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1.
Due to their small diameter-to-length-ratio, drill-strings are vulnerable to torsional vibrations. Moreover, the string is exposed to unknown or uncertain time-variant and nonlinear loads (e.g. friction with falling friction characteristics, contact with the borehole, differential sticking), which can result in severe torsional vibrations and stick-slip. The control law for the boundary controller at the top drive of the string needs to adapt to those unknown loads in order to stabilize the vibrations. The torsional vibrations of a drill-string are governed by the wave equation. Analytical solutions and control laws are often based on a separation of the dynamics into a time- and a space-dependent part (modal representation). Here, we decompose the vibrations into two traveling waves according to the D'Alembert solution, using only few measurements along the string. The wave which travels up the string is then compensated by the actuator at the top drive. With this compensation, the upward-traveling wave is no longer reflected back into the string and vibration energy is absorbed, thus stabilizing the torsional vibrations. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
2.
The dynamical behavior of a drill-string is defined by its small diameter-to-length ratio, which makes the string vulnerable to torsional vibrations. In combination with the nonlinear friction characteristic at the drill bit, this can lead to self-excited stick-slip vibrations which are detrimental to the drilling process. The string can be modeled by the Finite Element Method or as a Multi-Body system to represent the distributed character of the system. The analysis of the resulting high-dimensional model is, however, elaborate and time-consuming. We show that through Galerkin Projection onto the first two Characteristic Functions gained from Karhunen-Loève-Transformation, a reduced system can be obtained which reproduces the essential dynamical properties of the original system, e.g. the stick-slip motion. With the reduced system, the linear stability of the drill-string can be analyzed. We show that by reducing the inertia of the rotary table the system can be stabilized. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
3.
4.
Here we are concerned about uniform stability of damped nonlinear transverse vibrations of an elastic string fixed at its
two ends. The vibrations governed by nonlinear integro-differential equation of Kirchoff type, is shown to possess energy
uniformly bounded by exponentially decaying function of time. The result is achieved by considering an energy-like Lyapunov
functional for the system. 相似文献
5.
E. I. Moiseev 《Differential Equations》2011,47(11):1675-1679
We consider the optimal boundary control of string vibrations and study the case of boundary force control at one end of the
string, the other end being free. We show that if the initial and terminal data are arbitrary, then the dependence of the
optimal boundary control on these data may be nonlinear. Conditions under which the dependence is linear are indicated. An
example is considered in which the optimal control can be found in closed form. 相似文献
6.
S. A. Sergeev 《Differential Equations》2011,47(5):746-757
We find necessary and sufficient conditions for the existence of a boundary control of vibrations of a string or a spherical
layer for critical and subcritical times. We completely analyze the existence of a boundary control of vibrations of a spherical
layer by a force on two spheres. We find necessary and sufficient existence conditions for the control. Along with the control
problem for vibrations of a spherical layer, we consider a similar control problem for string vibrations. 相似文献
7.
Donald Greenspan 《BIT Numerical Mathematics》1971,11(4):399-408
A new approach to the study of linear and nonlinear string vibrations is developed by means of discrete simulation. Computer examples using different laws of tension are described and compared. The method is exceptionally fast and allows for a wide range of parameter choices. 相似文献
8.
《Journal of Applied Mathematics and Mechanics》2003,67(2):243-252
Two problems of the vibrations of strings are considered using the approach described previously in [1]: the vibrations of the string of a plucked musical instrument, drawn out at one of the points and at rest at the initial instant of time (Problem 1), and the vibrations of the string of a keyboard musical instrument, the points of which are given an initial velocity at the initial instant of time by a hammer of small width (Problem 2). It is established that forced longitudinal oscillations of the string occur at frequencies of the transverse vibrations, the condition for possible resonance of the longitudinal vibrations is derived, and the nature of the vibrations at the point where the string is fastened due to elasticity and the related shift in the frequency of transverse vibrations is established. 相似文献
9.
A. A. Kholomeeva 《Differential Equations》2008,44(5):717-721
We consider the optimization problem for a nonlocal boundary control of vibrations of an elastic string with fixed right end for an arbitrary sufficiently large time interval that is a multiple of twice the string length. We prove the existence and uniqueness of a generalized solution of the first boundary value problem and indicate an explicit expression for the solution. The optimal control is found in closed form. 相似文献
10.
L. N. Znamenskaya 《Differential Equations》2010,46(5):748-752
We solve state observation problems for string vibrations, i.e., problems in which the initial conditions generating the observed
string vibrations should be reconstructed from a given string state at two distinct time instants. The observed vibrations
are described by the boundary value problem for the wave equation with homogeneous boundary conditions of the first kind.
The observation problem is considered for classical and L
2-generalized solutions of this boundary value problem. 相似文献
11.
Reflection of short rectangular pulses in the ideal string attached to strongly nonlinear oscillator
The system under consideration is ideal elastic string attached to strongly nonlinear oscillator with cubic nonlinearity by two different ways – immediately and by weak linear spring. The reflection of short rectangular pulses from the oscillator is accompanied by excitation of vibrations. The type of mode excited determines the amount of energy transferred to the oscillator as well as the structure of the reflected wave. 相似文献
12.
Based on the exact solution of a coupled problem of the magnetoelastic vibrations of an elastic electrically conductive layer
(for a finite value of its electrical conductance) in a longitudinal magnetic field, we have established that, in the case
of such vibrations of a thin plate, one can obtain correct results proceeding from Kirchhoff’s hypothesis and the hypothesis
of magnetoelasticity of thin bodies. 相似文献
13.
Vibrations in long torsional strings result in spatio‐temporal dynamics. In order to actively damp these vibrations the system has to be analysed analytically, numerically and experimentally. Stick‐slip‐effects result in torsional selfexcited vibrations of the string. These vibrations are coupled with bending vibrations which are constrained by the borehole. The straight string was modelled in an experimental setup. The control of the straight string and the unilateral constrained bending vibrations were treated seperately. The dynamics of straight strings were controlled using three different approaches: firstly, a simple PD‐controller with the parameters calculated based on a one‐degree‐of‐freedom oscillator, secondly, the parameters were optimized using a simplex‐method, thirdly, the Karhunen‐Loeve‐transformation was used in order to reduce the dimension of the system. A controller based on the reduced system was implemented and the parameters were optimized with the same simplex algorithm. The unilateral constrained bending motion were examined at a cantilever beam which was assumed to be constrained in one direction in the middle of the beam. First, the beam was modelled analytically as a continuous system. The two states (contact and no contact) were described separately. The transition between these states were modelled with energy assumptions. Second, the beam was modelled as a Finite‐Element‐System. The numerical results of both methods were compared with experimental data. 相似文献
14.
The boundary problem is considered which occurs in the theory of small transversal vibrations of a smooth inhomogeneous string. The ends of the string assumed to be fixed and the midpoint of the string is damped by a pointwise force. The problem is reduced to a spectral problem for a nonmonic quadratic operator pencil. The spectrum of the pencil (i. e. the set of normal eigenvalues) can be presented as a union of two subsequences. One of the subsequences approaches the real axis. Under an additional condition the second branch approaches a horizontal line located in the upper half–plane. The basis properties of the sets of projections (onto the corresponding subintervals) of eigenfunctions corresponding to each of the subsequences are investigated. 相似文献
15.
N. Yu. Kapustin 《Differential Equations》2014,50(10):1391-1394
We study a classical problem that arises in the analysis of natural vibrations of a loaded string with a free endpoint. We assume that the coefficient occurring in the boundary condition of the third kind with a spectral parameter instead of a physical parameter can take complex values. We discuss the traditional aspects of the completeness, minimality, and basis property of the system of root functions. Special attention is paid to the structure of root subspaces. 相似文献
16.
N. Yu. Kapustin 《Differential Equations》2012,48(10):1341-1347
We consider a classical problem that arises when studying natural vibrations of a loaded string. We assume that the coefficient playing the role of a physical parameter can take complex values. We discuss the completeness, minimality, and basis property of the system of root functions. 相似文献
17.
We present a numerical-analytic method for solution of the problem of nonlinear periodic oscillation of a string pendulum
(unstretched string suspended from a point in a gravitational field). The method is based on successive linearization of the
problem with the Lindstadt-Poincare method, separation of time dependencies with Fourier’s method, reduction of differential
equations to recurrence relations for the coefficients of power series in the spatial coordinate, and computer realization
of the convergence of the power series.
Translated fromDinamicheskie Sistemy, Vol. 12. pp. 44–51, 1993. 相似文献
18.
We study a problem of optimal boundary control of vibrations of a one-dimensional elastic string, the objective being to bring the string from an arbitrary initial state into an arbitrary terminal state. The control is by the displacement at one end of the string, and a homogeneous boundary condition containing the time derivative is posed at the other end. We study the corresponding initial-boundary value problem in the sense of a generalized solution in the Sobolev space and prove existence and uniqueness theorems for the solution. An optimal boundary control in the sense of minimization of the boundary energy is constructed in closed analytic form. 相似文献
19.
20.
H. B. Thompson 《Applicable analysis》2013,92(1-2):27-43
Explicit upper and lower bounds are constructed for a functional defined on the difference of solutions to a class of nonlinear hyperbolic problems modelling string vibrations. Such bounds indicate how solutions are affected by the input data over a finite time interval. The logarithmic convexity method is used to obtain two second order ordinary differential inequalities for the appropriate functional. These inequalities are then related to two systems of first order equations whose solutions are the desired bounds. 相似文献