Discussion: “Natural Frequencies and Normal Modes of a Spinning Timoshenko Beam With General Boundary Conditions” (Zu, J. W. Z., and Han, R. P. S., 1992, ASME J. Appl Mech., 59, pp. S197–S204)

Vibration analysis of externally damped spinning Timoshenko beams with general boundary conditions is performed analytically. Exact solutions for natural frequencies and normal modes for the six classical boundary conditions are derived for the first time. In the numerical simulations, the trend between the complex frequencies and the damping coefficient is investigated, and complex mode shapes are presented in three-dimensional space.

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Natural Frequencies and Normal Modes of a Spinning Timoshenko Beam With General Boundary Conditions

Journal of Applied Mechanics ◽

10.1115/1.2899488 ◽

1992 ◽

Vol 59 (2S) ◽

pp. S197-S204 ◽

Cited By ~ 72

Author(s):

Jean Wu-Zheng Zu ◽

Ray P. S. Han

Keyword(s):

Boundary Conditions ◽

Mode Shape ◽

Timoshenko Beam ◽

Natural Frequencies ◽

Normal Modes ◽

Single Mode ◽

Mode Shapes ◽

Simulation Studies ◽

Classical Boundary ◽

First Time

A free flexural vibrations of a spinning, finite Timoshenko beam for the six classical boundary conditions are analytically solved and presented for the first time. Expressions for computing natural frequencies and mode shapes are given. Numerical simulation studies show that the simply-supported beam possesses very peculiar free vibration characteristics: There exist two sets of natural frequencies corresponding to each mode shape, and the forward and backward precession mode shapes of each set coincide identically. These phenomena are not observed in beams with the other five types of boundary conditions. In these cases, the forward and backward precessions are different, implying that each natural frequency corresponds to a single mode shape.

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Free Vibration and Stability Analysis of Internally Damped Rotating Shafts With General Boundary Conditions

Journal of Vibration and Acoustics ◽

10.1115/1.2893897 ◽

1998 ◽

Vol 120 (3) ◽

pp. 776-783 ◽

Cited By ~ 29

Author(s):

J. Melanson ◽

J. W. Zu

Keyword(s):

Boundary Conditions ◽

Equations Of Motion ◽

Normal Modes ◽

Beam Theory ◽

General Boundary ◽

General Boundary Conditions ◽

Rotating Shaft ◽

Classical Boundary ◽

Rotating Shafts ◽

The Stability

Vibration analysis of an internally damped rotating shaft, modeled using Timoshenko beam theory, with general boundary conditions is performed analytically. The equations of motion including the effects of internal viscous and hysteretic damping are derived. Exact solutions for the complex natural frequencies and complex normal modes are provided for each of the six classical boundary conditions. Numerical simulations show the effect of the internal damping on the stability of the rotor system.

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Dynamic Response of a Spinning Timoshenko Beam With General Boundary Conditions and Subjected to a Moving Load

Journal of Applied Mechanics ◽

10.1115/1.2901390 ◽

1994 ◽

Vol 61 (1) ◽

pp. 152-160 ◽

Cited By ~ 32

Author(s):

J. W.-Z. Zu ◽

R. P. S. Han

Keyword(s):

Dynamic Response ◽

Boundary Conditions ◽

Timoshenko Beam ◽

Moving Load ◽

Original System ◽

Adjoint System ◽

General Boundary ◽

General Boundary Conditions ◽

Adjoint Systems ◽

Analysis Technique

The dynamic response of a spinning Timoshenko beam with general boundary conditions and subjected to a moving load is solved analytically for the first time. Solution of the problem is achieved by formulating the spinning Timoshenko beams as a non-self-adjoint system. To compute the system dynamic response using the modal analysis technique, it is necessary to determine the eigenquantities of both the original and adjoint systems. In order to fix the adjoint eigenvectors relative to the eigenvectors of the original system, the biorthonormality conditions are invoked. Responses for the four classical boundary conditions which do not involve rigidbody motions are illustrated. To ensure the validity of the method, these results are compared with those from Euler-Bernoulli and Rayieigh beam theories. Numerical simulations are performed to study the influence of the four boundary conditions on selected system parameters.

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Orthogonal Bases for Eigenvalue Sensitivities—Torsional Vibration

Volume 3C: 15th Biennial Conference on Mechanical Vibration and Noise — Vibration Control, Analysis, and Identification ◽

10.1115/detc1995-0648 ◽

1995 ◽

Author(s):

S. A. Nayfeh ◽

H. Asada

Keyword(s):

Boundary Conditions ◽

Natural Frequency ◽

Torsional Vibration ◽

Natural Frequencies ◽

General Boundary ◽

General Boundary Conditions ◽

Orthogonal Bases ◽

Design Changes

Abstract The sensitivity of the natural frequencies of torsional vibration of an initially uniform shaft to nonuniform changes in its radius is studied. For the particular example of a fixed-fixed shaft, a set of design changes are found that modify a single natural frequency without modifying any of the others. The possibility of finding such a set of design changes for general boundary conditions is then investigated, and conditions for the existence of such a set are determined.

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A Unified Approach of Free Vibration Analysis for Stiffened Cylindrical Shell with General Boundary Conditions

Mathematical Problems in Engineering ◽

10.1155/2019/4157930 ◽

2019 ◽

Vol 2019 ◽

pp. 1-14 ◽

Cited By ~ 1

Author(s):

Xue-Qin Li ◽

Wei Zhang ◽

Xiao-Dong Yang ◽

Lu-Kai Song

Keyword(s):

Cylindrical Shell ◽

Free Vibration ◽

Boundary Conditions ◽

Vibration Analysis ◽

Natural Frequencies ◽

Free Vibration Analysis ◽

General Boundary ◽

General Boundary Conditions ◽

Unified Approach ◽

Stiffened Cylindrical Shell

A unified approach of free vibration analysis for stiffened cylindrical shell with general boundary conditions is presented in this paper. The vibration of stiffened cylindrical shell is modeled mathematically involving the first-order shear deformation shell theory. The improved Fourier series is selected as the admissible displacement function while the arbitrary boundary conditions are simulated by adjusting the equivalent spring stiffness. The natural frequencies and modal shapes of the stiffened shell are obtained by solving the dynamic model with the Rayleigh-Ritz procedure. Various numerical results of free vibration analysis for stiffened cylindrical shell are obtained, including natural frequencies and modes under simply supported, free, and clamped boundary conditions. Moreover, the effects of stiffener on natural frequencies are discussed. Compared with several state-of-the-art methods, the feasibility and validity of the proposed method are verified.

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