Abstract: Three structure-dependent integration methods have been developed for solving equations of motion, which are second-order ordinary differential equations, for structural dynamics and earthquake engineering applications. Although they generally have the same numerical properties, such as explicit formulation, unconditional stability and second-order accuracy, a different performance is found in solving the free vibration response to either linear elastic or nonlinear systems with high frequency modes. The root cause of this different performance in the free vibration responses is analytically explored herein. As a result, it is verified that a weak instability is responsible for the different performance of the integration methods. In general, a weak instability will result in an inaccurate solution or even numerical instability in the free vibration responses of high frequency modes. As a result, a weak instability must be prohibited for time integration methods.
Abstract: It has been shown that a load discontinuity at the end of
an impulse will result in an extra impulse and hence an extra amplitude
distortion if a step-by-step integration method is employed to yield the
shock response. In order to overcome this difficulty, three remedies
are proposed to reduce the extra amplitude distortion. The first remedy
is to solve the momentum equation of motion instead of the force
equation of motion in the step-by-step solution of the shock response,
where an external momentum is used in the solution of the momentum
equation of motion. Since the external momentum is a resultant of the
time integration of external force, the problem of load discontinuity
will automatically disappear. The second remedy is to perform a single
small time step immediately upon termination of the applied impulse
while the other time steps can still be conducted by using the time step
determined from general considerations. This is because that the extra
impulse caused by a load discontinuity at the end of an impulse is
almost linearly proportional to the step size. Finally, the third remedy
is to use the average value of the two different values at the integration
point of the load discontinuity to replace the use of one of them for
loading input. The basic motivation of this remedy originates from the
concept of no loading input error associated with the integration point
of load discontinuity. The feasibility of the three remedies are
analytically explained and numerically illustrated.