Austin, MarkThis paper describes a new class of algorithms for integrating linear second order equations, and those containing smooth nonlinearities. The algorithms are based on a combination of ideas from standard Newmark integration methods, and extrapolation techniques. For the algorithm to work, the underlying Newmark method must be stable, second order accurate, and produce asymptotic error expansions for response quantities containing only even ordered terms. It is proved that setting the Newmark parameter t to 1/2 gives a desirable asymptotic expansion, irrespective of the setting for ݮ Numerical experiments are conducted for two linear and two nonlinear applications.en-USalgorithmsnonlinear systemsnumerical analysisextrapolationdynamicsIntelligent ServomechanismsTechnical Report