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Variations of the first three modes with axial mean velocity for locations η 1 = 0.1 and η 2 = 0.9 and for different v f values (ω 1 :-, ω 2 :-, ω 3 :-·-)
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This study represents the transverse vibrations of an axially accelerating Euler–Bernoulli beam resting on multiple simple supports. This is one of the examples of a system experiencing Coriolis acceleration component that renders such systems gyroscopic. A small harmonic variation with a constant mean value for the axial velocity is assumed in the...
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The stability of an axially moving string system subjected to parametric excitation resulting from speed fluctuations has been examined in this paper. The time-dependent velocity is assumed to be a harmonically varying function around a (low) constant mean speed. The method of characteristic coordinates in combination with the two timescales pertur...
Citations
... Dynamic behavior and stability of an axially moving beam with internal nonlinear supports were investigated by several scholars in Refs. [15][16][17][18][19] examined the dynamic response of a multi-span beam under periodic impact excitation and observed complicated nonlinear behavior of such beam system in experiment. Ghayesh 21 established a general solution framework to investigate the dynamic behavior and stability of beam structure supported by nonlinear supports. ...
Dynamic analysis of an Euler–Bernoulli beam with nonlinear supports is receiving greater research interest in recent years. Current studies usually consider the boundary and internal nonlinear supports separately, and the system rotational restraint is usually ignored. However, there is little study considering the simultaneous existence of axial load, lumped mass and internal supports for such nonlinear problem. Motivated by this limitation, the dynamic behavior of an axially loaded beam supported by a nonlinear spring-mass system is solved and investigated in this paper. Modal functions of an axially loaded Euler–Bernoulli beam with linear elastic supports are taken as trail functions in Galerkin discretization of the nonlinear governing differential equation. Stable steady-state response of such axially loaded beam supported by a nonlinear spring-mass system is solved via Galerkin truncation method, which is also validated by finite difference method. Results show that parameters of nonlinear spring-mass system and boundary condition have a significant influence on system dynamic behavior. Moreover, appropriate nonlinear parameters can switch the system behavior between the single-periodic state and quasi-periodic state effectively.
... Also, Bağdatli et al. investigated continuous media as an axially moving beam with middle support [10]. The combined longitudinal-transverse nonlinear dynamics of an axially moving beam were described in Ghayesh et al. [11] and they discussed the stability of an axially moving beam that has middle spring support. ...
In this study, the axially moving string with spring-loaded middle support is discussed. The supports assumed as simple support on the string both ends. The intermediate support shows the characteristics of the spring. The string velocity is accepted as harmonically varying around a mean value. The Hamiltonian principle is used to find the equations of motion. The equations of motion become nonlinear, considering the nonlinear effects caused by string extensions. The equations of motion and boundary conditions are become dimensionless by nondimensionalization. Approximate solutions were found by using multiple time scales which is one of the perturbation methods. By solving the linear problem that is obtained by the first terms of the perturbation series, the exact natural frequencies were calculated for the different locations of the mid-support, various spring coefficients, and various axial velocity values. The second-order nonlinear terms reveal the correction terms for the linear problem. Stability analysis is carried out for cases where the velocity change frequency is away from zero and two times the natural frequency. Stability boundaries are determined for the principal parametric resonance case.
... Zhang et al. introduced Fourier differential quadrature method [29]. However, there is another nonlinear model, an integro-partial-differential model, also was adopted by many researchers for discovering the free vibration [30], the nonlinear dynamics [31,32], nonlinear dynamics and bifurcations [33,34], the steadystate periodic responses [35][36][37][38][39] of the axially moving materials. In order to distinguish the two kinds of nonlinear models, the free vibration frequencies [40] and the steady-state periodic solutions [41] are compared to couple planar vibration, because these nonlinear models of the transverse vibration both can be derived from the transverse and longitudinal coupled model. ...
The most important issue in the vibration study of an engineering system is dynamics modeling. Axially moving continua is often discussed without the inertia produced by the rotation of the continua section. The main goal of this paper is to discover the effects of rotary inertia on the free vibration characteristics of an axially moving beam in the sub-critical and super-critical regime. Specifically, an integro-partial-differential nonlinear equation is modeled for the transverse vibration of the moving beam based on the generalized Hamilton principle. Then the effects of rotary inertia on the natural frequencies, the critical speed, post-buckling vibration frequencies are presented. Two kinds of boundary conditions are also compared. In super-critical speed range, the straight configuration of the axially moving beam loses its stability. The buckling configurations are derived from the corresponding nonlinear static equilibrium equation. Then the natural frequencies of the post-buckling vibration of the super-critical moving beam are calculated by using local linearization theory. By comparing the critical speed and the vibration frequencies in the sub-critical and super-critical regime, the effects of the inertia moment due to beam section rotation are investigated. Several interesting phenomena are disclosed. For examples, without rotary inertia, the study overestimates the stability of the axially moving beam. Moreover, the relative differences between the super-critical fundamental frequencies of the two theories may increase with an increasing beam length.
... By using the 4th-order Runge-Kutta method by discretizing the temporal variable, u(x j ,t) (j ¼ 2,3,…,N-1) can be numerically calculated for a given system parameters. In addition, the initial conditions for the first calculation are given as follows u À x j ; 0 Á ¼ 0 ; u; t À x j ; 0 Á ¼ 0:0001; j ¼ 2; 3; :::; N À 1 (38) In the following numerical examples, the number of sampling points for quadrature is set as N ¼ 17. u(x (Nþ1)/2 ,t) represents the transverse nonlinear vibration displacement of the midpoint of the traveling belt. Therefore, in the steady-state phase, the maximum deformation of the midpoint of the traveling belt can be solved by the following equation ...
Free and forced nonlinear vibrations of a traveling belt with non-homogeneous boundary conditions are studied. The axially moving materials in operation are always externally excited and produce strong vibrations. The moving materials with the homogeneous boundary condition are usually considered. In this paper, the non-homogeneous boundaries are introduced by the support wheels. Equilibrium deformation of the belt is produced by the non-homogeneous boundaries. In order to solve the equilibrium deformation, the differential and integral quadrature methods (DIQMs) are utilized to develop an iterative scheme. The influence of the equilibrium deformation on free and forced nonlinear vibrations of the belt is explored. The DIQMs are applied to solve the natural frequencies and forced resonance responses of transverse vibration around the equilibrium deformation. The Galerkin truncation method (GTM) is utilized to confirm the DIQMs’ results. The numerical results demonstrate that the non-homogeneous boundary conditions cause the transverse vibration to deviate from the straight equilibrium, increase the natural frequencies, and lead to coexistence of square nonlinear terms and cubic nonlinear terms. Moreover, the influence of non-homogeneous boundaries can be exacerbated by the axial speed. Therefore, non-homogeneous boundary conditions of axially moving materials especially should be taken into account.
... Moreover, frictional con-tact [35], boundary damping [36], and "cantilever" type [37] were investigated by scientists. Dynamics of axially transporting materials with multiple supports was of concern to many researchers [38][39][40][41][42]. One thing should be noted that the above-mentioned boundary conditions are all homogeneous boundary conditions. ...
This paper focuses on the free and forced nonlinear vibration of a viscoelastic transporting belt supported by pulleys for the first time. The transporting belt boundary conditions are usually specified as simply supported or fixed. Therefore, the static equilibrium of the belt is supposed to remain straight. However, the belt wraps around two pulleys in most practical mechanical systems. The round support of the pulleys causes nonhomogeneous terms in the boundary condition. A nonlinear dynamic model is established to describe the transverse vibration of the transporting belt. The static equilibrium configuration caused by the nonhomogeneous boundary conditions is determined by developing a numerical iteration scheme. By means of coordinate transformation, the influence of nonhomogeneous boundary conditions is eliminated. Then, the natural frequencies of transverse vibration of the belt around the equilibrium configuration are obtained by adopting the differential quadrature method. Furthermore, the inertia excitation of the belt system is taken from the periodic motion of the foundation. In dynamic mechanical systems, this kind of fluctuation induced by the engine is very common. By applying the differential quadrature method and the Runge–Kutta time discretization, the time histories of the forced vibration are numerically calculated. Resonance areas of the belt are studied by using the frequency sweep. Moreover, parametric studies are carried out to understand the influences of the system parameters. It is concluded that pulley support ends cause nonhomogeneous boundaries and the static equilibrium configuration, and consequences are exacerbated by transmission speed.
... Eksenel hareketli kirişlerin, içerisinde hareketli akışkan taşıyan kiriş sistemleri ile benzer dinamik davranış gösterdiği bilinmektedir. Daha önce yapılan çalışmalarda, eksenel olarak hareketli çok mesnetli kirişlerin dinamik davranışları [4] ve eksenel hareketli kirişlerin lineer olmayan titreşimlerine ait doğal frekansları verilmiştir [5]. Zamana bağlı değişken hızla eksenel olarak hareket eden sabit mesnetli kirişlerin titreşimleri [6][7] incelenmiştir. ...
... Also, natural frequencies of non-linear vibrations of axially moving beams are presented by Ding and Chen (2011). Axially moving beams with multiple supports are analyzed by Bağdatlı et al. (2013). In addition, non-linear vibrations of springsupported axially moving string have been studied by Kesimli et al. (2015). ...
This study aimed to present the effects of nonideal boundary conditions (BCs) on fundamental parametric
resonance behavior of fluid conveying clamped microbeams. Non-ideal BCs are modelled by using the weighting factor (k). Equations of motion are obtained by using the Hamilton’s Principle. A perturbation technique, method of multiple scales, is applied to solve the non-linear equations of motions. In this study, frequency-response curves of fundamental parametric resonance are plotted and the effects
of non-ideal BCs are shown. Besides, instability areas of microbeams under ideal and non-ideal BCs are investigated by considering different system parameters. Numerical results show that instability areas significantly changed by the effect of non-ideal BCs.
... It is known that axially moving beams show similar dynamic behaviors with fluid-conveying beams. In previous studies, axially moving beams with multiple supports are investigated by Bağdatlı et al. (2013). Also, Ding and Chen (2011) Bağdatlı and Uslu (2015) studied the axially moving string under non-ideal boundary conditions. ...
In this study, vibration analysis of fluid conveying microbeams under non-ideal boundary conditions (BCs) is
performed. The objective of the present paper is to describe the effects of non-ideal BCs on linear vibrations of fluid conveying microbeams. Non-ideal BCs are modeled as a linear combination of ideal clamped and ideal simply supported boundary conditions by using the weighting factor (k). Non-ideal clamped and non-ideal simply supported beams are both considered to show the effects of BCs. Equations of motion of the beam under the effect of moving fluid are obtained by using Hamilton principle. Method of multiple scales which is one of the perturbation techniques is applied to the governing linear equation of motion. Approximate solutions of the linear equation are obtained and the effects of system parameters and non-ideal BCs on natural frequencies are presented. Results indicate that, natural frequencies of fluid conveying microbeam changed significantly by varying the weighting factor k. This change is more remarkable for clamped microbeams rather than simply supported ones.
... Yang et al. [12] investigated the free vibration of an axially moving elastic beam with simple supports resting on elastic foundation. Baǧatli et al. [13][14] examined the nonlinear vibration of the accelerating beam with single-spring support and multi-spring support. Kural andÖzkaya [15] investigated the transverse vibrations of an axially accelerating flexible beam resting on multiple supports. ...
In this paper, transverse vibration of an axially moving beam supported by a viscoelastic foundation is analyzed by a complex modal analysis method. The equation of motion is developed based on the generalized Hamilton’s principle. Eigenvalues and eigenfunctions are semi-analytically obtained. The governing equation is represented in a canonical state space form, which is defined by two matrix differential operators. The orthogonality of the eigenfunctions and the adjoint eigenfunctions is used to decouple the system in the state space. The responses of the system to arbitrary external excitation and initial conditions are expressed in the modal expansion. Numerical examples are presented to illustrate the proposed approach. The effects of the foundation parameters on free and forced vibration are examined.
... In another study about axially accelerating viscoelastic Timoshenko beams, periodic responses of system are investigated by using Galerkin method and fourth order Runge-Kutta algorithm (Yan et al. 2014). Bağdatli et al. (2011) obtained natural frequencies of simply supported two span Euler-Bernoulli beam by using perturbation techniques. In this study, the influences of axial speed, flexural rigidity and intermediate support on dynamic behaviour of the beam are investigated. ...
In this study, the free vibration analysis of axially moving beams is investigated according to Reddy-Bickford beam theory (RBT) by using dynamic stiffness method (DSM) and differential transform method (DTM). First of all, the governing differential equations of motion in free vibration are derived by using Hamilton's principle. The nondimensionalised multiplication factors for axial speed and axial tensile force are used to investigate their effects on natural frequencies. The natural frequencies are calculated by solving differential equations using analytical method (ANM). After the ANM solution, the governing equations of motion of axially moving Reddy-Bickford beams are solved by using DTM which is based on Finite Taylor Series. Besides DTM, DSM is used to obtain natural frequencies of moving Reddy-Bickford beams. DSM solution is performed via Wittrick-Williams algorithm. For different boundary conditions, the first three natural frequencies that calculated by using DTM and DSM are tabulated in tables and are compared with the results of ANM where a very good proximity is observed. The first three mode shapes and normalised bending moment diagrams are presented in figures.
