L.-W. Chen

Dynamic stability of a rotating beam with a constrained damping layer

The dynamic behavior and dynamic instability of the rotating sandwich beam with a constrained damping layer subjected to axial periodic loads are studied by the finite element method. The influences of rotating speed, thickness ratio, setting angle and hub radius ratio on the resonant frequencies and modal system loss factors are presented. The regions of instability for simple and combination resonant frequencies are determined from the Mathieu equation that is obtained from the parametric excitation of the rotating sandwich beam. The regions of dynamic instability for various parameters are presented.

Dynamic stability analysis of a rotating shaft by the finite element method

Abstract The dynamic stability of a rotating shaft subjected to axial periodic forces varying with time is studied by the finite element method. By using Timoshenko beam theory the transverse shear effect is included in the shape functions. Bolotins method is used to construct the dynamic instability diagrams for various values of the rotating speed and different end conditions of the shaft. Due to the effect of the gyroscopic moment, the boundaries of the regions of dynamic instability are shifted out and the sizes of these regions are increased as the rotational speed increases.

Eigenvalue sensitivity in the stability analysis of beck's column with a concentrated mass at the free end

Abstract The stability of a uniform cantilever column carrying a concentrated tip mass and subjected to a follower force at the free end is investigated by using the finite element method and the technique of eigenvalue sensitivity. In the present finite element formulation, the shape functions, which contain a shear deformation parameter, are developed from the static deflection of a Timoshenko beam, and the effects of both transverse shear deformation and rotatory inertia are included. The advantage of such a formulation is that if a suitable value of the shear deformation parameter is given, a specific beam model is then simulated without too much additional work to modify the original formulation. In order to obtain promptly the critical flutter load, a simple and direct method which utilizes the eigenvalue sensitivity with respect to the follower force is proposed instead of the conventional trial-and-error technique. Numerical results demonstrate the excellent agreement of the present solutions and the rapid rate of convergence of the iterative procedure. They also show that the critical flutter load is more affected by the effect of transverse shear deformation than by that of rotatory inertia.

Vibrations of initially stressed thick circular and annular plates based on a high-order plate theory

The average stress method is used to derive the governing equations for circular and annular plates in a general state of initial stress based on a high-order plate theory. The initial stress is taken to be a combination of an extensional stress plus a pure bending stress in the plane of the plate. These equations are solved by the Galerkin method. Natural frequencies of various boundary conditions are obtained for axisymmetric circular and annular plates. The effects of some parameters on the natural frequencies are studied. The results are compared with those of the Mindlin theory.

Axisymmetric dynamic stability of polar orthotropic thick circular plates

Abstract Dynamic stability of polar orthotropic thick circular plates subjected to a periodic uniform radial stress is studied by the finite element method. An annular element based on the Mindlin plate theory is employed. The regions of dynamic instability for both clamped and simply supported edges are determined by Bolotins method. The effects of various parameters on the dynamic stability are investigated.

Vibration and damping analysis of annular plates with constrained damping layer treatments

The natural frequencies and modal loss factors of annular plates with fully and partially constrained damping treatments are considered. The equations of free vibration of the plate including the transverse shear effects are derived by a discrete layer annular finite element method. The extensional and shear moduli of the viscoelastic material layer are described by the complex quantities. Complex eigenvalues are then found numerically, and from these, both frequencies and loss factors are extracted. The effects of viscoelastic layer stiffness and thickness, constraining layer stiffness and thickness, and treatment size on natural frequencies and modal loss factors are presented. Numerical results also show that the longer constrained damping treatment in radial length does not always provide better damping than the shorter ones.

Axisymmetric dynamic stability of transversely isotropic Mindlin circular plates

Dynamic stability of transversely isotropic Mindlin circular plates subjected to periodic radial loads is studied. The periodic radial loads are taken to be the combination of a pulsating compressive stress and a pulsating bending stress. The Galerkin method and Bolotins method are used to obtain the regions of dynamic instability. The effects of some parameters on the dynamic instability regions are investigated.

Asymmetric dynamic stability of thick annular plates based on a high-order plate theory

Abstract The asymmetric dynamic stability of thick annular plates subjected to a periodic uniform radial loading along the outer edge is studied by the finite element method. An annular element based on a high-order plate theory is used. The regions of dynamic instability are determined by Bolotins method. The effects of various parameters on the dynamic stability are investigated.

Axisymmetric vibration and damping analysis of rotating annular plates with constrained damping layer treatments

The axisymmetric vibration and damping of rotating annular plates with constrained damping layer treatments are analyzed in the present paper. The discrete layer annular finite element method is employed to derive the equations of motion for the rotating composite plate where the geometry stiffness matrix induced by rotation is evaluated from solutions of static problems. The viscoelastic material in the central layer is assumed to be incompressible, and the extensional and shear moduli are described by complex quantities. Complex-eigenvalued problems are then solved, and the frequencies and modal loss factors of the composite plate are extracted. The effects of stiffness and thickness of the viscoelastic and constraining layers on the natural frequencies and modal loss factors of rotating composite plates are presented. Also, the composite plates with various boundary conditions and partial treatments are discussed.

Stability Behavior of a Shaft -Disk System Subjected to Longitudinal Force

The stability behavior of a spinning Timoshenko shaft with an intermediate attached disk subjected to a longitudinal force is analytically studied. The expressions of frequency (whirl speed) equations for hinged‐hinged, hinged ‐clamped, clamped ‐hinged, and clamped ‐clamped rotors are given. By using the numerical technique, the critical axial and follower loads are sought. Numerical results reveal that the instability mechanisms are complex. Eight different types of instability mechanisms are presented. The critical load jump is possible when the instability mechanism changes from one to another. Furthermore, in some special cases, the critical longitudinal loads are almost zero; therefore, such a rotor combination is extremely unstable and should be avoided for design purposes. Nomenclature A = area of cross section of the shaft ai, bi = coefe cients given by Eqs. (21) and (22), respectively Ci, ¯ Ci = integration constants E = Young’ s modulus G = shear modulus hi = coefe cients given by Eq. (26) I = moment of inertia of the shaft cross section ID, IP = diameter and polar mass moments of inertia of the disk ¯D = nondimensional diameter mass moment of inertia of the disk Id, Ip = diameter and polar moments of inertia per unit length of the shaft j = 1 L = length of the shaft L1 = distance between the left end and the intermediate disk