MODAL ANALYSIS OF ROTATING COMPOSITE CANTILEVER PLATES

A modelling method for the modal analysis of a rotating composite cantilever plate is presented in this paper. A set of linear ordinary differential equations of motion for the plate is derived by using the assumed mode method. Two in-plane stretch variables are employed and approximated to derive the equations of motion. The equations of motion include the coupling terms between the in-plane and the lateral motions as well as the motion-induced stiffness variation terms. Dimensionless parameters are identified and the explicit mass and the stiffness matrices for the modal analysis are obtained with the dimensionless parameters. The effects of the dimensionless angular velocity and the fiber orientation angles of rotating composite cantilever plates on their modal characteristics are investigated. Natural frequency loci veering and crossing along with associated mode shape variations are observed.     

Modal analysis of a rotating multi-packet blade system

A modeling method for the modal analysis of a multi-packet blade system undergoing rotational motion is presented in this paper. Blades are idealized as tapered cantilever beams that are fixed to a rotating disc. The stiffness coupling effects between blades due to the flexibilities of the disc and the shroud are modeled with discrete springs. Hybrid deformation variables are employed to derive the equations of motion. To obtain more general information, the equations of motion are transformed into a dimensionless form in which dimensionless parameters are identified. The effects of the dimensionless parameters and the number of packets on the modal characteristics of the rotating multi-packet blade system are investigated with numerical examples.     

DYNAMIC ANALYSIS OF A ROTATING CANTILEVER BEAM BY USING THE FINITE ELEMENT METHOD

A finite element analysis for a rotating cantilever beam is presented in this study. Based on a dynamic modelling method using the stretch deformation instead of the conventional axial deformation, three linear partial differential equations are derived from Hamilton's principle. Two of the linear differential equations are coupled through the stretch and chordwise deformations. The other equation is an uncoupled one for the flapwise deformation. From these partial differential equations and the associated boundary conditions, are derived two weak forms: one is for the chordwise motion and the other is for the flapwise motion. The weak forms are spatially discretized with newly defined two-node beam elements. With the discretized equations, the behaviours of the natural frequencies are investigated for the variation of the rotating speed. In addition, the time responses and distributions of the deformations and stresses are computed when the rotating speed is prescribed. The effects of the rotating speed profile on the vibrations of the beam are also investigated.     

Magneto-elastic dynamics and bifurcation of rotating annular plate

In this paper, magneto-elastic dynamic behavior, bifurcation, and chaos of a rotating annular thin plate with various boundary conditions are investigated. Based on the thin plate theory and the Maxwell equations, the magneto-elastic dynamic equations of rotating annular plate are derived by means of Hamilton's principle. Bessel function as a mode shape function and the Galerkin method are used to achieve the transverse vibration differential equation of the rotating annular plate with different boundary conditions. By numerical analysis, the bifurcation diagrams with magnetic induction, amplitude and frequency of transverse excitation force as the control parameters are respectively plotted under different boundary conditions such as clamped supported sides, simply supported sides, and clamped-one-side combined with simply-anotherside. Poincare′ maps, time history charts, power spectrum charts, and phase diagrams are obtained under certain conditions,and the influence of the bifurcation parameters on the bifurcation and chaos of the system is discussed. The results show that the motion of the system is a complicated and repeated process from multi-periodic motion to quasi-period motion to chaotic motion, which is accompanied by intermittent chaos, when the bifurcation parameters change. If the amplitude of transverse excitation force is bigger or magnetic induction intensity is smaller or boundary constraints level is lower, the system can be more prone to chaos.     

VIBRATION ANALYSIS OF NON-CIRCULAR CURVED PANELS BY THE DIFFERENTIAL QUADRATURE METHOD

Free-vibration characteristics of cantilever non-circular curved panels are analyzed by using the differential quadrature method (DQM) in this paper. The equations of motion of a curved panel are based on the Love's hypothesis and are expressed in an orthogonal curvilinear co-ordinate system. By applying the differential quadrature formulation and the proposed modified relationships for specified boundary conditions, the free-vibration equations of motion of the curved panel are transformed to a set of algebraic equations. Natural frequencies of a cantilever flat plate and a circular curved panel are obtained for verifying the applicability of the present approach. Good convergent trend and accuracy are observed. Effects of shallowness, thickness and aspect ratios on the natural frequencies of a cantilever curved panel are also investigated. Furthermore, natural frequencies of parabolic curved panels are obtained. In all cases studied, the efficiency and convenience of the DQM are illustrated.     

Dynamic stiffness method for inplane free vibration of rotating beams including Coriolis effects

The paper addresses the in-plane free vibration analysis of rotating beams using an exact dynamic stiffness method. The analysis includes the Coriolis effects in the free vibratory motion as well as the effects of an arbitrary hub radius and an outboard force. The investigation focuses on the formulation of the frequency dependent dynamic stiffness matrix to perform exact modal analysis of rotating beams or beam assemblies. The governing differential equations of motion, derived from Hamilton's principle, are solved using the Frobenius method. Natural boundary conditions resulting from the Hamiltonian formulation enable expressions for nodal forces to be obtained in terms of arbitrary constants. The dynamic stiffness matrix is developed by relating the amplitudes of the nodal forces to those of the corresponding responses, thereby eliminating the arbitrary constants. Then the natural frequencies and mode shapes follow from the application of the Wittrick–Williams algorithm. Numerical results for an individual rotating beam for cantilever boundary condition are given and some results are validated. The influences of Coriolis effects, rotational speed and hub radius on the natural frequencies and mode shapes are illustrated.     

Stability of cantilever plates subjected to biaxial subtangential loading

The stability of cantilever plates which, mathematically, comprises a non-self-adjoint problem is investigated. It is assumed that the plate is acted upon by a subtangential biaxial edge load embodying the dead loading and the follower type loading as its limiting states. The scheme of modal expansions, containing the constrained rigid modes, together with Galerkin's method is employed and the stability of the plate in terms of subtangency and load parameters is analysed. As an example the kinetic stability analysis of a square cantilever plate is carried out in detail.     

旋转内接悬臂梁的刚柔耦合动力学特性分析

对固结于旋转刚环上内接柔性梁的刚柔耦合动力学特性进行了研究. 在精确描述柔性梁非线性变形基础上, 利用Hamilton变分原理和假设模态法, 在计入柔性梁由于横向变形而引起的轴向变形二阶耦合量的条件下, 推导出一次近似耦合模型. 忽略柔性梁纵向变形的影响,给出一次近似简化模型,引入无量纲变量, 对简化模型做无量纲化处理. 首先分析在非惯性系下内接悬臂梁的动力学响应, 并与外接悬臂梁进行比较; 其次研究内接悬臂梁的稳定性;最后分析内接悬臂梁失稳临界转速的收敛性. 研究发现, 与外接悬臂梁存在动力刚化效应不同,内接悬臂梁存在着动力柔化效应; 给出了内接悬臂梁无条件稳定的临界径长比以及失稳的临界转速的计算方法; 若第一阶固有频率随转速增大而减小,则该内接悬臂梁处于有条件稳定; 随着模态截断数的增加,内接悬臂梁失稳的临界转速减小且有收敛值. 关键词： 内接悬臂梁 一次近似简化模型 动力柔化 临界转速     

Vibration characteristics of thin rotating cylindrical shells with various boundary conditions

An analysis is presented for the vibration characteristics of thin rotating cylindrical shells with various boundary conditions by use of Fourier series expansion method. Based on Sanders’ shell equations, the governing equations of motion which take into account the effects of centrifugal and Coriolis forces as well as the initial hoop tension due to rotating are derived. The displacement field is expressed as a product of Fourier series expressions which represents the axial modal displacements and trigonometric functions which represents the circumferential modal displacements. Stokes’ transformation is employed to derive the derivatives of the Fourier series expressions. Then, through the process of formula derivation, an explicit expression of the exact frequency equation can be obtained for a thin rotating cylinder with classical boundary conditions of any type. Once the frequency equation has been determined, the frequencies are calculated numerically. To validate the present analysis, comparisons between the results of the present method and previous studies are performed and very good agreement is achieved. Finally, the method is applied to investigate the vibration characteristics of thin rotating cylindrical shells under various boundaries, and the results are presented.     

Heat and mass transfer analysis on the flow of a second grade fluid in the presence of chemical reaction

The laminar flow problem of convective heat transfer for a second grade fluid over a semi-infinite plate in the presence of species concentration and chemical reaction is investigated. The governing equations are transformed into a dimensionless system of three non-linear coupled partial differential equations. These equations have been solved analytically subject to the relevant boundary conditions by employing a homotopy analysis method (HAM). It is noted that for the arising system, the HAM performs extremely well in terms of efficiency and simplicity. The influence of dimensionless pertinent parameters on the velocity, temperature and concentration fields has been examined carefully.     

Impact of nanoparticles and radiative heat flux in von Kármán swirling flow of Maxwell fluid

In this paper, the classical von Kármán swirling flow problem due to a rotating disk is modeled and studied for the rate type Maxwell nanofluid together with heat and mass transfer mechanisms. The model under consideration predicts the relaxation time characteristics. The novel aspects of thermophoresis and Brownian motion features due to nanoparticles are investigated by employing an innovative Buongiorno’s model. The analysis further explores the impact of linear Rosseland radiation on heat transfer characteristics. The concept of boundary layer approximations is utilized to formulate the basic governing equations of Maxwell fluid. The dimensionless form of a system of ordinary differential equations is obtained through similarity approach adopted by von Kármán. The system of equations is integrated numerically in domain [0,∞) by using bvp midrich scheme in Maple software. The obtained results intimate that higher rotation raises the radial and angular velocity components. The nano-particles concentration enhances with Brownian motion parameter. Further, the heat transfer rate at the disk surface diminishes with thermophoresis parameter. The achieved numerical computations of velocity profiles, friction coefficient and Nusselt number are matched in limiting cases with previously published literature and an outstanding agreement is observed.     

DYNAMIC RESPONSE OF TWO ROTORS CONNECTED BY RIGID MECHANICAL COUPLING WITH PARALLEL MISALIGNMENT

The effect of parallel misalignment on the lateral and torsional responses of two rotating shafts (Jeffcott rotors) is examined with theoretical and numerical analysis. The general equations of motion are derived and given in dimensionless form to represent the general case. The equations of motion revealed that parallel misalignment couples the translation and angular deflections through the stiffness matrix and the force vector. The non-linear equations are solved numerically using a combination of Newmark and Newton-Raphson methods to determine the dimensionless frequency and transient responses in terms of misalignment magnitude. The numerical results show that the system natural frequencies are excited at transient condition due to the presence of pure parallel misalignment. At steady state condition, the 1×-rotational speed excitation is present in the translation and angular directions, which indicates that parallel misalignment can be a source of both torsional and lateral excitations.     

Hydromagnetic flow of a variable viscosity nanofluid in a rotating permeable channel with hall effects

The flow, heat and mass transfer of water-based nanofluid are examined between two horizontal parallel plates in a rotating system. The effects of Brownian motion, thermophoresis, viscosity and Hall current parameters are considered. The governing partial differential equations are reduced to ordinary differential equations that are then solved numerically using the Runge–Kutta–Fehlberg method. Validation of numerical solution is achieved with an exact solution of primary velocity and found to be in good agreement. Results show that both surfaces experience opposite behavior regarding skin friction, Nusselt and Sherwood numbers in both primary and secondary flows. These physical quantities depend upon dimensionless parameters and numbers.     

FREE VIBRATION ANALYSIS OF A SPINNING FLEXIBLE DISK-SPINDLE SYSTEM SUPPORTED BY BALL BEARING AND FLEXIBLE SHAFT USING THE FINITE ELEMENT METHOD AND SUBSTRUCTURE SYNTHESIS

Free vibration of a spinning flexible disk-spindle system supported by ball bearing and flexible shaft is analyzed by using Hamilton's principle, FEM and substructure synthesis. The spinning disk is described by using the Kirchhoff plate theory and von Karman non-linear strain. The rotating spindle and stationary shaft are modelled by Rayleigh beam and Euler beam respectively. Using Hamilton's principle and including the rigid body translation and tilting motion, partial differential equations of motion of the spinning flexible disk and spindle are derived consistently to satisfy the geometric compatibility in the internal boundary between substructures. FEM is used to discretize the derived governing equations, and substructure synthesis is introduced to assemble each component of the disk-spindle-bearing-shaft system. The developed method is applied to the spindle system of a computer hard disk drive with three disks, and modal testing is performed to verify the simulation results. The simulation result agrees very well with the experimental one. This research investigates critical design parameters in an HDD spindle system, i.e., the non-linearity of a spinning disk and the flexibility and boundary condition of a stationary shaft, to predict the free vibration characteristics accurately. The proposed method may be effectively applied to predict the vibration characteristics of a spinning flexible disk-spindle system supported by ball bearing and flexible shaft in the various forms of computer storage device, i.e., FDD, CD, HDD and DVD.     

Dynamic characteristics of traveling waves for a rotating laminated circular plate with viscoelastic core layer

The dynamic governing equations and the corresponding boundary conditions for a rotating thin laminated circular plate with a viscoelastic core layer are derived in this paper based on the Hamilton principle. The analysis on dynamic features of the forward and Backward Traveling Waves for the rotating laminated plate is performed by means of Galerkin's method. The frequency-dependent complex modulus model for describing the constitutive behavior of the viscoelastic core layer is employed. The dynamic characteristics of frequencies and dampings of traveling waves for the rotating plate are obtained numerically. The effects of geometrical and material parameters on the critical speed of the rotating laminated plate with viscoelastic core are discussed in detail.     

The role of non-linear theories in transient dynamic analysis of flexible structures

Transient dynamic analysis of flexible structures undergoing large motions is considered. For rotating structures, it is explicitly shown that appropriate account of the influence of centrifugal force on the bending stiffness requires the use of a geometrically non-linear (at least second-order) beam theory. Use of a first-order (linearized) linear beam theory results in a spurious loss of bending stiffness. For a rotating plane beam, a set of linear partial differential equations of motion—that includes all inertia effects (Coriolis, centrifugal, acceleration of revolution) and coupling between extensional and flexural deformations—is derived from the fully non-linear beam theory by consistent linearization. The analysis is subsequently extended to the more general case of a plate, accomodating shear deformation, and undergoing a general three-dimensional rotating motion. The discretization process of the resulting linear equations of motion for the beam and the plate is also discussed.     

The effects of rotation,tip mass,and hub radius on the stability of beck's column

The stability of a cantilever beam subjected to a follower force at its free end and rotating at a uniform angular velocity is investigated. The beam is assumed to be offset from the axis of rotation, carries a tip mass at its free end, and undergoes deflection in a direction perpendicular to the plane of rotation. The equations of motion are formulated within the Euler-Bernoulli and Timoshenko beam theories for the case of a Kelvin model viscoelastic beam. The associated adjoint boundary value problems are derived and appropriate adjoint variational principles are introduced. These variational principles are used for the purpose of determining approximately the values of the critical flutter load of the system as it depends upon its damping parameters, tip mass and its rotary inertia, hub radius, and speed of rotation. The variation of the critical flutter load with these parameters is revealed in a series of several graphs. The numerical results show that the critical load can be reduced significantly due to (a) the transverse and rotary inertia of the tip mass and (b) increasing values of the internal damping parameter associated with the transverse shear deformation of the rotating beam.     

Finite-amplitude waves in isotropic elastic plates

The propagation of finite-amplitude waves in a homogeneous, isotropic, stress-free elastic plate is investigated theoretically. Geometric and weak material non-linearities are included, and perturbation is used to obtain solutions of the non-linear equations of motion for harmonic generation in the waveguide. Solutions for the second-harmonic, sum, and difference-frequency components are obtained via modal decomposition. Ordinary differential equations for the modal amplitudes in the expansion of the second-order solution are obtained using a reciprocity relation. There are no restrictions on the modes or frequencies of the primary waves. Two conditions for internal resonance are quantified: phase matching, and transfer of power from the primary to the secondary wave.     

Dynamic stability of thermoelastic coupling moving plate subjected to follower force

The dynamic characteristics and stability of the moving thermoelastic coupling rectangular plate subjected to uniformly distributed tangential follower force are investigated. Based on the heat conduction equation containing the thermoelastic coupling term and the thin plate theory, the thermoelastic coupling differential equation of motion of the rectangular plate under the action of uniformly distributed tangential follower force is established. Dimensionless complex frequencies of the moving thermoelastic coupling rectangular plate with four edges simply supported, two opposite edges simply supported and other two edges clamped are calculated by the differential quadrature method. The effects of the dimensionless thermoelastic coupling factor and dimensionless moving speed on the stability and critical load of the moving plate are analyzed. The results show that the divergence loads of the first order mode increase with the increase of the dimensionless thermoelastic coupling factor, and decrease with increasing the dimensionless moving speed.     

AEROELASTIC FLUTTER OF CIRCULAR ROTATING DISKS: A SIMPLE PREDICTIVE MODEL

This paper is an attempt to predict aeroelastic flutter of a rotating disk in an unbounded fluid. In the first part of the paper, the linear vibration of a rotating, potential fluid driven by transverse, harmonic motion of a rotating disk is solved. We extend the existing solution for a rigid disk to include flexible disks and compare alternative numerical evaluation schemes. Our principal interest in this problem is the identification of possible physical mechanisms for aeroelastic flutter. In the forced vibration problem considered here, fluid rotation renders the governing equations hyperbolic for low-frequency oscillation. As a result, the fluid motion may be discontinuous along the two characteristics that emanate from the rim of the disk. These discontinuities suggest the presence of previously unrecognized boundary layers near the rim of the disk that may be important for aeroelastic flutter. This idea is used to develop a simple mathematical model for predicting aeroelastic flutter. The model and its dependence on the dimensionless parameters describing the system are derived from first principles except for the compressible boundary layer, which is described by a simple function whose magnitude is empirically determined by fitting experimental data. Although the model is simple, its predictions are quantitatively similar to the experimental evidence and gives analytic predictions of aeroelastic flutter that are within an order of magnitude of the experimental values.