Nonlinear Forced Vibration Analysis Research Articles

A SGM-IHB approach for nonlinear free and forced vibration analysis of FG-GPLRC beams rested on viscoelastic foundation

A SGM-IHB approach for nonlinear free and forced vibration analysis of FG-GPLRC beams rested on viscoelastic foundation

Nonlinear forced vibration analysis of doubly curved shells via the parameterization method for invariant manifold

Nonlinear forced vibration analysis of doubly curved shells via the parameterization method for invariant manifold

Nonlinear vibration behaviours of foam-filled honeycomb sandwich cylindrical shells: Theoretical and experimental investigations

In this study, the nonlinear vibration behaviours of foam-filled honeycomb sandwich cylindrical shells (FHSCSs) are investigated theoretically and experimentally. Initially, a time-domain minimum residual (TDMR) method is first adopted for the analysis of nonlinear free and forced vibrations of the FHSCSs, which is based on a dynamical model developed, with the effective material parameters of the foam-filled honeycomb core being obtained by employing the Gibson method and the Hamilton equivalence technique. Based on the high-order shear deformation shell principle and von Karman's theory, the nonlinear natural frequencies and resonant responses are solved using the Broyden iterative approach and TDMR method. Furthermore, several literature results are utilized for the rough verification of such a method. Meanwhile, the FHSCS specimens are made manually and comprehensive tests with various base excitation energies are undertaken to provide a thorough validation of the current method. It can be found that the largest calculation errors of natural frequencies and resonant responses are 4.9 and 12.3%, respectively. Finally, the influences of the core form and base excitation amplitudes on the nonlinear vibration behaviours of the FHSCSs are examined. It can be noticed that the foam-filled FHSCS structure exhibits a more excellent anti-vibration capability compared to the non-foam-filled structure. The solving method, fabrication technique, and valuable findings in the current study highlight the path for the design and implementation of such advanced shells.

Nonlinear Forced Vibration Analysis using ‘Elmer’ FEM Package to Develop Poincaré Map and Correlation Method-based Damage Indicators

With the rapid advancement of sensor technology, structural health monitoring (SHM) is also blossoming with prowess in the field of civil, mechanical and aerospace engineering. It deals with the initiation of damage and its characteristic behavior in a deterministic or predictive way. The time-dependent fluctuation associated with the dynamic motion of structures was addressed with few efficient indicators in the past to identify the presence of damage as well as to localize the faulty area. The formulation of all those indicators is validated on specific structural arrangements with certain assumptions. This limitation is considered in this article to project the global applicability of the method along with the inherent constraints. The damage detection in the time-domain is considered to utilize direct accessibility of responses from the sensors attached to the members. It reduces the computational difficulty in transforming response to another domain and also associated approximations that emerge through numerical calculations. Broadly, two types of methods are discussed in this article along with modelling and analysis procedure; methods developed from cross-correlation of response and graphical method or Poincaré map. The model of cantilever beam and axial bar is developed in FreeCad where damage is induced by reducing either thickness or Young’s modulus of damaged elements to simulate cracked member. Meshing of member is done in Gmsh. To generate response history of state variables a newly developed finite element (FE) analysis software package Elmer is then employed for the dynamic analysis of the member with geometric nonlinearity. At last, application of Teager energy operator (TEO) in noise reduction is explained with a general example.

Nonlinear forced vibration analysis of PFG-GPLRC conical shells under parametric excitation considering internal and external resonances

Comprehending the vibration dynamics of porous functionally graded-graphene platelet reinforced composite (PFG-GPLRC) structures is vital for accurate predictions and reliability in practical applications. This study addresses gaps in nonlinear dynamics, instability, and frequency response research within truncated PFG-GPLRC conical shells under parametric loading and ½ subharmonic and 1:1 internal resonance. To achieve this, three porosity distributions in metal foam (uniform, non-uniform symmetric, and non-uniform asymmetric) are considered, along with various graphene platelet dispersion patterns (GPL-O, GPL-V, GPL-U, GPL-A, and GPL-X) within the matrix. These considerations lead to a comprehensive conical shell model. Utilizing the first-order shear deformation theory and von-Karman's assumptions, stress-strain relations are extracted, yielding nonlinear motion equations for the truncated conical shell. Employing Galerkin's method and considering simply supported boundaries, two-degree-of-freedom equations of motion are derived. The research culminates in steady state frequency responses obtained through perturbation theory and the multiple scales method, encompassing ½-subharmonic excitation resonance and 1:1 internal resonance. Bifurcation points are analysed to highlight the influence of parametric excitation and system instabilities. A parametric study underscores the significance of porosity and graphene platelets within the metal foam in relation to system instability, revealing their intricate impact on PFG-GPLRC structure behavior.

Nonlinear Forced Vibration Analysis of Two-Directional Functionally Graded Truncated Conical Shells

It is crucial to comprehend the nonlinear vibration behavior of sophisticated engineering constructions to aid in the examination, creation and production procedures. The primary objective of this study, therefore, is to focus on exploring the nonlinear forced vibrational characteristics of two-directional functionally graded (TDFG) cone-shaped shell for the first time. To accomplish this, the motion equations of the TDFG cone-shaped shell are derived based upon the framework of first-order shear deformation theory (FSDT) in conjunction with von Kármán assumption. TDFG cone-shaped shell models have mechanical properties that can change smoothly and continuously across the thickness and length of the shell, following a power law distribution pattern. To discretize the partial differential equations and transform them into ordinary differential equations, the Galerkin approach was used and ultimately, these equations were solved using the multiple time scales method. The accuracy and effectiveness of this analytical model are indicated through comparison with existing solutions. Finally, some comprehensive parametric investigations are carried out to gain insight into the impacts of several factors on the nonlinear behavior of the shell, highlighting the significant influence of transverse and longitudinal gradient indices on the nonlinear frequency response, which is not considered before. The new insights gained from this research could serve as valuable benchmark outcomes as well as contribute to a more comprehensive understanding of the nonlinear vibrations in future analysis and design processes.

Nonlinear free and forced vibration analysis of the sandwich composite cylindrical panel with auxetic core and GPL-reinforced facing sheets subjected to the temperature gradient

This study deals with the nonlinear free and forced vibration of a sandwich panel, composed of Graphene platelet-reinforced composite (GPLRC) skins and re- entrant auxetic core subjected to heat conduction and harmonic force excitation. The nonlinear equations are obtained by applying Hamilton’s principles in the framework’s higher-order shear deformation theory and von-Karman’s nonlinear theory. Linear and nonlinear equations of motion are solved by applying the differential quadrature method and homotopy perturbation technique, respectively. In the numerical illustration, at first validation of the present formulation is carried out by comparing the numerical results with those available in the open literature. Then the effects of several parameters such as geometric parameters of auxetic core, force excitation, GPL volume fraction, different boundary conditions and temperature gradient on the nonlinear frequencies- amplitude of sandwich panel are studied. Finally, the important findings of this research indicate that the ratio of core thickness, inclined angle, and thickness to inclined length have significant effects on nonlinear frequencies’ amplitude response.

Nonlinear forced vibration analysis of micro-rotating shaft–disk systems through a formulation based on the nonlocal strain gradient theory

Nonlinear forced vibration analysis of micro-rotating shaft–disk systems through a formulation based on the nonlocal strain gradient theory

Nonlinear forced vibration analysis of a multi-cracked Euler-Bernoulli curved beam with inclusion of damping

Curved beams that have advantages of excellent bearing capacity and beautiful appearance are widely applied in various engineering constructions. This paper studies nonlinear forced vibration of a multi-cracked Euler-Bernoulli curved beam (ECB) with damping effects and derives the closed-formed analytical solution of steady-state forced vibration of the multi-cracked curved beam by means of Green’s functions. Based on the mathematical structure of Green’s functions, a new transfer matrix method called the opposite coordinate two-step transfer matrix (OCTM) method is proposed. By employing the OCTM method and Laplace transform method, Green’s functions of ECBs with one crack and multiple cracks are obtained. The vibration model of the multi-cracked ECB degenerates into a healthy curved beam vibration model when depths of cracks are set to zero. The healthy curved beam vibration model can be reduced to a straight beam vibration model bysetting the radius of curvature to infinity. The present analytical solution is verified by experimental results, finite element results, and analytical solutions in the literature in this work. Effects of some important physical parameters, such as the crack depth and location, on vibration of a curved beam with one crack are investigated. Interaction between two cracks of a curved beam is also investigated.

Nonlinear forced vibration analysis of laminated composite doubly-curved shells enriched by nanocomposites incorporating foundation and thermal effects

Assessment of the geometrically nonlinear forced vibration of the doubly-curved laminated composite shallow shell panels enriched by nanocomposites have been carried out in this paper. The proposed shells incorporate some layers composed of fibres and nanocomposite-reinforced resin, resting on the two types Winkler and Pasternak elastic foundations. Since the thin and moderately thick shells are investigated, the first-order transverse shear deformation theory (FSDT) is employed for extending the governing equations. The achieved system of differential equations incorporating the geometrical nonlinear terms is solved by applying the principles of the Galerkin method. This approach yields a closed-form semi-analytical solution for motion differential equation of the shell which is solved by fourth-order Runge-Kutta method. The consequence of the present approach is a strong and robust tool without any time-consuming computational process. The applicability of the proposed method is verified by considering some benchmark problems from the literature and comparing the relevant results with those are achieved in this study. Finally, by solving several examples, the effects of various factors on the forced-vibration of proposed shells are evaluated.

Nonlinear free and forced vibration analysis of sandwich cylindrical panel with auxetic core and GPLRC facing sheets in hygrothermal environment

In this study, the free and forced vibration analysis of composite sandwich panel subjected to the harmonic force excitation in the hygrothermal environment is investigated. The sandwich panel is composed of three-phase composites with polymer/Graphene platelet/fiber skins at the top and bottom surfaces and double-V auxetic honeycombs core layer with a negative Poisson’s ratio. Governing equations of motion within the framework of higher-order shear deformation theory (HSDT) and von Kármán nonlinearity are obtained. Applying the generalized differential quadrature method (GDQM) and homotopy perturbation technique, respectively the linear and nonlinear equations of motion results in solving the vibration problem. In the numerical illustration, at first validation of the present formulation is carried out by comparing the numerical results with those available in the open literature. Then the effects of several parameters such as geometric parameters of double-V auxetic core, force excitation, GPL volume fraction, and different boundary conditions on the nonlinear frequencies–amplitude of sandwich panel is studied. Finally, the important findings of this research indicate that the ratio of core thickness, inclined angle, and thickness to inclined length have significant effects on nonlinear frequencies–amplitude response. Based on the results of this article designers can development of different parts of aircraft.

Nonlinear forced vibration and stability analysis of nonlinear systems combining the IHB method and the AFT method

The incremental harmonic balance (IHB) method is a very powerful tool for analyzing nonlinear forced vibrations in structures subjected to periodic forces. In this paper, a methodology is proposed to interpret the stability of the calculated solutions and the nonlinear vibrations of nonlinear systems by combining the IHB method and the Alternating Frequency/Time (AFT) method. The main purpose of combining the IHB method and the AFT method is to analyze the nonlinear vibration of systems with complex nonlinearities. For the mass matrix, damping matrix, and linear stiffness matrix, the Galerkin procedure is performed according to the classical IHB method. The nonlinear force matrix and its derivative matrix are calculated in the time domain and transformed into the frequency domain by Fourier transform. By multiplying these matrices with a transformation matrix, they are transformed into matrices that can be used in the IHB method. Three example problems are proposed to prove the effectiveness of the suggested method and the results compared with the results calculated using numerical integration (NI) and the results presented in previous literature. The results are very close. The proposed method can analyze nonlinear vibrations of systems with complex nonlinearities as well as geometric nonlinearities.

Nonlinear forced vibration analysis of composite beam considering internal damping

Nonlinear forced vibration analysis of composite beam considering internal damping

Nonlinear Forced Vibration Analysis of Smart Curved CNTs Conveying Fluid

In this study, the nonlinear vibration of a curved carbon nanotube conveying fluid is analyzed. The nanotube is assumed to be covered by a piezoelectric layer and the Euler–Bernoulli beam theory is employed to establish the governing equations of motion. The influence of carbon nanotube curvature on structural modeling and fluid velocity vector is considered and the slip boundary conditions of CNT conveying fluid are included. The mathematical modeling of the structure is developed using Hamilton’s principle and then, the Galerkin procedure is employed to discretize the equation of motion. Furthermore, the frequency response of the system is extracted by applying the multiple scales method of perturbation. Finally, a comprehensive study is carried out on the primary resonance and piezoelectric-based parametric resonance of the system. It is shown that consideration of nanotube curvature may lead to an increase in nonlinearity. Implementing the fluid velocity vector in which nanotube curvature is included highly affects the maximum amplitude of the response and should not be ignored. Furthermore, different system parameters have evident impacts on the behavior of the system and therefore, selecting the reasonable geometrical and physical parameters of the system can be very useful to achieve a favorable response.

Non-linear forced vibration analysis of piezoelectric functionally graded beams in thermal environment.

This work proposes a geometrically non-linear vibratory study of a functional gradation beam reinforced by surface-bonded piezoelectric fibers located on an arbitrary number of supports and subjected to excitation forces and thermoelectric changes. The non-linear formula is based on Hamilton's principle combined with spectral analysis and developed using Euler-Bernoulli's beam theory. In the case of a non-linear forced response, numerical results of a wide range of amplitudes are given based on the approximate multimodal method close to the predominant mode. In order to test the methods implemented in this study, examples are given and the results are very consistent with those of the literature. It should also be noted that the thermal charge, the electrical charge, the volume fraction of the structure, the thermal properties of the material, the harmonic force and the number of supports have a great influence on the forced non-linear dynamic response of the piezoelectrically functionally graded structure.

Nonlinear forced vibration analysis of functionally graded non-uniform cylindrical microbeams applying the semi-analytical solution

The linear and nonlinear forced vibration response of axially functionally graded (AFG) cylindrical truncated conical and imperfect microbeam subjected to the dynamic harmonically load carried out in the presented research. Based on a couple of modified couple stress theory, the Euler-Bernoulli beam theory and von-Kármán theory, the linear and nonlinear governing equations and related boundary conditions for dynamic response of micro-size tubes are derived employing the Hamilton principle. We considered the uniform and nonuniform functions for the cross-section, in which the convex, linear and exponential functions are the nonuniform sections, and the porosity is regarded as an imperfection. The generalized differential quadrature method (GDQM) is used to prepare the initial conditions for homotopy perturbation (HP) techniques as the semi-analytical approach to calculate the linear and nonlinear results of dynamic responses. The obtained linear and nonlinear results of the free and forced vibration response show the negative and positive effects of some parameters such as the porosity parameter, the nonlinear amplitude, the small-scale parameter, AFG parameter, and different cross-section impact on the dynamic deflection and natural frequency of micro-scale tube and beams with both clamped and simply-supported boundary conditions.

Nonlinear forced vibration and resonance analysis of rotating stiffened FGM truncated conical shells in a thermal environment

In this paper, a semi-analytical method is presented for nonlinear vibration analysis of rotating stiffened truncated conical shells with functionally graded materials (FGM) in the thermal environment subjected to harmonic excitation. Considering Coriolis acceleration and the centrifugal force, the governing equations of stiffened rotating FGM truncated conical shells are extracted utilizing the classical shell theory regarding the nonlinear von Karman relationships and the smeared stiffeners technique. The partial differential equations are discretized to a set of ordinary nonlinear equations by employing Galerkin’s approach. The condensation method is utilized to reduce the equations of motion into the transverse direction. For analyzing the nonlinear resonance behavior, we use the multiple scales method. According to this approach, the primary resonance of the nonlinear vibration of the stiffened rotating FGM truncated conical shells is examined. Besides, linear free vibration of the stiffened rotating shell is analyzed to investigate the natural circular frequency. Numerical results are studied to investigate the influence of shell rotating speed on the frequency response curve. Also, the effects of force amplitude, volume–fraction index, temperature changes, and different vertex angles on the frequency response curve are examined for various rotating speeds.

Forced nonlinear vibrations of circular cylindrical sandwich shells with cellular core using higher-order shear and thickness deformation theory

A geometrically nonlinear forced vibration analysis of circular cylindrical sandwich shells with open/closed cellular core using higher-order thickness and shear deformation theory is presented. The proposed sandwich shell comprises two thin face-sheets perfectly bonded to a functionally graded porous core. The face-sheets are made of carbon nanotubes (CNTs) reinforced composites. Three different types of porosity distribution, including two non-uniform and a uniform variation through the thickness, are considered. The effective mechanical properties of the core material, having open cells or closed cells, are determined using the Gibson and Ashby model. The rule-of-mixture, which includes efficiency parameters to account for scale-dependent properties of nanocomposite media, is adopted to obtain the mechanical properties of the face-sheets. The in-plane and transverse displacements of a generic point are assumed as a third and fourth-order polynomials of the through-the-thickness coordinate. The model is derived within the framework of an equivalent single layer (ESL) theory. Hamilton's principle is employed to obtain the nonlinear governing differential equations and further discretised by adopting the Galerkin method. Finally, the incremental harmonic balance (IHB) method, in conjunction with the arc-length continuation method, is used to solve the nonlinear system of coupled ordinary differential equations and compute the frequency-amplitude response. An extensive numerical study is carried out to examine the effects of the porosity coefficient, porosity distribution, core-to-face ratio and the volume fraction of CNTs in the face-sheets on the frequency-amplitude response of circular cylindrical sandwich shells.

Nonlinear forced vibration analysis of the composite shaft-disk system combined the reduced-order model with the IHB method

In this paper, the internal resonance phenomena of a composite shaft-disk system with multi-degrees-of-freedom are analyzed. The force caused by the unbalanced mass of the disk is considered as an external excitation force. The shaft is simply supported. Shear deformation and gyroscopic effects are considered. The strain–displacement relationship of the shaft element is expressed using the Timoshenko beam theory. Each node has 5 degrees of freedom. SHBT (simplified homogenized beam theory) is applied to calculate the stiffness of the composite shaft. WQEM (weak form quadrature element method) is used to construct the element matrices, and the system matrices are established using the element matrix assembly rule of the FEM (finite element method). The reduced-order model is applied to reduce the calculation time. IHB (incremental harmonic balance) method is utilized to solve the nonlinear equations of motion of the composite shaft-disk system. The nonlinear vibration characteristics of the Jeffcott rotor are analyzed using the proposed method and compared with the results of previous researches, and the results are very similar. Based on these considerations, the nonlinear vibration phenomena of the composite shaft-disk system with multi-degrees-of-freedom are considered at the several resonance points.

Nonlinear finite element study on forced vibration of cylindrical micro-panels based on modified strain gradient theory

This research deals with the nonlinear forced vibration analysis of cylindrical micro-panels (CMP) under the modified strain gradient shell theory using the higher-order finite element method. The main objective is to study the impacts of length scale and geometrical parameters on the forced vibration behavior of CMP. The mechanical model based on the modified strain gradient theory (SGT), shear deformation shell theory (SDST), and von Kármán nonlinear kinematic relations is developed using Hamilton’s principle and the hermitian finite elements are employed to solve the problem. Variations of the frequency responses are presented to analyze the nonlinear forced vibration of CMP.