Free Vibration Frequencies Research Articles

Fatigue Analysis for Shaft of Inland River Ship Under Ice Load

Inland river ships navigating in an ice area cannot avoid contact between the propeller and ice block. In addition to ensuring the safety of propeller blades, the fatigue strength of the propulsion shaft system under ice load excitation must also be considered. This paper first studies how to calculate the natural frequency of free torsional vibration of the system, then uses Newmark integral programing to calculate the maximum torsional stress of shaft system under ice load at resonance speed. Low cycle stress and high cycle stress are studied according to fatigue analysis theory. The method of determining S–N curve and ice load stress spectrum is given and the cumulative damage ratio is calculated based on Palmgren–Miner linear cumulative damage theory. Finally, taking a real inland river vessel propulsion shaft system as an example, the fatigue strength of the shaft system under different working conditions of ice load excitation is studied. Therefore, this study has practical significance and engineering application value in conducting fatigue research on the propulsion shaft system of an inland waterway vessel sailing in an ice area.

Frequency and stress optimization of an engineered wood portal frame suporting a rotating machine

We consider a portal frame with 3 prismatic bars. The horizontal beam, of length L, is articulated at the top of the vertical columns, of height h, embedded in the base. The cross sections are rectangular with thickness b, constant, and width d, which can be different in the beam (dv) and in the columns (dc), always greater than or equal to the thickness. The material is considered linear and homogeneous elastic, with given modulus of elasticity (E) and density. In the middle of the beam span, a motor of mass M is mounted, rotating at a frequency of N rpm.Considering only the Service Limit State, we present the minimization of the mass of the structure so that the motor rotation frequency is always at least 20% above any of the first 2 undamped free vibration frequencies of the frame.

Free vibrational mode shapes and frequencies of sandwich folded plates standing on folded foundation and structured by auxetic honeycomb core and FG-GPLRC coating layers

In this article, for the first time, the free vibration and modal shapes of sandwich folded plate with honeycomb core are studied. The current article investigates the free vibrational mode shapes and frequencies of sandwich folded plate made of auxetic honeycomb core and graphene platelet-reinforced functionally graded composite (FG-GPLRC) coating layers which standing on the normal-shear folded supports. The structures are supported by normal-shear folded supports. The negative Poisson’s ratio exhibited by honeycomb materials results in distinct physical and mechanical properties, profoundly impacting the behavior of structures constructed from this material. While structures with basic shapes like rectangular, circular, and spherical configurations find limited applications across various industries, the versatility of folded structures makes them highly suitable for a wide range of industrial applications, given their more complex geometries. Dynamic equations are formulated within the framework of the first-order shear deformation theory (FSDT) for each flat section of the folded plate. The continuous equality conditions for displacements, rotations, and stresses at the junction edges of the two flat sections are rigorously satisfied. To effectively solve these equations, a robust numerical technique known as the generalized differential quadrature method (GDQM) is employed specifically tailored for the 2D analysis of folded plates. Furthermore, the study investigates the influence of various parameters, including material properties and geometry, on the frequencies and mode shapes of the folded structures, shedding light on the intricate interplay between these factors in shaping the dynamic response of the structures.

New method for predicting the free vibration characteristics of composite laminates based on generalized cell and spectral‐Tchebychev methods

AbstractThis paper presents a new method for analyzing the free vibration characteristics of composite laminates under arbitrary boundary conditions by integrating the generalized method of cells (GMC) and the Two‐Dimensional Spectral‐Tchebychev (2D‐ST) method. First, a meso‐macro correlation matrix is constructed based on the GMC, and the macroscopic performance parameters of the composite laminates are predicted. Then, virtual boundary springs are introduced to simulate the arbitrary boundary conditions, and the energy equation of the laminate is derived based on the first‐order shear deformation theory (FSDT) and Hamilton's principle. Furthermore, an approximate displacement function of the laminate is derived by using the 2D‐ST Tchebychev polynomials, and the Gauss‐Lobatto points are selected for meshless discretization of the structure to obtain the discretized characteristic equation, which is solved to obtain the free vibration frequency of the laminate. Based on Matlab programming, the free vibration frequencies of the laminate under different conditions are calculated, and the convergence, accuracy and computational efficiency of the calculation method are discussed, and the effects of boundary conditions, stacking angle, geometrical parameters, and material parameters on the free vibration frequencies of the laminate are analyzed. The numerical results show that the method has the advantages of fast convergence, high accuracy and high efficiency, and the method can be convenient and fast for parametric studies.Highlights The free vibration characteristics of composite laminates are studied. Combining generalized cell method with spectral‐Tchebychev method. The method has the characteristics of fast convergence, high accuracy and high efficiency. A parametric study is conducted on the free vibration of laminated panels.

Forced vibration assessment of moderately thick carbon nanotube-reinforced composite plates

Carbon nanotubes, distributed in composite sheets with various arrangements, can lead to the development of materials with functionally graded properties. In the present study, the method of spectral components was used to investigate the free vibration frequencies and dynamic responses of single-layer sheets reinforced with carbon nanotubes. The spectral component method offers significant advantages for high-frequency dynamic problems, as it requires fewer fine elements to solve boundary conditions. Based on the first-order shear theory, the governing equations of sheet vibration are derived. The discrete Fourier transform is applied to convert the differential equations from the time domain to the frequency domain. Using dynamic shape functions obtained from the exact solutions, the stiffness matrix of the spectral element is constructed. Dynamic frequency responses are then derived, and time-domain responses are obtained using the inverse Fourier transform. The study extracted the exact natural frequencies of functionally graded sheets and verified them with results from the literature, showing high accuracy with a minimal number of elements. The spectral finite element method of fundamental natural frequencies for different ratios of modulus of elasticity and width-to-thickness ratios obtained were investigated and compared with the results of research. The forced vibration was addressed, demonstrating that the method efficiently captures dynamic responses under different carbon nanotube distributions, with spectral displacements verified through numerical comparison. These findings demonstrate the capability and efficiency of the spectral finite element method for analyzing carbon nanotube-reinforced composite plates.

Thermomechanical Nonlinear Vibration of Axially Loaded Functionally Graded Material Cylindrical Panels Including Porosity

For the first time, the combined influences of axial compressive load, porosity, geometric imperfection, elastic edge restraint, and elevated temperature on the nonlinear free vibration of functionally graded material (FGM) cylindrical panels are investigated in this paper. Unlike previous studies, the structural model considered in this paper includes practical situations and conditions arising in the design process and application of FGM cylindrical panels. The properties of material constituents are assumed to be temperature-dependent, and the effective properties of porous FGM are determined using a modified rule of mixture. Governing equations in terms of deflection and stress function are established based on first-order shear deformation theory, taking into account von Kármán–Donnell nonlinearity and initial geometric imperfection. Analytical solutions are assumed to satisfy simply supported boundary conditions, and the Galerkin method is applied to derive a time differential equation containing both quadratic and cubic nonlinear terms. This differential equation is numerically integrated employing the fourth-order Runge–Kutta scheme to determine the frequencies of nonlinear free vibration. Parametric studies are carried out to assess numerous effects on both linear and nonlinear frequencies of porous FGM cylindrical panels. The study reveals that natural frequencies are strongly decreased and that frequency nonlinearity is more significant due to axial compressive loads.

A new semi-analytical approach for nonlinear dynamic responses of functionally graded porous graphene platelet-reinforced circular plates and spherical shells

This paper presents the semi-analytical approach for nonlinear dynamic responses of porous graphene platelet-reinforced circular plates (CiPs) and spherical shells (SpSs) resting on the Pasternak foundation in thermal environment. The graphene platelets (GPLs) are distributed in the functionally graded (FG) distribution patterns and uniform distribution pattern through the CiP and SpS thickness direction. The governing equations of the GPL-reinforced structures subjected to time-dependent external pressure respected to the harmonic and linear functions of time are established based on the geometrically nonlinear higher-order shear deformation theory. A simple and effective semi-analytical solution for the considered problem is established using the Euler-Lagrange equations, and the damping viscosity of the foundation can be counted using the Rayleigh dispassion function. The Runge-Kutta method is employed to determine the dynamic responses of the GPL-reinforced structures, while the fundamental frequencies of free and linear vibration, and frequency-amplitude curves are obtained in explicit form. The numerical results obtained reveal that the GPL and porous distributions, geometrical and material properties can significantly affect the frequency-amplitude curves, and dynamic responses of harmonically loaded and linearly loaded structures, specially, the dynamic buckling phenomenon is clearly observable with the SpSs but not with the CiPs.

Nonlinear buckling and free vibration analysis of auxetic graphene origami composite beams under nonuniform thermal environment

This study examines the thermo-mechanical behavior of auxetic metamaterial beams enhanced by graphene origami (GOri) under spatially varying nonuniform temperature distributions (SVTD). Utilizing Timoshenko beam theory considering von-Kármánn type nonlinear strain–displacement relationship, GOri beams are modeled as layered structures. The Ritz method is employed to solve equilibrium equations, analyzing the impact of GOri distribution patterns, content, and folding degree on post-buckling and vibration paths. The effects of five SVTDs, three end conditions, and three GOri distribution patterns on buckling, post-buckling behavior, and nonlinear free vibration characteristics are explored. Findings reveal that the parabolic temperature distribution with peak temperatures at beam ends (P-MAE) results in higher critical temperatures and nonlinear free vibration frequencies. This research provides crucial insights into the design and optimization of GOri-enabled metamaterial structures in complex thermal environments, highlighting the significant influence of nonuniform temperature distributions along the beam’s length.

Accurate and efficient analytical simulation of free vibration for embedded nonlocal CNTRC beams with general boundary conditions

This study aims to evaluate the free vibrational response of embedded restrained nanobeams enriched by nanocomposites based on an exact Fourier series approach. In order to capture the small-scale effects on the dynamical response, Eringen's differential form of nonlocal elasticity is used which employs a scale (nonlocal) parameter. Within the framework of Rayleigh and Bernoulli-Euler beam theories, including the effect of nonlocality and employing the Fourier sine series together with Stokes' transformation, systems of linear equations are obtained and solved using the coefficient matrices. The combined effects of elastic boundary conditions, elastic foundation, dispersion patterns and volume fractions of carbon nanotubes, and nonlocal parameter are examined by solving eigenvalue problems constructed with Fourier infinite series. Free vibration frequencies are calculated for carbon nanotube-reinforced nanobeams under different rigid or restrained boundary conditions, including Winkler-Pasternak type elastic foundation. A comprehensive parametric study is performed, focusing on various effects for the free vibrational response of the composite nanobeam reinforced with carbon nanotubes. It is concluded that adding a small amount of carbon nanotube material can reinforce the stiffness of the composite nanobeam, and its free vibration performance is significantly affected by the distribution patterns, elastic medium, and boundary conditions.

Molecular structure, thermodynamic and spectral characteristics of chloroboron(III) subphthalocyanine and its dodecafluorinated derivative

A comprehensive study of the structural, spectral and energetic properties of chloroboron(III) complexes of subphthalocyanine (H[Formula: see text]SubPc, for the first time) and dodecafluorosubphthalocyanine (F[Formula: see text]SubPc, reinvestigation) was performed by mass-spectroscopy (MS), gas-phase electron diffraction (GED), IR spectroscopy and quantum-chemical (QC) calculations. A synchronous GED/MS method showed that at T = 630 K and T = 540 K, thermally stable H[Formula: see text]SubPc and F[Formula: see text]SubPc molecular forms are present in the gas phase. The geometric structure of free molecules has been determined, in which the N3-B-Cl fragment has the structure of a distorted tetrahedron, and the phthalocyanine skeleton has a dome shape. QC calculations of the geometry agree well with the GED results. The similarities and differences of the molecular structure in the gas and solid phases were discussed. It is shown, that despite the similarity of most geometric parameters, the H[Formula: see text]SubPc and F[Formula: see text]SubPc have significant differences in electronic characteristics, which determines the differences in their physicochemical properties. The interpretation of the experimental IR spectra was carried out. The distribution of potential energy of normal vibrations over the internal vibrational coordinates has been analyzed. The sublimation enthalpies of H[Formula: see text]SubPc ([Formula: see text]H[Formula: see text](589 K) = 135(5) kJ ⋅ mol[Formula: see text] and F[Formula: see text]SubPc ([Formula: see text]H[Formula: see text](516 K) = 189(3) kJ ⋅ mol[Formula: see text] were determined by the mass spectrometric Knudsen effusion method. The obtained data is important for designing processes employed in fabricating optoelectronic devices based on subphthalocyanines by PVD. Experimental geometric parameters for free molecules and vibrational frequencies can be used to calculate the thermodynamic functions of gaseous H[Formula: see text]SubPc and F[Formula: see text]SubPc.

Anisotropic elasticity measurements of the retina using optical coherence elastography

Anisotropic elasticity measurements of the retina are essential for retinal disease diagnosis and function assessment. Optical coherence elastography (OCE) is a high-resolution imaging technique for mapping the elasticity distribution of tissues. However, previous OCE measurements quantified the tissue elasticity in a single direction, resulting in a biased estimation of the elastic properties. In this study, we propose an OCE method with acoustic radiation force (ARF) excitation to map the retinal anisotropic elasticity in the depth and lateral directions. The axial elasticity was analyzed using the natural frequency of free vibration, and the lateral elasticity was quantified using the elastic wave velocity. After evaluating the feasibility of the OCE method on the phantoms, the anisotropic elasticity of ex vivo porcine retinas was mapped. The results show that the OCE method with ARF excitation can assess the elasticity in orthogonal directions and provide a comprehensive understanding of the elasticity of the anisotropic tissues.

Vibrational analysis of polyurethane-sandwiched bridge decks with variable skew angles

Bridge decks are the surface of bridges that carry the weight of the vehicles and pedestrians crossing over them. The design of bridge decks varies depending on the span, traffic volume, and material availability. But nowadays, the need for a sustainable approach is required. So, use of a sustainable material for construction and retrofitting purposes is the need of the hour. In the present study, a novel synthetic material polyurethane has been used in bridges. The study deals with the variation in skew angles to determine the response of the bridge deck. The response of natural frequencies on the bridge deck due to the variation in skewness and thickness of steel are analysed under simply supported and clamped boundary conditions. Further, the bridge deck is sandwiched using steel and polyurethane having different thicknesses, and the responses are recorded. Afterwards, a bridge deck is modelled using polyurethane as a special case, to pursue sustainability and justify the RRR (reduce, reuse, and recycle) concept of waste management. A comparative study is also performed between the isotropic steel deck and sandwiched deck by varying the skewness. The skew angle is varied from 0° to 60° with a difference of 10°, i.e., 0°, 10°, 20°, 30°, 40°, 50°, and 60°. The frequencies of the isotropic steel and sandwiched decks are increasing when the skewness is increased. Also, the decks may be modelled in a way to enhances the vibration behaviour by sandwiching the existing steel decks. The free vibration frequencies of the sandwiched decks are comparable to the steel deck of similar thickness, which shows the use of polyurethane as the core material does not affect the vibrational characteristics of the deck, while at the same time reducing the cost significantly. The research lays the groundwork for the creation of engineering recommendations that practitioners can use.

Multiscale analysis of carbon nanotube-reinforced curved beams: A finite element approach coupled with multilayer perceptron neural network

This paper presents a comprehensive investigation into the structural response of curved composite beams enhanced with carbon nanotube (CNT). Employing a multiscale framework, our analysis leverages the finite element method (FEM) to account for both bending and shear deformations across six degrees of freedom. The inquiry encompasses diverse mechanical, geometrical, and boundary configurations to assess these composite beams' natural vibration features. Moreover, we introduce a multilayer perceptron (MLP) neural network architecture designed to forecast such beams' dimensionless first natural frequency. Trained on a meticulously curated dataset derived from FEM simulations, the neural network model exhibits promising predictive capabilities concerning the free vibration frequency. To ascertain the efficacy and precision of our proposed methodology, we conduct a comparative analysis between FEM results and employ statistical metrics to evaluate the neural network's predictive performance. The findings of this study reveal an impressive predictive accuracy of over 95 % with regards to the initial natural frequency of the composite beams, thereby emphasizing the potential effectiveness of neural network methodologies in engineering analyses. This study significantly contributes to advancing our comprehension of the vibrational dynamics inherent in carbon nanotube-reinforced composite beams, while concurrently underscoring the potential efficacy of neural networks in forecasting their dynamic attributes.

Nonlinear free vibration analysis of multi-directional functionally graded porous sandwich plates

Multi-directional functionally graded materials (MFGMs) have attracted significant research attention due to their advantages over one-directional FGMs. In MFGM structures, material properties can be tailored to grade in the required direction, overcoming practical problems like excessive temperature gradients or extreme deflections. This paper aims to investigate the nonlinear free vibration of MFG porous sandwich plates to improve their application in sandwich structures. The two outer layers of the plates are composed of three-directional functionally graded material (3D-FGM), with a bi-directional functionally graded material (2D-FGM) core layer. Additionally, the porosity distribution within the material matrix is assumed to be either even or uneven across the plate thickness. A higher-order finite element model based on Shi's plate theory and the von Kármán assumption is developed. The nonlinear free vibration frequencies are determined through the maximum vibrational amplitude using an iterative algorithm with a displacement control strategy. The accuracy and effectiveness of the proposed model are demonstrated through a comparison with published data. The results show that the vibration response is significantly influenced by various parameters. Specifically, increasing the material gradient indexes in the thickness direction enhances the frequency ratio, while increasing the gradient indexes in the length and width directions reduces it. Higher porosity coefficients in the core layer decrease the frequency ratio, whereas higher pore coefficients in the outer layers increase it. The additional knowledge gained from this study can help with future analysis and design procedures related to the nonlinear responses of these complex structures.

Dynamic analysis of the three-phase magneto-electro-elastic (MEE) structures with the finite element method enriched by proper enrichment functions

In this study, a carefully designed enriched finite element method (EFEM) is presented to improve the solution accuracy of the conventional FEM by analyzing the dynamic behavior of the magnetic-electric-elastic (MEE) composite structures, which are frequently used in designing various smart and intelligent devices. By formulating the proper EFEM with ideal numerical performance, different enrichment functions are considered and the corresponding solution quality of different versions of the EFEM is compared and examined in great detail. When the Lagrange polynomial basis functions together with the harmonic trigonometric functions are used as enrichment functions, the obtained EFEM shows extremely powerful and ideal numerical performance, which is obviously better than the other versions of EFEM and the conventional FEM, in studying the free vibration and harmonic frequency responses of the MEE structures. Nearly exact numerical solutions for three-phase physical fields of MEE structures can be generated by the proposed EFEM even if very coarse mesh patterns are used. Intensive numerical studies are conducted to confirm and verify the excellent properties of the proposed EFEM in performing dynamic analysis of the MEE structures.

Electro-mechanical vibration characteristics of imperfect sandwich FG piezoelectric honeycomb nanoplates resting on Kerr foundation

This study presents an innovative analytical approach to examine the electro-mechanical vibration characteristics of sandwich nanoplates featuring a honeycomb core and two surface layers composed of imperfect piezoelectric functionally graded (FG) material, incorporating porosities, atop a Kerr elastic foundation. Utilizing nonlocal elasticity theory and four-variable refined plate theory (HSDT-4), we account for nanoscale effects and model the electric potential variation across the piezoelectric layer thickness as a combination of cosine and linear variations satisfying Maxwell's equation. The proposed method's novelty lies in its comprehensive analysis of the interplay between porosity, elastic foundation parameters, material properties, geometrical dimensions, external voltage, and nonlocal parameters on the free vibration frequency of these nanostructures. Numerical simulations validate the model's reliability and assess the significant impacts of these factors. Our findings contribute to the optimization of nanostructural designs for advanced engineering applications, highlighting the method's added value in predicting the behavior of imperfect piezoelectric FG honeycomb nanoplates.

INFLUENCE OF FEED RATE ON THE DYNAMIC PROPERTIES OF THIN-WALLED PART DURING END-MILLING

When selecting cutting modes for end-milling of thin-walled parts from hard-to-machine materials, their influence on machining stability must be considered. To do this, it is necessary to understand the types of vibrations that operate and the conditions under which they arise. Research has shown that different types of vibrations operate sequentially during cutting and only for a certain of time. During end-milling, forced and accompanying free oscillations operates due to the short duration of cutting. Their influence on the stability of the cutting process depends on the speed zone in which the milling takes place. The most unfavorable is the third speed zone of vibrations, where accompanying free vibrations of high intensity occur. Cutting speeds for machining parts from hard-to-machine materials fall precisely within this zone. The intensity of vibrations is influenced by the dynamic properties of the part. During end-milling of thin-walled parts, these properties are characterized by the amplitude and frequency of accompanying free vibrations, which exceed the natural frequency of the part's free vibrations. In other words, cutting modes alter the dynamic properties of the part. Some researchers focus on utilizing the damping properties of the feed. Therefore, the aim of this paper was to determine how the feed affects the dynamic properties of the part during cutting. A universal stand was used for the investigation, which allows for the creation of various dynamic characteristics of the processed sample and recording its motion laws during milling on an oscillogram. The research results showed that the feed, as an element of cutting modes, influences the dynamic properties of the part during cutting. It creates a variable layer that is being cut. Therefore, during the cutting time, the frequency and amplitude of accompanying free vibrations change during both up- and down milling. The feed affects the dynamic stiffness. Increasing the feed increases the dynamic stiffness of the part. This leads to a reduction in the amplitude of accompanying free vibrations during milling in the third speed zone of vibrations. The results presented in the article have practical significance for technologists in the aviation manufacturing industry. They help to better understand the physical processes occurring during milling and develop optimal processing strategies for thin-walled parts, which have high requirements for dimensional accuracy and surface quality.

The Effect of Hyperelasticity and Nonlinearity on the Dynamic Behaviors of Hyperelastic Functionally Graded Beams on Nonlinear Elastic Foundation

Hyperelastic functionally graded materials have a wide range of application prospects in soft robotics and biomedical fields. This paper investigates the nonlinear free and forced vibrations of a hyperelastic functionally graded beam (HFGB) based on higher-order shear deformation beam theory. The geometrical nonlinearity is considered by using the von-Kármán’s nonlinear theory. The three-material-parameter free energy function named as Ishihara model is employed to characterize the hyperelastic material. The power-law gradient form along the thickness direction is adopted. The HFGB is resting on the elastic foundation. The Winkler, Pasternak and nonlinear stiffness coefficients are considered. The time-harmonic external force is applied to the HFGB. The nonlinear governing equations for the vibration of the HFGB are derived by using Hamilton’s principle, and are subsequently transformed into ordinary differential equations via Galerkin’s method. The nonlinear free vibration and primary resonance of the HFGB are investigated analytically by employing the extended Hamiltonian method and multiple scales method, respectively. The results indicate that the power-law index, slenderness ratio, material properties, and elastic foundation parameters have significant influences on the nonlinear frequency of free vibration as well as the frequency–response and force–response curves of forced vibration. The phase plane method is employed to analyze the system’s stability states under various excitation amplitudes. The relative error between the results of the current computational model and the published literature is less than 0.1 percent.

Nonlinear vibration analysis of geometrically imperfect FGM Timoshenko beams with porosity and elastically restrained ends in thermal environments

This paper aims to analyze the combined influences of pore imperfection, initial geometric imperfection, elasticity of tangential constraints of ends, elastic foundations and elevated temperature on the nonlinear vibration of functionally graded material (FGM) beams. The pores are assumed to distribute into FGM according to even and uneven patterns and effective properties of porous FGM are estimated using a modified rule of mixture. Motion equations of geometrically imperfect beams are established within the framework of Timoshenko beam theory incorporating von Kármán nonlinearity and foundation interaction. Analytical solutions are assumed to satisfy simply supported and clamped conditions of beam ends and Galerkin method is applied to derive a nonlinear time ordinary differential equation. The frequencies of nonlinear free vibration are determined employing fourth-order Runge-Kutta numerical integration scheme. Parametric studies are carried out to assess numerous effects on both linear and nonlinear frequencies. The results reveal that frequency nonlinearity is more significant for beams with tangentially restrained ends, especially at elevated temperatures. It is also found that geometric imperfection increases the linear frequency and lowers ratio of nonlinear-to-linear frequencies.

WITHDRAWN: Accurate and efficient analytical simulation of free vibration for embedded nonlocal CNTRC beams with general boundary conditions

This study aims to evaluate the free vibrational response of restrained nanobeam enriched by nanocomposites based on an exact Fourier series approach. In order to capture the small size effects on the dynamical response, Eringen’s differential form of nonlocal elasticity is used which employs the one size (nonlocal) parameter. Within the framework of Rayleigh and Bernoulli-Euler beam theories to include the effect of nonlocality and by employing Fourier sine series together with Stokes’ transformation, a system of linear equations is obtained then solved by using the coefficient matrix. The combined effects of elastic boundary conditions, elastic medium, dispersion patterns and nonlocal effects are examined by solving this eigen-value problem constructed with Fourier infinite series. Free vibration frequencies are calculated for carbon nanotube reinforced nanobeam under different rigid or restrained boundary conditions, including an elastic medium Winkler-Pasternak type. A comprehensive parametric study is performed, with the particular focus on the influences of distribution pattern, elastic boundary and medium parameter on the free vibrational response of the composite nanobeam with different graded distributions. It is concluded that the addition of a small amount of carbon nanotube material can reinforce the stiffness of the nanobeam, and its free vibration performance is significantly affected by the distribution patterns, elastic medium and boundary conditions.