Large Amplitude Free Vibration Research Articles

Large Amplitude Free Vibration of Spherical Dome Structures with Variable Thickness

Large Amplitude Free Vibration of Spherical Dome Structures with Variable Thickness

Large Amplitude Free Vibration of Elastically Restrained Tapered Beams Resting on Non-linear Elastic Foundation

Large Amplitude Free Vibration of Elastically Restrained Tapered Beams Resting on Non-linear Elastic Foundation

Nonlinear Vibration of Cracked Porous FG-GPL RC Cylindrical Panels Using a Phase-Field Crack Model

This study is concerned with the nonlinear free vibration of a cracked functionally graded porous cylindrical panel reinforced with graphene platelets by introducing a phase-field crack model. Conventional crack modeling by separating the grid nodes lying on the crack line is not only painstaking but also suffers from numerical instability. To overcome this problem, the internal crack is modeled by adopting the phase-field formulation and a virtual geometry rotation. The nonlinear numerical method is developed based on the first-order shear deformation theory incorporated with the von Kármán geometry nonlinearity in the framework of the 2-D extended natural element method, a recently introduced mesh-free method. The crack-induced singular field is represented by adopting the crack-tip singular functions, and the troublesome numerical locking is restrained by combining the MITC3+ shell concept and the shear stabilization factor. The curved shell surface is mapped to a 2-D rectangular NEM grid to avoid difficulty in defining the interpolation functions. The developed numerical method is verified through a comparison with the reference solutions, and the large-amplitude free vibration of porous cracked functionally graded grapheme platelet-reinforced cylindrical panels is profoundly examined by changing the major parameters.

Effect of a Four-Parameter Kerr-Type Elastic Foundation on the Large-Amplitude Free Vibration of the Nanocomposite Beams: A Dual-FG Modeling

This study examines the free vibration of a unique nanocomposite beam, characterized by large-amplitude motions. The composition of the material comprises functionally graded (FG) polymers fortified with graphene platelet (GPL) nanofillers, forming a nanocomposite structure with a dual FG designation. The arrangement of graphene platelet nanofillers within the dual-phase matrix follows one of five linear functionally graded models. To ascertain the behavior of the nanocomposite, the homogenization process employs the Voigt rule of mixtures for the power law-based FG matrix, while the Halpin–Tsai homogenization procedure is employed for the functionally graded graphene platelet (FG-GPL) reinforced nanocomposite. The beam is supported by a novel four-parameter nonlinear Kerr elastic foundation, located at the beam’s lower surface, which governs the linear and nonlinear frequencies. Employing the Karama higher-order shear deformation theory, the beam’s displacements are estimated, and the strains are described by the Von-Kármán geometrically nonlinear relations. Solving the problem necessitates the utilization of the generalized differential quadrature (GDQ) method, coupled with a bilateral iterative displacement control (BIDC) technique. Notably, the equations are rendered independent of the time variable implementing the weighted residual Galerkin procedure. The obtained frequencies of the GPL-reinforced beam with an isotropic homogeneous matrix are compared against those documented in existing literature, ensuring validation. Subsequently, parametric results are presented to assess the impact of the novel nanocomposite features on the linear and nonlinear frequencies of beams.

Large amplitude free vibration analysis of circular arches with variable thickness

This study aims to investigate the large amplitude nonlinear vibration of variable thickness arches with different cross-sectional shapes under different boundary conditions. The governing equations of the arches are derived using Hamilton's principle by considering their geometric nonlinearities. Then, the differential quadrature method (DQM) is employed for the discretization to obtain the ordinary differential equations of motion for the circular arches with variable thickness. The theoretical model is validated by comparing the linear and nonlinear natural frequencies of arches. The nonlinear vibration characteristics of arches are solved using the direct iteration scheme. The effects of the slenderness ratio, modified slenderness ratio of the arch, and variable thickness parameters under different cross-sectional shapes on the nonlinear to linear frequency ratios of the arches with clamped and pin-ended boundary conditions are investigated. The results indicate that appropriate variable thickness parameters can improve the linear frequency and the nonlinear to linear frequency ratio of arches, which offers some suggestions for the design of variable thickness arches.

Large amplitude free vibrations analysis of prismatic and non-prismatic different tapered cantilever beams

Large amplitude free vibrations analysis of prismatic and non-prismatic different tapered cantilever beams

Large amplitude free vibration of sandwich beams with flexible core and FG Graphene Platelet Reinforced Composite (FG-GPLRC) face sheets based on extended higher-order sandwich panel theory

The present article deals with the nonlinear free vibration analysis of sandwich beams with flexible core integrated with face sheets which are reinforced with graphene platelets (GPLs). Each layer of the composite media may have a different amount of graphene platelets which leads to a functionally graded (FG) distribution. The elastic modulus of the reinforced composite (RC) is calculated via the Halpin–Tsai model. Following the extended higher order sandwich panel theory (EHSAPT) which captures the core thickness compressibility effect and the axial stress of the core, the complete set of governing equations is obtained using the Lagrangian formulation. To analyze the effect of the inclusion of nonlinear terms in the core, hitherto not reported in the literature, results are computed considering the von Kármán type of geometrical nonlinearity assumptions for the core and face sheets. With the aid of the Ritz-based method, Chebyshev functions are implemented as admissible functions of the displacement fields. Furthermore, the nonlinear governing equations are solved using a direct displacement control technique. As the EHSAPT can be applied with any combination of the core and face sheets and not only the soft cores that the other theories demand, the results for the sandwich beam with stiff core and consist of functionally graded carbon nanotube-reinforced composite face sheets provided for validation. Since this study is the first in the open literature to develop nonlinear vibration of the sandwich beam based on EHSAPT, Timoshenko beam theory (TBT) is also applied to investigate the problem and to extract nonlinear frequencies with various stiffness of the core, meanwhile, the results are compared with those obtained by the EHSAPT. Parametric studies are conducted in detail to examine the influences of weight fraction and distribution pattern of graphene platelets, core to face sheet thickness ratio, boundary conditions, length to thickness ratio, and amplitude of vibrations. It is found that by increasing the stiffness of the core, the inclusion of nonlinear effects in the core as well as the face sheet strain nonlinearity is more significant. Finally, the nonlinear vibration analysis of sandwich beams can change the variation trend of frequency predicted in the linear analysis, and the different trends are observed depending on the vibration amplitude.

Nonlinear Large-Amplitude Oscillations of PFG Composite Rectangular Microplates Based Upon the Modified Strain Gradient Elasticity Theory

In this research work, the nonlinear large-amplitude free vibration characteristics of composite microplates made of a porous functionally graded (PFG) material are addressed numerically in the presence of different size-dependent strain gradient tensors as microscale. Accordingly, for the first time, the effect of each microstructural tensor is analyzed separately on the nonlinear free oscillations of PFG microplates with and without a central cutout. In order to fulfill this goal, the isogeometric computation approach is engaged to integrate the finite element approach into the nonuniform B-spline-based computer aided design tool. Accordingly, the geometry of the microplate with a central cutout is modeled smoothly to verify C−1 continuity based upon a refined higher-order plate formulations. In this regard, the microstructural-dependent frequency responses associated with the nonlinear free oscillations of microplates are traced. In both the cases of simply supported and clamped boundary conditions, it was revealed that the fundamental frequency is enhanced about 1.20% by considering only the symmetric rotation gradient tensor, about 3.27% by taking only the dilatation gradient tensor, and 9.43% by considering only the deviatoric stretch gradient tensor. On the other hand, the anisotropic character of PFG composite microplates results in an unsymmetrical frequency response curve, as the nonlinear frequencies associated with negative oscillation amplitudes are a bit higher than those of positive ones.

On the large-amplitude vibration of rotating pre-twisted graphene nanocomposite blades in a thermal environment

The large-amplitude free vibration behaviors of rotating pre-twisted functionally graded (FG) nanocomposite multilayered blade reinforced with graphene platelets (GPLs) in a thermal environment is comprehensively investigated in the paper. It is assumed that the GPLs volume fraction varies gradually from layer to layer across the blade thickness direction but remains constant in each individual GPLs-reinforced composite (GPLRC) layer. The modified Halpin-Tsai micromechanical model is exploited to obtain the effective Young’s modulus of each GPLRC layer while Poisson’s ratio, mass density, and thermal expansion coefficient are determined via Voigt’s rule. The nonlinear dynamic model of pre-twisted GPLRC blades at moderate rotational speed in a thermal environment is derived based on the first-order shear deformation theory (FSDT) combined with the improved version of the Novozhilov nonlinear shell theory. The nonlinear vibration characteristics of FG GPLRC blades are obtained by applying the element-free improved moving least-square Ritz (IMLS-Ritz) method in conjunction with a direct iterative scheme. Validation of obtained results is performed in detail and show effectiveness of used numerical approach and fast convergence of obtained results to exact values. The impacts of rotating speed, pre-twisted angle, presetting angle, and aspect ratio on nonlinear to linear frequency ratio of GPLRC blades are studied with emphasis on temperature rise and nano reinforcement parameters. Finally, new insights into dynamics of nanocomposite rotating blades with conclusions are presented in detail.

Experimental investigations on post-flutter performance of a bridge deck sectional model using a novel testing device

This paper studies the flutter performances of a streamlined bridge deck sectional model using a recently developed testing device for large-amplitude free vibration test. The reported torsional vibration amplitude of the sectional model exceeds 15°, and higher vibration amplitudes can be achieved if desired. The flutter performances at various initial angles of attack α0, the hysteresis behavior, and the energy budget property of the flutter response are analyzed. Experimental results show that the sectional model exhibits divergent vibrations right after the critical flutter velocities for α0 = 0°, −3°, and −5°, while it performs limit cycle oscillations (LCO) after the critical flutter velocities for α0 = +3° and +5°. Hysteresis phenomena are observed for all tested angles of attack. Both the critical flutter velocity and the vibration amplitudes around the critical velocity may be affected by the initial impulse excitation exerted on the bridge deck. The vertical-to-torsional amplitude ratio decreases with increasing the vibration amplitude for a coupled LCO, while the opposite rule is observed for a divergent vibration. The energy dissipated by the mechanical damping accumulates continuously while the total mechanical energy varies non-monotonously with increasing time. This study is expected to deepen the understanding of the flutter responses of a bridge deck in the presence of nonlinear aerodynamic effects. The experimental results can also serve as a database for validating computational fluid dynamics procedures and nonlinear self-excited force models.

Large Amplitude Free Vibration Analysis of Nanobeams Based on Modified Couple Stress Theory

In this study, the large amplitude free vibration of nanobeams based on the modified couple stress theory was developed by using Total Lagrangian finite element formulation. In this study, Timoshenko beam theory has been used in free vibration analysis of nanobeams. Minimal kinematic assumptions have been used to model nanobeams. With this model, free vibration of nanobeams with small to large amplitude and with arbitrary boundary conditions can be analyzed. The numerical results obtained for free vibration based on the modified couple stress theory with small amplitude and the results obtained for free vibration with large amplitude without considering the modified couple stress theory are in good agreement with the similar results reported in previous research. Effects of the dimensionless length scale parameter, slenderness ratio, vibration amplitude and different boundary conditions on the nonlinear frequency ratio of nanobeams have been investigated. The results show the importance of considering nonlinear and size effects in the free vibration analysis of nanobeams with large amplitude.

Large amplitude free vibration of porous skew and elliptical nanoplates based on nonlocal elasticity by isogeometric analysis

The current work presents an investigation on size-dependent nonlinear free vibration of porous skew and elliptical nanoplates. Formulations are based on a four-variable higher-order shear deformation plate theory to satisfy automatically the traction-free condition, von Kármán geometric nonlinearity to account for the mid-plane stretching, and Eringen's nonlocal elasticity theory to capture the scale effect. The isogeometric analysis and a displacement control strategy are employed synthetically to obtain large amplitude responses of the nanoplates. New results show the influences of porosity distribution pattern, porosity coefficient, and nonlocal parameter on the behaviors of vibration behaviors of the skew and elliptical nanoplates.

Large-amplitude vibration analysis of a strong nonlinear tapered beam using continuous piecewise linearization method

The large-amplitude free vibrations of an inextensible tapered beam with inertia and static nonlinearities were investigated using the continuous piecewise linearization method (CPLM). The CPLM solution was found to be in excellent agreement with exact numerical solutions and more accurate than other published approximate solutions. The study focuses on the large-amplitude vibrations where the nonlinear effects are strong and the response is characterized by anharmonicity. The present investigation showed that the CPLM solution predicted the anharmonic response due to strong nonlinearity whereas the other published solutions showed large errors. The effects of inertia and static nonlinearities were investigated and it was observed that an increase in the static nonlinear stiffness increases the vibration frequency whereas an increase in inertia nonlinearity reduces vibration frequency. Also, the combined effect of the inertia and static nonlinearities on the large-amplitude vibration response was investigated. It was observed that an increase in the ratio of the static to inertia nonlinearity increases the vibration frequency and vice versa.

Isogeometric analysis for size-dependent nonlinear free vibration of graphene platelet reinforced laminated annular sector microplates

The proposed work fills a gap of study on size-dependent large amplitude free vibration of functionally graded graphene platelets-reinforced composite (FG-GPLRC) annular sector microplates. Based on a four-variable higher-order shear deformation plate theory, the von Kármán large deflection assumption and the modified couple stress theory (MCST), the governing equations for the discrete nonlinear free vibration of the multilayer FG-GPLRC annular sector microplate without considering the external loads are established by the principle of virtual work. The NURBS-based isogeometric analysis (IGA) in conjunction with a displacement control strategy are utilized synthetically to acquire linear natural frequencies and nonlinear-to-linear natural frequency ratios for the microplates numerically. Comparisons and convergence study are carried out to verify accuracy and correctness of IGA and iteration method in present formulation. Detailed parametric studies are performed to present an insight into the influences of distribution pattern and weight fraction of the graphene platelets, sector angle and material length scale parameter on the behaviors of linear natural frequencies and large amplitude responses of the microplates under different boundary conditions.

Large amplitude vibrations of imperfect spider web structures

Due to the high-efficiency energy absorption and high-tension strength material properties of spider silk, many researchers have studied the mechanical properties and microstructure of the spider web. The concept of spider web structure has been recognized to be adopted for structural engineering aspect. The structure of spider web and its material properties have been studied for decades. However, the fundamental free vibration mode shapes and their corresponding frequencies have never been fully investigated. This study investigates the nonlinear characteristics in the large-amplitude free vibration of imperfect spider web structures using finite element analysis. The spider web applies the concept of elastic cables taking only axial deformation into account. The finite element models of a spider web considering geometric nonlinearities are employed. It should be noted that spider web could experience large deformation when the spider uses its silk to catch prey. This research aims at analyzing the linear and geometric nonlinear behaviour of imperfect spider web structure. Four different types of imperfect spider web: spiral imperfect spider web, radial imperfect spider web, central imperfect spider web, and circular rings imperfect spider web, are considered. It is found that pretension in spider silk plays a significant role in nonlinear vibration characteristics of the spider web. Moreover, the radial thread damaged tends to have a greater effect on structural free vibration of spider web in comparison with other imperfections. The outcome will help a structural engineer to adapt the concept of spider web, its properties, and damage patterns for any larger structures.

Large Amplitude Free Vibrations of FGM Beams on Nonlinear Elastic Foundation in Thermal Field Based on Neutral/Mid-plane Formulations

The present research deals with the small and large amplitude free vibrations of thick functionally graded material (FGM) beams. Beams are resting on an elastic foundation which consists of the Winkler layer, shear layer and nonlinear springs. The effect of uniform temperature elevation is also included into the formulation. A third-order shear deformation beam theory is adopted as the basic assumption of the formulation which satisfies the traction-free boundary conditions on the top and bottom surfaces. Using two different formulations, i.e. the mid-surface-based formulation and the neutral surface-based formulation, the governing equations of motion are established. For the case of beams with immovable edges in in-plane direction and simply supported in flexure, the governing equations are given in a new presentation. These equations are solved using the two-step perturbation technique. Closed-form expressions are provided to obtain the small and large amplitude natural frequencies of the FGM beam resting on nonlinear elastic foundation and operating in thermal environment. Numerical results are given to discuss the effects of thermal environment, elastic foundation parameters, different formulations, and the geometrical parameters.

Large-Amplitude Vibrations of Spider Web Structures

Spider silk, as a natural material, shows exceptional performance in its properties. The combination of the superior properties of spider silk and the geometry of spider structures make the spider web very resilient. A spider web structure can be considered as a cable-like structure with inappreciable torsional, bending and shear rigidities. An investigation emphasising on natural frequencies and corresponding mode shapes with and without the consideration of geometric nonlinearity is presented in this paper. This study is the world’s first discovery of large amplitude free vibration behaviours of spider web structures. Large deformable finite element 3D models of spider web structures have been developed and validated. By using the energy method, the variational model of the spider web structures have been established to further extend the finite element model, consisting of the strain energy due to axial deformation, kinetic energy due to the spider web movement and the virtual work caused by the self-weight per unit unstretched length. The emphasis of this study is placed on the linear and geometric nonlinear behaviour of the spider web structures considering different structural patterns and material properties. To determine the large-amplitude free vibrational behaviours, a series of pretension load is applied to the first step in Abaqus to initiate the nonlinear strain-displacement relationships enabling a precursor to free vibrations. The parametric studies stemming from structural patterns (the number of radial and capture threads), elastic modulus, density, and inertia moment have been highlighted. The insight will help engineers and scientists to adapt the concept of spider webs, their geometric properties, and damage patterns for the design of any structural membranes, preventing any failure from dynamic resonances and nonlinear phenomena.

Higher mode elliptical integral solution for the large amplitude free vibration of a rectangular plate backed by a cavity

In this article, the elliptic integral solution form is adopted to obtain the higher mode natural frequencies of the large amplitude rectangular plate backed by a cavity. The main advantage of the proposed method is the concise acoustic structural modal formulation which do not require heavy computation effort. A set of numerical results are presented for studying the higher mode natural frequencies.

Large amplitude free vibrations of FGM shallow curved tubes in thermal environment

In the current investigation, large amplitude free vibration behavior of shallow curved pipes (tubes) made of functionally graded materials is investigated. Properties of the tube are distributed across the radius of the tube and are obtained by means of a power law function. It is also assumed that all thermo-mechanical properties are temperature dependent. The governing equations of the tube are obtained using a higher order shear deformation tube theory, where the traction free boundary conditions are satisfied on the top and bottom surfaces of the tube. The von Karman type of geometrical non-linearity is included into the formulation to consider the large displacements and small strains. Uniform temperature elevation of the tube is also included into the formulation. For the case of tubes which are simply supported in flexure and axially immovable, the governing equations are solved using the two-step perturbation technique. Closed form expressions are provided to obtain the small and large amplitude fundamental natural frequencies of the FGM shallow curved tubes in thermal environment. Numerical results are given to explore the effects of thermal environment, radius ratio, and length to thickness ratio of the tube on the fundamental linear and non-linear frequencies.

Nonlinear vibration of piezoelectric graphene-reinforced composite laminated panels in thermal environment using Amabili-Reddy shear deformation theory

This work studies large amplitude free vibration of piezoelectric laminated doubly curved panels reinforced with zigzag/armchair graphene sheets under uniform temperature rise. Thermomechanical and kinematic relations are derived using refined Halpin-Tsai approach and Amabili-Reddy (fourth-order shear deformation) theory which remains all nonlinear terms in the in-plane displacements. The explicit form of the quadratic variation of electric potential with Green-Lagrange strains and pyroelectric effects under open and closed circuits are presented. As a first endeavor, the explicit forms of six-coupled PDEs of motions are extracted using Gibbs energy, Hamilton principle and Maxwell equation. For the first time, the eigenvalue problem is solved via the Galerkin and extended He variational method. To verify, linear frequencies of piezoelectric, graphene-reinforced composite and nonlinear frequencies of laminated composite plates are compared with available results and it is shown that Amabili-Reddy theory has less sensitivity to electric surface condition than first-order shear deformation theory. Parametric studies reveal that panels with low thickness of piezoelectric layer, closed circuit and high temperature show higher frequency ratios than other panels. Moreover, panels with BaTiO3 layers display more sensitivity to the amplitude and less sensitivity to electric surface condition than panels integrated with PZT-4 and PZT-5A layers in ambient temperature.