An Exact Solution for Whirling Speeds and Mode Shapes of Multi-span Rotating Shafts with Each Span Carrying a Number of Rigid Disks

2021 ◽  
Author(s):  
Jong-Shyong Wu ◽  
Jer-Jia Sheu
Keyword(s):  
Exact Solution ◽  
Mode Shapes ◽  
Rigid Disks ◽  
Rotating Shafts

Free Vibration of Damaged Frame Structures Considering the Effects of Axial Extension, Shear Deformation and Rotatory Inertia: Exact Solution

2017 ◽  
Vol 17 (10) ◽  
pp. 1750111
Author(s):  
Ugurcan Eroglu ◽  
Ekrem Tufekci
Keyword(s):  
Exact Solution ◽  
Free Vibration ◽  
Shear Deformation ◽  
Natural Frequencies ◽  
Plane Motion ◽  
Mode Shapes ◽  
Frame Structures ◽  
Crack Identification ◽  
Rotatory Inertia ◽  
Axial Extension

In this paper, a procedure based on the transfer matrix method for obtaining the exact solution to the equations of free vibration of damaged frame structures, considering the effects of axial extension, shear deformation, rotatory inertia, and all compliance components arising due to the presence of a crack, is presented. The crack is modeled by a rotational and/or translational spring based on the concept of linear elastic fracture mechanics. Only the in-plane motion of planar structures is considered. The formulation is validated through some examples existing in the literature. Additionally, the mode shapes and natural frequencies of a frame with pitched roof are provided. The variation of natural frequencies with respect to the crack location is presented. It is concluded that considering the axial compliance, and axial-bending coupling due to the presence of a crack results in different dynamic characteristics, which should be considered for problems where high precision is required, such as for the crack identification problems.

An Exact Solution for the Natural Frequencies of Flexible Beams Undergoing Overall Motions

2003 ◽  
Vol 9 (11) ◽  
pp. 1221-1229 ◽  
Author(s):  
Ali H Nayfeh ◽  
S.A. Emam ◽  
Sergio Preidikman ◽  
D.T. Mook
Keyword(s):  
Exact Solution ◽  
Natural Frequencies ◽  
Three Dimensional ◽  
Beam Theory ◽  
Free Vibrations ◽  
Mode Shapes ◽  
Flexible Beam ◽  
Bernoulli Beam ◽  
Flexible Beams ◽  
Euler Bernoulli Beam

We investigate the free vibrations of a flexible beam undergoing an overall two-dimensional motion. The beam is modeled using the Euler-Bernoulli beam theory. An exact solution for the natural frequencies and corresponding mode shapes of the beam is obtained. The model can be extended to beams undergoing three-dimensional motions.

EXACT SOLUTION OF FREE IN-PLANE VIBRATION OF SHALLOW CIRCULAR ARCHES

2001 ◽  
Vol 01 (03) ◽  
pp. 409-428 ◽  
Author(s):  
EKREM TÜFEKÇİ
Keyword(s):  
Exact Solution ◽  
Boundary Conditions ◽  
Shear Deformation ◽  
Slenderness Ratio ◽  
Mode Shapes ◽  
Rotatory Inertia ◽  
Shallow Arch ◽  
Inertia Effects ◽  
Plane Vibration ◽  
Axial Extension

The free in-plane vibration of a shallow circular arch with uniform cross-section is investigated by taking into account axial extension, shear deformation and rotatory inertia effects. The exact solution of the governing differential equations is obtained by the initial value method. By employing the same solution procedure, the solutions are also given for the other cases, in which each effect is considered alone, as well as no effect. The frequency coefficients are obtained for the lowest five vibration modes of arches with five combinations of classical boundary conditions, and various slenderness ratios and opening angles. The results show that the shear deformation and rotatory inertia effects are also very important as well as the axial extension effect, even if a slender shallow arch is considered. The discrepancies among the results of the five cases decrease, when opening angle increases for a constant radius and slenderness ratio. The effects of the boundary conditions and the slenderness ratio of the arch are investigated. The discrepancies among the results of the cases become much more important in higher modes. The mode shapes of a shallow arch are obtained for each case.

scholarly journals Analytical Solution for Whirling Speeds and Mode Shapes of a Distributed-Mass Shaft With Arbitrary Rigid Disks

2013 ◽  
Vol 81 (3) ◽  
Author(s):  
Jong-Shyong Wu ◽  
Foung-Tang Lin ◽  
Huei-Jou Shaw
Keyword(s):  
Free Vibration ◽  
Free Vibration Analysis ◽  
Mode Shapes ◽  
Natural Mode ◽  
Lumped Mass ◽  
Rotating Shaft ◽  
Disk System ◽  
Mass Moment ◽  
Rigid Disks ◽  
Equivalent Mass

The purpose of this paper is to present an approach for replacing the effects of each rigid disk mounted on the spin shaft by a lumped mass together with a frequency-dependent equivalent mass moment of inertia so that the whirling motion of a rotating shaft-disk system is similar to the transverse free vibration of a stationary beam and the technique for the free vibration analysis of a stationary beam with multiple concentrated elements can be used to determine the forward and backward whirling speeds, along with mode shapes of a distributed-mass shaft carrying arbitrary rigid disks. Numerical results reveal that the characteristics of whirling motions are significantly dependent on the slopes of the associated natural mode shapes at the positions where the rigid disks are located. Furthermore, the results obtained from the presented analytical method and those obtained from existing literature or the finite element method (FEM) are in good agreement.

A Numerical Method for the Dynamic Analysis of Rotating Flexible Blade-Disc-Shaft Assemblies

1995 ◽  
Author(s):  
Georges Jacquet-Richardet ◽  
Guy Ferraris ◽  
Pierre Rieutord
Keyword(s):  
Finite Element Analysis ◽  
Numerical Method ◽  
Natural Frequencies ◽  
Mode Shapes ◽  
Rotating System ◽  
Dynamic Interactions ◽  
Element Analysis ◽  
Modal Reduction ◽  
Rotating Shafts

Abstract A numerical method is proposed to compute the natural frequencies and mode shapes of rotating flexible blade-disc-shaft assemblies The formulation is based on a modal finite element analysis of the rotating system. The mode shapes used for the modal reduction are those associated to the global non-rotating undamped structure. To compute these modes, the size of the problem is reduced using both wave propagation and component mode techniques. An application is presented. The results obtained show that the chosen reductions do not reduce the quality of the model and illustrate its capability to deal with rotating shafts usually calculated using the rotordynamic approach. Finally, the possible dynamic interactions between shaft and disc components are described.

Natural Frequencies and Mode Shapes of Beams Carrying a Two Degree-of-Freedom Spring-Mass System

1993 ◽  
Vol 115 (2) ◽  
pp. 202-209 ◽  
Author(s):  
Ming Une Jen ◽  
E. B. Magrab
Keyword(s):  
Exact Solution ◽  
Natural Frequency ◽  
Natural Frequencies ◽  
Mode Shapes ◽  
Attached Mass ◽  
Approximate Methods ◽  
Mass System ◽  
The Laplace Transform ◽  
Rigid Mass ◽  
Two Degree Of Freedom

An exact solution for the natural frequencies and mode shapes for a beam elastically constrained at its ends and to which a rigid mass is elastically mounted is obtained. The attached mass can both translate and rotate. The general solution is obtained using the Laplace transform with respect to the spatial variable and yields the exact solutions to several previously published simpler configurations that were obtained using approximate methods. Numerous numerical results are presented for the natural frequency coefficients that extend previously reported results and that show the transition between various limiting cases. In addition, values are presented for the lowest two natural frequency coefficients for a beam that is clamped at both ends and is carrying a two dof spring-mass system. Representative mode shapes at selected values of the system’s parameters are also given.

Exact solution of a four-site crystal model for an intermediate-valence system

1986 ◽  
Vol 47 (6) ◽  
pp. 1029-1034 ◽  
Author(s):  
J.C. Parlebas ◽  
R.H. Victora ◽  
L.M. Falicov
Keyword(s):  
Exact Solution ◽  
Intermediate Valence ◽  
Crystal Model
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