A NEW MATRIX METHOD FOR SOLVING VIBRATION AND STABILITY OF CURVED PIPES CONVEYING FLUID

This paper proposes a new matrix method for calculating critical flow velocity of curved pipes conveying fluid, which have arbitrary centerline shape and spring supports. Its main advantage over other methods is that the corresponding characteristic equation can be reduced to a third order one, no matter how many elements are discretized in calculation. This will lead to saving computer time and obtaining a solution with good precision.     

BIFURCATIONS AND CHAOS OF AN IMMERSED CANTILEVER BEAM IN A FLUID AND CARRYING AN INTERMEDIATE MASS

The concern of this work is the local stability and period-doubling bifurcations of the response to a transverse harmonic excitation of a slender cantilever beam partially immersed in a fluid and carrying an intermediate lumped mass. The unimodal form of the non-linear dynamic model describing the beam-mass in-plane large-amplitude flexural vibration, which accounts for axial inertia, non-linear curvature and inextensibility condition, developed in Al-Qaisia et al. (2000Shock and Vibration7 , 179-194), is analyzed and studied for the resonance responses of the first three modes of vibration, using two-term harmonic balance method. Then a consistent second order stability analysis of the associated linearized variational equation is carried out using approximate methods to predict the zones of symmetry breaking leading to period-doubling bifurcation and chaos on the resonance response curves. The results of the present work are verified for selected physical system parameters by numerical simulations using methods of the qualitative theory, and good agreement was obtained between the analytical and numerical results. Also, analytical prediction of the period-doubling bifurcation and chaos boundaries obtained using a period-doubling bifurcation criterion proposed in Al-Qaisia and Hamdan (2001 Journal of Sound and Vibration244, 453-479) are compared with those of computer simulations. In addition, results of the effect of fluid density, fluid depth, mass ratio, mass position and damping on the period-doubling bifurcation diagrams are studies and presented.     

DYNAMICS BEHAVIOR OF AXISYMMETRIC AND BEAM-LIKE ANISOTROPIC CYLINDRICAL SHELLS CONVEYING FLUID

The flow-induced vibration characteristics of anisotropic laminated cylindrical shells partially or completely filled with liquid or subjected to a flowing fluid are studied in this work for two cases of circumferential wave number, the axisymmetric, where n=0 and the beam-like, where n=1. The shear deformation effects are taken into account in this theory; therefore, the equations of motion are determined with displacements and transverse shear as independent variables. The present method is a combination of finite element analysis and refined shell theory in which the displacement functions are derived from the exact solution of refined shell equations based on orthogonal curvilinear co-ordinates. Mass and stiffness matrices are determined by precise analytical integration. A finite element is defined for the liquid in cases of potential flow that yields three forces (inertial, centrifugal and Coriolis) of moving fluid. The mass, stiffness and damping matrices due to the fluid effect are obtained by an analytical integration of the fluid pressure over the liquid element. The available solution based on Sanders' theory can also be obtained from the present theory in the limiting case of infinite stiffness in transverse shear. The natural frequencies of isotropic and anisotropic cylindrical shells that are empty, partially or completely filled with liquid as well as subjected to a flowing fluid, are given. When these results are compared with corresponding results obtained using existing theories, very good agreement is obtained.     

NATURAL FREQUENCIES AND MODE SHAPES OF A FREE-FREE BEAM WITH LARGE END MASSES

An analytical solution is presented for the natural frequencies, mode shapes and orthogonality condition, of a free-free beam with large off-set masses connected to the beam by torsion springs. Results are given for a range of masses with various fixed orientations and the validity of the method is confirmed against established results for natural frequencies of beams with five different boundary conditions. The study lays the foundation for investigations into the dynamics and vibration control of multi-link articulated systems such as the Space Shuttle Remote Manipulator.     

NON-LINEAR TRANSVERSE VIBRATIONS OF A SIMPLY SUPPORTED BEAM CARRYING CONCENTRATED MASSES

An Euler-Bernoulli beam carrying concentrated masses is considered to be a beam-mass system. The beam is simply supported at both ends. The non-linear equations of motion are derived including stretching due to immovable end conditions. The stretching introduces cubic non-linearities into the equations. Forcing and damping terms are also included. Exact solutions for the natural frequencies are given for the linear problem. For the non-linear problem, an approximate solution using a perturbation method is searched. Non-linear terms of the perturbation series appear as corrections to the linear problem. Amplitude and phase modulation equations are obtained. Non-linear free and forced vibrations are investigated in detail. The effect of the positions, magnitudes and number of the masses are investigated.     

FREE VIBRATIONS OF A RECTANGULAR PLATE CARRYING A DISTRIBUTED MASS

In this paper, an analytical method is presented to find the eigenfrequencies of a rectangular plate carrying a uniformly distributed mass. Using the standard Galerkin procedure, the equation of motion is reduced to a set of ordinary differential equations. From this set, the frequency equation is obtained. This polynomial equation is solved numerically. Due to the significance of the fundamental frequency of the system, its variation with respect to the non-dimensional parameters associated with the location, the area density and the distribution area of the mass attached to the plate, is investigated. Furthermore, it is shown by a numerical example that the method can be used to study plates with concentrated mass as a special case. Finally, an analysis to obtain the modal surfaces and the related nodal lines is carried out. It is demonstrated that the location of the attachment significantly affects the nodal lines, and modal interchanges may occur.     

FORCED AND SELF-EXCITED VIBRATIONS OF PIPES CONTAINING MOBILE BOILING FLUID CLOTS

Numerical modelling of the dynamic behaviour of a pipe containing inner non-homogeneous flows of a boiling fluid has been carried out. Inasmuch as the efforts to solve this problem analytically are confronted by considerable difficulties connected with varying system mass, geometry and discontinuity of equation coefficients, computational techniques for simulating pipe dynamics have been developed based on using of numerical time integration methods and transfer matrix methods together with orthogonalization procedures relating to the space variables. The system vibrations at different values of the parameters of the flow non-homogeneity and its velocity are observed. The possibility of forming stable and unstable flows depending on the character of the non-homogeneity and the velocity of fluid clots has been found.     

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.     

THE EFFECT OF ROTATION UPON THE NATURAL FREQUENCIES OF A MASS-SPRING SYSTEM

When a mass-spring system vibrates it does so with frequencies characteristic of the system. If the system as a whole now undergoes a rotational motion then these characteristic frequencies will change from their non-rotational values. It is the purpose of this paper to show how these changes may be calculated for a specified system and, in particular, to investigate the role in these changes of both the system and the rotational parameters. A system of N masses linked sequentially by springs in tension is allowed to vibrate about an equilibrium configuration both radially and transversely upon a smooth turntable. If the turntable is stationary then the radial and transverse vibrations are independent of each other, provided the amplitudes of vibration are sufficiently small. There are then N natural frequencies of vibration for each mode. However, when the turntable rotates then the Coriolis effects give rise to an interaction between the two modes of vibration, and there are now 2 N natural frequencies for the combined vibrations. If the rate of rotation is “small” then the two modes are almost separated and it is possible to discuss the “essentially radial” or “essentially transverse” mode of vibration each of which has N natural frequencies. It is these natural frequencies which are considered in this work, in particular their dependence upon the rotation rate and upon the tension in the springs (when in the static configuration). In a previous paper, it was shown that if only radial vibrations are allowed (by admitting say a guide rail) then all the natural frequencies decrease, with increasing rotation rate, from their static values. It is shown that the opposite is the case here in that the “essentially radial” natural frequencies increase with increasing rotation rate. This is due to the Coriolis interaction with the transverse vibrations. The “essentially transverse” frequencies are also found and the nature of their dependence discussed. Also included in the analysis is the effect on the frequencies of the (weak) coupling between the motion of the masses and the rotation of the turntable as a consequence of the conservation of angular momentum. In addition to treating N being finite the limiting case of an infinite number of masses is considered to determine the natural frequencies of vibration of a continuous stretched string undergoing rotation.     

ON THE FREQUENCY RESPONSE FUNCTION OF A DAMPED CANTILEVER SIMPLY SUPPORTED IN-SPAN AND CARRYING A TIP MASS

This paper deals with the determination of the frequency response function of a cantilevered Bernoulli-Euler beam which is viscously damped by a single damper. The beam is simply supported in-span and carries a tip mass. The frequency response function is obtained through a formula that was established for the receptance matrix of discrete linear systems subjected to linear constraint equations, by considering the simple support as a linear constraint imposed on generalized co-ordinates. The comparison of the numerical results obtained via a boundary value problem formulation justifies the approach used here.     

低Prandtl数水平流体层自然对流的振荡和分歧

本文用具有QUICK方案的有限差分法对底部加热的低Prandtl数水平流体层自然对流换热进行了数值计算,研究了这种问题中存在的振荡和分歧问题。结果显示,在Ra的一定取值区间,有4涡型流场和5涡型流场两个解的分支。但在这个区间以外,最终的结果没有出现分歧。在所发现的两个解的分支中,问题由稳态转变为非稳态的临界Racr是不同的。     

EXISTENCE OF NATURAL FREQUENCIES OF SYSTEMS WITH ARTIFICIAL RESTRAINTS AND THEIR CONVERGENCE IN ASYMPTOTIC MODELLING

A major limitation of the Rayleigh-Ritz method for determining the natural frequencies of a system is the need to choose admissible functions that do not violate the geometric constraints of that system (Courant 1943 Bulletin of the American Mathematical Society49, 1-23). Several researchers have attempted to overcome this problem by asymptotically modelling the rigid constraints with artificial (imaginary) restraints of very large stiffness (Courant 1943Bulletin of the American Mathematical Society49 , 1-23; Warburton and Edney 1984 Journal of Sound and Vibration95, 537-552; Gorman 1989 Journal of Applied Mechanics56, 893-899; Kim et al. 1990 Journal of Sound and Vibration143, 379-394; Yuan and Dickinson 1992 Journal of Sound and Vibration153, 203-216; Yuan and Dickinson 1992 Journal of Sound and Vibration159, 39-55; Cheng and Nicolas 1992 Journal of Sound and Vibration155, 231-247; Yuan and Dickinson 1994Computers and Structures53 , 327-334; Lee and Ng 1994 Applied Acoustics42, 151-163; Amabili and Garziera 1999 Journal of Sound and Vibration224, 519-539; Amabili and Garziera 2000 Journal of Fluids and Structures14, 669-690). While the numerical results thus obtained for the systems considered in the literature were in close agreement with exact values for the natural frequencies corresponding to the first few modes, sample calculations show that the error introduced by the asymptotic modelling increases with mode number and therefore to obtain accurate results for higher modes the magnitude of stiffness should also be increased. In any event, the error due to the asymptotic modelling would remain uncertain, except when the correct frequency values are known. However, the use of artificial restraints with negative stiffness, a new concept which was introduced in a recent publication (Ilanko and Dickinson 1999 Journal of Sound and Vibration219, 370-378) paves the way for estimating the error due to asymptotic modelling. This is possible since in this work, the Rayleigh-Ritz frequencies of the constrained system were found to be bracketed by the frequencies of the asymptotic models with positive and negative restraints. However, the use of artificial restraints with negative stiffness has raised some important questions: would a system with a large negative restraint become unstable, and if so what is the guarantee that the frequencies of the asymptotic model would converge to that of the constrained system? This paper is the result of the author's attempt to answer these questions and gives a proof of existence of natural frequencies for systems with artificial restraints (springs) having positive or negative stiffness coefficients, and their convergence towards constrained systems. Based on Rayleigh's theorem of separation, it has been shown that a vibratory system obtained by the addition of h restraints to an n -degree-of-freedom (d.o.f.) system, where h<n, will have at least (n÷h) natural frequencies and modes and that as the magnitude of the stiffness of the added restraints becomes very large, these (n÷h) natural frequencies will converge to the (n÷h) natural frequencies of a constrained system in which the displacements restrained by the springs are effectively constrained.     

EXPRESSIONS FOR PREDICTING FUNDAMENTAL NATURAL FREQUENCIES OF NON-CYLINDRICAL HELICAL SPRINGS

Numerical and analytical studies are performed for the free vibration analysis of non-cylindrical (conical, barrel and hyperboloidal types) helical springs. The stiffness matrix method is used in the numerical analysis. A total of 12 degrees of freedom (six displacements and six rotations) is described for an element. The exact element stiffness matrix and the exact concentrated element inertia matrix are used in the formulation. The rotary inertia, the shear and extensional deformation effects are considered in the analysis. Comparison of the numerical results with the reported results obtained numerically and experimentally gives satisfactory values. After verification of the numerical frequencies, the non-dimensional fundamental frequencies of fixed-fixed non-cylindrical helical springs with circular section are expressed in a simple formula with a maximum absolute relative error of 5% using those numerical values for the constant helix pitch angles (5°, 10°, and 15°). These expressions restricted to the fundamental frequencies are also verified with ANSYS results.     

交联氮气泡沫压裂液的携砂性能研究

在大型高压泡沫压裂液实验回路上,对交联氮气泡沫压裂液的携砂性能进行了实验研究.分析了携砂泡沫压裂液流动时临界携砂流速的产生原因,并根据实验结果分析了温度,压力、泡沫质量及支撑剂浓度对临界携砂流速的影响.临界携砂流速随温度的升高而增大,随压力、泡沫质量和支撑剂体积浓度的增大而减小,研究发现当支撑剂浓度大于0.17时,开始...     

H -ADAPTIVE REFINEMENT STRATEGY FOR ACOUSTIC PROBLEMS WITH A SET OF NATURAL FREQUENCIES

The finite element method has been applied to the analysis of acoustic problems with several natural frequencies and mode shapes. First, a recovery-based error estimation is performed following the well-known procedures of structural problems. Then, an h -adaptive refinement strategy is proposed that leads to a finite element mesh with the minimum number of elements and with a specified error for each of the natural frequencies included in the analysis. The procedure provides a useful numerical tool, since the computational requirements are reduced. In addition, results obtained by means of the minimum element size procedure are shown for comparison purposes. The similarity of the meshes given by the two methods is justified on the basis of the equations that lead to the element size of the mesh. The procedure has been applied to some numerical examples to illustrate its validity.