MODAL CHARACTERISTIC OF A ROTATING RECTANGULAR CANTILEVER PLATE

Modal characteristics of a rotating cantilever plate are investigated in the present work. A dynamic modelling method for rectangular plates undergoing prescribed overall motion is employed to derive the equations of motion. The general equations are particularized for the modal analysis of a rotating cantilever plate and dimensionless parameters are identified through dimensional analysis. The effects of the dimensionless parameters on the modal characteristics of the rotating plate are investigated. Incidentally, eigenvalue loci veering and crossing phenomena along with the corresponding modeshape variations are exhibited and discussed.     

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.     

Vibration characteristics of a rotating flexible arm with ACLD treatment

In this paper, the vibration behavior and control of a clamped–free rotating flexible cantilever arm with fully covered active constrained layer damping (ACLD) treatment are investigated. The arm is rotating in a horizontal plane in which the gravitational effect and rotary inertia are neglected. The stress–strain relationship for the viscoelastic material (VEM) is described by a complex shear modulus while the shear deformations in the two piezoelectric layers are neglected. Hamilton's principle in conjunction with finite element method (FEM) is used to derive the non-linear coupled differential equations of motion and the associated boundary conditions that describe the rigid hub angle rotation, the arm transverse displacement and the axial deformations of the three-layer composite. This refined model takes into account the effects of centrifugal stiffening due to the rotation of the beam and the potential energies of the VEM due to extension and bending. Active controllers are designed with PD for the piezosensor and actuator. The vibration frequencies and damping factors of the closed-loop beam/ACLD system are obtained after solving the characteristic complex eigenvalue problem numerically. The effects of different rotating speed, thickness ratio and loss factor of the VEM as well as different controller gain on the damped frequency and damping ratio are presented. The results of this study will be useful in the design of adaptive and smart structures for vibration suppression and control in rotating structures such as rotorcraft blades or robotic arms.     

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.     

Bending,vibration and buckling of simply supported thick multilayered orthotropic plates: An evaluation of a new displacement model

Based upon a piecewise linear displacement field which allows the contact conditions for the displacements and the transverse shearing stresses at the interfaces to be satisfied simultaneously, the non-linear (in the von Kármán sense) equations of motion for thick multilayered orthotropic plates are developed. Successively, the equations are specified to the linear boundary value problem of the bending and to the linear eigenvalue problems of the undamped vibration and buckling of rectangular plates. In order to assess the accuracy of the proposed theory, the sample problem of the bending, free undamped vibration and buckling of a three-layered, symmetric cross-ply, square plate simply supported on all edges is investigated. For purposes of comparison, numerical results from the exact elasticity theory, the classical lamination (Kirchhoff) theory and the shear deformation theory (Timoshenko and Mindlin) with three different values of the shear correction factor are also presented. It is found that the proposed approach is very efficient in predicting the global responses (deflection, natural frequencies and buckling loads) of thick multilayered plates and models effects, such as the distortion of the deformed normals, not attainable from the classical lamination theory, as well as the shear deformation theory.     

FREE NON-LINEAR VIBRATION OF A ROTATING THIN RING WITH THE IN-PLANE AND OUT-OF-PLANE MOTIONS

Free non-linear vibration of a rotating thin ring with a constant speed is analyzed when the ring has both the in-plane and out-of-plane motions. The geometric non-linearity of displacements is considered by adopting the Lagrange strain theory for the circumferential strain instead of the infinitesimal strain theory. By using Hamilton's principle, the coupled non-linear partial differential equations are derived, which describe the out-of-plane bending and torsional motions as well as the in-plane bending and extensional motions. During deriving the equations of motion, we discuss how to model the circumferential stress and strain in order to consider the geometric non-linearity. Four models are established: three non-linear models and one linear model. For the four models, the linearized equations of motion are obtained in the neighbourhood of the steady state equilibrium position. Based on the linearized equations of the four cases, the natural frequencies are computed at various rotational speeds and then they are compared. Through the comparison, this study recommends which model is appropriate to describe the non-linear behaviour more precisely.     

Dynamic stiffness matrix of thin-walled composite I-beam with symmetric and arbitrary laminations

For the spatially coupled free vibration analysis of thin-walled composite I-beam with symmetric and arbitrary laminations, the exact dynamic stiffness matrix based on the solution of the simultaneous ordinary differential equations is presented. For this, a general theory for the vibration analysis of composite beam with arbitrary lamination including the restrained warping torsion is developed by introducing Vlasov's assumption. Next, the equations of motion and force–displacement relationships are derived from the energy principle and the first order of transformed simultaneous differential equations are constructed by using the displacement state vector consisting of 14 displacement parameters. Then explicit expressions for displacement parameters are derived and the exact dynamic stiffness matrix is determined using force–displacement relationships. In addition, the finite-element (FE) procedure based on Hermitian interpolation polynomials is developed. To verify the validity and the accuracy of this study, the numerical solutions are presented and compared with analytical solutions, the results from available references and the FE analysis using the thin-walled Hermitian beam elements. Particular emphasis is given in showing the phenomenon of vibrational mode change, the effects of increase of the modulus and the bending–twisting coupling stiffness for beams with various boundary conditions.     

Vibrational characteristics of rotating soft cylinders

Rotating structural components are omnipresent in engineering structures and natural world. This work investigates the effects of the centrifugal and Coriolis forces on the free vibrational characteristics of soft cylinders rotating with respect to the axis of symmetry based on the nonlinear elasticity and linear incremental theories. The formulations indicate that the biasing deformation, instantaneous elastic moduli, and incremental equations of motion strongly depend on the rotating speed. The characteristic equation for the natural frequency is derived using the state-space method and approximate laminate technique.The numerical examples included in this work demonstrate that the centrifugal and Coriolis forces might have significant effects on the vibrational characteristics of the cylinder. Results of this work will benefit the design and control of novel engineering systems with rotating soft cylinders or shafts.     

On the motion of the centers of inertia of light beams and pulses

It is shown that the velocity of the centers of inertia of localized structures of an electromagnetic field in vacuum does not exceed the speed of light in vacuum. The relation of this constraint with the properties of so-called X waves is discussed. For linear homogeneous anisotropic (dichroic) media, the transverse motion of the centers of Gaussian beams of monochromatic radiation is analyzed. It is found that, in the general case, the beam center moves along a hyperbola (curvilinearly), and, if the field envelope distribution is axisymmetric, this motion is rectilinear.     

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.     

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.     

Nonlinear dynamics of a support-excited flexible rotor with hydrodynamic journal bearings

The major purpose of this study is to predict the dynamic behavior of an on-board rotor mounted on hydrodynamic journal bearings in the presence of rigid support movements, the target application being turbochargers of vehicles or rotating machines subject to seismic excitation. The proposed on-board rotor model is based on Timoshenko beam finite elements. The dynamic modeling takes into account the geometric asymmetry of shaft and/or rigid disk as well as the six deterministic translations and rotations of the rotor rigid support. Depending on the type of analysis used for the bearing, the fluid film forces computed with the Reynolds equation are linear/nonlinear. Thus the application of Lagrange's equations yields the linear/nonlinear equations of motion of the rotating rotor in bending with respect to the moving rigid support which represents a non-inertial frame of reference. These equations are solved using the implicit Newmark time-step integration scheme. Due to the geometric asymmetry of the rotor and to the rotational motions of the support, the equations of motion include time-varying parametric terms which can lead to lateral dynamic instability. The influence of sinusoidal rotational or translational motions of the support, the accuracy of the linear 8-coefficient bearing model and the interest of the nonlinear model for a hydrodynamic journal bearing are examined and discussed by means of stability charts, orbits of the rotor, time history responses, fast Fourier transforms, bifurcation diagrams as well as Poincaré maps.     

Bending vibration of axially loaded Timoshenko beams with locally distributed Kelvin-Voigt damping

Utilizing the Timoshenko beam theory and applying Hamilton's principle, the bending vibration equations of an axially loaded beam with locally distributed internal damping of the Kelvin-Voigt type are established. The partial differential equations of motion are then discretized into linear second-order ordinary differential equations based on a finite element method. A quadratic eigenvalue problem of a damped system is formed to determine the eigenfrequencies of the damped beams. The effects of the internal damping, sizes and locations of damped segment, axial load and restraint types on the damping and oscillating parts of the damped natural frequency are investigated. It is believed that the present study is valuable for better understanding the influence of various parameters of the damped beam on its vibration characteristics.     

Vibration suppression of rotating beams using time-varying internal tensile force

A new strategy for vibration suppression of a rotating beam using a time-increasing internal tensile force is proposed in this paper. Nonlinear coupled longitudinal and bending equations of motion are derived in non-dimensional form using the Hamilton principle. The first-order analytical solution of the equations of motion is obtained using the Galerkin technique combined with the multiple scales method (MSM). Numerical simulations are then performed for various increasing rates of the internal tensile force and performance of the vibration suppression strategy is studied. A very close agreement between the simulation results obtained by the numerical integration and the first-order analytical solution is achieved. Forced vibrations of the system for input excitations of either a sinusoidal or a random function with white noise time history are considered. The simulation results and dynamic performance of the suppressed system for an externally excited rotating beam show an interesting phenomenon of the form of remarkable effectiveness of the proposed vibration reduction strategy.     

Vibration analyses of an inclined flat plate subjected to moving loads

The object of this paper is to present a moving mass element so that one may easily perform the dynamic analysis of an inclined plate subjected to moving loads with the effects of inertia force, Coriolis force and centrifugal force considered. To this end, the mass, damping and stiffness matrices of the moving mass element, with respect to the local coordinate system, are derived first by using the principle of superposition and the definition of shape functions. Next, the last property matrices of the moving mass element are transformed into the global coordinate system and combined with the property matrices of the inclined plate itself to determine the effective overall property matrices and the instantaneous equations of motion of the entire vibrating system. Because the property matrices of the moving mass element have something to do with the instantaneous position of the moving load, both the property matrices of the moving mass element and the effective overall ones of the entire vibrating system are time-dependent. At any instant of time, solving the instantaneous equations of motion yields the instantaneous dynamic responses of the inclined plate. For validation, the presented technique is used to determine the dynamic responses of a horizontal pinned–pinned plate subjected to a moving load and a satisfactory agreement with the existing literature is achieved. Furthermore, extensive studies on the inclined plate subjected to moving loads reveal that the influences of moving-load speed, inclined angle of the plate and total number of the moving loads on the dynamic responses of the inclined plate are significant in most cases, and the effects of Coriolis force and centrifugal force are perceptible only in the case of higher moving-load speed.     

DYNAMIC RESPONSE OF A ROTATING FLEXIBLE ARM CARRYING A MOVING MASS

A clamped-free flexible arm rotating in a horizontal plane and carrying a moving mass is studied in this paper. The arm is modelled by the Euler-Bernoulli beam theory in which rotatory inertia and shear deformation effects are ignored. The assumed mode method in conjunction with Hamilton's principle is used to derive the equation of motion of the system which takes into account the effect of centrifugal stiffening due to the rotation of the beam. The eigenfunctions of a cantilever beam which satisfy the prescribed geometric boundary conditions are used as basis functions in the assumed mode method. The equation of motion is expressed in non-dimensional matrix form. Pre-designed transformed cosine profiles are used as trajectory inputs for the hub angle and the moving mass. The equation of motion is solved numerically using the fourth order Runge-Kutta method. Graphical results are presented to show the influence of centrifugal stiffening effect, moving mass values, mass travelling time, hub angle and mass trajectory profile on the deflection of the beam.     

Bryan's effect and isotropic nonlinear damping

Bryan's effect is the phenomenon of the rotation of the vibration pattern within the body of a rotating, vibrating body. In this paper we generalise Rayleigh's dissipation function whereby we include isotropic, nonlinear damping in the Euler–Lagrange equations. We then derive the equations of motion of a slowly rotating, vibrating, symmetric body. In so doing we analyse the effect that such damping has on Bryan's effect. Using a combination of linear and nonlinear damping in the equations of motion of a slowly rotating symmetric body, a numerical experiment indicates that underdamping, critical damping and overdamping scenarios appear to exist. For the underdamped case we compare the effect on amplitude of vibration made by linear, quadratic and a combination of these types of damping. Our final result for light, isotropic, nonlinear damping, mimicks a known result for light, isotropic, linear damping, namely for a slowly rotating symmetric body, Bryan's effect is invariant and the rate of rotation of the damped pattern is the same as it would be for the pattern of an ideal slowly rotating symmetric body.     

Vibration analysis of horizontal self-weighted beams and cables with bending stiffness subjected to thermal loads

This paper aims at proposing an analytical model for the vibration analysis of horizontal beams that are self-weighted and thermally stressed. Geometrical nonlinearities are taken into account on the basis of large displacement and small rotation. Natural frequencies are obtained from a linearization of equilibrium equations. Thermal force and thermal bending moment are both included in the analysis. Torsional and axial springs are considered at beam ends, allowing various boundary conditions. A dimensionless analysis is performed leading to only four parameters, respectively, related to the self-weight, thermal force, thermal bending moment and torsional spring stiffness. It is shown that the proposed model can be efficiently used for cable problems with small sag-to-span ratios (typically , as in Irvine's theory). For beam problems, the model is validated thanks to finite element solutions and a parametric study is conducted in order to highlight the combined effects of thermal loads and self-weight on natural frequencies. For cable problems, solutions are first compared with existing results in the literature obtained without thermal effects or bending stiffness. Good agreement is found. A parametric study combining the effects of sag-extensibility, thermal change and bending stiffness is finally given.     

基于绝对节点坐标法的大变形柔性梁几种动力学模型研究

对在平面内大范围转动的大变形柔性梁动力学进行了研究, 基于绝对节点坐标法建立了一种新的大变形柔性梁的非线性动力学模型. 该动力学模型中考虑了柔性梁的轴向拉伸变形和横向弯曲变形, 利用Green-Lagrangian应变张量计算柔性梁的轴向应变及应变能, 利用曲率的精确表达式计算柔性梁的横向弯曲变形能. 运用拉格朗日恒等式给出了柔性梁横向弯曲变形能新的表达式, 该变形能表达式更加简洁, 通过新的变形能表达式得到了新的弹性力模型, 由此得到的动力学方程可以精确地描述柔性梁的几何大变形问题. 通过与高次耦合模型以及ANSYS中BEAM188非线性梁单元模型的比较, 验证了本模型在计算大变形时的正确性以及高次耦合模型在处理大变形问题时的不足. 进一步研究发现, 新的广义弹性力模型可以适当地简化, 给出了两种简化模型, 根据不同模型的计算效率以及计算精度的比较确定了不同模型的适用范围.