Page 319 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)
P. 319
300 SLENDER STRUCTURES AND AXIAL FLOW
(1985) will be described in abbreviated form. However, before doing so, let us first
distinguish between implicit and explicit forms of the equations to be solved.
Presuming that the equations to be solved are either discrete or discretized, they can
be expressed as a set of second-order implicit nonlinear equations of the type
MX + CX + Kx = F(x, X, X, t), (5.78)
with appropriate initial conditions, x(0) and x(0); M, C and K are N x N matrices and
all nonlinearities are included in F. This equation is said to be implicit because of the
presence in F of nonlinear inertial terms, Le. terms involving x, which cannot be removed
or transformed.
In many cases, it is possible to express (5.78) as an explicit relation
Y = F(y, t), Y(0) = yo, (5.79)
which renders solution easier. However, when nonlinear inertial terms are present in F,
this transformation into (5.79) may not be possible, and means for the direct solution of
(5.78) must be sought.
With this in mind, the various methods available for solving equations (5.78) or (5.79)
will now be described.
Irrational analytical methods entail the simplification of the equations to be solved by
neglecting or approximating various terms, e.g. by the use of small-deflection or small-
angle approximations. Hence, such solutions are valid over a small range of parameters
or for small deviations from the state of equilibrium.
Rational analytic methods, such as perturbation methods (Hagedorn 198 1; Nayfeh &
Mook 1979), the method of averaging (Hagedorn 1981; Nayfeh 1981) and its precursor
the Krylov & Bololiubov method (Minorsky 1962; Nayfeh 1973), and the method of
multiple scales (Nayfeh & Mook 1979; Nayfeh 1985), achieve solution by an asymptotic
expansion or perturbation of the original set of equations, in terms of a small parameter
E (E << 1) which is either present in the equations ab initio or artificially introduced.
Hence these methods are also known as ‘small-parameter techniques’, and they involve
the sequential solution of simplified sets of equations, in which terms of order em+’
are disregarded while constructing the mth approximation. The method of averaging is
described in Appendix F.4.
Numerical time-difference methods (Gear 1971; Lambert 1973; Press et al. 1992) are
based on approximating the solution by its value at a sequence of discrete times. These
methods have been developed mostly on the assumption that equation (5.78) may be
rewritten as (5.79).+ If this can be done, one can distinguish single-step and multistep
methods of solution. The Runge-Kutta method is an example of the former; it requires
the values of x and X at time t,, in order to compute the solution at t,+l. Multistep or
‘k-step’ methods, e.g. that of Adams-Bashford-Moulton, accumulate information for the
values of x and X at t,, tn-l, . . . , tn-k to proceed to the next step.
Combined analytical-numerical methods, such as the Rayleigh-Ritz, Galerkin
(Meirovitch 1967) or harmonic balance methods (Hagedorn 1981; Nayfeh 1981) require
an initial assumption as to the form of the approximate solution. The solution is typically
expressed in the form of series, e.g. power, Taylor, Chebyshev, Fourier or Legendre series.
+The slightly more approximate form of the equation of motion of Section 5.2.7(b) has specifically been
obtained to take advantage of this.