scholarly journals Multiple scales solution for free vibrations of a rotating shaft with stretching nonlinearity

2013 ◽  
Vol 20 (1) ◽  
pp. 131-140 ◽  
Author(s):  
S.A.A. Hosseini ◽  
M. Zamanian
Keyword(s):  
Multiple Scales ◽  
Free Vibrations ◽  
Rotating Shaft ◽  
Multiple Scales Solution

Propagation of unsteady disturbances in a slowly varying duct with mean swirling flow

2001 ◽  
Vol 445 ◽  
pp. 207-234 ◽  
Author(s):  
A. J. COOPER ◽  
N. PEAKE
Keyword(s):  
Turning Point ◽  
Multiple Scales ◽  
Swirling Flow ◽  
Coupled System ◽  
Mean Flow ◽  
Amplitude Variation ◽  
Small Shift ◽  
Swirl Velocity ◽  
Vorticity Equations ◽  
Multiple Scales Solution

The propagation of unsteady disturbances in a slowly varying cylindrical duct carrying mean swirling flow is described. A consistent multiple-scales solution for the mean flow and disturbance is derived, and the effect of finite-impedance boundaries on the propagation of disturbances in mean swirling flow is also addressed.Two degrees of mean swirl are considered: first the case when the swirl velocity is of the same order as the axial velocity, which is applicable to turbomachinery flow behind a rotor stage; secondly a small swirl approximation, where the swirl velocity is of the same order as the axial slope of the duct walls, which is relevant to the flow downstream of the stator in a turbofan engine duct.The presence of mean vorticity couples the acoustic and vorticity equations and the associated eigenvalue problem is not self-adjoint as it is for irrotational mean flow. In order to obtain a secularity condition, which determines the amplitude variation along the duct, an adjoint solution for the coupled system of equations is derived. The solution breaks down at a turning point where a mode changes from cut on to cut off. Analysis in this region shows that the amplitude here is governed by a form of Airy's equation, and that the effect of swirl is to introduce a small shift in the location of the turning point. The reflection coefficient at this corrected turning point is shown to be exp (iπ/2).The evolution of axial wavenumbers and cross-sectionally averaged amplitudes along the duct are calculated and comparisons made between the cases of zero mean swirl, small mean swirl and O(1) mean swirl. In a hard-walled duct it is found that small mean swirl only affects the phase of the amplitude, but O(1) mean swirl produces a much larger amplitude variation along the duct compared with a non-swirling mean flow. In a duct with finite-impedance walls, mean swirl has a large damping effect when the modes are co-rotating with the swirl. If the modes are counter-rotating then an upstream-propagating mode can be amplified compared to the no-swirl case, but a downstream-propagating mode remains more damped.

Subcritical Large-Amplitude Vibrations of Imperfect Rotating Shafts

1989 ◽  
Vol 42 (11S) ◽  
pp. S157-S160
Author(s):  
C. E. N. Mazzilli
Keyword(s):  
Dry Friction ◽  
Large Amplitude ◽  
Critical Speed ◽  
Multiple Scales ◽  
Viscous Damping ◽  
Rotating Shaft ◽  
Geometrical Imperfection ◽  
Rotating Shafts ◽  
Large Amplitude Vibrations ◽  
Practical Implications

The effect of a geometrical imperfection, such as the axis flexural deformation, on the large-amplitude vibrations of a horizontal rotating shaft is analyzed with the aid of the Multiple Scales Method. Internal viscous damping and linear elasticity are assumed in the model. It is then seen that no matter how small the imperfection is, a “critical” speed of the order of half the classical critical speed arises, with relevant practical implications. A number of reported large-amplitude cases may eventually be explained this way. It is possible that events such as this will not appear in systems with high dry friction.

Nonlinear Free Vibrations of a Timoshenko Beam Using Multiple Scales Method

Volume 2 ◽  
2004 ◽  
Author(s):  
Asghar Ramezani ◽  
Mehrdaad Ghorashi
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Free Vibration ◽  
Timoshenko Beam ◽  
Large Amplitude ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Free Vibrations ◽  
Partial Differential ◽  
Cantilevered Beam

In this paper, the large amplitude free vibration of a cantilever Timoshenko beam is considered. To this end, first Hamilton’s principle is used in deriving the partial differential equation of the beam response under the mentioned conditions. Then, implementing the Galerkin’s method the partial differential equation is converted to an ordinary nonlinear differential equation. Finally, the method of multiple scales is used to determine a second order perturbation solution for the obtained ODE. The results show that nonlinearity acts in the direction of increasing the natural frequency of the thick-cantilevered beam.

Nonlinear free vibrations analysis of overhung rotors under the influence of gravity

2019 ◽  
Vol 234 (2) ◽  
pp. 575-588
Author(s):  
M Moradi Tiaki ◽  
SAA Hosseini ◽  
H Shaban Ali Nezhad
Keyword(s):  
Natural Frequency ◽  
High Frequency ◽  
Multiple Scales ◽  
Perturbation Technique ◽  
Equations Of Motion ◽  
Beat Frequency ◽  
Dynamical Behavior ◽  
Free Vibrations ◽  
Beam Model ◽  
Rayleigh Beam

In this paper, nonlinear free vibration of a cantilever flexible shaft carrying a rigid disk at its free end (overhung rotor) is investigated. The Rayleigh beam model is used and the rotor has large amplitude vibrations. With the assumption of inextensibility, the effect of nonlinear curvature and inertia is considered. The effect of disk mass on the dynamical behavior of the system is studied in the presence and absence of gravity (horizontal and vertical rotors). By using perturbation technique (method of multiple scales), the main focus is on the influence of gravity on equations of motion and on quantities such as amplitude and damped natural frequency. Here, a different behavior is observed due to the rotor weight. Indeed, the combination effects of gyroscopic term, nonlinearity and gravity are studied on the modal behavior of the system. It is shown that the static deflection creates second order nonlinear terms and changes the nonlinear damped natural frequency. With considering of gravity, both beat and high frequency in beat phenomenon increase. With increasing of the rotor weight, the minimum value of amplitude is extremely amplified in the direction of gravity but in the other transverse direction, amplitude of vibrations decreases. In addition, it is found that the weight has directly influence on beat frequency, while the mass ratio between disk and beam affects the high frequency.

A multiple-scales solution of the undular hydraulic jump problem

PAMM ◽  
2007 ◽  
Vol 7 (1) ◽  
pp. 4120007-4120008 ◽  
Author(s):  
Richard Jurisits ◽  
Wilhelm Schneider ◽  
Yee Seok Bae
Keyword(s):  
Hydraulic Jump ◽  
Multiple Scales ◽  
Jump Problem ◽  
Multiple Scales Solution

scholarly journals Sound transmission in slowly varying circular and annular lined ducts with flow

1999 ◽  
Vol 380 ◽  
pp. 279-296 ◽  
Author(s):  
S. W. RIENSTRA
Keyword(s):  
Multiple Scales ◽  
Natural Extension ◽  
Sound Transmission ◽  
Attractive Alternative ◽  
Mean Flow ◽  
Modal Expansion ◽  
Present Solution ◽  
Impedance Wall ◽  
Lined Ducts ◽  
Multiple Scales Solution

Sound transmission through straight circular ducts with a uniform inviscid mean flow and a constant acoustic lining (impedance wall) is classically described by a modal expansion. A natural extension for ducts with axially slowly varying properties (diameter and mean flow, wall impedance) is a multiple-scales solution. It is shown in the present paper that a consistent approximation of boundary condition and isentropic mean flow allows the multiple-scales problem to have an exact solution. Since the calculational complexities are no greater than for the classical straight duct model, the present solution provides an attractive alternative to a full numerical solution if diameter variation is relevant. A unique feature of the present solution is that it provides a systematic approximation to the hollow-to-annular cylinder transition problem in the turbofan engine inlet duct.

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