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chap97 - Start-up of the Calculation Process Hence the...

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1 Start-up of the Calculation Process Hence the value D n +1 at instant n +1 is calculated from the known values D n and D n 1 at instants n and n 1, respectively. Initial displacement and velocity, are known (given). 0 0 and D D ± D 1 as well as D 0 must be known to calculate D 1 . D 1 is not a real quantity but algorithm demands it. So, we assume that D is a smooth function of time around the origin and extrapolate it to instant t = −∆ t . We do this by expanding D in a Taylor series about t=0 and evaluate it at t = −∆ t .

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2 Direct Integration Algorithms Now the acceleration at t =0 is needed, which is calculated from the discretized equation of motion at instant n =0: from which 0 0 0 0 R KD D C D M = + + ± ± ± [ ] r nonsingula 0 0 0 1 0 M KD D C R M D = ± ± ± Using this, D 1 is determined from which D 1 is calculated.
3 Numerical Stability of the Central Difference Method The method is conditionally stable : Accuracy decreases as t gets large and the method diverges if t is too large . To guarantee numerical stability, t must be chosen such that where ω max is the largest undamped natural frequency of the system and, therefore, T min is the smallest natural period . . Note that t indicates how often we sample the motion . On the other hand, T min indicates how fast the motion changes .

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