The global equation of motion in the time domain is given by:

(10–12)

where is the assembled structural and added mass matrix in the fixed reference axes, is the unknown acceleration vector, and is the total applied force vector on all of the element nodes.

Due to the high frequencies present in the higher modes of vibration, a semi-implicit two-stage predictor corrector scheme is used to integrate in time for velocity and displacement.

The higher frequencies, and hence the semi-implicit aspect of the formulation, involve the forces due to structural bending. We thus rewrite Equation 10–12 as

(10–13)

where the is assembled structural stiffness matrix, is the displacement vector over all nodes, and is a vector representing forces other than those due to structural stiffness.

At the first stage of the integration scheme, we write

(10–14)

where and are the known displacement and velocity vectors, respectively, of nodes at time t.

This equation leads to a solution for acceleration at the predictor stage, from which predictions and for nodal velocities and displacements at time may be made:

(10–15)

At the corrector stage, the acceleration is determined from

(10–16)

where the added mass and forces at are calculated using and .

The final solutions for velocity and displacement at time are then given by

(10–17)

For some extreme cases of loading, the above time-centered scheme may be unstable. A factor is therefore introduced to make the formulation non-time-centered, which increases the stability but reduces the accuracy. Introducing gives:

(10–18)

When the value of is 0.5, the expressions for and in Equation 10–18 equate to the formulas given by Equation 10–15 and Equation 10–16 respectively. Tests have shown that the optimum value for is 0.54, which gives the best trade-off of stability and accuracy when employing Equation 10–18.