Solution For Free Vibration Research Articles

Analytical free vibration solution of orthotropic thin plates with three edges rotationally restrained and one edge free

Abstract Orthotropic thin plates are essential in aerospace applications due to their high strength-to-weight ratio, which is critical for efficient structural performance. The mechanical properties of these structures are influenced by numerous factors, with free vibration characteristics being particularly significant. However, analytical solutions for free vibration are often limited to plates with simple boundary conditions (BCs). Few exact analytical solutions exist for plates with complex non-Lévy-type or rotationally restrained BCs. This study aims to analyze the free vibration behavior of orthotropic thin plates featuring three rotationally restrained edges and one free edge, employing the finite integral transform (FIT) method. A key advantage of this method is its simplicity and versatility, as it does not require the predefinition of a deflection function. By setting different values of the rotating fixed coefficient, the study offers analytical free vibration solutions for plates with various combinations of rotationally restrained, simply supported, clamped, and free BCs without additional derivations. The solutions derived using the FIT method are validated by comparison with numerical results obtained from ABAQUS software and available analytical solutions, confirming the method’s accuracy and reliability. Parameter analyses are conducted to examine the effects of aspect ratio, BCs, material properties, and rotating fixed coefficient on non-dimensional frequency parameters and corresponding vibration modes. The results reveal that these parameters significantly influence the frequency characteristics of orthotropic thin plates. The results can be used as a benchmark for validating other analytical and numerical methods.

Insight Into the Separation-of-Variable Methods for the Closed-Form Solutions of Free Vibration of Rectangular Thin Plates

Insight Into the Separation-of-Variable Methods for the Closed-Form Solutions of Free Vibration of Rectangular Thin Plates

Explicit Analytical Solution for Free Vibration of Restrained Orthotropic Plates

Explicit Analytical Solution for Free Vibration of Restrained Orthotropic Plates

Benchmark exact free vibration solutions of two-dimensional decagonal piezoelectric quasicrystal cylindrical shells

Abstract Quasicrystalline materials with piezoelectric effects show significant potential for advancing actuators, sensors and energy harvesters. In this paper, the free vibration characteristics of two-dimensional decagonal piezoelectric quasicrystal cylindrical shells (PQCSs) are investigated in the framework of symplectic mechanics system. By introducing an original vector and its dual variable vector as the fundamental unknowns, the governing equations are reduced into a set of low-order ordinary differential equations system, thus the free vibration analysis is transformed into an eigenvalue problem within the symplectic space. By using the symplectic mathematics, the exact solutions for free vibration of PQCSs are finally obtained and expanded as a series of symplectic eigensolutions. Finally, accurate natural frequency and analytical vibration mode shapes for arbitrary classical boundary conditions are obtained simultaneously. The accuracy of the obtained solutions is verified by comparing with existing results in open literature. In addition, the effects of geometrical parameters, temperature rise, external voltage and coupling fields on the natural frequency and vibration mode shapes are investigated in numerical examples. Results indicate that the phason field exhibits significant influences on the natural frequencies and cannot be neglected in free vibration analysis of PQCSs. Furthermore, all the results can be served as benchmarks for the development of new analytical and numerical approaches.

Series solutions for free in-plane vibration of composite plates with arbitrary shape

Series solutions for free in-plane vibration of composite plates with arbitrary shape

A boundary element solution for free vibration of FGM beams

The main purpose in this article is to derive a new solution based on Boundary Element Method (BEM) for dynamic analysis of Euler-Bernoulli beams made of composites such as Functionally Graded Materials (FGM) that have properties that vary continuously through the thickness direction according to the volume fraction of constituents. In contrast, laminated composite materials are oriented plies bonded together, where each ply may be made of a different material. A boundary element solution for any problem is obtained using integral equations and fundamental solutions defined on continuum, and algebraic equations for the discretized problem. In this paper, original discussions on deriving both integral equations and fundamental solutions for dynamic composite beam problems are properly made. Algebraic representation is also obtained, incorporating rigid and elastic end supports into the BEM solution. Numerical results of different cases of mechanical properties and boundary conditions are considered and compared to other published works.

Vibration Characteristic Analysis of Sandwich Composite Plate Reinforced by Functionally Graded Carbon Nanotube-Reinforced Composite on Winkler/Pasternak Foundation

This paper presents numerical investigations into the free vibration properties of a sandwich composite plate with two fiber-reinforced plastic (FRP) face sheets and a functionally graded carbon nanotube-reinforced composite (FG-CNTRC) core made of functionally graded carbon nanotube-reinforced composite resting on Winkler/Pasternak elastic foundation. The material properties of the FG-CNTRC core are gradient change along the thickness direction with four distinct carbon nanotubes reinforcement distribution patterns. The Hamilton energy concept is used to develop the equations of motion, which are based on the high-order shear deformation theory (HSDT). The Navier method is then used to obtain the free vibration solutions. By contrasting the acquired results with those using finite elements and with the previous literature, the accuracy of the present approach is confirmed. Moreover, the effects of the modulus of elasticity, the carbon nanotube (CNT) volume fractions, the CNT distribution patterns, the gradient index p, the geometric parameters and the dimensionless natural frequencies’ elastic basis characteristics are examined. The results show that the FG-CNTRC sandwich composite plate has higher dimensionless frequencies than the functionally graded material (FGM) plate or sandwich plate. And the volume fraction of carbon nanotubes and other geometric factors significantly affect the dimensionless frequency of the sandwich composite plate.

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.