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Theory Manual Version 3.6
 Chapter 6: Dynamics Up Chapter 6: Dynamics Section 6.2: Elastodynamics 

6.1 Newmark Integration

To solve a differential equation which is second-order in time, we need to perform a numerical integration in the time domain. Let denote the function of interest and let and represent consecutive time steps such that . The function may be represented at each time point as and . The Newmark integration formulas are used to evaluate and at time , assuming that they can be integrated from a judiciously selected in the time interval . Using the mean value theorem for definite integrals, we know that an exact solution may be found for the integral according to where is generally unknown a priori. In the Newmark integration scheme we let where is a user-selected parameter in the range to . It follows that
We can similarly integrate this function twice to obtain , where we let Here, represents a parameter that varies from to . It follows that or alternatively, from which we may re-evaluate (6.1-3) as Stability of this integration scheme is guaranteed when
 Chapter 6: Dynamics Up Chapter 6: Dynamics Section 6.2: Elastodynamics