores has been established based on classical thin shell theory. The free vibration and dynamic response solutions are obtained using the Galerkin method. The effects of axial velocity, half cone angle, ceramic material mass composition, material component index, and internal porosity on the free vibration and dynamic response of mentioned shells were analyzed and discussed. The results show that the increase of axial velocity, half cone angle, and material composition index all decrease the natural frequency of the truncated conical shell but amplify its dynamic response, and the rise of the mass fraction of the ceramic material increases the natural frequency of the truncated conical shell but reduces the dynamic response. The results also demonstrate that compared with non-uniformly distributed pores, the effects of uniformly distributed pores on the shells’ dynamic responses are more evident under axial motion.

Semi-analytical solutions for forced and free vibration of damped fluid-conveying pipe systems based on complex modal superposition method

In this paper, the response solutions of the damped fluid-conveying pipe system with elastic torsion constraints at both ends are analyzed. The pipe system considering gyroscopic effect induced by internal flow and damping effect is a typical damped gyroscopic system. This system cannot be decoupled in the modal space by the traditional modal analysis, and then the semi-analytical response solutions cannot be obtained by the classical modal superposition method. In order to remedy this problem, this paper proposes a new method based on complex modal superposition method and state-space method to give the semi-analytical response solutions of the damped fluid-conveying pipe system. The adjoint system is introduced through the concept of adjoint operator, and the orthogonality conditions of the damped fluid-conveying pipe system are derived in the state-space by using the eigenvalue relationship between the original system and the adjoint system. The Laplace transformation method is used to solve the eigenvalue problems of the original system and the adjoint system, and the implicit function equations about the eigenvalues and analytical eigenfunctions are obtained. According to the obtained orthogonality conditions, the semi-analytical response solutions of the system under arbitrary initial conditions and excitations are given, and the obtained solutions include transient and steady-state response solutions. In the numerical discussion part, the reliability and accuracy of the present method are verified. This research has positive significance for the dynamic analysis of fluid-conveying pipe systems, and the obtained orthogonality conditions have potential value for the nonlinear dynamics research.

Approximate closed-form solutions for free vibration of circular arches with discrete lateral braces

Discrete lateral braces are commonly employed to enhance the out-of-plane stability of arches. However, these braces generate coupling effects among vibration modes, which significantly impact the vibration characteristics of the arches. This paper presents an analytical solution for the free vibration analysis of two-hinged arches with equally-spaced lateral translational and/or rotational braces. A simplified method of evaluating the fundamental frequency for the out-of-plane vibration of arches with equally-spaced lateral braces is introduced. The method relies on the conventional approach of representing arch braces through elastic springs, but here the out-of-plane deflection of the braced arch is constructed by adding related coupled sinusoidal functions to that of the unbraced arch, satisfying both the boundary and kinematic conditions. Subsequently, the fundamental frequency and threshold bracing stiffness or stiffness requirement are derived using the Rayleigh–Ritz method. Finite element analysis (FEA) is applied to investigate the flexural-torsional vibration mode and fundamental frequency of arches with discrete braces, and the results from FEA closely match the theoretical predictions. The effects of number of lateral translational and/or rotational brace, out-of-plane slenderness, included angle, bracing stiffness and bracing position on the fundamental frequency are comprehensively analyzed. It is found that when the fundamental frequency of the braced arch increases due to changes in bracing stiffness, the mode shape displays more intricate behavior, rather than following a straightforward low-to-high mode growth pattern. Furthermore, it is observed that in some cases, the addition of rotational bracing stiffness can markedly enhance the maximum fundamental frequency of the arch with both translational and rotational braces.

On the variable length scale parameter in functionally graded non-porous and porous microplate/nanoplate

In this article, the solution for free vibration and static bending of nonporous and porous square microplate/nanoplate is studied by practicing the modified couple stress theory (MCST). The impact of incorporating the length scale parameter of MCST is confirmed by comparing the results to that of molecular dynamics simulations. This study is aimed at investigating the influence of implementing a variable length scale parameter on the static and dynamic behavior of the small-scaled plates. Results are obtained adopting the analytical technique and compared to cases of presupposing constant numerical values of material length scale throughout each plate. The novelty of this theoretical study is the approach employed to achieve the formulation of the material length scale parameter, which varies along the thickness of both porous and nonporous plates. The results indicate that the inclusion of the variable length scale parameter increases the fundamental frequency and stiffness of plates. When comparing two cases of variable length scale parameter, one approximating the scale parameter of each constituent based on the bending parameter and Poisson’s ratio of the material, and the other adopting the numerical values estimated by prior experimental studies, reveals that the latter gives higher results and is chosen correspondingly.

New Analytic Free Vibration Solutions of L-Shaped Moderately Thick Plates by Symplectic Superposition

L-shaped moderately thick plates have widespread applications in diverse engineering structures. Exploring the benchmark analytic solutions for the free vibration of L-shaped moderately thick plates is important in order to accurately analyze and efficiently design structures. Nevertheless, analytical solutions, which serve as the benchmarks, have been rarely documented in previous literature due to the challenge of finding suitable solutions that satisfy both the governing higher-order partial differential equations (PDEs) and the boundary conditions of the plates. The symplectic superposition method was employed in our recently published study to present the benchmark vibration solutions of clamped rectangular moderately thick plates. In this study, we expand upon this method to solve the free vibration problem of L-shaped moderately thick plates. By employing domain decomposition, we construct an irregular domain by combining multiple rectangular domains. First, the construction of the superposition system is carried out, followed by the import of the sub-problems into the Hamiltonian system, utilizing the fundamental governing equations of the plate. Then, the sub-problems are resolved through the application of the symplectic geometry methodology in an analytical manner. Ultimately, the analytical solutions for frequencies and mode shapes are derived through ensuring the equivalence between the initial problem and the combination of sub-problems. The finite element method is used to validate and present a comprehensive analysis of the natural frequencies and mode shapes obtained from this method. This method possesses the benefits of rapid convergence and accurate precision, rendering it well suited for the analytic modeling of a broader range of plate-related problems.

Numerical and Analytical Solution for Nonlinear Free Vibration of Tapered Beams

Numerical and Analytical Solution for Nonlinear Free Vibration of Tapered Beams

Analytical and numerical solution for free vibrations of laminated composite plates

Analytical and numerical solution for free vibrations of laminated composite plates

Analysis of free vibration in thin-walled plates using an enhanced polygonal plate element with selective interpolation approach

A highly efficient selective element-interior interpolation approach is proposed that impressively enhances the performance of a spatially isotropic polygonal element (SI-ARS-Poly) and provides outstanding solutions in the displacement and energy norms for the static analysis of thin-walled plate structures (Nguyen and Phan (2023) [22]). In this approach, the element-edge rotations and shear strains are respectively interpolated into interior of polygonal element using linear and quadratic approximations. In order to comprehensively explore the advantages of this element, we extend it to free vibration analysis. Numerical results demonstrate that the proposed approach is stable in the free vibration analysis of plates with smoothed mode shapes. Notably, its solutions for free vibration remain robust in the face of mesh distortion. In comparison with other existing polygonal elements, it is superior with the lowest frequencies closely approximate to the exact/reference ones.

Differential-integral quadrature numerical solution for free and forced vibration of bidirectional functionally graded Terfenol-D curved beam

The dynamic behaviour of a bidirectional functionally graded Terfenol-D curved beam under a moving load is the focus of this study. Combined differential and integral quadrature solve the curved beam boundary value problem. A bidirectional functionally graded Terfenol-D curved beam's damping, damped frequencies for different boundary conditions, and mode shapes were examined in free vibration investigations. Based on accessible literature data, the findings are reliable. Furthermore, DQ-IQ numerical technique findings examine the impact of many factors, including control gain, thickness, moving load velocity, bidirectional gradation index, and multi-moving load.

Analytical and isogeometric solutions of flexoelectric microbeams based on a layerwise beam theory

The theory of layerwise beam, which considers both microstructure and flexoelectric effects, is used in the curvature-based flexoelectricity theory. The displacement field function of the layerwise beam model is given. The governing equations and boundary conditions are obtained using a variational formulation based on Hamilton's principle. Analytical solutions for static bending and free vibration of a simply supported layerwise beam are derived. Isogeometric analysis (IGA) with higher-order continuity considering the flexoelectric effect is presented to numerically solve the static bending and free vibration problems. The results obtained from the analytical and IGA approaches are compared through several examples in which the microstructure and flexoelectric effects on the mechanical responses of layerwise nanobeams are analyzed.

Magneto-thermo-elastic coupled free vibration and nonlinear frequency analytical solutions of FGM cylindrical shell

The magneto-thermo-elastic coupled free vibration of functionally graded materials (FGM) cylindrical shell is investigated. Based on the physical neutral surface theory and Kirchhoff-Love theory, the expressions of internal force and internal torque are obtained by considering the constitutive relationship that includes thermal stress. According to the electromagnetic elasticity theory, the model of Lorentz forces of the shell in magnetic field are derived. The expressions of potential energy, kinetic energy and its variational operators are given by introducing geometric nonlinearity, respectively. Based on Hamilton principle, the magneto-thermal-elastic coupled vibration equation of FGM cylindrical shell in multi-physical field is established. The displacement functions under simply supported boundary conditions are solved, which are combined with Galerkin method for derivation of nonlinear ordinary differential equations. The second-order approximate analytical solution of natural frequency is obtained by applying the multi-scale method. The characteristic curves of natural frequency with different parameters are drawn through numerical examples. The results show that reducing the volume fraction index and increasing the initial amplitude can lead to the increase of the natural frequency under transient conditions. With the increase of temperature and magnetic induction intensity, the natural frequency decreases. In addition, the natural frequency is also influenced by the shell size and volume fraction index.