A simplified implicit Newmark integration scheme for finite rotations

Implicit Newmark schemes for integration of finite rotations in structural and continuum mechanics typically are more complicated than those used for translational motion. Using Euler’s representation of the rotation tensor Λ in terms of a rotation vector ϕ, numerical integration connecting the values {ϕn,Λn} and {ϕn+1,Λn+1} at the beginning and end of the time step is expressed in terms of an incremental rotation vector θ and the associated incremental rotation tensor Λ(θ). Here, it is shown using backward differentiation that neglecting third order terms in θ, the approximation for the angular velocity ω in terms of θ and its time derivative has the same form as that between velocity and displacement. Consequently, the simplified Newmark scheme in terms of {θ,ω,ω̇} has the same form as that for updating translations. Details of the Newmark scheme and an analytical expression for the tangent stiffness tensor for the associated Newton–Raphson iteration procedure have been presented for rigid body dynamics. The resulting integration scheme has been tested on a nontrivial problem of three-dimensional motion of a rigid body using a constant time step. The results justify the use of the simplified Newmark scheme for finite rotations.