Improving the two

The vibration of the engineering systems with distributed delay is governed by delay integro-differential equations. Two-stage numerical integration approach was recently proposed for stability identification of such oscillators. This work improves the approach by handling the distributed delay—that is, the first-stage numerical integration—with tensor-based higher order numerical integration rules. The second-stage numerical integration of the arising methods remains the trapezoidal rule as in the original method. It is shown that local discretization error is of order O ( Δ t ) 3 irrespective of the order of the numerical integration rule used to handle the distributed delay. But ( Δ t ) 3 is less weighted when higher order numerical integration rules are used to handle the distributed delay, suggesting higher accuracy. Results from theoretical error analyses, various numerical rate of convergence analyses, and stability computations were combined to conclude that—from application point of view—it is not necessary to increase the first-stage numerical integration rule beyond the first order (trapezoidal rule) though the best results are expected at the second order (Simpson’s 1/3 rule)