Model Of Massless Rotational Spring Research Articles

Stability Analyses of Cracked Functionally Graded Graphene-Platelets Reinforced Composite Beam Covered with Piezoelectric Layers

As cracks are unavoidable and always reduce structural local stiffness and strength, this paper pays attention to the effect of cracks on the stability of the cracked functionally graded (FG) graphene-nanoplates reinforced composite (GRC) beam covered with piezoelectric layers. Both the critical buckling loads and postbuckling paths of the novel structures with cracks are considered. The massless rotational spring model is employed to calculate the bending stiffness of the cracked section. Three different graphene platelets (GPLs) distribution patterns along the thickness direction of the FG-GRC core beam are studied. The effective material properties of the FG-GRC core beam are calculated by Halpin–Tsai model and the rule of mixture. The governing equations of stability of the cracked FG-GRC piezoelectric beam are established within the framework of the first-order shear deformation beam theory, von Kármán geometric nonlinearity and Ritz method. The direct iteration method is used to examine the effects of boundary conditions, crack parameters, piezoelectric layers and GPL parameters on the critical buckling loads and postbuckling responses of the cracked FG-GRC piezoelectric beams. Results clearly illustrate that GPLs can significantly improve the stability of the cracked FG-GRC piezoelectric beams, while the increasing crack depth has the opposite effect.

Nonlinear free vibration of cracked functionally graded graphene platelet-reinforced nanocomposite beams in thermal environments

The present paper studies the nonlinear free vibration of edge-cracked graphene nanoplatelet (GPL)-reinforced composite laminated beams resting on a two-parameter elastic foundation in thermal environments. GPL nanofillers are assumed to be randomly oriented and are functionally graded distributed in a layer-wise pattern along the beam thickness. The temperature field considered is assumed to be a uniformly distributed in the domain of the beam. Effective material properties of the GPL-reinforced nanocomposite (GPLRC) are estimated by micromechanical models. Stress intensity factors are calculated based on the finite element methods. In the framework of the first order shear deformation beam theory, the equations of motion of cracked GPLRC beams are established including the von Kármán-type geometric nonlinearity. The bending stiffness of the cracked section is evaluated by the massless rotational spring model. The differential quadrature method is used to calculate the linear and nonlinear natural frequencies of the cracked beams. Numerical results illustrate the effects of GPL distribution pattern and concentration, GPL geometry and size, crack length, foundation stiffnesses and temperature variation on the linear and nonlinear free vibration characteristics of the cracked GPLRC beams.

Vibration simulation for a cantilever beam including a slant vertical crack

Purpose Modelling methods can be helpful for understanding vibrations of beam structures including cracks, as well as for early detection of crack. This study aims to provide an analytical modelling approach for a cantilever beam considering a slant vertical crack along its height. However, previous uniform crack methods cannot be used for describing this case. The results from the analytical, finite element (FE) and experimental methods are compared to verify the vibration problem. Design/methodology/approach A massless rotational spring model is adopted to describe the crack. An extended method based on the calculation method for a uniform vertical edge crack is proposed to obtain the stiffness of the slant case. The beam is divided into a series of independent thin slices along the beam height. An Euler–Bernoulli beam model is applied to formulate each slice. The crack in each slice is considered as a uniform one. The transfer matrix method in the literature is used to obtain the beam vibration frequencies and mode shapes. Influences of crack location and sizes on the natural frequencies for the cantilever beam, as well as the mode shapes, are analysed. An established FE model and test results in the listed references are used to validate the developed method. Findings The numerical results show that the rotational stiffness at the cracked section and the natural frequencies of the beam decrease by increasing the crack sizes; the natural frequencies for the beam are greatly influenced by the crack sizes and location; the first natural frequency decreases with the distance from the beam fixed end to the crack location; the value of the first natural frequency reaches a minimum value when the crack is at the beam fixed end; the value of the second natural frequency is a minimum value when the crack is at the beam middle; and the value of the third natural frequency is a minimum value when the crack is at the beam free end. Saltation is observed in some mode shapes at the crack location, especially for larger crack depths; but, the mode shapes of the beam are slightly influenced by the vertical crack. Originality/value This study gives a useful analytical modelling method for free vibration analysis for the cantilever beam with a vertical crack, which can overcome the disadvantages of the previous uniform crack methods.

Boundary element method based sensitivity analysis of the crack detection in beams

A sensitivity analysis of natural frequencies to detect defects in beams is presented. The performance of the Boundary element method (BEM) as a means of the forward problem solver is compared with that of the Finite element method (FEM) with massless rotational spring model. Experimentally measured natural frequencies are input as the reference frequencies. As for the crack cases considered in the present study BEM shows better performance than FEM with massless rotational spring model. Because all the computations are restricted to the boundary BEM is particularly useful when subsequent changes in the meshes are made, which makes it ideal for the defect detection problem that requires the re-meshing many times. BEM may be an alternative to FEM with massless rotational spring model in the defect detection of a beam.

Identification of a crack in a beam by the boundary element method

A method to detect a crack in a beam is presented. The crack is not modeled as a massless rotational spring, and the forward problem is solved for the natural frequencies using the boundary element method. The inverse problem is solved iteratively for the crack location and the crack size by the Newton-Raphson method. The present crack identification procedure is applied to the simulation cases which use the experimentally measured natural frequencies as inputs, and the detected crack parameters are in good agreements with the actual ones. The present method enables one to detect a crack in a beam without the help of the massless rotational spring model.

Vibratory Characteristics of Axially-Loaded Timoshenko Beams With Arbitrary Number of Cracks

A simple and efficient analytical approach is presented to determine the vibrational frequencies and mode shape functions of axially-loaded Timoshenko beams with an arbitrary number of cracks. The local compliance induced by a crack is described by a massless rotational spring model. A set of boundary conditions are used as initial parameters to define the mode shape of the segment of the beam before the first crack. Using this, the remaining set of boundary conditions and recurrence formula developed in the study, the mode shape function of vibration of the beam containing multiple cracks can be easily determined. Four different classical boundary conditions (pinned-pinned, clamped-pinned, clamped-free, and clamped-clamped) are considered. Elastically-restrained support condition with concentrated masses is also considered. Three crack depths and five axial force levels representing the conditions under service loads are used. A parametric study is carried out for each case of support conditions to investigate the effect of crack and axial load on the vibrational properties of cracked Timoshenko beams. The influence of crack on the buckling load of the beam is also studied statically. Part of the results obtained is checked against the published values. The study concludes that the crack location, crack severity, and axial force level strongly affect the eigenfrequencies.

Stability of Nonuniform Cracked Bars Under Arbitrarily Distributed Axial Loading

This study is concerned with the stability analysis of nonuniform bars with arbitrary number of cracks and with arbitrary distribution of flexural stiffness or arbitrarily distributed axial loading. The homogeneous solutions of the governing differential equation for buckling of a nonuniform uncracked bar are derived for several important cases. A model of massless rotational spring is adopted to describe the local flexibility induced by cracks. Then a new approach that combines the exact buckling solution of a nonuniform uncracked bar, the model of massless rotational spring, and the transfer matrix method is presented for the title problem. The main advantage of the proposed method is that the eigenvalue equation for buckling of a nonuniform bar with an arbitrary number of cracks, arbitrary distribution of flexural stiffness, or arbitrarily distributed axial loading can be conveniently determined from a second-order determinant. The decrease in the determinant order as compared with other methods leads to significant savings in the computational effort. A numerical example is given to illustrate the reliability of the proposed approach through comparisons with numerical solutions and to study the effect of cracks on the stability of a nonuniform bar. Nomenclature ai = depth of the ith crack Ci =fl exibility of the ith rotational spring Ci1, Ci2 = constants of integration 1F2(a1; a2; a3; a4) =h ypergeometric series f () =fl exibility function hi = height of cross section at xi Jν(x) = first Bessel function of the vth order Ki(x) =fl exural stiffness at x in the ith

Classes of exact solutions for buckling of multi-step non-uniform columns with an arbitrary number of cracks subjected to concentrated and distributed axial loads

The governing differential equation for buckling of a multi-step non-uniform column subjected to combined concentrated and distributed axial loads, each step of which has an arbitrary number of cracks with or without spring supports, is expressed in the form of bending moment. A model of massless rotational spring is adopted to describe the local flexibility induced by cracks in the column. In this paper, the distribution of flexural stiffness of a non-uniform column is arbitrary, and the distribution of axial forces acting on the column is expressed as a functional relation with the distribution of flexural stiffness and vice versa. The governing equation for buckling of a one-step non-uniform column is reduced to a differential equation of the second-order without the first-order derivative by means of functional transformation. Then, this kind of differential equation is reduced to Bessel equations and other solvable equations for six important cases. The exact buckling solutions of one-step non-uniform columns are thus found. Then a new approach that combines the exact buckling solution of a one-step column and the transfer matrix method is presented to establish the eigenvalue equation for buckling of a multi-step non-uniform column with spring supports. The main advantage of the proposed method is that the eigenvalue equation for buckling of a non-uniform column with an arbitrary number of cracks, any kind of two end supports and various spring supports at intermediate points can be conveniently determined from a second order determinant. Due to the decrease in the determinant’s order as compared with previously developed procedures the computational time required by the present method can be reduced significantly. A numerical example is given to examine the accuracy of the proposed method and to investigate the effect of cracks on buckling of a multi-step non-uniform column.

FREE VIBRATION ANALYSIS OF NON-UNIFORM BEAMS WITH AN ARBITRARY NUMBER OF CRACKS AND CONCENTRATED MASSES

An exact approach for free vibration analysis of a non-uniform beam with an arbitrary number of cracks and concentrated masses is proposed. A model of massless rotational spring is adopted to describe the local flexibility induced by cracks in the beam. Using the fundamental solutions and recurrence formulas developed in this paper, the mode shape function of vibration of a non-uniform beam with an arbitrary number of cracks and concentrated masses can be easily determined. The main advantage of the proposed method is that the eigenvalue equation of a non-uniform beam with any kind of two end supports, any finite number of cracks and concentrated masses can be conveniently determined from a second order determinant. As a consequence, the decrease in the determinant order as compared with previously developed procedures leads to significant savings in the computational effort and cost associated with dynamic analysis of non-uniform beams with cracks. Numerical examples are given to illustrate the proposed method and to study the effect of cracks on the natural frequencies and mode shapes of cracked beams.

Vibratory characteristics of multi-step beams with an arbitrary number of cracks and concentrated masses

A new exact approach for free vibration analysis of a multi-step beam with an arbitrary number of cracks and concentrated masses is proposed. A model of massless rotational spring is adopted to describe the local flexibility induced by cracks in the beam. Based on the rotational spring modeling, the fundamental solutions and recurrence formulas developed in this paper, the mode shape function of vibration of a multi-step beam, each step beam has an arbitrary number of cracks and concentrated masses, can be easily determined. The main advantage of the proposed method is that the eigenvalue equation of a multi-step beam with any kind of two end supports, any finite number of cracks and concentrated masses can be conveniently determined from a second order determinant. As a consequence, due to the decrease in the determinant order as compared with previously developed procedures, the computational time required by the present method for solving the title problem can be reduced significantly. A numerical example is given to illustrate the application of proposed methods and to investigate the effect of cracks on natural frequencies and mode shapes of a multi-step beam.

Eigenvalue Sensitivity of Cracked Beam Structures

This paper presents an analytical method for vibration eigenvalue sensitivity of cracked beam structures on the basis of variational principles. A physical model of massless rotational spring is adopted to describe the discontinuity of rotation caused by the crack. The discontinuity is further represented as a distributed loading along the beam in form of singular functions. The matrix characteristic equation and eigenvalue sensitivity equation are derived based on variational principles along with the orthogonal series for beams of arbitrary boundary conditions. Analogous analysis is performed on the axial vibrations of a beam. Meanwhile, an exact analytical solution of eigenparameters for a cracked simply supported beam is provided to qualitatively study the effect of crack height on the accuracy of the sensitivity equation. The significance of the proposed method is that it can be easily extended to the case where the beam has multiple cracks by superposition of loads. A simply supported beam is analysed by the proposed method and the exact method. The comparison of results is satisfactory and the factors affecting the accuracy are discussed.