Transverse Vibration Analysis Research Articles

Nonlinear coupled longitudinal–transverse vibration analysis of a beam subjected to a moving mass traveling with variable speed

In this paper, transverse and coupled longitudinal–transverse vibration of a beam subjected to a moving mass traveling with variable speed is considered. The speed change can be due to either an external force (traction or braking force) or the sliding friction force created immediately after the braking. The equation of motions for both cases was extracted and discretized using Galerkin decomposition method. The dynamic response of the beam and the moving mass was calculated and compared for different loading scenario. Next, coupled longitudinal–transverse vibration of a beam under the friction force was studied and the effect of friction coefficient on the dynamic response of the system was evaluated. A comparison between transverse and coupled longitudinal–transverse vibration of the beam subjected to a moving mass was also made.

Transverse vibration analysis of Euler-Bernoulli beam carrying point masse submerged in fluid media

In the present paper, an analytical method is developed to investigate the effects of added mass on natural frequencies and mode shapes of Euler-Bernoulli beams carrying concentrated masse at arbitrary position submerged in a fluid media. A fixed-fixed beams carrying concentrated masse vibrating in a fluid is modeled using the Bernoulli-Euler equation for the beams and the acoustic equation for the fluid. The symbolic software Mathematica is used in order to find the coupled vibration frequencies of a beams with two portions. The frequency equation is deduced and analytically solved. The finite element method using Comsol Multiphysics software results are compared with present method for validation and an acceptable match between them were obtained. In the eigenanalysis, the frequency equation is generated by satisfying all boundary conditions. It is shown that the present formulation is an appropriate and new approach to tackle the problem with good accuracy.

Transverse vibration analysis of concentrated mass-loaded thin circular plate based on strcutural circumferential periodicity

The structural circumferential periodicity of inertial excitation produced by concentrated mass was utilized to establish the mathematical model of thin circular plate carrying eccentric concentrated mass and to analyze its transverse vibration. The fundamental frequency coefficient, natural frequency and mode shape function are determined by this method. A clamped thin circular plate was taken as an example to study the mass effect on the vibrating system. Comparison between the present results and published ones exhibits excellent agreement, which shows that the analytical method in this paper can be used to predict the transverse vibration parameters accurately.

Analiza drgań poprzecznych płyty pierścieniowej o złożonym kształcie z uwzględnieniem własności cyklicznej symetrii układu

Analiza drgań poprzecznych płyty pierścieniowej o złożonym kształcie z uwzględnieniem własności cyklicznej symetrii układu

3-node Basic Displacement Functions in Analysis of Non-Prismatic Beams

Purpose– Analysis of non-prismatic beams has been focused of attention due to wide use in complex structures such as aircraft, turbine blades and space vehicles. Apart from aesthetic aspect, optimization of strength and weight is achieved in use of this type of structures. The purpose of this paper is to present new shape functions, namely 3-node Basic Displacement Functions (BDFs) for derivation of structural matrices for general non-prismatic Euler-Bernoulli beam elements. Design/methodology/approach– Static analysis and free transverse vibration of non-prismatic beams are extracted studied from a mechanical point of view. Following structural/mechanical principles, new static shape functions are in terms of BDFs, which are obtained using unit-dummy-load method. All types of cross-sections and cross-sectional dimensions of the beam element could be considered in this method. Findings– According to the outcome of static analysis, it is verified that exact results are obtained by applying one or a few elements. Furthermore, it is observed that results from both static and free transverse vibration analysis are in good agreement with the previous published once in the literature. Research limitations/implications– The method can be extended to structural analysis of curved and Timoshenko beams as well as plates and shells. Furthermore, exact dynamic shape functions can be derived using BDFs by solving the governing equation for transverse vibration of beams. Originality/value– The present investigation introduces new shape functions, namely 3-node Basic Displacement Functions (BDFs) extended from 2-node functions, and then compares its performance with previous element.

Transverse vibration analysis of single-layered graphene sheet under magneto-thermal environment based on nonlocal plate theory

In the present paper, the effect of magneto-thermal environment on the transverse vibration of magnetically sensitive single-layered graphene sheets (SLGS) has been analyzed based on nonlocal plate theory. Governing differential equations for the analysis of vibration characteristics of SLGS under magneto-thermal environment are derived considering the Lorentz magnetic force obtained from Maxwell's relationship and thermal elasticity. The governing differential equations are solved employing differential quadrature method. Convergence and validation study are performed. Moreover, the influences of SLGS geometrical properties, nonlocal parameter, in-plane magnetic field and environmental temperature change on the vibration characteristics of SLGS are studied and reported.

Transverse vibration analysis of rotating porous beam with functionally graded microstructure using the differential transform method

This study presents free vibration analysis of rotating functionally graded Timoshenko beam made of porous material using the semi-analytical differential transform method.The material properties are supposed to vary along the thickness direction of the beam according to the rule of mixture, which is modified to approximate the material properties with the porosity phases. The frequency equation is obtained using Hamilton’s principle. It is demonstrated that the DTM has high precision and computational efficiency in the vibration analysis of porous FG rotating beams. The good agreement between the results of this article and those available in literature validated the presented approach. Detailed mathematical derivations are presented and numerical investigations are performed, while emphasis is placed on investigating the effect of the several parameters such as porosity, functionally graded microstructure, thickness ratio, rotational speed and hub radius on the normalized natural frequencies of porous FG rotating beams in detail.

Research on Vibration Characteristics of Large Scale Belt Conveyor and its Application

High speed and large capacity belt conveyor is the main development trend. In the design, calculated and used of belt conveyor must be considered the high speed, large capacity, dynamic load. This paper starts from the analysis of conveyor belt transverse vibration. Through calculate transverse vibration natural frequency of conveyor belt, and analyze the lateral stability of belt conveyor.

Dynamics of multiple viscoelastic carbon nanotube based nanocomposites with axial magnetic field

Nanocomposites and magnetic field effects on nanostructures have received great attention in recent years. A large amount of research work was focused on developing the proper theoretical framework for describing many physical effects appearing in structures on nanoscale level. Great step in this direction was successful application of nonlocal continuum field theory of Eringen. In the present paper, the free transverse vibration analysis is carried out for the system composed of multiple single walled carbon nanotubes (MSWCNT) embedded in a polymer matrix and under the influence of an axial magnetic field. Equivalent nonlocal model of MSWCNT is adopted as viscoelastically coupled multi-nanobeam system (MNBS) under the influence of longitudinal magnetic field. Governing equations of motion are derived using the Newton second low and nonlocal Rayleigh beam theory, which take into account small-scale effects, the effect of nanobeam angular acceleration, internal damping and Maxwell relation. Explicit expressions for complex natural frequency are derived based on the method of separation of variables and trigonometric method for the “Clamped-Chain” system. In addition, an analytical method is proposed in order to obtain asymptotic damped natural frequency and the critical damping ratio, which are independent of boundary conditions and a number of nanobeams in MNBS. The validity of obtained results is confirmed by comparing the results obtained for complex frequencies via trigonometric method with the results obtained by using numerical methods. The influence of the longitudinal magnetic field on the free vibration response of viscoelastically coupled MNBS is discussed in detail. In addition, numerical results are presented to point out the effects of the nonlocal parameter, internal damping, and parameters of viscoelastic medium on complex natural frequencies of the system. The results demonstrate the efficiency of the suggested methodology to find the closed form solutions for the free vibration response of multiple nanostructure systems under the influence of magnetic field.

Free transverse vibration analysis of an underwater launcher based on fluid-structure interaction

A pneumatic launcher is theoretically investigated to study its natural transverse vibration in water. Considering the mass effect of the sealing cover, the launcher is simplified as a uniform cantilever beam with a top point mass. By introducing the boundary and continuity conditions into the motion equation, the natural frequency equation and the mode shape function are derived. An iterative calculation method for added mass is also presented using the velocity potential function to account for the mass effect of the fluid on the launcher. The first 2 order natural frequencies and mode shapes are discussed in external flow fields and both external and internal flow fields. The results show good agreement with both natural frequencies and mode shapes between the theoretical analysis and the FEM studies. Also, the added mass is found to decrease with the increase of the mode shape orders of the launcher. And because of the larger added mass in both the external and internal flow fields than that in only the external flow field, the corresponding natural frequencies of the former are relatively smaller.

Analytical approach for transverse vibration analysis of castellated beams

This paper presents an analytical study on the dynamic characteristics of castellated beams. The study focuses on the effect of web shear on the free vibration frequencies of castellated beams. By using the Hamilton's principle, a simple closed-form solution for determining the free vibration frequencies of simply supported castellated beams is developed. The results show that the shear effect on the free vibration frequencies increases with the cross-sectional area and distance between the centroids of the two tee sections of castellated beams, but decreases with respect to increasing web thickness or increasing beam length. The shear effect is also found to be greater in higher vibration modes.

Closed-Form Formula of the Transverse Dynamic Stiffness of a Shallowly Inclined Taut Cable

The segmented vibration-governed equations and their general solutions for cables acted upon by intermediate transverse forces are derived by applying Hamilton’s principle. Including the effects of sagging, flexible stiffness, clamped boundary conditions, and inclination angle of the cable, the element-wise dynamic stiffness for each cable segment, split into segments having unique transverse forces, is derived. By using methods from the global stiffness assembly process of FEM, the global level of the cables’ dynamic equilibrium equation is obtained, and, as a result, the final closed-form formula of transverse dynamic stiffness is derived. Additionally, the corresponding analytic form, without considering sagging effects, is also obtained. Case studies are conducted on the aspects of accuracy, rationality of the distribution on the spatial field, and frequency domains of dynamic stiffness calculations. By comparison with the Guyan-based static FEM reduction method, it is shown that the result obtained from the proposed closed-form solution, which includes sagging effects, is exact and rational, thus creating a powerful tool in transverse vibration analysis.

Free Vibration Analysis of an Un-cracked & Cracked Fixed Beam

The presence of cracks causes changes in the physical properties of a structure which introduces flexibility,and thus reducing the stiffness of the structure with an inherent reduction in modal natural frequencies. Consequently it leads to the change in the dynamic response of the beam. This paper focuses on the theoretical analysis of transverse vibration of a fixed beam and investigates the mode shape frequency. All the theoretical values are analyzed with the numerical method by using ANSYS software and co relate the theoretical values with the numerical values to find out percentage error between them.Also in this paper, a model for free vibration analysis of a beam with an open edge has been presented. Variations of natural frequencies due to at various locations and with varying depths have been studied. A parametric study has been carried out. The analysis was performed using ANSYS software. Most of the members of engineering structures operate under loading conditions, which may cause damages or cracks in overstressed zones. The presence of cracks in a structural member, such as a beam, causes local variations in stiffness, the magnitude of which mainly depends on the location and depth of the cracks. The presence of cracks causes changes in the physical properties of a structure which in turn alter its dynamic response characteristics. The monitoring of the changes in the response parameters of a structure has been widely used for the assessment of structural integrity, performance and safety. Irregular variations in the measured vibration response characteristics have been observed depending upon whether the is closed, open or during vibration. The vibration behavior of cracked structures has been investigated by many researchers. The majority of published studies assume that the in a structural member always remains open during vibration. However, this assumption may not be valid when dynamic loadings are dominant. In such case, the breathes (opens and closes) regularly during vibration, inducing variations in the structural stiffness. These variations cause the structure to exhibit non-linear dynamic behavior. Christides and Barr (1) developed a one- dimensional cracked beam theory at same level of approximation as Bernoulli-Euler beam theory. Liang, Choy and Jialou Hu (3) presented an improved method of utilizing the weightless torsional spring model to determine the location and magnitude in a beam structure. Dimaragonas (4) presented a review on the topic of vibration of cracked structures. His review contains vibration of cracked rotors, bars, beams, plates, pipes, blades and shells. Shen and Chu (5) and Chati, Rand and Mukherjee (6) extended the cracked beam theory to account for opening and closing of the crack, the so called breathing crack model. Kisa and Brandon (7) used a bilinear stiffness model for taking into account the stiffness changes of a cracked beam in the location. They have introduced a contact stiffness matrix in their finite element model for the simulation of the effect of the closure which was added to the initial stiffness matrix at the location in a half period of the beam vibration. Saavedra and Cuitino (8) and Chondros, Dimarogonas and Yao (9) evaluated the additional flexibility that the generates in its vicinity using fracture mechanics theory. Zheng et al (10) the natural frequencies and mode shapes of a cracked beam are obtained using the finite element method. An overall additional flexibility matrix, instead of the local additional flexibility matrix, is added to the flexibility matrix of the corresponding intact beam element to obtain the total flexibility matrix, and therefore the stiffness matrix. Zsolt huszar (11) presented the quasi periodic opening and closings of cracks were analyzed for vibrating reinforced concrete beams by laboratory experiments and by numeric simulation. The linear analysis supplied lower and upper bounds for the natural frequencies. Owolabi, Swamidas and Seshadri (12) carried out experiments to detect the presence of in beams, and determine its location and size. Behzad, Ebrahimi and Meghdari (14) developed a continuous model for flexural vibration of beams with an edge perpendicular to the neutral plane. The model assumes that the displacement field is a superposition of the classical Euler- Bernoulli beam's displacement and of a displacement due to the crack. Shifrin (16) presented a new technique is proposed for calculating natural frequencies of a vibrating beam with an arbitrary finite number of transverse open cracks. Most of the researchers studied the effect of single on the dynamics of the structure. A local flexibility will reduce the stiffness of a structural member, thus reducing its natural frequency. Thus most popular parameter applied in identification methods is change in natural frequencies of structure caused by the

A DQEM for transverse vibration analysis of multiple cracked non-uniform Timoshenko beams with general boundary conditions

In this paper, a differential quadrature element method (DQEM) for free transverse vibration analysis of multiple cracked non-uniform Timoshenko beams with general boundary conditions is proposed. Governing equations, the compatibility conditions at the damaged cross-sections and implementation of the external boundary conditions are derived and formulated by the differential quadrature analogue. The accuracy, convergence, and versatility of the proposed method are confirmed by the exact solution of the uniform beam which has been presented by other authors, and 2D finite element method (FEM) numerical results for non-uniform beam. After the validation of the presented method, the effect of quantity, depth and location of the cracks on the frequency values of vibrations are investigated. The achieved results show that the existence of the crack leads to a decrease in the frequencies of the vibrations through decrease in the stiffness of the beam. Meanwhile, the compatibility conditions at the damaged section is considered as a discontinuity in slope and vertical displacement where the effect of the discontinuity in the slope is more considerable as many authors have neglected the discontinuity in the vertical displacement. As it will be shown, consideration of the discontinuity in the vertical displacement causes more decrease in the frequencies.

Free vibration analysis of a non-uniform cantilever Timoshenko beam with multiple concentrated masses using DQEM

A B S T R A C T In this paper, a differential quadrature element method (DQEM) is developed for free transverse vibration analysis of a non-uniform cantilever Timoshenko beam with multiple concentrated masses. Governing equations, compatibility and boundary conditions are formulated according to the differential quadrature rules. The compatibility conditions at the position of each concentrated mass are assumed as the continuity in the vertical displacement, rotation and bending moment and discontinuity in the transverse force due to acceleration of the concentrated mass. The effects of number, magnitude and position of the masses on the value of the natural frequencies are investigated. The accuracy, convergence and efficiency of the proposed method are confirmed by comparing the obtained numerical results with the analytical solutions of other researchers. The two main advantages of the proposed method in comparison with the exact solutions available in the literature are: 1) it is less time-consuming and subsequently moreefficient; 2) it is able to analyze the free vibration of the beams whose section varies as an arbitrary function which is difficult or sometimes impossible to solve with analytical methods. }}

A new finite element for transverse vibration of rectangular thin plates under a moving mass

In this paper a new finite element which can be used in the analysis of transverse vibrations of the plates under a moving point mass is presented. In this technique, which allows for the inclusion of inertial effects of the moving mass, the load is replaced with an equivalent finite element. By means of using the relations between nodal forces and nodal deflections of 16 DOF conforming plate element with C(1) continuity, on the one hand, and shape functions, on the other hand, mass, stiffness, and damping matrices of the new finite element are determined by the transverse inertia force, Coriolis force and centrifuge force, respectively. This method was first applied on a simply supported beam so as to provide a comparison with the previous studies in the literature, and it was proved that the results were within acceptable limits. Second, it was applied on a cantilevered plate so as to determine the dynamic response of the planer entry plate of a high-speed wood-cutting machine.

Transverse free vibration of non uniform rotating Timoshenko beams with elastically clamped boundary conditions

In the present paper the Differential Quadrature Method, DQM, and the domain decomposition are used to carry out the free transverse vibration analysis of non-uniform multi-span rotating Timoshenko beams with perfect and not perfect boundary conditions. The cross section could vary in a continuous or discontinuous fashion along the beam length. The material of the beam could be different in each beam span. The influence of elastically clamped boundary conditions at hub end are studied and discussed. The effect of an arbitrary hub radius is considered. The governing differential equations of motion for rotating Timoshenko beams come from the derivation of Hamilton’s principle. The first six natural frequencies of vibration are obtained for many particular situations and for some of them the mode shapes are also available. The examples of applications of the method indicated its effectiveness. The results for particular cases are in excellent agreement with published results and results obtained by means of the finite element method.

Nonlinear thermo-nonlocal vibration and instability analysis of an embedded single-layered graphene sheet conveying nanoflow using differential quadrature method

In this study, transverse nonlinear vibration and instability analysis of a viscous-fluid-conveyed single-layered graphene sheet (SLGS) subjected to thermal gradient are investigated. The small-size effects on bulk viscosity and slip boundary conditions of nanoflow through Knudsen number ( Kn), as a small size parameter is considered. Viscopasternak model is considered to simulate the interaction between the graphene sheet and the surrounding elastic medium. Continuum orthotropic plate model and relations of classical plate theory are used. The nonlocal theory of Eringen is employed to incorporate the small-scale effect into the governing equations of the graphene sheet. Differential quadrature method is employed to solve the governing differential equations for simply supported edges. The convergence of the procedure is shown and the effects of flow velocity, temperature change and aspect ratio on the frequency of the single-layered graphene sheet are investigated. Moreover, the critical flow velocities and the instability characteristic are determined. It is evident from the results that the natural frequency of nanosheet increases with rising temperature.

Free transverse vibration analysis of thin rectangular plates locally suspended on elastic beam

Free transverse vibration of thin rectangular plates locally suspended on deformable beam is presented using generalized differential quadrature method. The plate is completely free at all edges except a local region which is attached to a thin beam with rectangular cross section. The other side of the beam is fixed and the whole system is subjected to free transverse vibrations. According to classical plate theory and Euler–Bernoulli beam assumption, two coupled partial differential equations of the system are obtained. The governing equations and solution domain are discretized based on the generalized differential quadrature method and natural frequencies of the plate attached to the beam are obtained. Accuracy of the predictions is investigated using four simplified cases which show reasonably good agreement. For the general case, however, due to lack of data in the literature, predictions are compared with finite element results, which also demonstrate close agreement.

Free transverse vibration analysis of circular and annular graphene sheets with various boundary conditions using the nonlocal continuum plate model

In this article, the free vibration behavior of circular and annular graphene sheet is studied. Using the nonlocal elasticity theory, the governing equations are derived for single-layered graphene sheets (SLGS) and nonlocal parameter appears into arguments of Bessel functions. Analytical frequency equations for circular and annular graphene sheets are obtained based on different cases of boundary conditions. New version of differential quadrature method has been used to solve the governing equations for circular nanoplate. The results of new version of differential quadrature method are successfully verified with those of the analytical method. The results are subsequently compared with valid results reported in the literature. The effects of the small scale on natural frequencies are investigated considering various parameters such as the radius of the plate, radius ratio, nodal circle, nodal diameter number, boundary conditions and mode numbers.