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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.
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