Effects of Nonconstant Spin Rate on the Vibration of a Rotating Beam

1987 ◽  
Vol 54 (2) ◽  
pp. 305-310 ◽  
Author(s):  
D. C. Kammer ◽  
A. L. Schlack
Keyword(s):  
Angular Velocity ◽  
Perturbation Technique ◽  
Equations Of Motion ◽  
Approximate Solutions ◽  
Time Dependent ◽  
Euler Beam ◽  
Kbm Method ◽  
Oriented Parallel ◽  
Small Periodic ◽  
Small Periodic Perturbation

The effects of a time-dependent angular velocity upon the vibration of a rotating Euler beam are presented. It is assumed that the angular velocity can be expressed as the sum of a steady-state value and a relatively small periodic perturbation. Equations of motion are derived for a beam oriented parallel to the spin axis. Terms with time-dependent coefficients appear in the equations of motion due to the nonconstant spin rate resulting in a nonautonomous system possessing parametric instabilities. A perturbation technique called the KBM method is used to derive general expressions for approximate solutions and instability region boundaries. A simple perturbation function is assumed for the purpose of illustrating the use of the derived general expressions.

Dynamic Response of a Radial Beam With Nonconstant Angular Velocity

1987 ◽  
Vol 109 (2) ◽  
pp. 138-143 ◽  
Author(s):  
D. C. Kammer ◽  
A. L. Schlack
Keyword(s):  
Angular Velocity ◽  
Perturbation Technique ◽  
Equations Of Motion ◽  
Parametric Instability ◽  
Approximate Solutions ◽  
Euler Beam ◽  
Radial Beam ◽  
Kbm Method ◽  
Small Periodic ◽  
Small Periodic Perturbation

The effects of a nonconstant angular velocity upon the vibration of a rotating Euler beam are investigated. It is assumed that the angular velocity can be written as the sum of a steady-state value and a small periodic perturbation. The time-dependence of the angular velocity results in the appearance of terms in the equations of motion which cause the system to be nonautonomous. These terms result in the existence of regions of parametric instability within which the amplitude grows exponentially. A perturbation technique called the KBM method is used to derive approximate solutions and expressions for the boundaries between stable and unstable motion. A simple perturbation function is assumed to illustrate the use of the derived general equations.

Nonlinear vibrations of fluid-filled functionally graded cylindrical shell considering a time-dependent lateral load and static preload

2015 ◽  
Vol 230 (1) ◽  
pp. 102-119 ◽  
Author(s):  
Frederico Martins Alves da Silva ◽  
Roger Otávio Pires Montes ◽  
Paulo Batista Gonçalves ◽  
Zenón José Guzmán Nuñez Del Prado
Keyword(s):  
Cylindrical Shell ◽  
Shell Thickness ◽  
Nonlinear Vibrations ◽  
Volume Fraction ◽  
Perturbation Technique ◽  
Equations Of Motion ◽  
Functionally Graded ◽  
Time Dependent ◽  
Nonlinear Frequency ◽  
Power Law Distribution

In the recent years, functionally gradient materials (FGMs) have gained considerable attention with possible applications in several engineering fields, especially in a high-temperature or hazardous environment. In this work, the nonlinear vibrations of a simply supported fluid-filled functionally graded cylindrical shell subjected to a lateral time-dependent load and axial static preload are analyzed. To model the shell, the Donnell nonlinear shell theory is used. The fluid is assumed to be incompressible, nonviscous, and irrotational. A new function to describe the variation in the volume fraction of the constituent material through the shell thickness is proposed, extending the concept of sandwich structures to a functionally graded material. Material properties are graded along the shell thickness according to the proposed volume fraction power law distribution. A consistent reduced order model derived from a perturbation technique is used to describe the displacements of the shell and, the Galerkin method is applied to derive a set of coupled nonlinear ordinary differential equations of motion. Results show the influence of the variation of the two constituent materials along the shell thickness, internal fluid, static preload, and shell geometry on the natural frequencies, nonlinear frequency–amplitude relation, resonance curves, and bifurcation scenario of the FG cylindrical shell.

On the Dynamics of a Beam Rotating at Nonconstant Speed

1991 ◽  
Author(s):  
Yii-Mei Huang ◽  
Ming-Shang Lin
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Equations Of Motion ◽  
Multiple Scale ◽  
Angular Speed ◽  
Partial Differential ◽  
Rotating Speed ◽  
Discrete Equations ◽  
Small Periodic ◽  
Small Periodic Perturbation

Abstract The response and its stability of a beam rotating at nonconstant angular speed are studied. The rotating speed is assumed to be the combination of a constant angular speed and a small periodic perturbation. The axial and flexural deformations due to rotation are considered simultaneously. Thus, the rotating team at nonconstant speed yields a set of parametric excited partial differential equations of motion. Extended Galerkin’s method is employed for obtaining the discrete equations of motion. Then, the solution and the its stability are found by using the method of multiple scale.

Dynamic Stability of Disks With Periodically Varying Spin Rates Subjected to Stationary In-Plane Edge Loads

2004 ◽  
Vol 71 (4) ◽  
pp. 450-458 ◽  
Author(s):  
T. H. Young ◽  
M. Y. Wu
Keyword(s):  
Dynamic Stability ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Parametric Instability ◽  
Stress Distributions ◽  
The Finite Element Method ◽  
Nodal Diameter ◽  
The Stability ◽  
Small Periodic ◽  
Small Periodic Perturbation

This paper presents an analysis of dynamic stability of an annular plate with a periodically varying spin rate subjected to a stationary in-plane edge load. The spin rate of the plate is characterized as the sum of a constant speed and a small, periodic perturbation. Due to this periodically varying spin rate, the plate may bring about parametric instability. In this work, the initial stress distributions caused by the periodically varying spin rate and the in-plane edge load are analyzed first. The finite element method is applied then to yield the discretized equations of motion. Finally, the method of multiple scales is adopted to determine the stability boundaries of the system. Numerical results show that combination resonances take place only between modes of the same nodal diameter if the stationary in-plane edge load is absent. However, there are additional combination resonances between modes of different nodal diameters if the stationary in-plane edge load is present.

scholarly journals Effect of Thrust on the Structural Vibrations of a Nonuniform Slender Rocket

2020 ◽  
Vol 25 (2) ◽  
pp. 29
Author(s):  
Desmond Adair ◽  
Aigul Nagimova ◽  
Martin Jaeger
Keyword(s):  
Natural Frequencies ◽  
Equations Of Motion ◽  
Approximate Solutions ◽  
Mode Shapes ◽  
Algebraic Equations ◽  
Structural Vibrations ◽  
Unknown Parameters ◽  
One Dimensional ◽  
Vibration Problems ◽  
Nonuniform Beam

The vibration characteristics of a nonuniform, flexible and free-flying slender rocket experiencing constant thrust is investigated. The rocket is idealized as a classic nonuniform beam with a constant one-dimensional follower force and with free-free boundary conditions. The equations of motion are derived by applying the extended Hamilton’s principle for non-conservative systems. Natural frequencies and associated mode shapes of the rocket are determined using the relatively efficient and accurate Adomian modified decomposition method (AMDM) with the solutions obtained by solving a set of algebraic equations with only three unknown parameters. The method can easily be extended to obtain approximate solutions to vibration problems for any type of nonuniform beam.

OBSERVATION OF THE TRANSVERSE STERN–GERLACH EFFECT IN NEUTRAL POTASSIUM

1967 ◽  
Vol 45 (4) ◽  
pp. 1481-1495 ◽  
Author(s):  
Myer Bloom ◽  
Eric Enga ◽  
Hin Lew
Keyword(s):  
Magnetic Field ◽  
Angular Velocity ◽  
Reference Frame ◽  
Inhomogeneous Magnetic Field ◽  
Time Dependent ◽  
Effective Magnetic Field ◽  
Classical Analysis ◽  
At Resonance ◽  
Rotating Reference Frame

A successful transverse Stern–Gerlach experiment has been performed, using a beam of neutral potassium atoms and an inhomogeneous time-dependent magnetic field of the form[Formula: see text]A classical analysis of the Stern–Gerlach experiment is given for a rotating inhomogeneous magnetic field. In general, when space quantization is achieved, the spins are quantized along the effective magnetic field in the reference frame rotating with angular velocity ω about the z axis. For ω = 0, the direction of quantization is the z axis (conventional Stern–Gerlach experiment), while at resonance (ω = −γH0) the direction of quantization is the x axis in the rotating reference frame (transverse Stern–Gerlach experiment). The experiment, which was performed at 7.2 Mc, is described in detail.

Investigation of the stability of boundary layers by a finite-difference model of the Navier—Stokes equations

1976 ◽  
Vol 78 (2) ◽  
pp. 355-383 ◽  
Author(s):  
H. Fasel
Keyword(s):  
Finite Difference ◽  
Stability Theory ◽  
Stokes Equations ◽  
Time Dependent ◽  
Linear Stability Theory ◽  
Navier Stokes ◽  
Experimental Measurements ◽  
Navier Stokes Equations ◽  
The Stability ◽  
Small Periodic

The stability of incompressible boundary-layer flows on a semi-infinite flat plate and the growth of disturbances in such flows are investigated by numerical integration of the complete Navier–;Stokes equations for laminar two-dimensional flows. Forced time-dependent disturbances are introduced into the flow field and the reaction of the flow to such disturbances is studied by directly solving the Navier–Stokes equations using a finite-difference method. An implicit finitedifference scheme was developed for the calculation of the extremely unsteady flow fields which arose from the forced time-dependent disturbances. The problem of the numerical stability of the method called for special attention in order to avoid possible distortions of the results caused by the interaction of unstable numerical oscillations with physically meaningful perturbations. A demonstration of the suitability of the numerical method for the investigation of stability and the initial growth of disturbances is presented for small periodic perturbations. For this particular case the numerical results can be compared with linear stability theory and experimental measurements. In this paper a number of numerical calculations for small periodic disturbances are discussed in detail. The results are generally in fairly close agreement with linear stability theory or experimental measurements.

Vibration and Stability of an Elastic Beam Subjected to a Periodic Axial Load With Time-Dependent Displacement Excitation at Both Ends

1991 ◽  
Author(s):  
Xiao-Feng Wu ◽  
Adnan Akay
Keyword(s):  
Axial Load ◽  
Perturbation Technique ◽  
Harmonic Balance Method ◽  
Elastic Beam ◽  
Time Dependent ◽  
Transverse Load ◽  
Order Partial Differential Equation ◽  
Weighted Residual Method ◽  
Hill’S Equation ◽  
Hill's Equation

Abstract This paper concerns the transverse vibrations and stabilities of an elastic beam simultaneously subjected to a periodic axial load, a distributed transverse load, and time-dependent displacement excitations at both ends. The equation of motion derived from Bernoulli-Euler beam theory is a fourth-order partial differential equation with periodic coefficients. To obtain approximate solutions, the method of assumed-modes is used. The unknown time-dependent function in the assumed-modes method is determined by a generalized inhomogeneous Hill’s equation. The instability regions possessed by this generalized Hill’s equation are obtained by both the perturbation technique up to the second order and the harmonic balance method. The dynamic response and the corresponding spectrum of the transversely oscillating elastic beam are calculated by the weighted-residual method.

A Method for Fatigue Analysis of Pipelines on Topsides of FPSO Systems

2005 ◽  
Author(s):  
Pol Spanos ◽  
Alba Sofi ◽  
Juan Wang ◽  
Berry Peng
Keyword(s):  
Fatigue Life ◽  
Random Vibration ◽  
Equations Of Motion ◽  
Plug Flow ◽  
Fatigue Curve ◽  
Sea Waves ◽  
Random Loading ◽  
Euler Beam ◽  
Stress Spectrum ◽  
The Galerkin Method

Pipelines located on the decks of FPSO systems are exposed to damage due to sea waves induced random loading. In this context, a methodology for estimating the fatigue life of conveying-fluid pipelines is presented. The pipeline is subjected to a random support motion which simulates the effect of the FPSO heaving. The equation of motion of the fluid-carrying pipeline is derived by assuming small amplitude displacements, modeling the empty pipeline as a Bernoulli-Euler beam, and adopting the so-called “plug-flow” approximation for the fluid (Pai¨doussis, 1998). Random vibration analysis is carried out by the Galerkin method selecting as basis functions the natural modes of a beam with the same boundary conditions as the pipeline. The discretized equations of motion are used in conjunction with linear random vibration theory to compute the stress spectrum for a generic section of the pipeline. For this purpose, the power spectrum of the acceleration at the deck level is determined by using the Response Amplitude Operator of the FPSO hull. Finally, the computed stress spectrum is used to estimate the pipeline fatigue life employing an appropriate S-N fatigue curve of the material. An illustrative example concerning a pipeline simply-supported at both ends is included in the paper.

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