Analysis of Stability and Bifurcation of an Asymmetrical Rotor

2015 ◽  
Author(s):  
Majid Shahgholi ◽  
S. E. Khadem ◽  
Mahsa Asgarisabet
Keyword(s):  
Multiple Scales ◽  
Equations Of Motion ◽  
Multiple Scales Method ◽  
Moments Of Inertia ◽  
Rotor Systems ◽  
Unequal Mass ◽  
Principal Axes ◽  
Analysis Of Stability ◽  
Stability And Bifurcations ◽  
The Stability

The effect of shaft and disk asymmetry on the harmonic resonances of a rotor system with the in-extensional nonlinearity and large amplitude are investigated. Two rotor systems, one of which has been comprised of a symmetrical shaft and an asymmetrical disk (SA), and the other one has been comprised of an asymmetrical shaft and an asymmetrical disk (AA) are investigated. The shaft in the AA rotor has unequal mass moments of inertia and flexural rigidities in the direction of principal axes. Also, in the AA system the rigid disk is asymmetric with unequal mass moments of inertia. The equations of motion are derived by the Hamiltonian principle. The stability and bifurcations are obtained using the multiple scales method. The influences of asymmetry of shaft, asymmetry of disk, inequality between two eccentricities corresponding to the principal axes, disk position and external damping on the stability and bifurcations of SA and AA rotors are investigated. The results achieved from multiple scales method show a good agreement with those of numerical simulations.

Nonlinear Transverse Vibration of a Hyperelastic Beam Under Harmonically Varying Axial Loading

2021 ◽  
Vol 16 (3) ◽  
Author(s):  
Yuanbin Wang ◽  
Weidong Zhu
Keyword(s):  
Governing Equation ◽  
Transverse Vibration ◽  
Harmonic Balance ◽  
Fourier Transforms ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Multiple Scales Method ◽  
Frequency Responses ◽  
Runge Kutta ◽  
Nonlinear Transverse Vibration

Abstract Nonlinear transverse vibration of a hyperelastic beam under a harmonically varying axial load is analyzed in this work. Equations of motion of the beam are derived via the extended Hamilton's principle, where transverse vibration is coupled with longitudinal vibration. The governing equation of nonlinear transverse vibration of the beam is obtained by decoupling the equations of motion. By applying the Galerkin method, the governing equation transforms to a series of nonlinear ordinary differential equations (ODEs). Response of the beam is obtained via three different methods: the Runge–Kutta method, multiple scales method, and harmonic balance method. Time histories, phase-plane portraits, fast Fourier transforms (FFTs), and amplitude–frequency responses of nonlinear transverse vibration of the beam are obtained. Comparison of results from the three methods is made. Results from the multiple scales method are in good agreement with those from the harmonic balance and Runge–Kutta methods when the amplitude of vibration is small. Effects of the material parameter and geometrical parameter of the beam on its amplitude–frequency responses are analyzed.

Nonlinear vibration, stability, and bifurcation analysis of unbalanced spinning pre-twisted beam

2019 ◽  
Vol 24 (11) ◽  
pp. 3514-3536
Author(s):  
Mohsen Tajik ◽  
Ardeshir Karami Mohammadi
Keyword(s):  
Nonlinear Vibration ◽  
Bifurcation Analysis ◽  
Multiple Scales ◽  
Damping Ratio ◽  
Twist Angle ◽  
Equations Of Motion ◽  
Multiple Scales Method ◽  
Vibration Stability ◽  
Stability And Bifurcation ◽  
Stability And Bifurcation Analysis

In this paper, an Euler–Bernoulli model has been used for nonlinear vibration, stability, and bifurcation analysis of spinning twisted beams with linear twist angle, and with large transverse deflections, near the primary and parametric resonances. The equations of motion, in the case of pure single mode motion are analyzed by two methods: directly applying multiple scales method and using multiple scales method after discretization by Galerkin’s procedure. It is observed that the same final relations are obtained in the two methods. Effects of twist angle, damping ratio, longitudinal to transverse stiffness ratio, and eccentricity on the frequency responses are investigated. Then, the results are compared with the results obtained from Runge–Kutta numerical method on ODEs in a steady state, and confirmed with some previous research. Finally, the results show a good correlation, and it shows that with increasing the twist angle from 0 to 90°, the natural frequencies increase in the first two modes.

The shape of the Moon, its internal structure and moments of inertia

1967 ◽  
Vol 296 (1446) ◽  
pp. 254-265 ◽  
Keyword(s):  
Convective Motion ◽  
The Moon ◽  
Moments Of Inertia ◽  
Lunar Interior ◽  
A Value ◽  
Principal Axes ◽  
Thermoelastic Effects ◽  
The Mean ◽  
The Stability ◽  
Sufficient Strength

Harmonic analysis of the Moon’s shape based on all available sets of hypsometric data disclose that the surface of the Moon, far from being a mere spheroid or ellipsoid, contains many significant harmonic terms, the single largest of which are of fourth order (being about three times as large as the second harmonics). Their sum makes the Moon to deviate from a mean sphere by ± 2 km over extensive regions; and local differences attaining 8 to 9 km in eleva­tion have been noted on the limb. These facts reveal that the lunar globe must possess sufficient strength to sustain stress differences of the order of 10 9 dyn/cm 2 ; and this could scarcely be the case if the large part of the Moon’s interior were molten. As melting should be expected if the Moon contained the same proportion of radioactive elements as chondritic meteroites, it is concluded that the mean radioactive content of the lunar interior must be less than that found in stony meteorites, or the terrestrial crust. The moments of inertia about the principal axes of inertia of the lunar globe, as determined from the Moon’s physical librations, are seriously at variance with a state of hydrostatic equilibrium—for any distance between the Earth and the Moon—of a homogeneous body, and can be accounted for only by assuming an asymmetric nonhomogeneity of the lunar globe, or the existence of internal processes which could support nonequilibrium from hydrodynamically. However, an application of Chandrasekhar’s theory of viscous convection in fluid globes reveals that, if such a globe is to possess the same difference, C – A , of momenta as the Moon, the velocity of convective motion should be of the order of 10 –8 cm/s (i. e. too small for the establishment of steady flow in 10 9 y); and the 'observed' value of the Rayleigh number characteristic of the Moon is several hundred times as large as that required theoretically for the stability of the respective flow. Thermoelastic effects due to secular insolation of the lunar globe, considered recently by Levin, are shown incapable to account for a value of the ratio (C – A)/B exceeding 0∙00005; while its empirical value deduced from librations is close to 0∙00063.

Nonconservative Stability of Gyroscopic Rotor Systems

1993 ◽  
Author(s):  
Hwang-Kuen Chen ◽  
Der-Ming Ku ◽  
Lien-Wen Chen
Keyword(s):  
Direct Method ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Follower Force ◽  
The Finite Element Method ◽  
Nonconservative System ◽  
Rotor Systems ◽  
Eigenvalue Sensitivity ◽  
The Stability ◽  
Flutter Load

Abstract The stability behavior of a cantilevered shaft, rotating at a constant speed and subjected to a follower force at the free end, is studied by the finite element method. The equations of motion for such a gyroscopic system are formulated by using deformation shape functions developed from Timoshenko beam theory. The effects of translational and rotatory inertia, gyroscopic moments, bending and shear deformations are included. In order to determine the critical load of the present nonconservative system more quickly and efficiently, a simple and direct method that utilizes the eigenvalue sensitivity with respect to the follower force is introduced. The numerical results show that for the present nonconservative system, the onset of flutter instability occurs when the first and second backward whirl speeds are coincident. And also, due to the effect of the gyroscopic moments, the critical flutter load decreases as the rotational speed increases.

Dynamic Stability of a Spinning Timoshenko Beam Subjected to a Moving Mass-Spring-Damper Unit

2010 ◽  
Author(s):  
T. H. Young ◽  
M. S. Chen
Keyword(s):  
Dynamic Stability ◽  
Timoshenko Beam ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Axial Direction ◽  
Longitudinal Axis ◽  
Moving Mass ◽  
The Stability ◽  
The Galerkin Method ◽  
Mass Spring

This paper investigates the dynamic stability of a finite Timoshenko beam spinning along its longitudinal axis and subjected to a moving mass-spring-damper (MSD) unit traveling in the axial direction. The mass of the moving MSD unit makes contact with the beam all the time during traveling. Due to the moving MSD unit, the beam is acted upon by a periodic, parametric excitation. In this work, the equations of motion of the beam are first discretized by the Galerkin method. The discretized equations of motion are then partially uncoupled by the modal analysis procedure suitable for gyroscopic systems. Finally the method of multiple scales is used to obtain the stability boundaries of the beam. Numerical results show that if the displacement of the MSD unit is equal to only one of the two transverse displacements of the beam, very large unstable regions may appear at main resonances.

Dynamic Behavior and Stability of a Rotor Under Base Excitation

2006 ◽  
Vol 128 (5) ◽  
pp. 576-585 ◽  
Author(s):  
M. Duchemin ◽  
A. Berlioz ◽  
G. Ferraris
Keyword(s):  
Dynamic Behavior ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Ritz Method ◽  
Base Excitation ◽  
Flexible Rotor ◽  
Lagrange Equations ◽  
Rotor Systems ◽  
Simple Systems ◽  
Flexible Rotor Systems

The dynamic behavior of flexible rotor systems subjected to base excitation (support movements) is investigated theoretically and experimentally. The study focuses on behavior in bending near the critical speeds of rotation. A mathematical model is developed to calculate the kinetic energy and the strain energy. The equations of motion are derived using Lagrange equations and the Rayleigh-Ritz method is used to study the basic phenomena on simple systems. Also, the method of multiple scales is applied to study stability when the system mounting is subjected to a sinusoidal rotation. An experimental setup is used to validate the presented results.

scholarly journals Dynamic and Stability of Harmonic Driving Flexible Cartesian Robotic Arm with Bolted Joints Based on the Sensitivity and Multiple Scales Method

2015 ◽  
Vol 2015 ◽  
pp. 1-12 ◽  
Author(s):  
Yufei Liu ◽  
Wei Li ◽  
Xuefeng Yang ◽  
Yuqiao Wang
Keyword(s):  
Multiple Scales ◽  
Virtual Work ◽  
Multiple Scales Method ◽  
Bolted Joints ◽  
Parametric Stability ◽  
Robotic Arms ◽  
Average Motion ◽  
Duhamel Integral ◽  
Vibration Responses ◽  
The Stability

Flexible Cartesian robotic arms (CRAs) are typical multicoupling systems. Considering the elastic effects of bolted joints and the motion disturbances, this paper investigates the dynamic and stability of the flexible CRA. With the kinetic energy and potential energy of the comprising components, Hamilton’s variational principle and Duhamel integral are utilized to derive the dynamic equation and vibration differential equation. Based on the proposed elastic restraint model of the bolted joints, boundary conditions and mode equations of the flexible CRA are determined with using the principle of virtual work. According to the mode frequencies and sensitivities analysis, it reveals that the connecting stiffness of the bolted joints has significant influences, and the mode frequencies are more sensitive to the tensional stiffness. Moreover, describing the motion displacement of the driving base as combination of an average motion displacement and a harmonic disturbance, the vibration responses of the system are studied. The result indicates that the motion disturbance has obvious influence on the vibration responses, and the influence enhances under larger accelerating operations. The multiple scales method is introduced to analyze the parametric stability of the system, as well as the influences of the tensional stiffness and the end-effector on the stability.

Dynamics of Axially Accelerating Beams With an Intermediate Support

2011 ◽  
Vol 133 (3) ◽  
Author(s):  
S. M. Bağdatli ◽  
E. Özkaya ◽  
H. R. Öz
Keyword(s):  
Natural Frequency ◽  
Axial Velocity ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Intermediate Support ◽  
Approximate Analytical Solutions ◽  
Fluctuation Frequency ◽  
Simple Supports ◽  
The Stability ◽  
Euler Bernoulli Beam

The transverse vibrations of an axially accelerating Euler–Bernoulli beam resting on simple supports are investigated. The supports are at the ends, and there is a support in between. The axial velocity is a sinusoidal function of time varying about a constant mean speed. Since the supports are immovable, the beam neutral axis is stretched during the motion, and hence, nonlinear terms are introduced to the equations of motion. Approximate analytical solutions are obtained using the method of multiple scales. Natural frequencies are obtained for different locations of the support other than end supports. The effect of nonlinear terms on natural frequency is calculated for different parameters. Principal parametric resonance occurs when the velocity fluctuation frequency is equal to approximately twice of natural frequency. By performing stability analysis of solutions, approximate stable and unstable regions were identified. Effects of axial velocity and location of intermediate support on the stability regions have been investigated.

scholarly journals Resonance in the Cart-Pendulum System—An Asymptotic Approach

2021 ◽  
Vol 11 (23) ◽  
pp. 11567
Author(s):  
Wael S. Amer ◽  
Tarek S. Amer ◽  
Roman Starosta ◽  
Mohamed A. Bek
Keyword(s):  
Multiple Scales ◽  
Equilibrium Points ◽  
Equations Of Motion ◽  
Approximate Solutions ◽  
Pendulum System ◽  
Damped System ◽  
Stability And Instability ◽  
Parametric Pendulum ◽  
The Stability ◽  
Fundamental Equations

The major objective of this research is to study the planar dynamical motion of 2DOF of an auto-parametric pendulum attached with a damped system. Using Lagrange’s equations in terms of generalized coordinates, the fundamental equations of motion (EOM) are derived. The method of multiple scales (MMS) is applied to obtain the approximate solutions of these equations up to the second order of approximation. Resonance cases are classified, in which the primary external and internal resonance are investigated simultaneously to establish both the solvability conditions and the modulation equations. In the context of the stability conditions of these solutions, the equilibrium points are obtained and graphically displayed to derive the probable steady-state solutions near the resonances. The temporal histories of the attained results, the amplitude, and the phases of the dynamical system are depicted in graphs to describe the motion of the system at any instance. The stability and instability zones of the system are explored, and it is discovered that the system’s performance is stable for a significant number of its variables.

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