Free Vibration Frequencies Research Articles

Interconnection of maximum deflections and frequencies of free vibrations of composite two-layeroval plates on pliable connections

The paper considers the relationship between the maximum deflections and the frequencies of natural oscillations of oval isotropic composite plates, as well as the dependence of the stiffness coefficient of the seam on the stiffness of the shift connections and the frequency of the natural oscillations of the plate. At the first stage, the system of differential equations of transverse bending of a plate under the action of a uniformly distributed load and the differential equation of free transverse oscillations of the plate is solved. The composite plate at this stage is represented in the form of a solid structure with cylindrical stiffness Ds, which is equivalent to the cylindrical rigidity of the composite plate on the pliable connections. After the transformations, the following regularity is obtained, connecting the maximal deflection of the plate with its frequency of transverse oscillations. Oval two-layer composite plates with a ratio of small to major axis from 1.0 to 0.2 with transverse connections and shift connections of variable stiffness were studied over the entire surface of the plate for rigid pinching and pivoting along the contour. The determination of oscillation frequencies and deflections was carried out by a numerical method. As a result of the research it was established that the oval composite plates, regardless of the support scheme and the stiffness of the shift connections, comply with the regularities (1) with an accuracy of 5%. Based on the results of the study, the graphs of the dependence of the maximum deflection and the natural oscillation frequency of the plate on the stiffness of the shift connections at hinged support and pinching along the contour are constructed. On the second stage of the research, based on A.P. Rzhanitsyn’s theory of compound rods, an analytical dependence of the stiffness of the seam on the rigidity of the shift connections was obtained. Based on the results of numerical studies, the dependence of the frequency of natural oscillations on the stiffness coefficient of the seam was obtained. The graphs of the dependence of the frequency of the natural oscillations of the composite plate and the stiffness coefficient of the seam upon the stiffness of the shift connections were constructed.

Effect of initial strain and material nonlinearity on the nonlinear static and dynamic response of graphene sheets

Computationally efficient multiscale modelling based on Cauchy–Born rule in conjunction with finite element method is employed to study static and dynamic characteristics of graphene sheets, with/without considering initial strain, involving Green–Lagrange geometric and material nonlinearities. The strain energy density function at continuum level is established by coupling the deformation at continuum level to that at atomic level through Cauchy–Born rule. The atomic interactions between carbon atoms are modelled through Tersoff–Brenner potential. The governing equation of motion obtained using Hamilton's principle is solved through standard Newton–Raphson method for nonlinear static response and Newmark's time integration technique to obtain nonlinear transient response characteristics. Effect of initial strain on the linear free vibration frequencies, nonlinear static and dynamic response characteristics is investigated in detail. The present multiscale modelling based results are found to be in good agreement with those obtained through molecular mechanics simulation. Two different types of boundary constraints generally used in MM simulation are explored in detail and few interesting findings are brought out. The effect of initial strain is found to be greater in linear response when compared to that in nonlinear response.

Vibration and bending behavior of functionally graded nanocomposite doubly-curved shallow shells reinforced by graphene nanoplatelets

Abstract Free vibration and static bending of functionally graded (FG) graphene nanoplatelet (GPL) reinforced composite doubly-curved shallow shells with three distinguished distributions are analyzed. Material properties with gradient variation in the thickness aspect are evaluated by the modified Halpin-Tsai model. Mathematical model of the simply supported doubly-curved shallow shells rests upon Hamilton Principle and a higher order shear deformation theory (HSDT). The free vibration frequencies and bending deflections are gained by taking into account Navier technique. The agreement between the obtained results and ANSYS as well as the prior results in the open literature verifies the accuracy of the theory in this article. Further, parametric studies are accomplished to highlight the significant influence of GPL distribution patterns and weight fraction, stratification number, dimensions of GPLs and shells on the mechanical behavior of the system.

Nonlinear dynamic responses of an axially moving laminated beam subjected to both blast and thermal loads

The nonlinear dynamic responses of an axially moving laminated beam subjected to a blast load in thermal environment is studied considering large-displacement. Firstly, the nonlinear dynamic equilibrium equation is established based on the large-displacement theory and the constitutive relation of the single layer material in thermal environment. Based on the Galerkin method, a set of ordinary differential equations is obtained. Secondly, the multiple scales method is adopted to get the nonlinear free vibration frequency. Then, the stability region of the axial velocity and temperature is derived and the truncation order is approximated by the convergence calculation of the natural frequencies. Finally, numerical calculations are performed to discuss the effects of different kinds of blast loads, axial velocity and temperature on the nonlinear dynamic responses adopting the Runge–Kutta technique.

Free vibration analysis of a single‐walled carbon nanotube embedded in an elastic matrix under rotational restraints

Free vibration analysis of a restrained carbon nanotube in an elastic matrix subjected to rotationally restrained boundary conditions is investigated based on Eringen's non-local elasticity. The analytical solution for free vibration frequencies and corresponding mode shapes of single-walled carbon nanotubes are established. Using Fourier series and Stokes’ transformation, a useful coefficient matrix is derived. The present analytical formulation permits to have more efficient coefficient matrix for calculating the vibration frequencies of carbon nanotubes with different boundary conditions (rigid or restrained). The eigen values of this matrix give the vibration frequencies. Comparisons between the free vibration frequency results of the present solutions and previous works in the literature are performed. The calculated results show an excellent agreement with other solutions available in the literature.

Buckling and Free Vibration Behavior of a Temperature Dependent FG-CNTRC Cylindrical Panel Under Thermal Load

Present study deals with the buckling and free vibration behavior of functionally graded carbon-nanotubes reinforced composite (FG-CNTRC) cylindrical panel exposed to uniform thermal load. Stresses setup due to thermal load and temperature dependent properties influences the buckling and free vibration behavior of the heated structures. Approach employed in the present study consists of static analysis to compute thermal stresses, eigen-value buckling analysis to compute critical buckling temperature and finally modal analysis, taking thermal stresses into account. Influence of different CNTs grading pattern, CNTs volume fraction, geometric parameters, boundary constraints and temperature dependent properties on the buckling strength are investigated. It is observed that hybrid CNTs distribution pattern gives comparatively higher buckling strength and free vibration frequencies. Investigation on free vibration characteristics of the FG-CNTRC panel at elevated temperature signifies that the decline in free vibration frequencies is very drastic at a temperature close to buckling temperature along with temperature dependent properties. c 2017 Elsevier Ltd. All rights reserved

A New Static Analysis Approach for Free Vibration of Beams

This study deals with a new method for the free vibration analysis of beams under different boundary conditions. We show that it is possible to apply a static approach for solving free vibration systems, i.e., we obtain natural frequencies for free vibration of beams by analyzing static beam bending problems. Specifically, the basic governing equation for beams with harmonic loadings and resting on an elastic foundation is solved and the solutions are used directly to yield the beam free vibration solutions. In the free vibration analysis, the natural frequency can be a real number or an imaginary number while in the static analysis, the foundation stiffness can be either positive or negative. We show that one can solve the deflection of a beam subjected to a given concentrated force and subsequently deduce the possible infinite deflection when the stiffness becomes zero or negative. In such cases, there exists an equivalent relationship between the free vibration frequencies and the negative stiffness. Consequently, determining the natural frequencies becomes a problem of determining an appropriate negative foundation elastic constant. In general, the numerical vibration solutions can be obtained by analyzing the relationship between loadings and frequencies. For comparison, a comparison with the classical free vibration solutions is presented and excellent agreement is illustrated. We further show that this static approach for free vibration solutions has a clear edge over the classical free vibration approach in computational beam vibration solutions. Very accurate and convergent numerical solutions can be obtained using a very simple numerical solution method. This static approach for free vibration problems can be extended for plate, shell and other structural systems.

Large amplitude forced vibration analysis of an axially functionally graded tapered beam resting on elastic foundation

A displacement based semi-analytical method is utilized to study large amplitude forced vibration of a linearly tapered axially functionally graded (AFG) beam resting on an elastic foundation. The beam is considered to be slender to neglect the rotary inertia and shear deformation effects. The external excitation is assumed as harmonic and the system response is considered to have the same frequency as that of the external excitation. The frequency of excitation and the maximum amplitude of the harmonic excitation are supplied and the corresponding unknown coefficients are determined. At the peak load value it is assumed that the system satisfies the force equilibrium condition. Thus the dynamic problem is solved as a static one through an iterative scheme using a relaxation parameter. Energy principles are used for the formulation of the problems. The free vibrational frequencies are tabulated for various boundary conditions and foundation stiffness. The dynamic behavior of the system is presented in the form frequency response curve.

Application of the vibration method for damage identification of a beam with a variable cross-sectional area

The subject of the paper is an application of the non-destructive vibration method for identifying the location of two cracks occurring in a beam. The vibration method is based on knowledge of a certain number of vibration frequencies of an undamaged element and the knowledge of the same number of vibration frequencies of an element with a defect. The analyzed beam, with a variable cross-sectional area, has been described according to the Bernoulli-Euler theory. To determine the values of free vibration frequencies the analytical solution, with the help of the Green’s function method, has been used.

On the dynamic buckling of very slender structures to self-weight

We present a mathematical and experimental study of the dynamic buckling of very slender structures due to their self-weight. Modern materials and powerful new analysis methods are leading to the design of very slender tall structures that may be prone to instability issues. Elastic stability of such structures is a problem inside the scope of the Non-Linear Dynamics Analysis Methods. An indicator of instability is when the structure’s free vibration frequency approaches null value. Two main factors affect these frequency results. First the stiffness, composed of elastic stiffness, always positive and non-zero, that diminishes rapidly with height, and the geometric stiffness, negative for compressive forces, whose absolute value grows as the structure gets taller and heavier. Second, the mass, that also grows with the height of the structures. To access this behaviour, we first present a simple one-degree-of-freedom mathematical model derived with Rayleigh’s Method, adopting a cubic polynomial as shape function. Next, comparisons are made with results of an experimental set up composed of a variable length cantilever vertical aluminium bar. These models reasonably agree with analytical close solutions available in the literature.

Numerical study of temperature dependent eigenfrequency responses of tilted functionally graded shallow shell structures

The free vibration frequency responses of the graded flat and curved (cylindrical, spherical, hyperbolic and elliptical) panel structures investigated in this research considering the rectangular and tilted planforms under unlike temperature loading. For the numerical implementation purpose, a micromechanical model is prepared with the help of Voigt\'s methodology via the power-law type of material model. Additionally, to incur the exact material strength, the temperature-dependent properties of each constituent of the graded structure included due to unlike thermal environment. The deformation kinematics of the rectangular/tilted graded shallow curved panel structural is modeled via higher-order type of polynomial functions. The final form of the eigenvalue equation of the heated structure obtained via Hamilton\'s principle and simultaneously solved numerically using finite element steps. To show the solution accuracy, a series of comparison the results are compared with the published data. Some new results are exemplified to exhibit the significance of power-law index, shallowness ratio, aspect ratio and thickness ratio on the combined thermal eigen characteristics of the regular and tilted graded panel structure.

Exact radial free vibration frequencies of a thick-walled sphere made of an isotropic and homogeneous material – A case study with a carbon nanofiller reinforced aluminum hollow sphere

Exact radial free vibration frequencies of a thick-walled sphere made of an isotropic and homogeneous material – A case study with a carbon nanofiller reinforced aluminum hollow sphere

Vibration-based estimation of tension for an axially travelling web in roll-to-roll manufacturing

Precise estimation of web tension in roll-to-roll manufacturing is critical to ensure product quality. A vibration-based method to estimate web tension is proposed in this paper. By employing the Hamilton principle, the governing equations of motion are derived, which are discretized and solved by the Galerkin method. The natural frequencies are computed from the eigenvalue equation. This study attempts to derive an approximate fitting formula among the axial tension, the travelling speed and the natural frequency, from which the web tension can be calculated conveniently and quickly by employing the measured natural frequencies. A laser displacement sensor is used to measure the transverse vibration displacements of the travelling web and detect free vibration frequency. A major advantage of the proposed method is its contactless, which is more useful under conditions where load cells are not available. An experimental test is carried out to confirm the effectiveness and accuracy of the proposed approach. The web tensions estimated by the vibration method are compared with the measured tensions by load cells.

Free Vibrations of Cantilever Bars with Linear and Nonlinear Variable Cross-Section

The topic of this study is the first mode of natural transverse vibrations of isotropic, homogeneous and elastic bars (columns or beams) with clamped bottom and free head. The columns of the first group are shaped as truncated solid cones or as tubes with linearly variable wall thickness and with different inclination of lateral faces, from cylinder to cone. The columns of the second group were shaped in similar way, but the generatrices of the solids of revolution were curvilinear ?in the shape of a parabola. The first frequency of free vibrations was determined using the Rayleigh method. The deflection line of the column axis during the vibration was assumed in form of the bending line of the column axis subjected to a uniform load. Resulting frequencies (or periods) were compared with these obtained with the use of FEM (ANSYS) and a good compliance of results was observed. As the expression for the energy of an elementary slice of material was integrated over the length of the rod, the formula for the frequency was obtained in form of an integral equation. In some cases an exact solution of integral equation was obtained, however in other cases only a numerical solution was possible.

Evaluation of the Free-Vibration Frequency and the Variation of the Bending Rigidity of Graphene Nanoplates: The Role of the Shape Geometry and Boundary Conditions

Evaluation of the Free-Vibration Frequency and the Variation of the Bending Rigidity of Graphene Nanoplates: The Role of the Shape Geometry and Boundary Conditions

Noncoaxial vibration and buckling analysis of embedded double-walled carbon nanotubes by using doublet mechanics

Abstract Free vibration and buckling of double-walled carbon nanotubes embedded in an elastic medium with simply supported boundary conditions are studied. Doublet Mechanics (DM) is used in the analysis. Macro level strains and stresses are defined in terms of micro level strains and stresses in the DM theory. These micro deformations and micro stresses are expanded in Taylor series and the number of terms in the Taylor series defines degree of the approach. Double-walled carbon nanotubes are modelled as Euler-Bernoulli beams embedded in an elastic medium. Critical buckling loads and free vibration frequencies are obtained by using DM and compared with the classical elasticity solutions. It is obtained that for some frequencies carbon nanotubes move noncoaxially. Noncoaxial vibration and buckling affect the physical properties of carbon nanotubes. The present results show that a length scale dependent DM can be used in the design of nano electro-mechanical systems.

Effects of End Conditions of Cross-Ply Laminated Composite Beams on Their Dimensionless Natural Frequencies

The effects of the end conditions of cross-ply laminated composite beams on their dimensionless natural frequencies of free vibration is investigated. The problem is analyzed and solved by using the energy approach, which is formulated by a finite element model. Various end conditions of beams are used. Each beam has either movable ends or immovable ends. Numerical results are verified by comparisons with other relevant works. It is found that more constrained beams have higher values of natural frequencies of transverse vibration. The values of the natural frequencies of longitudinal modes are found to be the same for all beams with movable ends because they are generated by longitudinal movements only.

Closed-Form Exact Solutions for Viscously Damped Free and Forced Vibrations of Longitudinal and Torsional Bars

This paper studies the viscously damped free and forced vibrations of longitudinal and torsional bars. The method is exact and yields closed form solution for the vibration displacement in contrast with the well-known eigenfunction superposition (ES) method, which requires expression of the distributed forcing functions and displacement response functions as infinite series sums of free vibration eigenfunctions. The viscously damped natural frequency equation and the critical viscous damping equation are exactly derived for the bars. Then the viscously damped free vibration frequencies and corresponding damped mode shapes are calculated and plotted, aside from the undamped free vibration and corresponding mode shapes typically computed and used in vibration problems. The longitudinal or torsional amplitude versus forcing frequency curves showing the forced response to distributed loadings are plotted for various viscous damping parameters. It is found that the viscous damping affects the natural frequencies and the corresponding mode shapes of longitudinal and torsional bars, especially for the fundamental frequency.

Dynamic behaviour and seismic design method of a single-layer reticulated shell with semi-rigid joints

Most of the existing studies on reticulated shells with a semi-rigid joint system have been focused on the mechanical properties under static loads. Taking material and geometric nonlinearities into account, the finite element analysis (FEA) model of a single-layer reticulated shell with semi-rigid joints was established using the software ABAQUS and then validated through comparison with the experimental result. Based on the bending stiffness of a bolt-column (BC) joint obtained through experiments, the dynamic behaviour and a seismic design method for single-layer reticulated shells with semi-rigid joints were investigated in this paper. First, analysis of the free vibration frequency of the single-layer latticed domes with semi-rigid bolt-column (BC) joints was conducted based on several different parameters, including joint stiffness, ratio of rise to span, initial geometric imperfection. Second, the seismic internal force coefficient of the members of the semi-rigidly jointed spherical single-layer reticulated shells of different parameters was studied in detail. Finally, the seismic internal force coefficients for spherical single-layer reticulated shells with semi-rigid joints under a common earthquake were derived.

Transient vibration of wind turbine blades

This article aims to the transient vibration of wind turbine blades. We firstly introduce transient vibration and previous studies in this area. The report then shows the fundamental equations and derivation of Euler Equation. A 3-D beam are created to compare the analytical and numerical result. In addition we operate the existing result and Patran result of a truncation wedge beam, especially the frequencies of free vibration and transient vibration. Transient vibration cannot be vanished but in some case it can be reduced.