Dynamic Instability of Cantilever Beams With Open Cross-Section

2016 ◽  
Author(s):  
Júlio C. Coaquira ◽  
Paulo B. Gonçalves ◽  
Eulher C. Carvalho
Keyword(s):  
Composite Structures ◽  
Dynamic Instability ◽  
Nonlinear Oscillations ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Torsional Stiffness ◽  
Phase Portraits ◽  
Large Displacements ◽  
Torsional Coupling ◽  
Thin Walled

Structural elements with thin-walled open cross-sections are common in metal and composite structures. These thin-walled beams have generally a good flexural strength with respect to the axis of greatest inertia, but a low flexural stiffness in relation to the second principal axis and a low torsional stiffness. These elements generally have an instability, which leads to a flexural-flexural-torsional coupling. The same applies to the vibration modes. Many of these structures work in a nonlinear regime, and a nonlinear formulation that takes into account large displacements and the flexural-flexural-torsional coupling is required. In this work a nonlinear beam theory that takes into account large displacements, warping and shortening effects, as well as flexural-flexural-torsional coupling is adopted. The governing nonlinear equations of motion are discretized in space using the Galerkin method and the discretized equations of motion are solved by the Runge-Kutta method. Special attention is given to the nonlinear oscillations of beams with low torsional stiffness and its influence on the bifurcations and instabilities of the structure, a problem not tackled in the previous literature on this subject. Time responses, phase portraits and bifurcation diagrams are used to unveil the complex dynamic.

scholarly journals Vibration and Instability of Rotating Composite Thin-Walled Shafts with Internal Damping

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Ren Yongsheng ◽  
Zhang Xingqi ◽  
Liu Yanghang ◽  
Chen Xiulong
Keyword(s):  
Virtual Work ◽  
Numerical Study ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Dynamical Analysis ◽  
Design Parameters ◽  
Internal Damping ◽  
Analysis Method ◽  
Governing Equations ◽  
Thin Walled

The dynamical analysis of a rotating thin-walled composite shaft with internal damping is carried out analytically. The equations of motion are derived using the thin-walled composite beam theory and the principle of virtual work. The internal damping of shafts is introduced by adopting the multiscale damping analysis method. Galerkin’s method is used to discretize and solve the governing equations. Numerical study shows the effect of design parameters on the natural frequencies, critical rotating speeds, and instability thresholds of shafts.

Optimization of Nonlinear Unbalanced Flexible Rotating Shaft Passing Through Critical Speeds

2021 ◽  
Author(s):  
Sadegh Amirzadegan ◽  
Mohammad Rokn-Abadi ◽  
R. D. Firouz-Abadi
Keyword(s):  
Degrees Of Freedom ◽  
Nonlinear Oscillations ◽  
Nonlinear Differential Equations ◽  
Equations Of Motion ◽  
Angular Acceleration ◽  
Beam Theory ◽  
Rotor System ◽  
Rotating Shaft ◽  
Critical Speeds ◽  
Applied Torque

This work studies the nonlinear oscillations of an elastic rotating shaft with acceleration to pass through the critical speeds. A mathematical model incorporating the Von-Karman higher-order deformations in bending is developed to investigate the nonlinear dynamics of rotors. A flexible shaft on flexible bearings with springs and dampers is considered as rotor system for this work. The shaft is modeled as a beam and the Euler–Bernoulli beam theory is applied. The kinetic and strain energies of the rotor system are derived and Lagrange method is then applied to obtain the coupled nonlinear differential equations of motion for 6 degrees of freedom. In order to solve these equations numerically, the finite element method (FEM) is used. Furthermore, for different bearing properties, rotor responses are examined and curves of passing through critical speeds with angular acceleration due to applied torque are plotted. Then the optimal values of bearing stiffness and damping are calculated to achieve the minimum vibration amplitude, which causes to pass easier through critical speeds. It is concluded that the value of damping and stiffness of bearing change the rotor critical speeds and also significantly affect the dynamic behavior of the rotor system. These effects are also presented graphically and discussed.

Dynamic Stability of a Cantilever Shaft-Disk System

1992 ◽  
Vol 114 (3) ◽  
pp. 326-329 ◽  
Author(s):  
Lien-Wen Chen ◽  
Der-Ming Ku
Keyword(s):  
Dynamic Stability ◽  
Dynamic Instability ◽  
Equations Of Motion ◽  
Beam Theory ◽  
The Finite Element Method ◽  
Disk System ◽  
Stability Behavior ◽  
Gyroscopic Moment ◽  
The Mathematical Model ◽  
Cantilever Shaft

The dynamic stability behavior of a cantilever shaft-disk system subjected to axial periodic forces varying with time is studied by the finite element method. The equations of motion for such a system are formulated using deformation shape functions developed from Timoshenko beam theory. The effects of translational and rotatory inertia, gyroscopic moment, bending and shear deformation are included in the mathematical model. Numerical results show that the effect of the gyroscopic term is to shift the boundaries of the regions of dynamic instability outwardly and, therefore, the sizes of these regions are enlarged as the rotational speed increases.

Nonorthogonal solution for thin-walled members – a finite element formulation

2006 ◽  
Vol 33 (4) ◽  
pp. 421-439 ◽  
Author(s):  
R Emre Erkmen ◽  
Magdi Mohareb
Keyword(s):  
Finite Element ◽  
Computational Cost ◽  
Beam Theory ◽  
Torsional Stiffness ◽  
Element Formulation ◽  
Field Equations ◽  
Cross Sectional ◽  
Special Cases ◽  
Thin Walled ◽  
Shell Finite Element

Conventional solutions for the equations of equilibrium based on the well-known Vlasov thin-walled beam theory uncouple the equations by adopting orthogonal coordinate systems. Although this technique considerably simplifies the resulting field equations, it introduces several modelling complications and limitations. As a result, in the analysis of problems where eccentric supports or abrupt cross-sectional changes exist (in elements with rectangular holes, coped flanges, or longitudinal stiffened members, etc.), the Vlasov theory has been avoided in favour of a shell finite element that offer modelling flexibility at higher computational cost. In this paper, a general solution of the Vlasov thin-walled beam theory based on a nonorthogonal coordinate system is developed. The field equations are then exactly solved and the resulting displacement field expressions are used to formulate a finite element. Two additional finite elements are subsequently derived to cover the special cases where (a) the St.Venant torsional stiffness is negligible and (b) the warping torsional stiffness is negligible. Key words: open sections, warping effect, finite element,thin-walled beams, asymmetric sections.

scholarly journals Numerical/Experimental Validation of Thin-Walled Composite Box Beam Optimal Design

Aerospace ◽  
2020 ◽  
Vol 7 (8) ◽  
pp. 111
Author(s):  
Enrico Cestino ◽  
Giacomo Frulla ◽  
Paolo Piana ◽  
Renzo Duella
Keyword(s):  
Optimal Design ◽  
Composite Structures ◽  
Experimental Validation ◽  
Torsional Stiffness ◽  
Beam Model ◽  
Angle Distribution ◽  
Structural Configuration ◽  
Box Beam ◽  
Thin Walled ◽  
Composite Box Beam

Thin-walled composite box beam structural configuration is representative of a specific high aspect ratio wing structure. The optimal design procedure and lay-up definition including appropriate coupling necessary for aerospace applications has been identified by means of “ad hoc” analytical formulation and by application of commercial code. The overall equivalent bending, torsional and coupled stiffness are derived and the accuracy of the simplified beam model is demonstrated by the application of Altair Optistruct. A simple case of a coupled cantilevered beam with load at one end is introduced to demonstrate that stiffness and torsion angle distribution does not always correspond to the trends that one would intuitively expect. The maximum of torsional stiffness is not obtained with fibers arranged at 45° and, at the maximum torsional stiffness, there is no minimum rotation angle. This observation becomes essential in any design process of composite structures where the constraints impose structural couplings. Furthermore, the presented theory is also extended to cases in which it is necessary to include composite/stiffened hybrid configurations. Good agreement has been found between the theoretical simplified beam model and numerical analysis. Finally, the selected composite configuration was compared to an experimental test case. The numerical and experimental validation is presented and discussed. A good correlation was found confirming the validity of the overall optimization for the optimal lay-up selection and structural configuration.

Nonplanar Dynamics of Fixed-Free Beams With Low Torsional Stiffness

2012 ◽  
Author(s):  
Eulher Chaves Carvalho ◽  
Paulo Batista Gonçalves ◽  
Zenon J. G. N. del Prado
Keyword(s):  
Nonlinear Dynamics ◽  
Internal Resonance ◽  
Nonlinear Oscillations ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Harmonic Excitation ◽  
Torsional Stiffness ◽  
Complex Dynamic ◽  
Runge Kutta Method ◽  
The Galerkin Method

The three-dimensional motions of a clamped-free, inextensible beam subject to lateral harmonic excitation are investigated in this paper. Special attention is given to the nonlinear oscillations of beams with low torsional stiffness and its influence on the bifurcations and instabilities of the structure, a problem not tackled in the previous literature on this subject. For this, the nonlinear integro-differential equations describing the flexural-flexural-torsional couplings of the beam are used, together with the Galerkin method, to obtain a set of discretized equations of motion, which are in turn solved by numerical integration using the Runge-Kutta method. Both inertial and geometric nonlinearities are considered in the present analysis. By varying the beam stiffness parameters, and using several tools of nonlinear dynamics, a complex dynamic behavior of the beam is observed near the region where a 1:1:1 internal resonance occurs. In this region several bifurcations leading to multiple coexisting solutions, including planar and nonplanar motions are obtained. Finally, the paper shows how the tools of nonlinear dynamics can help in the understanding of the global integrity of the model, thus leading to a safe design.

Torsional Analysis of a Steel Cord-Rubber Tire Belt Structure

1996 ◽  
Vol 24 (4) ◽  
pp. 339-348 ◽  
Author(s):  
R. M. V. Pidaparti
Keyword(s):  
Finite Element ◽  
Finite Element Model ◽  
Composite Structures ◽  
Three Dimensional ◽  
Element Model ◽  
Torsional Stiffness ◽  
Element Analysis ◽  
Rubber Belt ◽  
Steel Cord ◽  
Torsional Analysis

Abstract A three-dimensional (3D) beam finite element model was developed to investigate the torsional stiffness of a twisted steel-reinforced cord-rubber belt structure. The present 3D beam element takes into account the coupled extension, bending, and twisting deformations characteristic of the complex behavior of cord-rubber composite structures. The extension-twisting coupling due to the twisted nature of the cords was also considered in the finite element model. The results of torsional stiffness obtained from the finite element analysis for twisted cords and the two-ply steel cord-rubber belt structure are compared to the experimental data and other alternate solutions available in the literature. The effects of cord orientation, anisotropy, and rubber core surrounding the twisted cords on the torsional stiffness properties are presented and discussed.

Post-buckling inelastic analysis of thin-walled metal members using the Generalised Beam Theory

2019 ◽  
Author(s):  
Miguel Abambres ◽  
Dinar Camotim ◽  
Miguel Abambres
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Beam Theory ◽  
Flow Theory ◽  
Promising Alternative ◽  
Inelastic Analysis ◽  
Post Buckling ◽  
Thin Walled ◽  
Shell Finite Element ◽  
Generalised Beam Theory

A 2nd order inelastic Generalised Beam Theory (GBT) formulation based on the J2 flow theory is proposed, being a promising alternative to the shell finite element method. Its application is illustrated for an I-section beam and a lipped-C column. GBT results were validated against ABAQUS, namely concerning equilibrium paths, deformed configurations, and displacement profiles. It was concluded that the GBT modal nature allows (i) precise results with only 22% of the number of dof required in ABAQUS, as well as (ii) the understanding (by means of modal participation diagrams) of the behavioral mechanics in any elastoplastic stage of member deformation .

First and Second Order Elastoplastic Analyses of Thin-Walled Metal Members using Generalized Beam Theory

2018 ◽  
Author(s):  
Miguel Abambres
Keyword(s):  
Finite Element ◽  
Stainless Steel ◽  
Cross Section ◽  
Beam Theory ◽  
Second Order ◽  
Flow Rule ◽  
Non Linear ◽  
Thin Walled ◽  
Shell Finite Element ◽  
Generalized Beam Theory

Original Generalized Beam Theory (GBT) formulations for elastoplastic first and second order (postbuckling) analyses of thin-walled members are proposed, based on the J2 theory with associated flow rule, and valid for (i) arbitrary residual stress and geometric imperfection distributions, (ii) non-linear isotropic materials (e.g., carbon/stainless steel), and (iii) arbitrary deformation patterns (e.g., global, local, distortional, shear). The cross-section analysis is based on the formulation by Silva (2013), but adopts five types of nodal degrees of freedom (d.o.f.) – one of them (warping rotation) is an innovation of present work and allows the use of cubic polynomials (instead of linear functions) to approximate the warping profiles in each sub-plate. The formulations are validated by presenting various illustrative examples involving beams and columns characterized by several cross-section types (open, closed, (un) branched), materials (bi-linear or non-linear – e.g., stainless steel) and boundary conditions. The GBT results (equilibrium paths, stress/displacement distributions and collapse mechanisms) are validated by comparison with those obtained from shell finite element analyses. It is observed that the results are globally very similar with only 9% and 21% (1st and 2nd order) of the d.o.f. numbers required by the shell finite element models. Moreover, the GBT unique modal nature is highlighted by means of modal participation diagrams and amplitude functions, as well as analyses based on different deformation mode sets, providing an in-depth insight on the member behavioural mechanics in both elastic and inelastic regimes.

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