For central differences corresponding to γ = , β = the displacement increment is approximated by (9.8b) as

where w is the acceleration vector. This formula suggests the following three displacement increment predictors:

The predictor ∆uA disregards the effect of the acceleration, ∆uB corresponds to neglecting the acceleration at the end of the interval, while ∆uC corresponds to the assumption of constant acceleration. Use the implementation from Exercise 9.5 of the acceleration vector wn in the energy-conserving Algorithm 9.3 to investigate the efficiency of the three displacement increment predictors by application to the elastic pendulum treated in Example 9.2.

Exercise 9.5:

In integrated state-space based algorithms the acceleration does not appear directly. However, the acceleration may be of independent interest, either as a result of the analysis or as part of an improved predictor in non-linear problems. In the energy-conserving Algorithm 9.3 the inertial term is represented via the velocity increment ∆v, which may be related to the acceleration w by the central difference ∆v = h(wn+1 + wn). When the initial acceleration w0 is evaluated from the equation of motion at time t = 0, this formula permits subsequent updates of the acceleration of the form wn+1 = (2/h)∆v − wn. Implement the acceleration vector wn in the energy-conserving Algorithm 9.3 and recalculate the elastic pendulum problem of Example 9.2 including the acceleration.

Example 9.2:

The elastic pendulum – Energy conservation. In this example the motion of the elastic pendulum introduced in Example 9.1 is integrated by the energy-conserving algorithm. The mass matrix M, the internal force vector g and the external load vector f are as defined before. However, the stiffness matrix now appears in two roles: as the incremental geometric stiffness

and as the stiffness matrix needed in the Newton iterations

Example 9.1:

The elastic pendulum. The pendulum shown in Fig. 9.4 consists of a concentrated mass m suspended in a hinged elastic bar with negligible mass and stiffness EA and length l0 in the unloaded state.