Because acceleration also varies with time, however, the extended mean value theorem must also be extended to the second time derivative to obtain the correct displacement. Thus,
where again
The discretized structural equation becomes
Explicit central difference scheme is obtained by setting and
Average constant acceleration (Middle point rule) is obtained by setting and
Stability Analysis
A time-integration scheme is said to be stable if there exists an integration time-step so that for any , a finite variation of the state vector at time induces only a non-increasing variation of the state-vector calculated at a subsequent time . Assume the time-integration scheme is
The linear stability is equivalent to , here is the spectral radius of the update matrix .
For the linear structural equation
here is the stiffness matrix. Let , the update matrix is , and
For undamped case (), the update matrix can be decoupled by introducing the eigenmodes of the structural system, which are solved by the generalized eigenvalue problem
For each eigenmode, the update matrix becomes
The characteristic equation of the update matrix is
As for the stability, we have
Explicit central difference scheme ( and ) is stable when .
Average constant acceleration (Middle point rule) ( and ) is unconditionally stable.
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