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Direct Numerical Integration Methods ———  323

                                                                                                     {}


                   where [M], [C] and [K] are the mass, damping and stiffness matrices for the system and  {} {}and X
                                                                                              , X
                                                                                           X
                   refer to the acceleration, velocity and displacement vectors, respectively. {F(t)} is the force vector. Several
                   numerical direct integrating schemes are available to determine the approximate solution of a system of
                   equations of motion. For a linear dynamic system, matrices [M], [C] and [K] are independent of time and
                   therefore remain unchanged during the integration procedure. These matrices vary with time for a non-
                   linear dynamic system and must be modified during the integration of equations of motion. For the solution
                   of equations of motion for a linear dynamic system, either the normal mode superposition method of dynamic
                   analysis or direct numerical integration methods can be used. However, for the solution of non-linear
                   equations of motion, direct numerical integration methods are generally recommended.
                       In a direct integration method, the system of equations of motion is integrated successively by using
                   a step by step numerical procedure. No transformation of the equations of motion is needed prior to
                   integration and using difference formulas that involve one or more increments of time usually approximates
                   time derivatives. Basically there are two principal approaches used in the direct integration method: explicit
                   and implicit schemes. In an explicit scheme, the response quantities are expressed in terms of previously
                   determined values of displacement, velocity and acceleration. In an implicit scheme, the difference equations
                   are combined with the equations of motion, and the displacements are calculated directly by solving the
                   equations.
                       In this section, only selected numerical integration schemes widely used for linear and non-linear
                   dynamic analyses are considered. Three explicit and four implicit direct integration schemes are examined.
                   A brief description of these schemes is presented and their application is illustrated. The explicit schemes
                   presented are the central difference method, two-cycle iteration with trapezoidal rule and fourth-order Runge-
                   Kutta. The implicit schemes include the Houbolt, Wilson-Theta, Newark-Beta and Park Stiffly stable methods.
                   The accuracy, stability and efficiency of these schemes are examined by comparing the results for sample
                   problems.


                    6.4 EXPLICIT SCHEMES

                   As mentioned earlier, in an explicit formulation, the response quantities are expressed in terms of previously
                   determined values of displacement, velocity and acceleration.

                   6.4.1 Central Difference Method
                   The procedure indicated for the case of a single degree of freedom system can be directly extended to this
                   case. Consider a displacement time history curve as shown in Fig. 6.2. At the middle of the time interval t∆ ,
                   the velocity is given by


                                  X    1  =  X i+ 1 −  X i                                          ...(6.14)
                                   i+     ∆t
                                    2
                   and            X    =  X −  X i− 1                                               ...(6.15)
                                         i
                                   i− 1   ∆t
                                    2
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