The direct integration methods seek the history of the dynamic response in time. The response is not obtained in continuous form but rather for a predetermined series of points in time t i .
One of the most popular direct integration methods is the Newmark method. Starting-point is the governing equation of a structure with various degrees of freedom:
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For time t =t i , this equation takes on the following form:
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The velocity and the acceleration are expressed as follows:
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Substituting these expressions into the equation of motion, we obtain
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where,
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Usually the initial conditions are that the structure posesses neither displacement nor velocity.