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 Stability of Poroelastic Columns

1999 ◽  
Vol 67 (2) ◽  
pp. 360-362 ◽  
Author(s):  
G. Cederbaum
Keyword(s):  
Dynamic Stability ◽  
Multiple Scales ◽  
Transversely Isotropic ◽  
Axial Direction ◽  
Periodic Force ◽  
Stability Boundaries ◽  
Coupled Equations ◽  
Column Material ◽  
The Stability ◽  
Fluid Diffusion

The dynamic stability of a poroelastic column subjected to a longitudinal periodic force is investigated. The column material is assumed to be transversely isotropic with respect to the column axis, and the pore fluid flow is possible in the axial direction only. The motion of the column is governed by two coupled equations, for which the stability boundaries are determined analytically by using the multiple-scales method. It is shown that due to the fluid diffusion the stability regions are expanded, relative to the elastic (drained) case. The critical (minimum) loading amplitude, for which instability occurs, is also given. [S0021-8936(00)00902-8]

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.

The Coupled Longitudinal and Bending Vibrations on Stability in a Cracked Blade-Disk

2001 ◽  
Author(s):  
B. W. Huang ◽  
J. H. Kuang
Keyword(s):  
Dynamic Stability ◽  
Coriolis Force ◽  
Multiple Scales ◽  
Coupling Effect ◽  
Disk System ◽  
Bending Vibrations ◽  
Mode Localization ◽  
The Stability ◽  
The Galerkin Method ◽  
Blade Crack

The effect of coriolis force on the stability in a rotational blade-disk with a cracked blade was presented in this paper. A disk comprising of periodically shrouded blades was used to simulate the weakly coupled periodic structure. The mode localization phenomenon introduced by the blade crack on the longitudinal and bending vibrations on the rotating blades are considered. The Galerkin method was used to derive the unperturbation equations for the system. The boundaries of instability zones of the mistuned system were approximated by employing the so called multiple scales method. The effects of coriolis force and the magnitude of crack on the variation of the dynamic stability zones in a cracked blade-disk system are investigated numerically. Numerical results indicate that the coriolis force and the coupling effect between longitudinal and bending vibrations could affect the dynamic stability in a mistuned system significantly.

A Stability Analysis of Constrained Rotating Disks With Different Boundary Conditions

2007 ◽  
Author(s):  
Ramin M. H. Khorasany ◽  
Stanley G. Hutton
Keyword(s):  
Boundary Conditions ◽  
High Speed ◽  
Equations Of Motion ◽  
Order Partial Differential Equation ◽  
Rotating Disks ◽  
Different Boundary Conditions ◽  
The Difference ◽  
The Stability ◽  
The Galerkin Method ◽  
Mass Spring

The vibration behavior of constrained high speed rotating disks is of interest in industries as diverse as: aerospace, computer disk manufacture and saw design and usage. The purpose of this study is to investigate the stability behavior of guided circular disks with different boundary conditions. The equations of motion are developed for circular rotating disks constrained by space fixed linear, mass, spring, damper systems. The resulting equation of motion is a two dimensional fourth order partial differential equation that requires numerical solution. The Galerkin Method is employed using the eigenfunctions of the stationary non-constrained disk as approximation functions. Of interest is the effect on stability of conditions at the inner boundary. In particular the difference in behavior for centrally clamped, and splined disks (those disks that run on a spline arbor) is investigated. Also discussed is the effect of constraints on the flutter and divergence instability boundaries. Preliminary experimental results are presented for constrained splined disks, and these results are compared with the analytical predictions.

Combination Resonances of Traveling Unidirectional Plates Partially Immersed in Fluid with Time-Dependent Axial Velocity and Axially Varying Tension

2021 ◽  
Author(s):  
Hongying Li ◽  
Xibo Wang ◽  
Shumeng Zhang ◽  
Jian Li
Keyword(s):  
Nonlinear Vibration ◽  
Multiple Scales ◽  
Complex Dynamics ◽  
Axial Direction ◽  
Mean Velocity ◽  
Velocity Amplitude ◽  
Study Results ◽  
Axially Moving ◽  
The Stability ◽  
The Galerkin Method

Abstract Nonlinear vibrations of axially moving plates partially immersed in fluid are investigated in this paper. The system has time dependency in velocity as well as tension in axial direction. The Galerkin method is used to solve the nonlinear vibration differential equation. The method of multiple scales and Runge-Kutta method are applied to solve the nonlinear vibration response of the system. Additionally, the stability conditions of trivial and nontrivial solutions are analyzed using the Routh-Hurwitz criterion. The effects of mean velocity, amplitude of pulsating velocity, mean tension, amplitude of pulsating tension and pulsating frequency on the complex dynamics of the system are obtained. The study results reveal rich dynamic behaviors of fluid-structure coupling system.

On the Dynamic Stability of Self-Aligning Journal Gas Bearings

1977 ◽  
Vol 99 (4) ◽  
pp. 434-440 ◽  
Author(s):  
M. J. Cohen
Keyword(s):  
Dynamic Stability ◽  
Journal Bearing ◽  
Equations Of Motion ◽  
Stability Boundary ◽  
Initial Condition ◽  
Small Motion ◽  
Gas Bearings ◽  
Loading Case ◽  
The Stability ◽  
Small Disturbances

The report presents an investigation of the dynamic stability behaviour of self-aligning journal gas bearings when subjected to arbitrary small disturbances from an initial condition of operational equilibrium. The method is based on an approach similar to the nonlinear-ph solution of the author for the quasi-static loading case but the equations of motion of the journal are the linearized forms for small motion in the two degrees (translational) of freedom of the journal center. The stability domains for the infinite journal bearing are presented for the whole of the eccentricity (ε) and rotational speed (Λ) ranges for any given bearing geometry, in the shape of stability boundaries in that domain. It is shown that a given bearing will be stable within a corridor in the (ε, Λ) parametral domain having as its lower bound the so called “half-speed” whirl stability boundary and as its upper bound another whirling instability at a higher characteristic (relative) frequency, the instability occurs generally at the higher eccentricities and lower rotational speeds.

scholarly journals Nonlinear tuning of microresonators for dynamic range enhancement

2015 ◽  
Vol 471 (2179) ◽  
pp. 20140969 ◽  
Author(s):  
M. Saghafi ◽  
H. Dankowicz ◽  
W. Lacarbonara
Keyword(s):  
Nonlinear Response ◽  
Dynamic Range ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Equations Of Motion ◽  
Path Following ◽  
Critical Amplitude ◽  
Inertia Effects ◽  
Response Characteristics ◽  
The Galerkin Method

This paper investigates the development of a novel framework and its implementation for the nonlinear tuning of nano/microresonators. Using geometrically exact mechanical formulations, a nonlinear model is obtained that governs the transverse and longitudinal dynamics of multilayer microbeams, and also takes into account rotary inertia effects. The partial differential equations of motion are discretized, according to the Galerkin method, after being reformulated into a mixed form. A zeroth-order shift as well as a hardening effect are observed in the frequency response of the beam. These results are confirmed by a higher order perturbation analysis using the method of multiple scales. An inverse problem is then proposed for the continuation of the critical amplitude at which the transition to nonlinear response characteristics occurs. Path-following techniques are employed to explore the dependence on the system parameters, as well as on the geometry of bilayer microbeams, of the magnitude of the dynamic range in nano/microresonators.

Parametric Vibrations in Discs: Point-Wise and Distributed Loads, Including Rotating Friction

1995 ◽  
Author(s):  
H. Ouyang ◽  
S. N. Chan ◽  
J. E. Mottershead ◽  
M. I. Friswell ◽  
M. P. Cartmell
Keyword(s):  
Multiple Scales ◽  
Equations Of Motion ◽  
Transverse Load ◽  
Transverse Mass ◽  
Follower Load ◽  
Parametric Resonances ◽  
Spring System ◽  
Multiple Scales Analysis ◽  
Mass Spring ◽  
Rotating Load

Abstract This paper is concerned with the parametric resonances in a stationary annular disc when excited by a rotating load system. Two forms of the load system are considered. In the first, the load consists of a discrete transverse mass-spring-damper system and a frictional follower load. Secondly, a distributed mass-spring system (without friction) is studied. In both cases the transverse load is rotated at a uniform speed around the disc. Equations of motion are developed for the two cases, and the results of a multiple scales analysis are presented. The disc is found to exhibit many parametric resonances at subcritical speeds when friction is present.

Dynamic Stability Characteristics of Axially Accelerated Beams

2006 ◽  
Vol 321-323 ◽  
pp. 1654-1658 ◽  
Author(s):  
Hong Hee Yoo ◽  
Sung Jin Eun
Keyword(s):  
Dynamic Stability ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Stability Diagram ◽  
Accelerated Motion ◽  
Stability Diagrams ◽  
Divergence Type ◽  
The Stability ◽  
Direct Numerical Integration ◽  
Free Beam

Dynamic stability of axially accelerated beams is investigated in this paper. The equations of motion of a fixed-free beam undergoing axially accelerated motion are derived. Unstable regions due to the acceleration are obtained by using the Floquet’s theory. Stability diagrams are presented to illustrate the influence of the acceleration characteristics. Large unstable regions of flutter type instability exist around the first, twice the first, and twice the second bending natural frequencies. Divergence type instability also occurs when the acceleration exceeds a certain value. The validity of the stability diagram is confirmed by direct numerical integration of the equations of motion.

scholarly journals Dynamic Modeling and Cutting Stability of Rotating Tapered Composite Cutter Bar considering Material Damping

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Yuhuan Zhang ◽  
Ren Yongsheng ◽  
Bole Ma ◽  
Jinfeng Zhang
Keyword(s):  
Constitutive Relations ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Chatter Stability ◽  
Differential Motion ◽  
Chatter Suppression ◽  
Fixed Type ◽  
The Stability ◽  
The Galerkin Method ◽  
Cutting System

Traditional milling cutter bars are generally made up of metals and exhibit poor capacity of chatter suppression. This study proposes an anisotropic composites tapered cutter bar for increasing natural frequency and damping and finally achieves the goal of enhancing chatter stability. Based on Hamilton principle and Euler–Bernoulli beam theory, the partial differential motion equations of the cutting system with a 3D rotating tapered composite cutter bar are established. Next, using the Galerkin method, the equations of motion are discretized so as to derive ordinary differential equations. In the model, damping modeling of the composite cutter bar is achieved theoretically by using damping dissipation constitutive relations for viscoelastic composites. Moreover, by introducing the rotating effect of the 3D cutter bar in the 2-DOF analytical model of stability analysis first proposed for a fixed-type cutter bar, an improved prediction model is developed and used to solve the stability lobes of the cutting system in the frequency domain analytically. Furthermore, the influences of the gyroscopic effect, material, ply angle, stacking sequence, and taper ratio on chatter stability are also discussed.

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