  Time-integration for lumped parameter systems obeying implicit Bingham-Kelvin
constitutive models is studied. The governing system of equations describing
the lumped parameter system is a non-linear differential-algebraic equation and
needs to be solved numerically. The response of this system is non-smooth and
the kinematic variables can not be written in terms of the dynamic variables,
explicitly. To gain insight into numerical time-integration of this system, a
new time-integration scheme based on the trapezoidal method is derived. This
method relies on two independent parameters to adjust for damping and is
stable. Numerical examples showcase the performance of the proposed
time-integration method and compare it to a benchmark algorithm. Under this
scheme, implicit-explicit integration of the governing equations is possible.
Using this new method, the limitations of the trapezoidal time-integration
methods when applied to a non-smooth differential-algebraic equation are
highlighted.
