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.

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 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.

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.

Nonlinear Transverse Vibrations and Stability of Spinning Disks With Nonconstant Spinning Rate

1992 ◽  
Vol 114 (4) ◽  
pp. 506-513 ◽  
Author(s):  
T. H. Young
Keyword(s):  
Periodic Solutions ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Parametric Instability ◽  
Angular Speed ◽  
Equation Of Motion ◽  
Numerical Results ◽  
Transverse Vibrations ◽  
Small Periodic ◽  
Small Periodic Perturbation

This paper studies nonlinear transverse vibrations of spinning disks with nonconstant spinning rate. Here the angular speed of the disk is characterized as a small, periodic perturbation superimposed upon a constant speed. Due to this perturbation in angular speed, nonautonomous terms appear in the equation of motion, which results in the existence of parametric instability. In this paper, Galerkin’s method is first applied to yield a discretized system, and the method of multiple scales is used to obtain periodic solutions. All types of possible resonant combinations are investigated, and numerical results are shown for a simple harmonic speed perturbation.

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.

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.

scholarly journals Perturbation Solutions For Magnetohydrodynamics (Mhd) Flow of in a Non-Newtonian Fluid Between Concentric Cylinders

2019 ◽  
Vol 24 (1) ◽  
pp. 199-211
Author(s):  
M. Yürüsoy ◽  
Ö.F. Güler
Keyword(s):  
Analytical Solutions ◽  
Numerical Solutions ◽  
Variable Viscosity ◽  
Perturbation Technique ◽  
Equations Of Motion ◽  
Mhd Flow ◽  
Model Space ◽  
Viscosity Model ◽  
Concentric Cylinders ◽  
Approximate Analytical Solutions

Abstract The steady-state magnetohydrodynamics (MHD) flow of a third-grade fluid with a variable viscosity parameter between concentric cylinders (annular pipe) with heat transfer is examined. The temperature of annular pipes is assumed to be higher than the temperature of the fluid. Three types of viscosity models were used, i.e., the constant viscosity model, space dependent viscosity model and the Reynolds viscosity model which is dependent on temperature in an exponential manner. Approximate analytical solutions are presented by using the perturbation technique. The variation of velocity and temperature profile in the fluid is analytically calculated. In addition, equations of motion are solved numerically. The numerical solutions obtained are compared with analytical solutions. Thus, the validity intervals of the analytical solutions are determined.

Vibration Analysis of a Partially Covered Double Sandwich Cantilever Beam With Concentrated Mass at the Free End

1993 ◽  
Author(s):  
C. Levy ◽  
Q. Chen
Keyword(s):  
Loss Factor ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Physical Parameters ◽  
Mass Ratio ◽  
Euler Beam ◽  
Concentrated Mass ◽  
Sandwich Type ◽  
Euler Beam Theory ◽  
Two Parameters

Abstract The partially covered, sandwich-type cantilever with concentrated mass at the free end is studied. The equations of motion for the system modeled via Euler beam theory are derived and the resonant frequency and loss factor of the system are analyzed. The variations of resonance frequency and system loss factor for different geometrical and physical parameters are also discussed. Variation of these two parameters are found to strongly depend on the geometrical and physical properties of the constraining layers and the mass ratio.

The Role of Modal Coupling on the Non-Linear Response of Cylindrical Shells Subjected to Dynamic Axial Loads

2000 ◽  
Author(s):  
Paulo B. Gonçalves ◽  
Zenón J. G. N. Del Prado
Keyword(s):  
Dynamic Instability ◽  
Equations Of Motion ◽  
Cylindrical Shells ◽  
Parametric Instability ◽  
Compressive Load ◽  
Basins Of Attraction ◽  
Dynamic Component ◽  
Linear Ordinary Differential Equations ◽  
Non Linear ◽  
Low Dimensional

Abstract This paper discusses the dynamic instability of circular cylindrical shells subjected to time-dependent axial edge loads of the form P(t) = P0+P1(t), where the dynamic component p1(t) is periodic in time and P0 is a uniform compressive load. In the present paper a low dimensional model, which retains the essential non-linear terms, is used to study the non-linear oscillations and instabilities of the shell. For this, Donnell’s shallow shell equations are used together with the Galerkin method to derive a set of coupled non-linear ordinary differential equations of motion which are, in turn, solved by the Runge-Kutta method. To study the non-linear behavior of the shell, several numerical strategies were used to obtain Poincaré maps, stable and unstable fixed points, bifurcation diagrams and basins of attraction. Particular attention is paid to two dynamic instability phenomena that may arise under these loading conditions: parametric instability and escape from the pre-buckling potential well. The numerical results obtained from this investigation clarify the conditions, which control whether or not instability may occur. This may help in establishing proper design criteria for these shells under dynamic loads, a topic practically unexplored in literature.

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