Non-linear vibration of variable speed rotating viscoelastic beams

Non-linear vibration of a variable speed rotating beam is analyzed in this paper. The coupled longitudinal and bending vibration of a beam is studied and the governing equations of motion, using Hamilton’s principle, are derived. The solutions of the non-linear partial differential equations of motion are discretized to the time and position functions using the Galerkin method. The multiple scales method is then utilized to obtain the first-order approximate solution. The exact first-order solution is determined for both the stationary and non-stationary rotating speeds. A very close agreement is achieved between the simulation results obtained by the numerical integration method and the first-order exact solution one. The parameter sensitivity study is carried out and the effect of different parameters including the hub radius, structural damping, acceleration, and the deceleration rates on the vibration amplitude is investigated.     

On the rotating rod with variable cross section

Summary Stability of a heavy rotating rod with a variable cross section is studied by energy method. Bifurcation points for the system of equilibrium equations are analyzed. It is shown that for the case when the rotation speed exceeds the critical one, the trivial solution ceases to be the minimizer of the potential energy, so that rod loses stability, according to the energy criteria. Also, a new estimate of the maximal rod deflection in the post-critical state is obtained. Accepted for publication 11 November 1996     

Nonlinear dynamics of a spinning shaft with non-constant rotating speed

Research on spinning shafts is mostly restricted to cases of constant rotating speed without examining the dynamics during their spin-up or spin-down operation. In this article, initially the equations of motion for a spinning shaft with non-constant speed are derived, then the system is discretised, and finally a nonlinear dynamic analysis is performed using multiple scales perturbation method. The system in first-order approximation takes the form of two coupled sets of paired equations. The first pair describes the torsional and the rigid body rotation, whilst the second consists of the equations describing the two lateral bending motions. Notably, equations of the lateral bending motions of first-order approximation coincide with the system in case of constant rotating speed, and considering the amplitude modulation equations, as it is shown, there are detuning frequencies from the Campbell diagram. The nonlinear normal modes of the system have been determined analytically up to the second-order approximation. The comparison of the analytical solutions with direct numerical simulations shows good agreement up to the validity of the performed analysis. Finally, it is shown that the Campbell diagram in the case of spin-up or spin-down operation cannot describe the critical situations of the shaft. This work paves the way, for new safe operational ‘modes’ of rotating structures bypassing critical situations, and also it is essential to identify the validity of the tools for defining critical situations in rotating structures with non-constant rotating speeds, which can be applied not only in spinning shafts but in all rotating structures.     

Aeroelastic dynamics stability of rotating sandwich annular plate with viscoelastic core layer

A dynamic model for a rotating sandwich annular plate with a viscoelastic core layer is developed. All fundamental equations and boundary conditions are established based on Hamilton’s principle, and the rotation effect and viscoelastic properties of the sandwich structure are taken into account. The aerodynamics force acting on the plate is described by a rotating damping model, and the constitutive behavior of the viscoelastic core layer is formulated by the frequency-dependent complex modulus. The effects of geometrical and material parameters on frequencies and damping of forward and backward traveling waves and the dynamic stability for the rotating sandwich plate are numerically analyzed by means of Galerkin’s method. The results show that the critical and flutter speeds of the rotating plate can be increased at some certain parameters of the viscoelastic core layer.     

On the dynamics of rotating and rolling structures

The dynamics of rotating flexible bodies is influenced by gyroscopic effects, which yields complex-valued eigenvectors and a split of eigenfrequencies in comparison with a resting body depending on rotational speed. These phenomena, which have been well investigated for simple structures like elastic rings, also affect more complicated systems such as rolling drums in printing machines. The aim of this contribution is to give a clear introduction into the physics of rotating bodies, starting with a rather simple analytical model, followed by some basic experimental investigations and ending with a finite element approach of more complicated three-dimensional structures. Special care is taken for the numerical solution of complex eigenvalue problems. The results obtained in each step are discussed in detail regarding their physical meaning.     

On the dynamics of rotating circular cable structures

The combined effect of gravity and a centrifugal field on the dynamics of a heavy cable with negligible bending stiffness is treated. The inextensible length of the cable is preassigned in such a way that the cross-sectional shape of the cable structure is circular if the contiguration rotates about its central axis with a constant speed and no external forces act. Under the additional influence of gravity, complicated nonlinear weight-excited vibrations occur.To understand the variety of vibrational phenomena. The dynamic system is studied by some experiments first. In order to explain the experimental results theoretically, the governing nonlinear boundary value problem is derived next. Subsequently, an appropriate variational formulation for application of a Rayleigh Ritz procedure is suggested. The resulting nonlinear ordinary differential equations approximately deseribe a part of the observed vibrational behaviour.Both the experiments and the calculations demonstrate that, for high speeds, nonlinear weight-excited vibrations about the circular reference configuration occur. On the other hand, for low velocitites, periodie and even chaos-like snap-through phenomena appear.     

A device to regulate rotational speed using a rotating regulator with electrorheological fluid

The authors propose a device consisting of a rotating regulator within an electrorheological fluid (ERF), in which the rotational speed of the rotating body can be regulated by supplying an electric field. This new type of device with ERF is relevant to engineering applications and can play a role in braking as well as in rotation regulation. The ERF used here consists of a colloidal suspension of smectite particles in a solvent. The experimental conditions are fixed in a range of comparatively small rotational speed. The authors clarified the hydrodynamic and electrical effects of the shape of the rotating disk, the kind of solvent contained in the ERF, the temperature, the electrode gap and the electric field strength on the steady characteristics of the torque, the electric current density and the electric power to a rotational speed. From the torque and rotational speed data, empirical equations were derived using nondimensional parameters for the purpose of convenient engineering design. The authors also clarified the correlation between the hydrodynamic and steady electric characteristics of the device and the transient characteristics of torque, the electric current and the rotational speed.     

On embedded material derivative and frame spin

In the present paper, the concept of embedded material derivative is proposed, the general expression for frame spin is derived, the difference between embedded material derivative and conventional one of trial unit vectors is clarified. Furthermore, the formula for strain-rate frame spin is obtained. And by means of this example, the distinction between the two kinds of material derivatives is discussed. Then, the relations between several frame spins and the rates of corresponding tensors co-rotating with the frame are elucidated. Now part-time Ph.D. student at Tsinghua University.     

Equi-strength design of nonhomogeneous variable thickness high speed rotating disk under steady temperature field

Considering the mutual interaction between the thickness ofthe disk and temperature distribution, and directly using results in [1].[2] and the iteration method stage sied in this paper, we obtain the equi-strength thickness distribution of nonhomogeneous variable thickness high speed rotating disk under axisymmetrieal steady temperature field.Projects Supported by the Science and Technic Fund of the National Education Committee     

Nonstationary vibration of a rotating shaft with nonlinear spring characteristics during acceleration through a critical speed (A critical speed of a summed-and-differential harmonic oscillation)

Nonstationary vibration of a flexible rotating shaft with nonlinear spring characteristics during acceleration through a critical speed of a summed-and-differential harmonic oscillation was investigated. In numerical simulations, we investigated the influence of the angular acceleration , the initial angular position of the unbalance _n and the initial rotating speed on the maximum amplitude. We also performed experiments with various angular accelerations. The following results were obtained: (1) the maximum amplitude depends not only on but also on _n and : (2) when the initial angular position _n changes. the maximum amplitude varies between two values. The upper and lower bounds of the maximum amplitude do not change monotonously for the angular acceleration: (3) In order to always pass the critical speed with finite amplitude during acceleration. the value of must exceed a certain critical value.Nomenclature O-xyz rectangular coordinate system - , ₁, ₁ inclination angle of rotor and its projections to thexy- andyz-planes - I _r polar moment of inertia of rotor - I diametral moment of inertia of rotor - i _r ratio ofI _r toI - dynamic unbalance of rotor - directional angle of fromx-axis - c damping coefficient - spring constant of shaft - N _nt ,N _nt nonlinear terms in restoring forees in ₁ and ₁ directions - ₄ representative angle - a small quantity - V. V _u .V _N potential energy and its components corresponding to linear and nonlinear terms in the restoring forees - directional angle - _n coefficients of asymmetrical nonlinear terms - _n coefficients of symmetrical nonlinear terms - coefficients of asymmetrical nonlinear terms experessed in polar coordinates - coefficients of symmetrical nonlinear terms expressed in polar coordinates - rotating speed of shaft - t time - _n initial angular position of att=0 - p natural frequency - p ₁.p _t natural frequencies of forward and backward precessions - , ₁, ₁ total phases of harmonic, forward precession and backward precession components in summed-and-differential harmonic oscillation - , ₁, ₁ phases of harmonic, forward precession and backward precession components in summed-and-differential harmonic oscillation - P, R _t ,R _b amplitudes of harmonic, forward precession and backward precession components in summed-and-differential harmonic oscillation - difference between phases ( = _f – _u) - acceleration of rotor - initial rotating speed - t _t ,r _b amplitudes of nonstationary oscillation during acceleration - (r _t )_max, (r _b )_max maximum amplitudes of nonstationary oscillation during acceleration - (r ₁ ¹ )_max, (r _b ¹ )_max maximum value of angular acceleration of non-passable case - ₀ critical value over which the rotor can always pass the critical speed - p ₁,p ₂,p ₃,p ₄ natural frequencies of experimental apparatus     

A new variable inclination shock tube for multiple investigations

A new variable inclination shock tube has been set up especially for the visualization of shock induced high speed flows. This device allows for the generation of shock waves of Mach numbers ranging from 1.05 to 4, moving horizontally, vertically from up to down or the opposite, or with any inclination angle. It is coupled with a suitable high-speed camera shadowgraph visualization system, and the whole facility has been realized so as to be both very easy to use and movable. We believe this example could be of interest to scientists interested in starting experiments on shock wave induced multi-phase complex flows.     

Nonlinear vibrations of blade with varying rotating speed

This paper investigates the nonlinear dynamic responses of the rotating blade with varying rotating speed under high-temperature supersonic gas flow. The varying rotating speed and centrifugal force are considered during the establishment of the analytical model of the rotating blade. The aerodynamic load is determined using first-order piston theory. The rotating blade is treated as a pretwist, presetting, thin-walled rotating cantilever beam. Using the isotropic constitutive law and Hamilton??s principle, the nonlinear partial differential governing equation of motion is derived for the pretwist, presetting, thin-walled rotating beam. Based on the obtained governing equation of motion, Galerkin??s approach is applied to obtain a two-degree-of-freedom nonlinear system. From the resulting ordinary equation, the method of multiple scales is exploited to derive the four-dimensional averaged equation for the case of 1:1 internal resonance and primary resonance. Numerical simulations are performed to study the nonlinear dynamic response of the rotating blade. In summary, numerical studies suggest that periodic motions and chaotic motions exist in the nonlinear vibrations of the rotating blade with varying speed.     

Analysis of functionally graded rotating disks with variable thickness

Elastic solutions for axisymmetric rotating disks made of functionally graded material with variable thickness are presented. The material properties and disk thickness profile are assumed to be represented by two power-law distributions. In the case of hollow disk, based on the form of the power-law distribution for the mechanical properties of the constituent components and the thickness profile function, both analytical and semi-analytical solutions are given under free–free and fixed-free boundary conditions. For the solid disk, only semi-analytical solution is presented. The effects of the material grading index and the geometry of the disk on the stresses and displacements are investigated. It is found that a functionally graded rotating disk with parabolic or hyperbolic convergent thickness profile has smaller stresses and displacements compared with that of uniform thickness. It is seen that the maximum radial stress for the solid functionally graded disk with parabolic thickness profile is not at the centre like uniform thickness disk. Results of this paper suggest that a rotating functionally graded disk with parabolic concave or hyperbolic convergent thickness profile can be more efficient than the one with uniform thickness.     

Large amplitude free flexural vibration of rings using finite element approach

Here, the large amplitude free flexural vibrations of isotropic/laminated orthotropic rings are investigated, using a shear flexible curved beam element based on field consistency principle. A laminated refined beam theory is introduced for developing the element, which satisfies the interface transverse shear stress and displacement continuity, and has a vanishing shear stress on the inner and outer surfaces of the beam. The formulation includes in-plane and rotary inertia effects, and the non-linearity due to the finite deformation of the ring. The governing equations obtained using Lagrange's equations of motion are solved through the direct integration technique. Amplitude-frequency relationships evaluated from the dynamic response history are examined. Detailed numerical results are presented considering various parameters such as radius-to-thickness ratio, circumferential wave number and ovality for isotropic and laminated orthotropic rings. The nature and degree of the participation of various modes in non-linear asymmetric vibration of oval ring brought out through the present study are useful for accurate modelling of the closed non-circular structures.     

Ball spin generation for oblique impacts with a tennis racket

In this paper, we describe an experimental investigation of the oblique impact between a tennis ball and head clamped tennis racket. It was found that the magnitude of the ball rebound spin was not a function of the material, gage or tension of the string used in the tennis racket. Furthermore, it was concluded that all strings exhibit a sufficiently large friction coefficient that the ball begins to roll during impact. There is anecdotal evidence from tennis players that suggests that a high string tension or a rough string surface enable them to impart more spin to the ball. For example, players have been quoted as saying that a high string tension makes the strings “bite” into the ball, giving more spin. The data reported in this study do not support these observations. Analysis of the experimental data has shown that the balls are rebounding from the surface with more spin than would typically be associated with rolling. A second experiment showed that the balls commenced rolling at the mid-point of the impact. This information was used in a theoretical model to show that the spin that acts on the ball during the impact can be higher than the value of the rolling spin at the end of the impact.     

On the complex variable displacement method in plane isotropic elasticity

A modified formulation of the complex variable displacement method in plane isotropic elasticity is presented. It makes use of two equations deduced from the planar Navier equations in terms of the complex variable, which differs from England’s original formulation based on only one equation. This formulation is more direct and complements the one by England.     

INTEGRATION METHOD FOR THE DYNAMICS EQUATION OF RELATIVE MOTION OF VARIABLE MASS NONLINEAR NONHOLONOMIC SYSTEM

1.IntroductionMoreandmoreattentionhasbeenpaidtothestudyofdynamicsofcomplicatedsystemwiththedevelopmentofmodernscienceandtechnology.Thestudyoftherelativemotionofvariablemasssystembyusingthetheoryandmethodofanalyticalmechanicsnotonlycanunifytheexpressionformbutalsocandisplayitssuperioritytothecomplicatedsystem.In1961,thedynamicsofrelativemotionofholonomicsystemwasderivedbyLur'ell].Inrecehtyears,LiulZIandLuol3'4]havegiventhedynamicsequationsofrelativemotionofvariablemassnonholonomicsystem.Howe…     

Elastic-plastic stress distribution in the rotating annular disk with variable thickness

Summary The plane state of stress in a rotating annular disk with variable thickness is studied. The analysis is based on Tresca's yield condition, its associated flow rule and linear strain hardening.

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Die elastisch-pastische Spannungsverteilung in einem rotierenden Kreisring mit veränderlicher Dicke

Übersicht Der ebene Spannungszustand in einem rotierenden Kreisring mit veränderlicher Dicke und aus Material mit linearer Verfestigung wird unter Zugrundelegung der Trescaschen Fließbedingung und der zugehörigen Fließregel untersucht.

     

Dynamic analysis of the variable stiffness support rotor system with elastic rings

Nonlinear Dynamics - Elastic rings are common rotor supporting structure, which have been widely used in aeroengine rotor support system. However, large inertia force and gyroscopic moment may...