Section 2. Some of the definitions used throughout the book are given here. A great deal if not all of this material is well known to most readers; yet, some unusual features e. The structure of Section 2. Some generalities on the various flow regimes of interest e. This is then followed by two examples, in which the fluid forces exerted on an oscillating structure are calculated, for: a two-dimensional vibration of coaxial shells coupled by inviscid fluid in the annulus; b two-dimensional vibration of a cylinder in a coaxial tube filled with viscous fluid.
Finally, in Section 2. Some systems, for example a simple pendulum, are sui generis discrete; i. However, what distinguishes a discrete system more precisely is that its configuration and position in space at any time may be determined from knowledge of a numerable set of quantities; i. Thus, the simple pendulum has one degree of freedom, and a double compound pendulum has two.
The quantities variables required to completely determine the position of the system in space are the generalized coordinates , which are not unique, need not be inertial, but must be equal to the number of degrees of freedom and mutually independent , of each of the pipes to the vertical may be utilized as the generalized coordinates [Figure 2. Contrast this to a flexible pipe [Figure 2.
Discrete systems are described mathematically by ordinary differential equations ODEs , whereas distributed parameter systems by partial differential equations PDEs. If a system is linear, or linearized, which is admissible if the motions are small e. This is very convenient, since computers understand matrices very well!
Fluid-Structure Interactions: Volume 2 : Slender Structures and Axial Flow
Press et al. Thus, a damped system subjected to a set of external forces may be described by. Figure 2. On the other hand, the form of PDEs tends to vary much more widely from one system to another. Although helpful classifications e. Also, the solutions are generally considerably more difficult, if the equations are tractable at all by other than numerical means.
Furthermore, the addition of some new feature to a known problem i. Consider, for instance the situation of the articulated pipe system which can be described by an equation such as 2. Then, contrast this to the difficulties associated with the addition of such a mass to a continuously flexible pipe: since the boundary conditions will now be different, this problem has to be solved from scratch, even if the solution of the problem without the mass i. Hence, it is often advantageous to transform distributed parameter systems into discrete ones by such methods as the Galerkin or Ritz-Galerkin or the Rayleigh—Ritz schemes, or by the finite element method Meirovitch , In this section, first the standard methods of analysis of discrete systems will be reviewed.
Then, the Galerkin method will be presented via example problems, as well as methods for dealing with the forced response of continuous systems. Along the way, a number of important definitions and classifications of systems, e. Special forms or interpretation of 2. For a conservative system, the equations of motion may be written as. Constraints are auxiliary kinematical conditions; e. The homogeneous form of equation 2.
Oscillatory solutions are sought, of the form. Substituting , leads to the standard eigenvalue problem,. Nontrivial solution of 2. The latter may be viewed as shape functions. Thus, for the double pendulum of , and similarly for second-mode motions. Other initial conditions generate motions which involve — can be synthesized from — both eigenvectors and both eigenfrequencies. Relations 2. For the forced response , equation 2.
This may be done in many ways, e. This latter will be reviewed briefly in what follows. First, the modal matrix is defined,. Hence, the coordinate transformation.
Fluid-Structure Interactions: Slender Structures and Axial Flow
Substituting leads to. The system 2. The response in terms of the original coordinates may then be obtained via 2. If damping is present, then the full form of equation are defined:. To that end, the adjoint of eigenvalue problem 2. As already mentioned, it is advantageous to analyse distributed parameter or continuous systems by transforming them into discrete ones by the Galerkin method, or, for that matter, by collocation or finite element techniques, or by the direct Lagrangian discretization method Section 2.
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The Galerkin method will be reviewed here by means of an example. The simplest equation describing its flexural motion is. The boundary conditions are. The solution of this problem is well known [e. Substitution into 2.
Fluid-Structure Interactions: Volume 2 | Angus & Robertson
This transcendental equation yields an infinite set of eigenvalues, the first three of which are:. Before proceeding further, an important note should be made. It is customary in vibration theory and in classical mathematics to define the eigenvalue as being essentially the square or, as in equation 2. This can lead to confusion, no doubt. However, these different meanings and notations are so deeply embedded in the field [cf. Instead, the context and occasional reminders are preferable to make the reader aware of which of the two notations for eigenvalue is being used.
Of course, for a simple problem like this, it is possible to proceed in the normal way and determine the eigenvalues and eigenfunctions of the modified problem. It will nevertheless be found convenient to transform such systems into discrete ones by the Galerkin method. To this end, for the problem at hand, the end-shear is transferred from the boundary conditions into the equation of motion, which may be re-written as. When approximation. Thus, in this example, substituting approximation , leads to.
Table 2. The exact values, by solving 2.
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Approximations to the lowest three eigenfrequencies of the modified cantilevered pipe for various in the case of. The eigenvalue problem associated with equations 2. In the case of equation. The equivalent to statement 2.
The elements of the mass and stiffness matrices [cf. Hence, strictly , yet still obtained the correct results. However, this is not always true, as will be seen in Section 2. As shown in Chapter 3, the equation of motion in this case is. The lowest two eigenfrequencies calculated by two different methods for different.
In method a the extra mass, , is included in the equation of motion via a Dirac delta function, while in b it is accounted for in the boundary conditions. That the system is nonconservative may be assessed by calculating the rate of work done by all the forces acting on the pipe. If it is zero, then there is no net energy flow in and out of the system, which must therefore be conservative; otherwise, the system is nonconservative.
In this case,. This action might not be possible to undo.
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