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.

Investigation on Free Vibration of Buckled Beams

2012 ◽  
Vol 433-440 ◽  
pp. 41-44 ◽  
Author(s):  
Ming Hsu Tsai ◽  
Wen Yi Lin ◽  
Kuo Mo Hsiao ◽  
Fu Mio Fujii
Keyword(s):  
Free Vibration ◽  
Virtual Work ◽  
Beam Theory ◽  
Taylor Series Expansion ◽  
Element Formulation ◽  
Governing Equations ◽  
Nonlinear Beam ◽  
Buckled Beams ◽  
Buckled Beam ◽  
D’Alembert Principle

The objective of this study is to investigate the deformed configuration and free vibration around the deformed configuration of clamped buckled beams by co-rotational finite element formulation. The principle of virtual work, d'Alembert principle and the consistent second order linearization of the nonlinear beam theory are used to derive the element equations in current element coordinates. The governing equations for linear vibration are obtained by the first order Taylor series expansion of the equation of motion at the static equilibrium position of the buckled beam. Numerical examples are studied to investigate the natural frequencies of clamped buckled beams with different slenderness ratios under different axial compression.

scholarly journals Investigation on the Dynamics of an On-Board Rotor-Bearing System

2012 ◽  
Author(s):  
Mzaki Dakel ◽  
Sébastien Baguet ◽  
Régis Dufour
Keyword(s):  
Dynamic Behavior ◽  
Fourier Transforms ◽  
Virtual Work ◽  
Equations Of Motion ◽  
Linear Equations ◽  
Beam Theory ◽  
Rigid Base ◽  
The Finite Element Method ◽  
Mass Unbalance ◽  
Motion Display

In ship and aircraft turbine rotors, the rotating mass unbalance and the different movements of the rotor base are among the main causes of vibrations in bending. The goal of this paper is to investigate the dynamic behavior of an on-board rotor under rigid base excitations. The modeling takes into consideration six types of base deterministic motions (rotations and translations) when the kinetic and strain energies in addition to the virtual work of the rotating flexible rotor components are computed. The finite element method is used in the rotor modeling by employing the Timoshenko beam theory. The proposed on-board rotor model takes into account the rotary inertia, the gyroscopic inertia, the shear deformation of shaft as well as the geometric asymmetry of shaft and/or rigid disk. The Lagrange’s equations are applied to establish the differential equations of the rotor in bending with respect to the rigid base which represents a noninertial reference frame. The linear equations of motion display periodic parametric coefficients due to the asymmetry of the rotor and time-varying parametric coefficients due to the base rotational motions. In the proposed applications, the rotor mounted on rigid/elastic bearings is excited by a rotating mass unbalance associated with sinusoidal vibrations of the rigid base. The dynamic behavior of the rotor is analyzed by means of orbits of the rotor as well as fast Fourier transforms (FFTs).

Kinematic Aspects in Modeling Large-Amplitude Vibration of Axially Moving Beams

2019 ◽  
Vol 11 (02) ◽  
pp. 1950021 ◽  
Author(s):  
Yuanbin Wang ◽  
Hu Ding ◽  
Li-Qun Chen
Keyword(s):  
Large Amplitude ◽  
Model Equation ◽  
Classical Model ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Governing Equations ◽  
Amplitude Vibration ◽  
Large Amplitude Vibration ◽  
Axially Moving Beams ◽  
Axially Moving

This paper clarified kinematic aspects of motion of axially moving beams undergoing large-amplitude vibration. The kinematics was formulated in the mixed Eulerian–Lagrangian framework. Based on the kinematic analysis, the governing equations of nonlinear vibration were derived from the extended Hamilton principle and the higher-order shear beam theory. The derivation considered the effects of material parameters on the beam deformation. The proposed governing equations were compared with a few previous governing equations. The comparisons show that proposed equations are with higher precision. Besides, the proposed equations can be viewed as the asymptotic governing equations of Lagrange’s equations of motion for large displacement. Finally, the corresponding boundary conditions and the comparison between the presented model equation and classical model equation were provided.

Mathematical model for predicting the blade behaviour of horizontal axis wind turbine

2008 ◽  
Vol 222 (9) ◽  
pp. 1681-1694 ◽  
Author(s):  
J Wang ◽  
D Qin ◽  
Q Zhang
Keyword(s):  
Mathematical Model ◽  
Wind Turbine ◽  
Governing Equation ◽  
Virtual Work ◽  
Beam Theory ◽  
Horizontal Axis ◽  
Structure Theory ◽  
Deformation Pattern ◽  
Horizontal Axis Wind Turbine ◽  
Thin Walled

A mathematical model using both beam finite element and thin-walled structure theory is developed to predict the natural frequency and blade behaviour of a horizontal axis wind turbine under constant wind speed and turbulence condition. First of all, the deformation pattern of the blade is defined on the basis of thin-walled structure theory and Timoshenko beam theory, and by considering the blade as a rotation cantilever beam, the governing equation is obtained using the principle of virtual work. Then, it is discreted by a beam element. Constraints are applied to define boundary conditions and coupling of flapwise, edgewise, and elongation deformations on the governing equation using the penalty method. Finally, natural frequencies of the blade are analysed. Detailed expressions for centrifugal and Coriolis forces are obtained. The stress on the root and displacement at the tip are also analysed in detail. The blade's deflection in turbulent conditions is simulated and shown to mostly influence flapwise blade deformation.

Unbalance Response of Flexible Rotors Coupled With Torsion

1989 ◽  
Vol 111 (2) ◽  
pp. 179-186 ◽  
Author(s):  
H. Diken ◽  
I. G. Tadjbakhsh
Keyword(s):  
Shear Deformation ◽  
Equations Of Motion ◽  
Continuous System ◽  
Internal Damping ◽  
Governing Equations ◽  
Disk System ◽  
Torsional Oscillations ◽  
Unbalance Response ◽  
Flexible Rotors ◽  
Mass Eccentricity

The effect of coupling with torsion on the unbalance response of flexible rotors, supported by isotropic flexible and damped bearings is investigated. Flexural vibrations of the shaft-disk system are coupled with torsional oscillations through mass eccentricity. The governing equations of motion of the continuous system are solved numerically with a modified Myklestad-Prohl method without the necessity of considering an equivalent lumped system. The cases of constant or harmonic torque applied to the disk are considered. Gyroscopic, rotary inertia, shear deformation, external and internal damping effects are taken into account.

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.

Dynamic Analysis of a Beam-Type Resonator With Off-Axis Attached Mass

2012 ◽  
Author(s):  
Pezhman A. Hassanpour ◽  
Kamran Behdinan
Keyword(s):  
Equations Of Motion ◽  
Center Of Mass ◽  
Beam Theory ◽  
Numerical Approach ◽  
Governing Equations ◽  
Bernoulli Beam ◽  
Comb Drive ◽  
Comb Drives ◽  
Beam Type ◽  
Euler Bernoulli Beam

In this paper, the model of a micro-machined beam-type resonator is presented. The resonator is a micro-bridge which is modeled using Euler-Bernoulli beam theory. A comb-drive electrostatic actuator is attached to the micro-bridge for the excitation/detection of vibrations. In the models presented in the literature, it is assumed that the center of mass of the comb-drive is located on the neutral axis of the beam. In this paper, it is demonstrated that this assumption can not be applied for asymmetric-shaped comb-drives. Furthermore, the governing equations of motion are derived by relaxing the above assumption. It has been shown that the off-axis center of mass of the comb-drive generates an amplitude-dependent transverse force in the beam, which is essentially a nonlinear effect. The governing equations of motion are solved using a hybrid analytical-numerical approach. The end application of the structure under investigation is in resonant sensing and energy harvesting applications.

Consistent Modeling of Rotating Timoshenko Shafts Subject to Axial Loads

1992 ◽  
Vol 114 (2) ◽  
pp. 249-259 ◽  
Author(s):  
S. H. Choi ◽  
C. Pierre ◽  
A. G. Ulsoy
Keyword(s):  
Torsional Vibration ◽  
Axial Load ◽  
Finite Strain ◽  
Virtual Work ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Flexural Vibration ◽  
Rotating Shaft ◽  
Principal Moments Of Inertia ◽  
Mass Eccentricity

The equations of motion of a flexible rotating shaft have been typically derived by introducing gyroscopic moments, in an inconsistent manner, as generalized work terms in a Lagrangian formulation or as external moments in a Newtonian approach. This paper presents the consistent derivation of a set of governing differential equations describing the flexural vibration in two orthogonal planes and the torsional vibration of a straight rotating shaft with dissimilar lateral principal moments of inertia and subject to a constant compressive axial load. The coupling between flexural and torsional vibration due to mass eccentricity is not considered. In addition, a new approach for calculating correctly the effect of an axial load for a Timoshenko beam is presented based on the change in length of the centroidal line. It is found that the use of either a floating frame approach with the small strain assumption or a finite strain beam theory is necessary to obtain a consistent derivation of the terms corresponding to gyroscopic moments in the equations of motion. However, the virtual work of an axial load through the geometric shortening appears consistently in the formulation only when using a finite strain beam theory.

Nonlinear Free Vibration of a Beam With Off-Axis Attached Mass

2013 ◽  
Author(s):  
Pezhman A. Hassanpour
Keyword(s):  
Free Vibration ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Center Of Mass ◽  
Beam Theory ◽  
Governing Equations ◽  
Attached Mass ◽  
Lumped Mass ◽  
Nonlinear Free Vibration ◽  
Euler Bernoulli Beam

A model of a clamped-clamped beam with an attached lumped mass is presented in this paper. The system is modeled using the Euler-Bernoulli beam theory. In the models presented in literature, it is assumed that the center of mass of the attached mass is located on the neutral axis of the beam. In this paper, this assumption is relaxed. The governing equations of motion are derived. It has been shown that the off-axis center of mass of the attached mass generates an amplitude-dependent transverse force in the beam, which introduces a quadratic nonlinearity. The nonlinear governing equations of motion are solved using the Multiple Scales method. The nonlinear free vibration frequencies are determined.

On the Free Vibration Analysis of a Sandwich Beam With Tip Mass

2018 ◽  
Author(s):  
E. F. Joubaneh ◽  
O. R. Barry
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Vibration Analysis ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Sandwich Beam ◽  
Free Vibration Analysis ◽  
Design Parameters ◽  
Governing Equations ◽  
Tip Mass

This paper presents the free vibration analysis of a sandwich beam with a tip mass using higher order sandwich panel theory (HSAPT). The governing equations of motion and boundary conditions are obtained using Hamilton’s principle. General Differential Quadrature (GDQ) is employed to solve the system governing equations of motion. The natural frequencies and mode shapes of the system are presented and Ansys simulation is performed to validate the results. Various boundary conditions are also employed to examine the natural frequencies of the sandwich beam without tip mass and the results are compared with those found in the literature. Parametric studies are conducted to examine the effect of key design parameters on the natural frequencies of the sandwich beam with and without tip mass.

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